The Conical Sundial Of The Archaeological Museum Of Athens
Evangelia Panou1, Efstratios Theodossiou1, Vassilios N. Manimanis1, and Peter Z. Mantarakis2 1
Department of Astrophysics-Astronomy and Mechanics, School of Physics, National and Kapodistrian University of Athens E-mail:
[email protected] 2 22127 Needles St, Chatsworth, California, USA. E-mail:
[email protected]. Abstract The ancient sundial described in this work is a marble one of the conical type; it is now in the National Archaeological Museum of Athens (index catalog number 3158). Characteristic parameters of this sundial, such as the angle formed between the cone’s axis and the generatrix, the geographical latitude of operation and the gnomon’s length, are calculated from measurements of its geometrical dimensions. Ι. Conical sundials Conical sundials are unique in that the dial is comprised of part of a conical surface with its axis parallel to the axis of the Earth. The upper part has a plane portion that is parallel to the horizon, while its frontal part has another flat surface, which is parallel to the equator. The hour lines and the date curves of conical sundials form a special grid of lines, which have been approximated with acceptable precision with arcs of conic sections that connect the respective hour points (Gibbs, 1976, p. 31). The shape of the grid depends on the position of the cone’s vertex, on the conic surface and on the position of the gnomon’s base. The orientation of the sundial is defined by the position of cone’s vertex. The sundial has a southern orientation when the vertex is above the horizontal surface and a northern orientation when the vertex is below the horizontal surface. Additionally, there is the possibility of the cone’s vertex being a point of that surface (‘celestial sundials’). By analyzing the grid of the lines of a conical sundial, the following parameters φ, ω and d can be determined: φ is the geographical latitude of the place of the sundial ω is the angle between the cone’s axis and the generatrix of the dial’s conical surface d is the gnomon’s length Correspondingly, a conical sundial can be described and constructed when the above three parameters are known. Conical sundials of southern orientation According to Gibbs (1976) from the antiquity to this day, about 90 conical sundials of southern orientation have been preserved. Their characteristics that confirm that the dial’s surface is a part of a right circular cone having its vertex above the horizon are the following: 1. The curve on the edge of the frontal surface of the dial is circular. 2. The carved date curves on the dial are circular arcs that are parallel to the edge frontal curve. 3. The hour points, which are the intersections of the date curves with the hour lines are at equal distances from one another. 4. The meridian line is a straight line and coincides with the generatrix of the conical surface. 5. The distance between the winter solstice curve and the equinoxes curve along the meridian line is shorter than the distance between the equinoxes curve and the summer solstice curve. The dial’s surface is a part of a cone, which is formed by a generatrix (Fig. 1) that is the hypotenuse (KI) of a right triangle and revolves around the cone’s axis; the cone’s axis is one of the triangle’s legs (the two sides of the triangle that meet at a right angle). The generatrix coincides with the meridian line KT, while the cone’s axis is parallel to the terrestrial axis OK. The vertex of the cone lies above the horizon, a fact that also testifies to its (original) southern orientation. The dial’s boundaries are on the upper side a plane that is parallel to the horizon (AZ), and on the frontal side another plane, which is parallel to the equator. The date curves of the solstices and the curve of the equinoxes are carved starting from the
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gnomon’s hole (O) and with radii of three different lengths. The radial hour lines are 11; they start from O and their intersections with the date curves indicate the hour points on the curve arcs. In Figure 1 the geometry of a conical sundial of southern orientation is shown, with the following characteristics: K: vertex of the cone O: hole (base) of the gnomon OK or KΔ: axis of the cone (parallel to the Earth’s axis) KΓ or KT: generatrix of the cone (coincides with the meridian line) AZ: upper surface of the sundial (parallel to the horizon) OΓ: length of the gnomon ω: angle between the cone’s axis and its generatrix φ: geographical latitude of the place ε: obliquity of the ecliptic Χ: winter solstice I: equinoxes Θ: summer solstice ΓΘ: dial’s surface ΟΧ: radius of the winter solstice’s circle Fig. 1 Geometry of a conical sundial of southern OI: radius of the circle of the equinoxes orientation. OΘ: radius of the summer solstice’s circle ΓX: distance along the meridian of dial’s upper edge to the curve of winter solstice XI: distance along the meridian of the curve of winter solstice to that of the equinoxes ΘI: distance along the meridian of the curve of the equinoxes to that of the summer solstice ΘT: distance along the meridian of the summer solstice to the base of the dial. II. Calculation of the characteristic parameters of conical sundials i) Calculation of the angle ω between the axis and the generatrix By applying the law of sines on the triangle IOX for the angles ε and IXO we can derive the following sin cos (1) relation: By applying the law of sines on the triangle IOΘ for the angles ε and OΘΙ we can derive the additional sin cos (2) relation:
cos cos From the relations (1) and (2) we obtain by division: This can then be converted to the following: tan tan
(3)
ii) Calculation of the length d of the gnomon By applying trigonometric formulae for the angle ω of the right triangle KOI we obtain: tan . By applying the law of cosines on the triangle OKΓ we have for the length of the gnomon d : 2 2 2 2 cos (4) Further, using triangle and acute angles and 90 ( ) , this becomes:
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d 2 2 2 cos , where
cos , cos (5) sin tan
The length d of the gnomon is thus a function of the distances ΓX, XI and of the angles ε and ω.
iii) Calculation of the geographical latitude φ The geographical latitude of the location where the sundial is to be mounted corresponds to the angle KOΓ. By applying the law of sines on the triangle KΓO for the angles and we sin , recalling that and d are both functions of ε , ω , ΓX and XI. (6) obtain: sin d The ancient Greek conical sundial of the National Archaeological Museum of Athens The ancient Greek conical sundial with a gnomon of the National Archaeological Museum of Athens has index catalog No. 3158. It was recovered from the ruins of the ancient Dionysus Theater, which is located under the southern side of the Acropolis and is the oldest stone theater in the world. This dial dates from the Roman period (Kraus, 1991, p. 96). It is made of Pentele marble and it incorporates a marble base which rests on lion’s feet. The whole construction has a cube-like appearance 22.2 cm high; its weight is 16.40 Kg.
Figure 2: The conical sundial of the National Archaeological Museum of Athens (No. 3158).
The conical surface, i.e. the “plate” of the sundial, 1.8 cm thick, is preserved in very good condition. All the lines carved upon it are easily discernible: the 11 hour lines and the curves of the summer solstice, the winter solstice and the equinoxes. The gnomon of the sundial, a horizontal gnomon, was also recovered, a rare case for an ancient bronze artifact. The gnomon has a pyramid shape with an equilateral triangular base, whose side measures 1.4 cm. The present-day length of the gnomon is 5.9 cm. The distance from the gnomon’s base to the lower point of the conical surface is 11.0 cm, to the leftmost point of the curve is 15.0 cm and to the rightmost point of the curve is 15.5 cm. A characteristic of this sundial is a hole under the 4th hour line (measuring from the left) with a maximum width of 2.2 cm and a maximum height of 0.5 cm. This hole extends to the other side, where its opening is of larger dimensions (4.0 cm × 5.6 cm × 5.5 cm) and it is visible on the left-hand side of the whole The Compendium - Volume 20 Number 4
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construction. It was probably used for the more convenient mounting of the sundial at different sites, or for its transportation. The lengths of the hour lines (from the left to the right), measuring from the curve of the winter solstice to the curve of the equinoxes and from the curve of the equinoxes to the curve of the summer solstice are given in Table Ι. The lengths of the successive arcs formed upon the curves of the solstices and the equinoxes by their intersection with the hour lines are listed in Table II, along with the values of the angles formed between the successive hour lines according to the formula: l l where lx and lθ are the lengths of the arcs of the winter and the summer solstice curves corresponding to angle θ. Table Ι Lengths of the hour lines of sundial No. 3158
Number of hour line (left-to-right)
From the winter solstice curve to the equinoctial curve ΧΙ (cm)
1st
3.7
2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th
3.7 3.4 3.3 3.2 3.1 3.2 3.2 3.3 3.5 3.6
Between the hour lines: Edge – 1st 1st – 2nd 2nd – 3rd 3rd – 4th 4th – 5th 5th – 6th 6th – 7th 7th – 8th 8th – 9th 9th – 10th 10th – 11th 11th – 12th
From the equinoctial curve to the summer solstice curve ΙΘ (cm) ~6.6 (a part is missing) 6.6 6.4 6.3 6.2 6.2 6.2 6.3 6.4 6.7 6.9
From the winter solstice curve to the summer solstice curve ΧΘ (cm) ~10.3 (a part is missing) 10.3 9.8 9.6 9.4 9.2 9.4 9.5 9.7 10.2 10.5
Table ΙI Lengths of arcs of the curves Lengths of arcs Lengths of arcs on Lengths of arcs on the winter on the summer the equinoctial solstice curve lx solstice curve lθ curve lΙ (cm) (cm) (cm) 1.6 0.9 broken (slightly broken) 1.4 1.8 2.9 0.9 1.8 3.4 1.1 1.8 3.1 1.1 1.9 3.5 1.0 1.9 3.1 1.2 1.9 3.2 1.1 1.8 3.1 1.0 1.8 3.0 1.2 1.8 3.3 1.0 1.7 2.8 1.0 1.4 (broken) 3.5
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Angle θ (degrees) 8.34 14.62 11.94 14.63 12.94 12.19 12.06 11.81 11.80 9.82 Page 30
Other characteristic distances upon the conical surface along the meridian line are: a) the distance from the upper side of the sundial to the curve of the winter solstice (2.8 cm), b) the distance from the lower side of the gnomon’s base to the curve of the winter solstice (1.7 cm), and c) the distance from curve of the summer solstice to the base of the conical surface (0.4 cm). In order to determine the present-day length of the gnomon (d = 5.9 cm) the distances of the points of intersection of the curves with the meridian hour line from the gnomon’s apex were measured using a pair of compasses; these are given in Table III. By applying the law of cosines on the triangles ΟΧΙ and ΙΟΘ (Figure 1) we have: Figure 3: Schematic representation of a part of the conical surface for the measurement of the angles formed between successive hour lines.
cos 1
2 2 2 2 2 2 , cos 1 . 2 2
Table III Distances of the gnomon’s apex for a length of 5.9 cm from the points of intersection of the curves with the meridian hour line Curve Distance Winter solstice ΟΧ = 5.7 cm Equinoxes ΟΙ = 6.6 cm Summer solstice ΟΘ = 10.9 cm
From these relations the values of the obliquity of the ecliptic were calculated (Table IV). The values calculated for the obliquity of the ecliptic do not correspond to the modern value of εm = 23° 27΄, or to the ancient one for any given period of antiquity. The reason for this difference is most probably the loss of a part of the gnomon’s apex: therefore, an extrapolation was made of the side faces of the gnomon’s pyramid and the point of their intersection was determined. Table IV Values of the obliquity of the ecliptic for the modern gnomon length (d = 5.9 cm) Values of the obliquity of the ecliptic obtained from Between the curves: the measurements of the gnomon (degrees) Winter solstice – equinoxes ε1 = 28.48 Equinoxes – summer solstice ε2 = 30.08
The values for the obliquity of the ecliptic were calculated for various extrapolations Δd, the lengths of which did not exceed the point of intersection of the faces of the gnomon’s pyramid. These values are given in Table V. From the extrapolations of Table 5 it was expected that ε1 = ε2 ~ 23°. However, only from the pair of values (ΟX, OI) = (7.50 cm, 7.80 cm) a value ε1 = 23.27 ° was found, which approximates the true value of ε = 23° 27΄ (or approximately 23° 41΄ for the 1st century A.D.). For this value the characteristic parameters ω, d and φ of the sundial were computed from the formulae (3), (5) and (6), and they are given in Table VI.
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Figure 4: View of the existing part of the gnomon with a drawing of its extrapolation and the meridian hour line of the sundial No. 3158. Table V Values of the obliquity of the ecliptic for various extrapolations of the gnomon’s length Δd Δd d + Δd ΟΧ ΟΙ ΧΟΙ = ε1 ΙΟΘ = ε2 No. ΟΘ (cm) (cm) (cm) (cm) (cm) (degrees) (degrees) 1 0 5.90 5.60 6.80 10.85 26.78 31.72 2 0.40 6.30 5.70 6.50 10.90 28.48 30.08 3 1.70 7.60 6.90 7.30 11.00 25.02 32.23 4 1.80 7.70 7.50 7.80 11.05 23.27 33.04 5 2.55 8.45 7.75 8.00 11.30 22.63 32.05 6 2.70 8.60 7.95 8.20 11.20 22.06 32.88 Table VI Values of characteristic parameters ω, d and φ for obliquity ε = 23,27 ο Angle between the Gnomon’s axis and the Obliquity of the Geographic latitude φ length d generatrix ω (degrees) ecliptic ε (degrees) (cm) (degrees) 23.27 37.78 or 37ο 46.8΄ 6.13 40.50
However, in order to take into consideration the measured geometrical quantities of the sundial’s construction, the ω, d and φ were computed based on the two different values ε1 and ε2 that were obtained for the obliquity of the ecliptic. In this case, we rewrite equations (1) and (2) as: sin 1 cos 1 and sin 2 cos 2 . The only unknown quantity is ω, which can be calculated from the relation:
2 2 2 2 2 2 , 2 2 where sin 1 , sin 2 , cos 1 cos 2 , sin 1 sin 2 . cos 1, 2
For every pair of values (ε1, ε2), two solutions for the angle ω were obtained. The ω value with the greater difference from the one obtained for obliquity ε = 23° 27΄ was rejected. The other value of ω was
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inserted in the formulae (5) and (6) for the calculation of the gnomon’s length and of the geographical latitude. All values are given together in Table VII.
No. 1 2 3 4 5 6
Table VII The values for the characteristic parameters ω, d and φ for various extrapolations of the gnomon’s length Δd ΧΟΙ = ε1 ΙΟΘ = ε2 d2 d1 ω φ1 (degrees) (degrees) (degrees) (cm) (cm) (degrees) 26.78 31.72 28.72 6.56 6.01 37.99 28.48 30.08 29.83 6.24 6.06 34.85 25.02 32.23 25.94 7.12 6.19 41.83 23.27 33.04 22.56 7.80 6.38 45.69 22.63 32.05 23.92 7.82 6.37 46.39 22.06 32.88 20.73 8.27 6.53 48.16
φ2 (degrees) 30.54 32.42 31.07 31.41 32.16 32.34
Figure 5: Drawing of the line grid of the conical sundial of the National Archaeological Museum of Athens (No. 3158).
IV. Conclusions From the geometrical parameters of the winter solstice curve the values obtained for the obliquity of the ecliptic approximate the real value of ε = 23.27ο or 23ο 41΄. However, using the geometrical parameters
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of the summer solstice curve, even though the range of angle values is narrower than that of the angles ε1, the values obtained for the obliquity of the ecliptic do not approach the real value as closely. The sundial with gnomon’s length d = 6.13 cm functioned best for places with geographical latitude φ = 40.50°, if constructed for obliquity of the ecliptic ε1 = ε2 = ε = 23.27°. It would be fine at Dion (in Northern Greece), at latitude 40° 10΄ 16΄. Nevertheless, the obliquity angles calculated from the geometrical parameters of the curls of the conical surface do not coincide. For this reason, the relation (3) was amended in such a way that the value for ω can be calculated with better precision, by taking into consideration the geometrical characteristics of the whole conical surface. The values thus obtained for ω range from 20° to 30°, far from the value ω = 37.78° (see Table VI). The values for the gnomon’s length, d1, are not close to the theoretically expected value d = 6.13 cm, which is not the case with the d2 values. For these lengths, the sundial functions properly at latitudes from 38° to 48° and 30° to 32°, respectively. It should be noted that for the obliquity of the ecliptic value ε1 = 23.27 ° the gnomon length d1 = 7.82 cm and the geographical latitude value φ = 48.16 ° are greatly affected. Thus, the geometrical parameters of the conical surface between the winter solstice curve and the curve of the equinoxes are more representative, given that the obliquity values ε1 approximate well enough the real value. A significant difference exists when calculations based on parameters that have to do with the summer solstice curve are involved. In conclusion, the sundial with gnomon length d = 6.13 cm functioned best during the winter months in places with geographical latitude 40.50 °. The Balkan region with latitude ~40° is the one of northern Greece. For the same gnomon length the dial functioned best during the summer months in places with latitudes around 31° (e.g. in regions of northern Egypt). This fact reinforces our suspicion that the hole on the left-hand external side of the construction was probably used for mounting and transporting the sundial. What’s more, this sundial dates from the Roman period, an element that agrees with the different geographical latitudes at which it could be used. Yet, none of these values corresponds to the latitude of Athens (φ = 37° 58΄ 27΄΄), where the sundial was discovered. In the case that the sundial was used at a fixed position, its less satisfactory function during the summer months is because of its construction. There is the theory that ancient people did not care for the exact time during the summer months because the days were longer and so they had plenty of time for finishing their daily chores. In such a case the construction of the sundial as conical is not supported, since the angles ε1 and ε2 are unequal and different than ε = 23° 41΄. There is also the possibility that this sundial is a reconstruction of an original in a different location. References
Gibbs, Sharon, Greek and Roman Sundials, Yale University Press, New Haven and London, 1976. Kraus, Theodor, Katalog der Einzelfunde, Series Karthago I, 1991, p. 96. Acknowledgments We thank very much sculptor Mr. Athanassios Kalantzis, art conservator Mr. Michael Demetreadis, archaeologist Mrs. Hrysanthi Tsoulis and the supervisor of the Prehistoric and Classical Antiquities of the National Archaeological Museum of Athens Mrs. Helen Kourinos for their help during the collecting of data.
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