Solar Anomaly and Planetary Displays in the Antikythera Mechanism

Freeth T. Bitsakis Y. Moussas X. Seiradakis J. H. Tselikas A. Mangou H. Zafeiropolou M. Hadland R. Bate D. Ramsey A. Allen M. Crawley A. Hockley P. Malzbender T. Gelb D. Ambrisco W. , and Edmunds M. G. , “Decoding the ancient Greek astronomical calculator known as the Antikythera mechanism”, Nature, cdxliv (2006), 587–91; and Freeth T. Jones A. Steele J. M. Bitsakis Y. , “Calendars with Olympiad display and eclipse prediction on the Antikythera mechanism”, Nature, cdliv (2008), 2008–17. For each of these papers there is substantial “Supplementary information” linked to the online version of the paper at www.nature.com/nature. Important earlier work includes the following: Derek de Solla Price, Gears from the Greeks: The Antikythera mechanism — A calendar computer from ca. 80 B.C. (Transactions of the American Philosophical Society, n.s., lxiv/7 (1975)); Wright M. T. Bromley A. G. , and Magkou E. , “Simple x-ray tomography and the Antikythera mechanism”, PACT, xlv (1995), 1995–43; Wright M. T. , “A planetarium display for the Antikythera mechanism”, Horological journal, cxliv (2002), 2002–73 and 193; Wright M. T. , “In the steps of the master mechanic”, Ancient Greece and the modern world (Patras, 2003), conference paper version available at http://fsoso.free.fr/antikythera/DOCS/AG&MWOlympia2002texttablesnotes.pdf; Wright M. T. , “Epicyclic gearing and the Antikythera mechanism, Part I”, Antiquarian horology, xxvii, issue of March 2003, 270–9; and “Part II”, xxix, issue of September 2005, 51–63; Wright M. T. , “The Antikythera mechanism: A new gearing scheme”, Bulletin of the Scientific Instrument Society, lxxxv, issue of June 2005, 2–7; Wright M. T. , “Counting months and years: The upper back dial of the Antikythera mechanism”, Bulletin of the Scientific Instrument Society, lxxxvii, issue of December 2005, 8–13; Wright M. T. , “The Antikythera mechanism and the early history of the moon phase display”, Antiquarian horology, xxix (2006), 2006–29.

Wright , “A planetarium display” (ref. 1).

Freeth , “Decoding” (ref. 1).

“Supplementary information” to Freeth , “Decoding” (ref. 1), pp. 8 and 10–12.

“Supplementary information” to Freeth , “Calendars” (ref. 1), p. 5.

Wright , “A planetarium display” (ref. 1). A film of Wright's reconstruction in motion can be seen at http://www.youtube.com/watch?v=ZrfMFhrgOFc. This film was made in 2008 by the science writer Jo Marchant, who was then working for New scientist.

Freeth , “Calendars” (ref. 1).

The possibility of an Olympiad display was overlooked by all the principal investigators, but it was actually suggested by Victor J. Kean, in a popular book, The ancient Greek computer from Rhodes known as the Antikythera mechanism (Anixi, 1995), 77–84. Our thanks to Alexander Jones for this reference.

Heiberg J. L. , Claudii Ptolemaei opera quae exstant omnia, ii: Opera astronomica minora (Leipzig, 1907), 165. There is a French translation in [N.] Halma, Commentaire de Théon d'Alexandrie sur les tables manuelles astronomiques de Ptolemée, Première partie (Paris, 1822), Prolégomènes de Ptolemée, 7–8.

Manitius C. (ed.), Procli Diadochi Hypotyposis astronomicarum positionum (Leipzig, 1909), iii, 66–72 (pp. 72–77).

For example, see Roderick Webster Marjorie , Western astrolabes (Adler Planetarium and Astronomy Museum, Chicago, 1998), no. 1 (an English astrolabe of around a.d. 1250; and nos. 2 and 3 (astrolabes probably made by Johanne Fusoris at Paris around 1400).

See the Hartmann astrolabes nos. 5 and 6 (from 1532 and 1540) in Webster and Webster, Western astrolabes (ref. 11).

Price , Gears from the Greeks (ref. 1), 18.

The zodiac scale of the Antikythera mechanism reflects the common Greek nomenclature of its day by calling this zodiac sign XHAAI, Chelai, the “Claws” (of the Scorpion). Price , Gears from the Greeks (ref. 1), 17–18, mistakenly read XYAAI, an error that is still sometimes repeated. Our thanks to Michael Wright for pointing this out.

Price's discussion, in Gears from the Greeks (ref. 1), 18, contains an arithmetical or other error that probably prevented him from recognizing that, for the interval on the rings that he analysed, the number of degrees exceeds the number of days, which would be impossible on uniformly divided, concentric rings. Thus he missed concluding that one of the two rings was deliberately divided nonuniformly.

The holes underneath the Egyptian calendar scale, which are invisible to surface examination, were described by Price, Gears from the Greeks (ref. 1), 18, but were first explained by Wright, “In the steps of the master mechanic” (ref. 1), 4. The number of holes that existed on the original mechanism has been variously estimated as 360 by Price, 365 by Wright, and falling in the range 363–365 by Freeth , in “Decoding” (ref. 1, caption to Fig. 2). In principle, these holes should provide another method of estimating the solar anomaly intended by the designer of the Antikythera mechanism. But because of the errors in the spacing the holes, and the resulting uncertainty in their number, the question is a delicate one. We shall deal with it in a separate paper.

We reckon longitudes from the spring equinoctial point. We treat the long mark at the beginning of a zodiac sign as representing 0° of that sign, although we recognize that it is possible that an ancient user might have understood this as the first degree.

In the case of longitude 234, which falls between day marks 68 and 69, since day mark 69 cannot be seen, we extrapolated a value for 234 from day marks 67 and 68. This introduces negligible error, as longitude 234 lies only a very small distance beyond the day mark 68.

Evans James , The history and practice of ancient astronomy (New York, 1998), 234. Note that in this expression, q is given as a function of the true anomaly, X — A. Usually (as when calculating the longitude of the Sun for a given date) one wants q as a function of the mean anomaly, which involves a more complicated expression.

The best-fit eccentricity may be expressed in Ptolemy's terms as 2.142 parts out of 60, compared with the Hipparchos value of 2.5.

A similar argument applies to the holes drilled into the plate beneath the calendar scale: The holes cannot have been intended to be unequally spaced, as this would result in a non-uniform shift of the Egyptian calendar with respect to the zodiac signs.

Vitruvius explicitly informs his readers that the zodiac is divided nonuniformly on the anaphoric clock (On architecture, ix, 8.8).

Statistical methods confirm the common-sense impression from Fig. 9 that the best-fit eccentric model cannot be ruled out. First, we may estimate the standard deviation σ of the data distribution in Fig. 9 by temporarily assuming that System A is the underlying true theory. With N = 66 points, and the sum of squared residuals mentioned, this gives σ = 0.16°. We assume that this characterizes the original placement of the divisions as well as our measuring process. Now, we examine the possibility that the data are actually described by the best-fit eccentric. Then, for the eccentric model, χ² = sum of squared residuals/σ² = 103. The probability Q that χ² could be this large or larger simply by chance (assuming the validity of the best-fit eccentric) then turns out to be around 0.002 — A possibility that cannot be rejected. In model testing, “it is not uncommon to deem acceptable on equal terms any models with, say, Q > 0.001”. See Press William H. , Numerical recipes: The art of scientific computing (Cambridge, 1986), 503. By contrast, the Hipparchos value of e = 0.0417 has a sum of squared residuals of 5.09 deg², which gives χ² = 199 and a probability of 3 × 10⁻¹⁵. This, of course, means only that the Hipparchos value does not describe the actual data. But it does not prove that the mechanician was not aiming at the Hipparchos value. There could be, for example, a centre-placement error or some other systematic error.

Price , Gears from the Greeks (ref. 1), 16.

“Supplementary information” to Freeth , “Calendars” (ref. 1), p. 24.

Wright , “Early history of the moon phase display” (ref. 1).

Or perhaps the maker believed that the Moon's motion contains a zodiacal anomaly identical to the Sun's. We do not know of any examples of such a theory, so we do not seriously entertain this possibility.

Musée du Louvre, Département des Antiquités Egyptiennes, inv. n. 2325. (This is the papyrus sometimes called “The Art of Eudoxus”.) Text: Blass F. , Eudoxi ars astronomica (Kiel, 1887). There is a French translation by Paul Tannery, Recherches sur l'histoire de l'astronomie ancienne (Paris, 1893), 283–300.

Freeth , “Decoding” (ref. 1).

In our proposal, the solar input to the Moon phase display is based on the Sun's mean (rather than true) motion. But the solar anomaly only affects the times of new and full moons by about 4 hours — A quantity that would be imperceptible on the Moon phase display. (The Sun's maximum equation is about 2°. At the Moon's rate of motion of about 12° per day, it would take 1/6 day of motion to compensate for a 2° offset.).

The line numbers are from the preliminary text of the front cover inscription published in the “Supplementary information” to Freeth , “Decoding” (ref. 1), pp. 8 and 10–12. Agamemnon Tselikas and Yanis Bitsakis are in the course of preparing a new version of the front cover inscription, based on x-ray CT, and they presented a report on their progress in “The front cover plate of the Antikythera mechanism” (oral presentation), XXIII International Congress of History of Science and Technology, Budapest, 2009. Substantial changes in the text have been made (and the lines numbers changed by 1), so the preliminary text should be used with caution until the revised text is published. For this reason, we refrain from detailed discussion of the inscription.

Price , Gears from the Greeks (ref. 1), 20.

For an English translation of this text (British Museum, Shemtomb 135), see van der Waerden B. L. , Science awakening, ii: The birth of astronomy (New York, 1974), 107–8.

Ptolemy , Almagest ix, 3. Toomer G. J. , Ptolemy's Almagest (London, 1984), 424.

Neugebauer O. , Astronomical cuneiform texts (London, 1955), ii, 300.

We use the standard notation in which the integer and fractional parts of the number are separated by a semicolon, and the successive sexagesimal places are separated by commas.

But this (for r = 64/39) would result in x = 0;58,27 = 7 × 167/1200. A prime 167 gear would be cumbersome, though of course workable. But the truncation error is worse than for 63/40, so this option offers no advantage at all.

Freeth , “Decoding” (ref. 1), 589–90.

“Supplementary information” to Freeth , “Decoding” (ref. 1), p. 15.

Tselikas Bitsakis , “The front cover plate of the Antikythera mechanism” (oral presentation, ref. 31).

Neugebauer , Astronomical cuneiform texts (ref. 35), ii, 302.

Neugebauer , Astronomical cuneiform texts (ref. 35), ii, 313.

Neugebauer , Astronomical cuneiform texts (ref. 35), ii, 307.

Museum British , Sp. II, 985, published in Kugler F. X. , Sternkunde und Sterndienst in Babel , i (Münster, 1913), 49. Cited in Neugebauer, Astronomical cuneiform texts (ref. 35), ii, 307, n. 17.

P. Oxy. 4133, in Alexander Jones, Astronomical papyri from Oxyrhynchus ( Memoirs of the American Philosophical Society , ccxxxiii (2 vols. bound as one, Philadelphia, 1999)), ii, 2–5, with discussion in vol. i, 69–80.

Neugebauer , Astronomical cuneiform texts (ref. 35), ii, 287–8.

No winds are mentioned in the fragmentary parapegma text published by Price. Alexander Jones is working on a new edition of the parapegma inscription, and has kindly confirmed that there is “not the slightest trace of weather” (personal correspondence).

Greenwood Joseph Gouge Woodcroft Bennet (transl.), The pneumatics of Hero of Alexandria (London, 1851; reprinted London and New York, 1971).

Evans James Berggren J. Lennart , Geminos's Introduction to the Phenomena: A translation and study of a Hellenistic survey of astronomy (Princeton, 2006), 43–48 and 246–9. Sphairopoiïa is “sphere-making”, the branch of mechanics devoted to constructing images of the heavens, such as the Antikythera mechanism.

For example, the ivory astrological tablets of Grand (2nd century a.d.) have a central zodiac, with four winds in the corners; and this is true as well of the marble Bianchini Tablet: Abry J.-H. (ed.), Les tablettes astrologiques de Grand (Vosges) (Lyon, 1993). A Mithraic relief from Sidon (probably second century a.d.), now in the Louvre, has a prominent zodiac surrounding the tauroctony, with busts representing the four seasons in the corners: Giroire Cécile Roger Daniel , Roman art from the Louvre (New York, 2007), 243–5. Examples could readily be multiplied.

Freeth , “Decoding” (ref. 1), 590, caption to Fig. 6.

“Supplementary information” to Freeth , “Decoding” (ref. 1), p. 27.

Ptolemy , Almagest iv, 2.

See the discussion in Neugebauer Otto , A history of ancient mathematical astronomy (Berlin, 1975), 309–12.

For Venus, Ptolemy (Almagest ix, 2) uses a modification of 5 synodic cycles = 8 years, and for Mars, a modification of 37 synodic cycles = 79 years.

Ptolemy , Almagest ix, 2.

Note added while the article was in proof: The Antikythera Mechanism Research Project has recently estimated, by fresh study of the x-rays, that the interior width of the box housing the Antikythera mechanism was 164 mm. The reconstructions of Figs 15 and 16 will both fit inside such a width. The interior height is estimated at 314 mm, and the exterior dimensions at 190 × 340 mm. Our thanks to Tony Freeth for this information.