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Babylonian solar theory on the Antikythera mechanism

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Abstract

This article analyzes the angular spacing of the degree marks on the zodiac scale of the Antikythera mechanism and demonstrates that over the entire preserved 88° of the zodiac, the marks are systematically placed too close together to be consistent with a uniform distribution over 360°. Thus, in some other part of the zodiac scale (not preserved), the degree marks have been spaced farther apart. By contrast, the day marks on the Egyptian calendar scale are spaced uniformly, apart from minor errors. A solar equation of center is apparent which rises by nearly 2.7° over the preserved portion of the zodiac. The placement of the degree marks indicates that, in the preserved portion of the zodiac, the Sun was considered to run at a uniform pace of about 30° per synodic month, which is consistent with the Sun’s speed in the fast zone of the Babylonian solar theory of System A.

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Notes

  1. An Egyptian calendar year contains 365 days and would not span the whole 360° of the zodiac. In a strictly accurate representation, the calendar ring should show 365¼ days, but it is more likely that the ancient maker gave the ring 365 days for the sake of simplicity. In either case, the user would adjust the position of the calendar ring from time to time so that 1 Thoth corresponded to the currently correct degree of the zodiac. A ¼-day construction or round-off error would not be significant.

  2. Some writers have argued for a mechanical device, using gears, for producing the nonuniformity of the Sun’s motion. This would require two separate pointers, one indicating the day of the year and one indicating the longitude of the Sun (see Wright 2002). The same view is maintained in Freeth and Jones (2012), e.g., in the reconstruction of the front face of the mechanism in their Fig. 4. However, there is no material evidence for the “day pointer” in the remains of the mechanism. Also, because the two pointers could never be more than about 2° apart, one would often hide the other. Finally, the front of the mechanism was intended to present a working image of the cosmos: indeed, this display is referred to as the “cosmos” in one of the inscriptions (Freeth and Jones 2012, sec. 2.3.2). And, although date-indicating pointers are used on the Antikythera mechanism (as on the Metonic calendar on the back of the mechanism), a pointer for the date would likely have been out of place in the working image of the cosmos on the front.

  3. However, this decision does not in any way affect our analysis. If, as we suppose most likely, the maker of the Antikythera mechanism would have known to subtract 10° from the Babylonian longitudes for the zone boundaries (in order to express their positions in terms of the Greek 0° norm for the equinox), the fast zone would run from 153° to 347°. And if the maker had naively left the Babylonian longitudes for the zone boundaries unchanged, the fast zone would run from 163° to 357°. On the artifact, as we can see in Figs. 6 and 7, the preserved portion of the zodiac stretches from longitude 164° to 252°. Thus, in either case, the preserved portion of the zodiac would lie entirely within the Babylonian fast zone.

  4. To be sure, Theon of Smyrna (second century CE) mentions a theory in which the anomalistic year is 365½ days. According to Theon, the Sun returns to the same longitude in 365¼ days, to the same latitude in \(365{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 8$}}\) days, and to the same distance from the Earth in 365½ days (Dupuis 1892), sec. iii, 27, p. 281). Moreover, a papyrus of the second or third century CE carries a table of mean motions apparently based on the same theory (P. Oxy 4174a in Jones 1999, vol. 1, pp. 170–171 and vol. 2, pp. 164–167). So we should not automatically assume that the mechanic held the length of the anomalistic year to be 365¼ days, as Hipparchos and Ptolemy did. While no such theory is known to us, it is conceivable (though unlikely) that the mechanic believed the anomalistic year to be 365 days exactly, in which case the date on which the Sun reaches apogee would be fixed in the Egyptian calendar. Or perhaps he felt that making this approximation was good enough for practical purposes in his own lifetime. So the argument given in the paragraph above is not enough to settle the matter.

  5. The years chosen in the first column are all at the same position in the leap-year cycle. Thus, the time intervals between successive years are always evenly divisible by 4. Julian calendar dates for the 1st of Thoth are given in Bickerman (1980), pp. 115–122.

  6. For a convenient calculator for solar longitudes (and other quantities) see the Jet Propulsion Laboratory HORIZONS-Web Interface (https://ssd.jpl.nasa.gov/horizons.cgi).

  7. Image AK32a, available in high-resolution PTM (Polynomial Texture Map) may be downloaded from the server at Aristotle University of Thessaloniki: http://antikythera-mechanism.auth.gr/PTM Viewing the image requires PTM viewing software, available from a number of sites.

  8. The conversion factor was determined in the following way. We used the 5-cm scale that appears in a film photograph of Fragment C taken by Kostas Xenikakis for the AMRP (our thanks to Tony Freeth for providing a scan of this photograph, called Fragment C 4000 dpi No 14-15). On this photograph, we fitted three concentric circles simultaneously (the inner, middle and outer circles mentioned above) to the edges of the zodiac and calendar scales and determined their radii using the included 5-cm scale. These radii in centimeters were compared with the radii in Photoshop units determined from the X-ray images to establish the conversion factor. The three circles gave slightly different values for the conversion factor and the adopted value is the mean of the three. The conversion factor cannot be regarded as terribly precise, since we do not know how carefully the 5-cm scale was placed in the plane of the scales for the photograph.

  9. The values of the four radii resulting from fitting all four circles simultaneously are: Rinner = 32.2011 units. Rmiddle = 35.6410. Router = 39.1997. Rholes = 36.8718. The ratio (Rmiddle – Rinner)/Rinner (≅ 1/9.36) will be used in Sect. 7 below. The centers of the individual circles all lie within a circle of radius 0.68 mm about the principal center. The consequences of this modest spread in the centers for the estimate of the sun’s speed will be treated below.

  10. For simplicity, we have made the demonstration for a particular location of point P, with CP at right angles to CD. This is close to the situation when the error θ is largest. But it may be shown that with P in other locations, it still holds that the error associated with the comparative method is reduced with respect to the absolute error by the factor ΔR/R.

  11. Because we require the sine of the equation of center in terms of the true longitude λ, the function is a simple sinusoid. But sin q expressed in terms of the mean longitude \( \bar{\lambda } \) is more complex.

  12. In System B, the rising times of the signs almost form an arithmetic progression, but with jumps of double value between Gemini and Cancer as well as between Sagittarius and Capricorn. For all the echoes of the System A and System B schemes for rising times and day lengths in Greek and Roman sources see Neugebauer 1975, vol. 2, 712-724. An up-to-date collection and study of the Babylonian material are available in Steele (2017).

  13. Key evidence comes from Hypsikles’ own introduction to his Book XIV of the Elements (Vitrac and Djebbar 2011, pp. 89 and 106). Hypsikles’ father and a certain Basilides of Tyre studied together a treatise of Apollonius on the ratio of the dodecahedron and icosahedron inscribed in the same sphere, found its demonstrations defective, and wrote up their correction. But later Hypsikles himself came across a second, revised version of the treatise of Apollonius with correct proofs. Basilides of Tyre is usually identified with Basilides the Syrian who became head of Epicurean school in Athens about 200 BCE. Basilides is also mentioned in a papyrus Life of Philonides from Herculaneum (P. Herc. 1044) as one of the teachers of Philonides the Epicurean. A geometer named Philonides, who was a younger contemporary of Apollonius, is mentioned in the introduction to Book II of Apollonius’s Conics. Thus, Hypsikles’ dates are linked to those of Apollonius, which are linked to those of Philonides. Philonides’ dates are reasonably well constrained by his associations with the Seleucid court as well as by inscriptions. Thus it is plausible to make Hypsikles’ father and Basilides contemporaries of Apollonius and to make Hypsikles overlap with Philonides. The dating of Hypsikles is discussed briefly and cogently by Vitrac and Djebbar (2011), pp. 52–53. For a summary of the evidence on the dating of Philonides as it bears on the date of Apollonius, see Evans and Carman (2014), pp. 149–151. It should be noted that Neugebauer dates Hypsikles’ activities to 150-120 BCE (Neugebauer 1975, vol. 1, p. 715, n. 5), which is definitely too late.

  14. This is most easily confirmed by using the data for synodic phenomena in Carman and Duke (2019).

  15. However, this would not be true in situations in which the Sun was in one of the two zones and Venus was in the other.

  16. This differs only slightly from the Metonic relation 235 sm = 19 years, which implies that 1 year = 12 7/19 sm ≅ 12.36842… sm.

  17. This is slightly different from the synodic month deduced from the Metonic cycle and a year of 365.25d, which lead to 1 sm = 29.53085… d.

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Acknowledgements

We thank James Bernard, Alexander Jones and John Steele for careful reading of the paper and for suggestions that helped us improve it.

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Authors and Affiliations

  1. Program in Science, Technology and Society, University of Puget Sound, 1500 North Warner Street, Tacoma, WA, 98115, USA

    James Evans

  2. Centro de Estudios de Filosofía e Historia de la Ciencia (CEFHIC), Universidad Nacional de Quilmes (UNQ), Buenos Aires, Argentina

    Christián C. Carman

  3. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Buenos Aires, Argentina

    Christián C. Carman

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Correspondence to James Evans.

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Communicated by Alexander Jones.

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This paper is dedicated to the memory of our friend and colleague Alan S. Thorndike (1946–2018).

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This article is partly based on CT data (courtesy X-Tek Systems, now owned by Nikon Metrology, 2005) processed, with permission, from the archive of experimental investigations by the Antikythera Mechanism Research Project in collaboration with the National Archaeological Museum of Athens. Equipment loaned by X-Tek Systems Ltd (now owned by Nikon Metrology) was used to collect the data. We have also made use of PTM data (courtesy Hewlett-Packard 2005) obtained through the same collaboration between the National Archaeological Museum and the Antikythera Mechanism Research Project. The numerical data on which the analysis is based are posted at https://soundideas.pugetsound.edu/faculty_pubs/3376.

Appendix: System A and other piecewise constant-velocity models

Appendix: System A and other piecewise constant-velocity models

Let us define some quantities:

  • Sf speed of Sun in the fast zone of System A (30º/synodic month)

  • Ss speed in the slow zone \(({28}\, {\scriptstyle{1/8}}^{ \circ }{/}{\text{sm}})\)

  • Sm mean speed (360º/365.25d)

  • Wf width of fast zone (194º)

  • Ws width of slow zone (166º)

The times spent by the Sun in the fast and slow zones, respectively, are

$$ T_{f} = \frac{{W_{f} }}{{S_{f} }} = \frac{{194^{ \circ} }}{{30^{ \circ } /{\text{sm}}}} = 6.466{\bar{6}}\,{\text{sm}} $$
$$ T_{S} = \frac{{W_{S} }}{{S_{S} }} = \frac{{166^{ \circ } }}{{28}\, {\scriptstyle{1/8}}^{ \circ }{/}{\text{sm}}} = 5.902{\bar{2}}\,{\text{sm}} . $$

(The overbared final digit indicates that that digit repeats indefinitely.) Since Tf + Ts = 1 year, we have

$$ (194/30 + 166/28\,{\scriptstyle{{1/8}}} )\,{\text{sm}} = 1\,{\text{year}}, $$

which can be developed to yield

$$ 12\, {\scriptstyle{83/225}} \,{\text{sm}} = 1\,{\text{year}} $$

orFootnote 16

$$ 2783\,{\text{sm}} = 225\,{\text{years}}.$$

If we take the length of the year to be 365.25 days, which is suitable for a Greek instrument, the implied length of the synodic month is thenFootnote 17

$$ 1\,{\text{sm}} = 29.5297^{\text{d}} . $$

In the fast zone, the excess of the Sun’s displacement over the mean displacement during a certain time interval Δt is

$$ \left( {S_{f} {-}S_{m} } \right)\Delta t. $$

The change in the Sun’s longitude in the same time interval is

$$ \Delta \lambda = S_{f} \Delta t. $$

Thus, a graph of the “equation of center” (excess of true longitude over mean longitude) plotted against the Sun’s true longitude is a linear zigzag and has in the fast zone the constant slope

$$ \begin{aligned} m_{f} & = \frac{{\left( {S_{f} {-} S_{m} } \right)}}{{S_{f} \Delta t}} \, \Delta t \\ & = 1{-} S_{m} /S_{f} . \\ \end{aligned} $$

The total rise in the equation of center over the entire fast zone is

$$ \left( {S_{f} {-}S_{m} } \right)T_{f} = \, 5.7858^{ \circ } . $$

Half of this is about 2°54′ (a figure mentioned above), which can be thought of as the maximum (±) value of the equation of center.

In the same way, it may be shown that in the slow zone the slope of the graph (equation of center versus longitude) is

$$ m_{s} = \, 1 \, {-} S_{m} /S_{s} . $$

Conversely, if one has a straight-line equation of center versus longitude graph, trending upward, one may calculate the Sun’s speed (which is faster than average) from

$$ S_{f} = S_{m} /\left( {1 \, {-}m_{f} } \right). $$

And if one has a straight-line equation of center graph, trending downward, one may calculate the Sun’s speed (slower than average) from

$$ S_{s} = S_{m} /\left( {1 \, {-}m_{s} } \right). $$

Similar rules apply to any model in which the Sun has a piecewise constant velocity. The equation of center versus longitude graph would again be a linear zigzag, with the slope in each section given by m = 1 − Sm/S, where m is the slope for one section of the graph and S is the Sun’s constant speed in that section.

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Evans, J., Carman, C.C. Babylonian solar theory on the Antikythera mechanism. Arch. Hist. Exact Sci. 73, 619–659 (2019). https://doi.org/10.1007/s00407-019-00237-9

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