decoding the ancient greek astronomical calculator known as the antikythera mechanism

Decoding the Antikythera Mechanism, published in Nature, Volume 444, Issue 7119, pp. 587-591 (2006). 1

Decoding the ancient Greek astronomical calculator known as the

Antikythera mechanism

T. Freeth1,2 , Y. Bitsakis3,5 , X. Moussas3, J.H. Seiradakis4, A.Tselikas5, E. Magkou6, M.

Zafeiropoulou6, R. Hadland7, D. Bate7, A. Ramsey7, M. Allen7, A. Crawley7, P. Hockley7, T.

Malzbender8, D. Gelb8, W. Ambrisco9 and M.G. Edmunds1

1 Cardiff University, School of Physics and Astronomy, Queens Buildings, The Parade, Cardiff CF24 3AA, UK.

Mike Edmunds Mike.Edmunds @ astro.cf.ac.uk

2 Images First Ltd 10 Hereford Road, South Ealing, London W5 4SE, UK. Tony Freeth tony @ images-

first.com

3 National & Kapodistrian University of Athens, Department of Astrophysics, Astronomy and Mechanics,

Panepistimiopolis, GR15783, Zographos, Greece. Xenophon Moussas, xmoussas @ phys.uoa.gr

4 Aristotle University of Thessaloniki, Department of Physics, Section of Astrophysics, Astronomy and

Mechanics, GR-54124 Thessaloniki, Greece. John Seiradakis jhs @ astro.auth.gr

5 Centre for History and Palaeography, National Bank of Greece Cultural Foundation, P. Skouze 3, 10560

Athens, Greece. Yanis Bitsakis bitsakis @ gmail.com

6 National Archaeological Museum of Athens, 44 Patission Str, 106 82 Athens, Greece.

7 X-Tek Systems Ltd, Tring Business Centre, Icknield Way, Tring, Herts HP23 4JX, UK.

8 Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, CA 94304, USA.

9 Foxhollow Technologies Inc., 740 Bay Road, Redwood City, CA 94063, USA.

http://www.nature.com/nature/journal/v444/n7119/abs/nature05357.html


The Antikythera Mechanism is a unique Greek geared device, constructed around the

end of the 2nd Century BC. From previous work1,2,3,4,5,6,7,8,9 it is known that it calculated

and displayed celestial information, particularly cycles such as the phases of the moon

and a luni-solar calendar. Calendars were important to ancient societies10 for timing

agricultural activity and fixing religious festivals. Eclipses and planetary motions were

often interpreted as omens, while the calm regularity of the astronomical cycles must

have been philosophically attractive in an uncertain and violent world. Named after its

place of discovery in 1901 in a Roman shipwreck, the Mechanism is technically more

complex than any known device for at least a millennium afterwards. Its specific

functions have remained controversial11,12,13,14 because its gears and the inscriptions upon

its faces are only fragmentary. Here we report surface imaging and high-resolution X-

ray tomography of the surviving fragments, enabling us to reconstruct the gear function

and double the number of deciphered inscriptions. The Mechanism predicted lunar and

solar eclipses based on Babylonian arithmetic-progression cycles. The inscriptions

support suggestions of mechanical display of planetary positions9,14,16, now lost. In the

second century BC, Hipparchos developed a theory to explain the irregularities of the

Moon's motion across the sky caused by its elliptic orbit. We find a mechanical

realization of this theory in the gearing of the Mechanism, revealing an unexpected

degree of technical sophistication for the period.



The bronze Mechanism (Figure 1), probably hand-driven, was originally housed in a wooden-framed case1 of (uncertain) overall size 340 x 180 x 90 mm. It had front and back doors, with astronomical inscriptions covering much of the exterior of the Mechanism (Figure 2). Our new transcriptions and translations of the Greek texts are given in Supplementary Notes 2 (Glyphs & Inscriptions). The detailed form of the lettering can be dated to the second half of the 2nd Century BC, implying that that the Mechanism was constructed during the period 150-100 BC, slightly earlier than previously suggested1. This is consistent with a date of around 80-60 BC of the wreck1,20 from which the mechanism was recovered by some of the first underwater archaeology. We are able to complete the reconstruction1 of the Back Door inscription with text from fragment E, and characters from fragments A and F. The Front Door is mainly from fragment G. The text is astronomical with many numbers

that could be related to planetary motions. The use of “sterigmos [] – station or stationary point” means where a planet’s apparent motion changes direction and the numbers may relate to planetary cycles. The Back Door inscription mixes mechanical terms about construction (“trunnions”, “gnomon”, “perforations”) with astronomical periods. Of the periods, 223 is the Saros eclipse cycle (see Box for brief explanation of astronomical cycles and periods). We discover the inscription “spiral divided into 235 sections”, which is the key to understanding the function6 of the Upper Back Dial. The references to “golden little sphere” and “little sphere” probably refer to the front zodiac display for the Sun and Moon – including phase for the latter.

The text near the Lower Back Dial includes “Pharos” and “From South (about/around)….Spain ten”. These geographical references, together with previous readings1 of “Towards the East”, “West-North-West” and “West-South-West” suggest an eclipse function for the dial, since solar eclipses occur only at limited geographical sites, and winds were often recorded22,23,24 in antiquity with eclipse observations. Possibly this information was added to the Mechanism during use.

Turning to the dials themselves, the Front Dial displays the position of the Sun and Moon in the Zodiac, and a corresponding calendar1 of 365 days that could be adjusted for leap years. Previously1, it was suggested that the Upper Back Dial might have five concentric rings with 47 divisions per turn, showing the 235 months of the 19-year Metonic Cycle. A later proposal5 augments this with the Upper Subsidiary Dial showing the 76-year Callippic Cycle. Our optical and CT imaging confirm these proposals, with 34 scale markings discovered on the Upper Back Dial. Based on a statistical analysis analogous to that described for gear tooth counts below, we confirm the 235 total divisions. We also find from the CT that the subsidiary dial is indeed divided into quadrants1,6, as required for a Callippic Dial. In agreement with the Back Door inscription, we also substantiate the perceptive proposal5,25 that the dial is in fact a spiral, made from semicircular arcs displaced to two centres on the vertical midline. In the CT of fragment B we find a new feature that explains why the dial is a spiral: a “Pointer-Follower” device (see Figure 3) travelled around the spiral groove to indicate which month (across the five turns of the scale) should be read.

From our CT data of the 48 scale divisions observed in fragments A, E and F, we establish 223 divisions in the four-turn5,25 spiral on the Lower Back Dial, the spiral starting at the bottom of the dial. This is the Saros eclipse cycle, whose number is on the Back Door inscription. The 54-year Exeligmos cycle of three Saros cycles is shown on the Lower Subsidiary Dial.

Between the scale divisions of the Saros Dial we have identified 16 blocks of characters, or “glyphs” (see Supplementary Notes 2 (Glyphs & Inscriptions)) at intervals of one, five and six months. These are eclipse

predictions and contain either for a lunar eclipse (from , Moon) or H for a solar eclipse (from

, Sun) or both. A correlation analysis (analogous to DNA sequence matching) with historic eclipse data26 indicates that over a period of 400 – 1 BC the sequence of eclipses marked by the identified glyphs would be exactly matched by 121 possible start dates. The matching only occurs if the lunar month starts at first crescent and confirms this choice of month start in the Mechanism. The sequences of eclipses can then be used to predict the expected position of glyphs on the whole dial, as seen in Figure 4. The dial starts and finishes with an eclipse. Although Ptolemy indicates that the Greeks recorded eclipses in the 2nd century BC, the Babylonian Saros Canon22,23,24 is the only known source of sufficient data to construct the dial.



The functions of the Mechanism are determined by the tooth counts of the gears. These are based mainly on the CT, using angular measurement from a nominal centre to the remains of tooth tips. In a few cases all teeth can be seen, but many gears are incomplete. Counts are established by fitting models with regularly spaced teeth and minimising the r.m.s. deviation from the measurements—varying the centre in software (when unclear) to find the best-fit solution or solutions (see Supplementary Notes 3 (Gears)). We have adopted a systematic nomenclature of lower case letters for the axis of the gear, with numbering increasing with ordering from the front of the Mechanism. Hypothetical (lost) gears are denoted by italics.

Several models have been proposed for the gear trains1,2,4,5,6,8. We agree with the assumption of four missing gears (n1, n2, p1, p2) to drive the Metonic and Callippic Dials4. We propose a new reconstruction for the other trains, which uses all extant gears (except the lone r1 from the separate fragment D). The proposed model is shown in Figure 5. We require the assumption of only one further gear (m3) whose proposed shaft is clearly broken off in the CT. A detailed description is contained in the Supplementary Notes 3 (Gears).

Of particular note is the dual use of the large gear, e3, at the back of the Mechanism, which has found no use in previous models. In our model, it is powered by m3 as part of a fixed-axis train that turns the Saros and Exeligmos Dials for eclipse prediction and also doubles as the “epicyclic table” for the gears k1, k2. These are part of epicyclic gearing that calculates the theory of the irregular motion of the moon, developed by Hipparchos sometime between 146 and 128 BC28—the “first anomaly”, caused by its elliptical orbit about the Earth. The period of this anomaly is the period from apogee to apogee (the anomalistic month). To realize this theory, the mean sidereal lunar motion is first calculated by gears on axes c, d and e and this is then fed into the epicyclic system. As explained in Figure 6, a pin-and-slot device on the epicyclic gears k1 and k2, clearly seen in the CT, provides the variation. This was previously identified4, but rejected as a lunar mechanism. The remarkable purpose of mounting the pin-and-slot mechanism on the gear e3 is to change the period of variation from sidereal month (i.e. the time taken for the Moon to orbit the Earth relative to the zodiac), which would occur if k1 and k2 were on fixed axes, to anomalistic month—by carrying the gears epicyclically at a rate that is the difference between the rates of the sidereal and anomalistic months, i.e. at the rate of rotation of about 9 years of the Moon’s apogee.

53-teeth gears are awkward to divide. So it may seem surprising that the gearing includes two gears with 53 teeth (f1, l2), whose effects cancel in the train leading to the Saros Dial. But the gearing has been specifically designed so that the “epicyclic table” e3 turns at the rate of rotation of the Moon’s apogee—the factor 53 being derived from the calculation of this rotation from the Metonic and Saros cycles, which are the basis for all the prime factors in the tooth counts of the gears. The establishment of the 53-tooth count of these gears is powerful confirmation of our proposed model of Hipparchos’ lunar theory. The output of this complex system is carried from e6 back through e3 and thence, via e1 and b3, to the zodiac scale on the Front Dial and the lunar phase7 mechanism. Our CT confirms the complex structure of axis e that this model entails.

A major aim of this investigation is to set up a data archive to allow non-invasive future research, and access to this will start in 2007. Details will be available on www.antikythera-mechanism.gr.

The Antikythera Mechanism shows such great economy and ingenuity of design. It stands as a witness to the extraordinary technological potential of Ancient Greece, apparently lost within the Roman Empire.


http://www.antikythera-mechanism.gr/


Box

Astronomical Cycles known to the Babylonians

The lunar (or synodic) month is the interval between the Moon being at the same phase – e.g. full moon to full moon. The Metonic Cycle results from the close equality of 19 years to 235 lunar months. It represents the return to the same phase of the Moon on the same date in the year. After the Cycle the Sun, Moon and Earth are back in nearly the same relative orientations. The Moon appears to return to the same point in the sky relative to the zodiac in a sidereal month, and in 19 years there are 235 + 19 = 254 sidereal months. The 76-year Callippic Cycle is four Metonic Cycles minus one day - and improves the accuracy of reconciling solar years with whole numbers of lunar months.

The Saros is an eclipse repeat cycle. If either a solar or lunar eclipse occurs, a very similar eclipse will occur 223 lunar months later21. A record of past eclipses can thus be used to predict future occurrences. The cycle arises from the coincidence of three orbital periods of the Moon. These are (i) same phase to same phase, 223 synodic months, eclipses will of course only occur at new or full Moon in the month (ii) the lunar crossing of the Earth-Sun orbital plane, 242 draconitic months – eclipses can only occur near these points (nodes) of co-alignment (iii) similar Earth-Moon distances which occur on the period from apogee to apogee of the Moon’s orbit, 239 anomalistic months. The distance will determine the magnitude of the eclipse, ensuring the similarity of eclipses at the period of the cycle. The Saros Cycle is not an integer number of days (6585⅓), causing the eclipses in successive cycles to be displaced by eight hours in time (and solar eclipses, only visible at limited geographical locations, to be displaced by 120˚ in longitude). True repeats come after 3 Saros cycles, the 54-year Exeligmos cycle, but not with identical solar eclipse paths.

Bibliography

1Price, D. de S. Gears from the Greeks: The Antikythera Mechanism — A Calendar Computer from ca. 80 BC, Trans Am. Philos. Soc., New Series, 64, Part 7 (reprinted as Science History Publications, NY 1975), (1974)

2Wright, M.T. Epicyclic Gearing and the Antikythera Mechanism, Part I, Antiquarian Horology, Vol. 27 No. 3, pp. 270-279, March (2003)

3Wright, M.T., Bromley, A. G. and Magkou, E. Simple X-Ray Tomography and the Antikythera Mechanism, PACT 45 (1995) , Proceedings of the conference Archaeometry in South-Eastern Europe, pp. 531-543, April (1991)

4Wright, M.T. The Antikythera Mechanism: a New Gearing Scheme. Bulletin of the Scientific Instrument Society, No. 85, pp. 2-7, (2005)

5Wright, M.T. Epicyclic gearing and the Antikythera Mechanism, Part II. Antiquarian Horology, Vol. 29, No. 1, pp. 51-63, September (2005)

6Wright, M.T. Counting Months and Years: The Upper Back Dial of the Antikythera Mechanism. Bulletin of the Scientific Instrument Society, No. 87, pp. 8-13, (2005)

7Wright, M.T. The Antikythera Mechanism and the Early History of the Moon-Phase Display, Antiquarian Horology, Volume 29, No.3, March 2006, pp. 319-329, (2006)

8Wright, M.T. Understanding the Antikythera Mechanism, Proceedings 2nd International Conferenceon Ancient Greek Technology, Technical Chamber of Greece, Athens, pp 49-60, (2006)

9Wright, M.T. A Planetarium Display for the Antikythera Mechanism. Horological Journal, Volume 144, No.5, pp. 169-173; 144, No.6 p193 (2002)

10North, J.D. The Fontana History of Astronomy and Cosmology, Fontana Press, (1994)

11Bromley, A. G. The Antikythera Mechanism, Horological Journal, Vol 132, pp. 412-415, (1990)

12Bromley, A. G. Antikythera: An Australian-Made Greek Icon!, Bassernet, Vol. 2, No. 3, June 1993, Basser Department of Computer Science, University of Sydney, (1993)

13Freeth, T. The Antikythera Mechanism: 1. Challenging the Classic Research, Mediterranean Archaeology &



Archaeometry, Vol. 2 No. 1, pp. 21-35, (2002)

14Edmunds, M. and Morgan, P. The Antikythera Mechanism: still a mystery of Greek astronomy?, Astronomy & Geophysics, Vol. 41, pp. 6.10-6.17, (2000)

15Toomer, G. J. Ptolemy’s Almagest, translated by G. J. Toomer, with a foreword by Owen Gingerich, Princeton University Press, (1998)

16Freeth, T. The Antikythera Mechanism: 2. Is it Posidonius’ Orrery?, Mediterranean Archaeology & Archaeometry, Vol. 2 No. 2, pp. 45-58, (2002)

17X-Tek Systems Ltd, 3-D Computed Tomography (2006). http://www.xtek.co.uk/ct/

18Malzbender, T. and Gelb, D. Polynomial Texture Mapping, Hewlett-Packard Mobile and Media Systems Laboratory, (2006). http://www.hpl.hp.com/research/ptm/

19Brooks, M. Tricks of the Light, New Scientist, No 2285, 7 April (2001)

20Illsley, J.S. http://cma.soton.ac.uk/HistShip/shlect36.htm

21Britton, J.P. Scientific Astronomy in Pre-Seleucid Babylon. Chapter in H.D. Galter (ed.), Die Rolle der Astronomie in den Kulturen Mesopotamiens. Graz, (1993)

22Stephenson, F. R. Historical Eclipses and Earth’s Rotation, Cambridge University Press, (1997)

23Steele, J. M. Observations and Predictions of Eclipse Times by Early Astronomers, Kluwer Academic Publishers, Dordrecht, ISBN 0-7923-6298-5, (2000)

24Steele, J. M. Eclipse Prediction in Mesopotamia, Arch. Hist. Exact Sci. 54, pp. 421-454, (2000)

25Wright, M.T. The Scholar, the Mechanic and the Antikythera Mechanism. Bulletin of the Scientific Instrument Society, No. 80, pp. 4-11, (2003)

26Espenak F. NASA’s website on eclipses developed by Fred Espenak. All modern eclipse data and predictions in our work are by Fred Espenak, NASA/GSFC (2005). http://sunearth.gsfc.nasa.gov/eclipse/eclipse.html

27Chapman, A. Dividing the Circle, J. Wiley, Chichester, (1995)

28Jones, A. The Adaptation of Babylonian Methods in Greek Numerical Astronomy, Isis, Vol. 82, No. 3, pp. 440-453, Sep (1991)

Received ** August 2006; revised*****,accepted *******.

Supplementary Information accompanies the paper on www.nature.com/nature.

Acknowledgements The authors gratefully acknowledge finance from the Leverhulme Trust, the Walter Hudson Bequest, the University of Athens Research Committee and the Cultural Foundation of the National Bank of Greece. For essential support we thank the Ministry of Culture, Greece (Petros Tatoulis) and the National Archaeological Museum of Athens (Nikolaos Kaltsas). We acknowledge invaluable help and advice from Janet Ambers, Jim Austin, Geraint Dermody, Hazel Forsyth, Ian Freestone, Peter Haycock, Velson Horie, Alexander Jones, Mark Jones, Panagiotis Kipouros, Haralambos Kritzas, Josef Lossl, Gerasimos Makris, Andrew Ray, Christof Reinhart (Volume Graphics GmbH), Ruth Westgate, , Tim Whiteside, Stuart Wright and Costas Xenikakis

Author Contributions statement.

T.F. carried out the majority of the CT analysis of structure and its interpretation. Y.B., A.T. and X.M. read, transcribed and translated the inscriptions. E.M and M.Z. catalogued the fragments, provided guidance on X-ray examination, and measured the fragments with J.H.S. R.H. led the team (D.B., A.R., M.A., A.C. and P.H.) that built and operated the Bladerunner CT machine, and provided CT reconstructions and advice. T.M., D.G. and W.A. built, operated and provided software for the PTM. M.G.E. was academic lead, and undertook the statistical analysis. T.F. and Y.B. organised the logistics of the experimental work, with crucial inter-agency liaison by X.M. and J.H.S. The manuscript was written by T.F. and M.G.E. including material from Y.B., A.T., X.M., J.H.S., E.M. and M.Z. T.F. designed the illustrations.

Author Information The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to M.G.E. ([email protected]).


http://www.xtek.co.uk/ct/

http://www.hpl.hp.com/research/ptm/

http://sunearth.gsfc.nasa.gov/eclipse/eclipse.html

http://www.nature.com/nature


Figure 1: The surviving fragments of the Antikythera Mechanism. The 82 fragments that survive in the National Archaeological Museum in Athens are shown to scale. A key and dimensions are provided in Supplementary Notes 1 (Fragments). The major fragments A, B, C, D are across the top, starting at top left, with E, F, G immediately below them. 27 hand-cut bronze gears are in fragment A and one gear in each of fragments B, C and D. Segments of display scales are in fragments B, C, E and F. A schematic reconstruction is given in Figure 2.

It is not certain that every one of the remaining fragments (numbered 1-75) belong to the Mechanism. The distinctive fragment A, which contains most of the gears, is approximately 180 x 150 mm in size. We have used three principal techniques to investigate the structure and inscriptions of the Antikythera Mechanism. (i) 3-dimensional X-ray microfocus computed tomography17 (CT), developed by X-Tek Systems Ltd. The use of CT has been crucial in making the text legible just beneath the current surfaces. (ii) Digital optical imaging to reveal faint surface detail using Polynomial Texture Mapping (PTM)18,19, developed by Hewlett-Packard Inc. (iii) Digitised high quality conventional film photography.



Figure 2: A schematic view of the Mechanism to illustrate the position of major inscriptions and dials. The Front Dial has two concentric scales. The inner scale shows the Greek Zodiac with 360 divisions. There are occasional Greek letters denoting references to the Parapegma inscription, and we add three further reference

letters (Z, H, ) to Price’s description1. The Parapegma is a star almanac showing rising and settings at dawn or evening of particular stars or constellations, which we will discuss elsewhere. Its form is consistent with a late 2nd Century BC date. The outer (originally) movable scale is a Calendar carrying the Egyptian names of the months with Greek letters. The Egyptian Calendar of 365 days, with twelve 30-day months and 5 extra (epagomenai) days was in standard use in Greek astronomy. The effect of the extra quarter day in a year could be corrected by turning the scale one day every four years – and a sequence of holes to take a locking pin is observed under the scale. We find that spacing of the holes is indeed what would be expected for a total of 365 days, with a possible range 363-365. The position of the Sun and Moon would have been indicated by pointers across the dial scales, and a device7 showing the phase of the Moon was probably carried round on the lunar pointer. It is not clear whether the Sun position pointer would have been separated from a date pointer, or whether any planetary positions might have been displayed. The spiral Upper Back Dial displays the luni-solar Metonic sequence of 235 lunar months with a subsidiary dial showing the Callippic cycle, while the spiral Lower Back Dial displays the 223 lunar month Saros eclipse cycle with a subsidiary dial showing the Exeligmos cycle.



Figure 3: The “pointer-follower” lunar month indicator of the Upper Back Dial. On the left, false-colour sections through CT images, analysed with VGStudio Max software by Volume Graphics GmbH. These show two views at right angles of the pointer-follower in the Metonic dial in fragment B. On the right, a computer reconstruction of the device from two different angles (with the Metonic scale omitted for clarity). The pin was constrained to follow the groove between the spiral scales (the scale is shown in Figure 4), causing the device to slide along the month pointer to indicate which ring on the spiral scale specified the month. A similar pointer-follower would have been present on the Lower Back (Saros) Dial. The Metonic dial would have required re-setting every 19 years, the Saros dial after 18 years. The groove-pin may have been held in place by the small pin through the front of the device, enabling its removal for re-setting.



Figure 4: Reconstruction of the Back Dials. A composite of fragments A, B, E and F. The Metonic Calendar is at top, with its subsidiary Callippic dial. The Saros eclipse cycle is below, with its subsidiary Exeligmos dial. The 16 observed eclipse glyphs are shown in turquoise on the Saros dial, with 35 hypothetical glyphs in violet. The hypothetical glyphs are based on the criterion that 99% of the 121 sequences exactly matching the observed glyphs have an eclipse at the month position. Both main dials would have a “pointer-follower” (see Figure 3) to indicate the relevant lunar month on the spiral. The monthly divisions on the Metonic Upper Back Dial are not simply scribed directly across all five turns, as might be expected for simplicity of construction. There are small misalignments, implying a systematic attempt at marking full (30-day) and hollow (29-day) months. The incomplete data does not allow good analysis, other than a hint of bimodality in the interval distribution. If the marking out of the scale were carried out using the Mechanism’s gearing, then this would greatly predate known “dividing engines”27 by many centuries.



Figure 5: New Reconstruction of the Gear Trains. A schematic sectional diagram (not to scale) of the gearing, following the style of Price1 and Wright4. The viewpoint is looking down from the top right of the Mechanism, and is stretched in the direction of the main axes to show the structure. Features that are outlined or labelled in red are hypothetical. Gears are lettered with their shaft, and numbered with increasing distance from the Front Dial. The two-or-three digit number on the gear is its actual or assumed tooth count (See Supplementary Notes 3 (Gears)). Hypothetical gears n1, n2, p1, p2 have been proposed previously, the gear m3 on the broken-off shaft m is our addition. All gears, except the lone one in fragment D, are now accounted for in the Mechanism. The function of the trains is outlined in the text. We find no evidence in the CT for an idler wheel carried on e3 and between e5 and k1 or between k2 and e6, as has been previously proposed1,2,4. The CT shows a pin through axis e between gears e1 and e2. We believe its purpose is to retain the square-bossed e1 on the shaft, but its passage right through the axis rules out previous reconstructions1,2,4 where e1 and e2 were joined by an outer pipe rotating around the shaft e.



Figure 6: The “Hipparchos” Lunar Mechanism mounted on gear e3. The figure is based on a CT slice of part of fragment A, showing (top) shaft e and (bottom) shaft k. The complete geometry cannot be seen in a single CT slice. The two gears on the e axis (e5 and e6) are coaxial, while the two k gears rotate on slightly displaced axes. k1 has a pin on its face that engages with a radial slot in k2 (and this was previously reported5). In the figure the pitch circles of e5 and k1 are shown in turquoise and those of e6 and k2 in pink. The gear e5 drives k1, which drives k2 via the pin-and-slot introducing a quasi-sinusoidal variation in the motion, which is then transmitted to e6. Our estimate of the distance between the arbors on the k gears is about 1.1 mm, with a pin distance of 9.6 mm, giving an angular variation of 6.5o. According to Ptolemy15, Hipparchos made two estimates for a lunar anomaly parameter, based on eclipse data, which would require angular variations of 5.9˚ or 4.5˚ here – although estimates of the anomaly from Babylonian astronomy were generally larger. The difference from our estimated value is probably not significant given the difficulty of precise measurement of the axes in the CT. The harmonic variation, together with the effect of carrying the gears on e3 which rotates at the period of the Moon’s apogee around the Earth, would simulate the correct variation for the Moon’s mean (sidereal) rotation rate on the Front Dial. An (unexplained) regular pentagon is visible at the centre of gear e5. It is tempting to associate the conception of the Mechanism with Hipparchos himself, but he was not the first to assume eccentric or epicyclic models.



Symbols (glyphs) in the Saros spiral scale Lunar Glyphs

48 54 107 142 218

Solar Glyphs

41 53 100 106 147

Lunar & Solar Glyphs

153 159 165 200 206 212



Supplementary Information Guide There are three Supplementary Information sections: Supplementary Notes 1 (Fragments) giving a key to fragment identification for Figure 1 of the main text and the dimensions of the fragments. Supplementary Notes 2 (Glyphs & Inscriptions) giving details of the script of the characters, their dating and the Greek text and its provisional translation from (a) the Front Door inscriptions, (b) the Back Door inscriptions and (c) the Back Plate inscriptions near the Lower Back Dial. Supplementary Notes 3 (Gears) giving a table to compare gear nomenclature and the gear tooth count estimates with previous estimates and to tabulate measured radii. Some notes are given on the individual gears and on the tooth count estimation procedure, including the effects of uncertainty in determining the centres of the gears. The gear train ratios are explained on the basis of simple Babylonian period relations. The equivalence of the epicyclic gearing and pin-and-slot mechanism to Hipparchos’ theory of the moon is proved.



Supplementary Notes 1 (Fragments) Key to Figure 1 of the main text:



Fragment Area of maximum

section [cm2]

Weight [g]

Thickness of discernible layers (mm)

A 224.209 369.1

B 66.692 99.4

C 65.767 63.8

D 15.491 15.0

E 12.623 22.1

F 50.197 86.2

G 68.757 31.7 7.6 (6 layers)

1 39.189 62.5

2 16.018 15.3

3 14.154 23.5

4 12.195 9.6

5 8.041 6.2

6 7.166 10.9

7 5.846 7.0

8 5.383 3.2 2 (1 layer)

9 3.512 1.7 3.2 (3 layers)

10 2.296 1.2

11 1.262 0.7

12 1.878 0.6

13 1.062 02

14 1.091 0.2

15 0.733 0.1

16 0.629 0.3

17 0.658 0.2

18 0.438 0.1

19 12.822 5.2 1.58 (1 layer)

20 5.920 2.2 1.24(1 layer), 1.0 (1 layer)

21 5.651 2.0 1.0 (1 layer)

22 9.547 2.7 1.6 (1 layer)

23 7.570 5.8 6.9 (6 layers)

24 2.153 0.5 1.0 (1 layer)

25 1.945 0.6 1.0 (1 layer)

26 2.951 1.1 2.6 (1 layer)

27 2.873 1.5 5.3 (5 layers) 1 mm (1layer)

28 3.379 1.1 2.8 (2 layers)

29 3.402 1.0 2.1 (1 layer)

30 1.385 0.3 1.5 (1 layer)

31 9.414 15.8

32 8.585 14.9

33 2.170 1.1

34 0.286 >0.1

35 0.222 0.1

36 0.180 0.1

37 2.027 0.7 1.6 (1 layer)

38 1.575 0.5 1.5 (1 layer)

39 1.376 0.4

40 1.026 0.3

41 1.228 0.5 1.7 (1 layer)

42 0.724 0.2

43 1.079 0.3

44 0.954 0.4



45 1.660 0.6 1.5 (1 layer)

46 0.592 0.2

47 0.911 0.3

48 0.395 0.1

49 0.489 0.1

50 0.322 0.1

51 1.108 0.2 1.5 (1 layer)

52 0.781 0.3 1.9 (1 layer), 1.2 (1 layer)

53 0.849 0.3 2.1 (1 layer)

54 0.651 0.2 1.7 (1 layer)

55 0.881 0.2 1.0 (1 layer), 1.0 (1 layer)

56 0.497 0.2

57 0.346 0.1

58 0.565 0.2

59 0.285 0.1

60 0.604 0.1 1 (1 layer)

61 0.456 0.1

62 0.357 0.1

63 0.334 0.1

64 0.237 >0.1

65 0.266 >0.1

66 0.208 0.1

67 0.528 0.2

68 0.208 0.1

69 0.187 >0.1

70 0.238 >0.1

71 0.270 0.1

72 0.270 0.1

73 0.485 0.1

74 0.201 0.1

75 0.146 0.1

For the dimensions in column 2 we have used digital scans of photographs, taken for us by Costas Xenikakis. The surface area in column 2 is the surface area of the largest section of each fragment (horizontal section, after positioning it down flat on a horizontal surface). Areas were estimated from prints of A4 images by square-counting using transparent-millimeter-graph-paper. Image distortion was checked to be small from horizontal and vertical scales photographed with the fragments. The errors in area measurement are estimated as certainly no more than 0.01 cm2 The thickness of most of the metal sheets appears to be from 1 mm up to 2 mm, except for fragment 26, which is has a layer of 2.6 mm.



Supplementary Notes 2 (Glyphs and Inscriptions) Glyphs

The 16 observed “Glyphs” from the Lower Back Dial. The lunar month number around the Saros dial is shown below each glyph. The data for the glyphs is transcribed directly from the PTMs in the rare cases when it is visible on the surface (e.g. Glyph 206), or from the CT when it is not (e.g. Glyph 218). Glyph 206 was

noted by Price but not interpreted. Nearly all contain (lunar eclipse, from , Moon) or H (solar

eclipse, from , Sun). We classify the glyphs into lunar, solar and lunar & solar, making reasonable inferences where there is only partial information. In the period 400-1 BC there are 121 possible start dates where the month sequence of glyphs exactly match not only the eclipses but also eclipse type. Where there is a lunar & solar glyph, both types occur in the same month. The anchor-like symbol is probably the “omega-rho” denoting “hour” (hora) – probably indicating the predicted hour of the eclipse after sunrise or after

sunset. The hour is indicated by a Greek letter used as a numeral, including in its early form for the number 9. The same symbol also appears in the Parapegma inscription. The eta with mu above it (e.g. in the right hand column of Glyph 218) may be the standard abbreviation of “day” (hemera) – possibly indicating that the (predicted lunar) eclipse was diurnal. Inscriptions Mirror image script found on some fragments is probably due to the accretion of fine silt against the original inscriptions, which became infused with bronze corrosion products and set in a hard matrix against the original. The style of writing is almost identical on the different fragments, except for the text near the Lower Back Dial, whose variation could be due either to the smallness of the characters or to a different hand. Full



details of variant readings and translations of all the inscriptions will be published in due course

Size (mm) Characters in Price (1974)

Characters by this work

Fragment

Position Text Type A I OK ? Total

OK

? Total

A-2 Back door plate Astronomic Mirror 97 24 121

185

8 193

A-2 Lower back dial Misc. Direct 1.2 2.1 46 17 63 51 20 71

B-1 Back door plate Astro/Mech Mirror 2.0 3.4 157 41 198

239

105

344

B-1 Upper back scales Calendrical? - - - - - 5 10 15

C-1 Parapegma Calendrical Direct 2.8 5.0 95 12 107

105

10 115

C-1 Front scales Calendrical Direct - - 23 0 23 23 0 23

C-2 Parapegma Calendrical Mirror 2.7 6.0 13 0 13 16 0 16

E-1 Back door plate Mechanical Mirror 2.0 3.6 - - -

109

8 117

D Gear (internal) - - - - - - - 6 0 6

E-2 Lower back dial Misc. Direct 1.3 2.3 - - - 10 7 17

F Lower back dial Misc. Direct 1.6 2.7 - - - 77 10 87

G-1 Front door plate Astronomic Direct 1.9 2.5 153 27 180

785

147

932

19-1 Back door plate Astro/Mech Direct 2.3 3.5 117 10 127

124

1 125

20-1 Parapegma Calendrical Direct 2.6 - 6 0 6 5 0 5

21-1 Front door plate? Astronomic Mirror 1.9 2.5 45 10 55 39 16 55

22-1 Parapegma Calendrical Direct 2.4 5.0 21 0 21 24 8 32

24-1 Lower back dial Glyph

25-1 ? - Direct - - - 6 1 7

In addition to these, there are visible traces of inscriptions on fragments 23-2, 26-1, 28-1, 29-1, 37-1 to 44-1, 51-2, 53-2, 61-1 and 67-2. A classification of fragments with visible inscriptions could also be made based on the colour surface texture and colour, which, together with text size and type, will help in assembling texts from disparate minor fragments. The “?” indicates a doubtful transcription. Fragment D has the letters “ME” at three different places on a gear wheel. More isolated characters will become available as reconstruction from the CT scans of smaller fragments is completed Totals: Price 923, this work 2160. (A portion of 19-1 is counted twice in B-1, and some minor inscription is not included in the table above) Data from the PTM has proved to be invaluable in inspecting the surface of the fragments and the inscriptions. The CT, whose primary aim was to collect information about the internal structure of the Mechanism, has allowed the discovery of unknown characters within fragments A, B, C, D, E, F and G, and within some of the smaller fragments. The case of Fragment G is exemplary: Price (1974) notes that its inscription is “almost illegible’, reading only 180 characters. The CT images, viewed at various angles, enable us to read 932 characters. The inscription on the fragment F (newly discovered and identified by M. Zafeiropoulou in 2005) has characters whose height is often less than 1.6 mm, totally invisible because they are covered by sea accretions. We propose two reconstitutions: the text from the back door plate, where part of the gap in Price (1974) is completed with text from Fragment E. Based on the internal structure of the fragment, showing portions of the scales, we are able to establish where the first line from fragment E joins with line 28 from fragment B1 and where the last line from fragment E joins with line 34 belonging to the fragment A2 (line 30 in Price). Similar results were produced with the text near the lower back dial, at the right side of the Mechanism. We are able to join characters from fragments A, E and F. We also believe that some characters in smaller



fragments may join with the big and intriguing text from fragment G. The surviving part of the Front Door Plate probably comes from the middle of the original plate and we unfortunately lack the beginning and the end of phrases, and, because of this, possible planet names that would greatly aid interpretation. The mechanical terms of fragment E (trunnions, pointers and gears) are common in Heron’s “Dioptra”.

The frequency of the (Parapegma) key letters in the Zodiac signs on the Front Dial suggest that the 24 letters of the greek alphabet might have been used twice here.

The figure shows an example of part of the inscription from the Back Door Plate on fragment 19, enhanced by the PTM technique. According to Dr. Haralambos Kritzas (Director Emeritus of the Epigraphic Museum, Athens) the style of the writing could date the inscriptions to the second half of the 2nd Century BC and the beginning of the 1st Century BC, with an uncertainty of about one generation (50 years). Dates around 150 BC to 100 BC are a plausible range.

We give here a few examples of the epigraphic clues to the dating, but detailed analysis will be published elsewhere: Π pi has unequal legs - second half of 2nd century BC Σ sigma has the two lines not horizontal but at an angle - second half of 2nd century BC, beginning of 1st century BC Μ mu has the two lines not vertical but at an angle - second half of 2nd century BC. There is one M with vertical lines Y upsilon has the vertical line short - second half of 2nd century BC Α alpha - just post Alexander Ζ zeta is written like I with long horizontal lines - 2nd century BC Ω omega and not like ω - 2nd century BC Β beta unequal upper circle, compared with the lower circle - old Ο omicron very small - old Θ theta has a short line in the middle, in one case a dot - 2nd century BC Φ phi is arc like - old Ξ xi middle line short - old



Greek Text of Front Door Inscription Mainly from fragment G. Red indicates dubious characters

1 2 Ο 3 Ο Δ Ε Η Ο Σ 4 Ο Σ Α Π Ο Σ Τ Η Μ Α 5 Δ Ο Ν Ε Ξ Α Ρ Χ Η Σ Α 6 Ε Σ Π Ο Μ Ε Ν Α Ο Δ Ε 7 Ο Υ Ο Ε Σ Π Ο Α Π Ο Κ Α Τ Α Σ Τ Α Σ 8 Π Α Σ Τ Ο Υ Δ Α Π Ο Κ Ο Τ Α Σ Ν Ν Α Τ Α Ι Σ Ο 9 Λ Ο Π Ρ Ο Σ Τ Ο Ν Η Λ Ι Ο Ν Σ Ο Π Ο Δ Ε 10 ΙΑ Ξ Ι Α Λ Ι Σ Α Σ Κ [Α Ι] Π Ρ Ο Σ Α Γ Ε Ι Ν Ε Π [Ι] Σ Η Σ Τ Ο Ν Η Λ [Ι Ο Ν] 11 [Π Ρ] Ο Σ Α Γ Ε Τ Α Ι Ε Π Ι Τ Ο Ν [Η Λ] Ι Ο Ν Ε Λ Α Σ Σ Ο Ν Α Σ Τ Η Ρ Ι Γ Μ Ο Ι Σ Τ Υ Γ Χ Α Ν Η Α Π Ο Σ 12 [Π Ρ] Ο Σ Α Γ Ε Ι Π Ρ Ο Σ Τ Ο Ν [Η] Λ Ι Ο Ν Ε Ω Σ Η Σ Ε Ι Ο Ν Κ Α Ι Σ Υ Ν Ο Δ Ο Ν Α 13 Ε Π Ι Τ Ο Μ Ε Γ Ι Σ Τ Ο Ν Ε Π Ο Μ Ε Ν Ο Ο Ε Ν Α Λ Λ Α Ι Σ Η Μ Ε Ρ Α Ι [Σ] 14 [Σ Τ Η Ρ Ι Γ] Μ Ο Ν [Ω Σ] Ο Π Ρ Ο Η Γ [Ο ]Υ Μ Ε Ν Ο Σ Α Π Ο Σ Τ Α Λ Θ 15 Α Σ Ε Ν Η Μ Ε Ρ Α [Ν ] Π Ο Ι Ε Ι Π [Ρ Ο ]Ε Ν Ο Σ Ε Ι Σ Τ Η [Ν ]Κ Ε Π 16 [Δ Ι] Α Σ Τ Η Μ Α Τ Ο Σ Π [Ρ ]Ο Σ Α [Ο ] Π Ρ Ο Σ Τ 17 Κ Η Σ Ε Ι Ο Σ Χ [Ε ]Π Ι Τ Ε Μ Χ Α 18 Ο Σ Ι Η Τ Ο Ν Η Λ Ι Ο Ν Η Α Φ Ρ Ο [Δ Ι Τ Η] Η Ν Ο Ι Σ Ν 19 Σ Τ Η Π Ρ Ο Σ Α Γ Ε Ι Π Α Σ Ι Ν Γ Ω Ν Ι Α Ι Σ 20 Ο Ι [Η Μ Ε Ρ Α Σ] Π Ρ Ο Σ Α Γ Ε Ι Τ Η Ν Π Ι Π Ε Τ Α Ν Α Κ Α Σ Α Ε 21 Ε Σ Π Η Μ Ε Ρ Α Σ Κ Α Ι Υ Π Ο Λ Ε Ι Π Ε Τ Α Ι Μ Ε Χ Ρ Ι Τ Η Σ Ε Ω Ι Α Σ Σ Τ 22 Η [Μ Ε Ρ] Α Ι Τ Μ Η Μ Ε Ρ Α Σ Σ Ο Ν Ο Η Ο Ν Ο Τ Ο Υ Λ Α Κ Μ Η 23 Α Ν Ι Σ Α Τ Ο Ν Ε Π Ι Σ Ο Ν Ε Χ Ω Ν Σ Τ Η Ρ Ι Γ Μ Ο Ν Ε Π Ι Σ Χ Ω Ν Α Π Ο Τ Ο Υ Η Λ Ι Ο Υ Ν Ε Σ Λ Ι 24 Α Ε Σ Α Ι Π Ι Σ Ξ Ε Η Λ [Ι Α] Κ Η Ν Π Α Ρ Α Τ Ε Ι Ν Ε Τ [Η Ν] Α Π Ο Σ Τ Α Σ Ι Ν Ε Σ Α Π Ο 25 Ρ Ε Σ Α Σ Ξ Ε Η Λ Ι Α Κ Η Ν Ε Π Ε Τ Ε Ι Ν Ε Ν Τ Ε Σ Σ Α Ρ Α Κ / Ε Ν Α Ε Υ Δ Ο Μ Ο Ν Κ Α 26 Ν Ε Ξ Η Μ Ε Ρ Α Σ Η Γ Ε Ν [Ε] Σ Ε Ω Σ Ε Ν Ε Ν Ε Χ Ε Ω Α Ν Κ Α Τ Α Λ Ο 27 Ε Α Ε Ι Ν Π Δ Ι Α Σ Τ Α Σ Ι Ν Σ Η Μ Ε Ρ Α Ι Σ Μ Ε Ν Μ Ε Γ Α Λ Α Ι Σ 28 Α Κ Α Ι Δ Ω Δ Ε K Α Τ Η Μ Ο Ρ Ι Ο Σ Κ Α Α Φ Α Ι Λ Ε Τ Η Ν Υ Π Ο Λ Ο Ι Π Ο [Ν] 29 Α Μ Ο Ν Ε Ξ Α Π Ο Τ Ο Υ Ε Σ Π Ε Ρ Ι Ν Ο Υ Ε Ε Χ Σ Κ Α Ι Υ Π Ο Λ Ο Ι Π Ο Ν 30 Α Ν Τ Μ Η Μ Ε Ν Χ Ρ Ο Ν Ω Α Π Α Ν Α Ι Ο Ν Η Μ Ε Ρ Α Ι Σ Τ Ο 31 Ε Α Σ Ε Π Α Γ Ε Ι Σ Ρ Λ Θ Ε Π Ι Τ Ο Ν Η Λ Ι Ο Ν Τ Ο Ν Σ Τ Η Ρ Ι Γ Μ Ο Ν [Ε] 32 Ο Ο Ν Ο Μ Α Η Μ Ε Ρ Α Σ Λ Α Π Ρ Ο Η Γ Ε Ι Τ Α Ι Η Μ Ε Ρ Α Ι Σ Τ Ο 33 Ν Α Ν Α Τ Ο Λ Η Σ Ε Ι Ν Α Ι Ο Η Λ Ι Ο Σ Μ Η Μ Ε Ρ Α Σ Π Α Λ Ο Ν 34 Α Ν Α Η Μ Ε Ρ Α Ν Γ Ι Ν Ε Τ Α Ι Η Π Ε Ρ Α Ι Ε Ι Σ 35 [Η Μ] Ε Ρ Α Σ Σ Ε Ι Σ Α Η Μ Ε Ρ Α Ι Α Π Ο Ο Ε 36 Ρ Α Σ Ι Ν Ε Μ 37 T H

Greek Text of Back Door Inscription Black and blue letters are believed to be good, red and orange are dubious. Black and red are from fragments A and B, blue and orange from fragment E. The second column in line numbering in Price (1974: reference 1 of main paper) 1 1 Τ Α Υ Τ Η Ν Δ 2 2 Δ Ι Δ Υ Π Ο Λ Λ 3 3 Υ Π Ο Δ Ε Τ Ο Ν Τ 4 4 Α Τ 5 5 Ε 6 6 7 7 Ο 8 8 Ι Ρ Μ Ο Σ 9 9 Α Κ Ρ Ο Υ Δ 10 10 Μ Ε Ν Ο 11 11 Μ 12 12 Ο Λ Ν 13 13 Υ Π Ο Λ Α 14 14 Ο Υ Δ Σ Φ Α Ι Ρ Ι Ο Ν Φ Ε Ρ Ε 15 15 Π Ρ Ο Ε Χ Ο Ν Α Υ Τ Ο Υ Γ Ν Ω Μ Ο Ν Ι Ο Ν Σ 16 16 Φ Ε Ρ Ε Ι Ω Ν Η Μ Ε Ν Ε Χ Ο Μ Ε Ν 17 17 Τ Ο Σ Τ Ο Δ Ε Δ Ι Α Υ Τ Ο Υ Φ Ε Ρ Ο Μ Ε Ν 18 18 Τ Η Σ Α Φ Ρ Ο Δ Ι Τ Η Ε Ρ Ο Υ 19 19 Τ Ο Υ Σ Ψ Ο Ρ Ο Υ Ι Ε Ε Ρ Ε Τ Α Ν 20 20 Γ Ν Ω Μ Ω Κ Ε Ι Τ Α Ι Χ Ρ Υ Σ Ο Υ Ν Σ Φ Α Ι Ρ Ι Ο Ν 21 21 Η Λ Ι Α Κ Τ Ι Ν Υ Π Ε Ρ Δ Ε Τ Ο Ν Η Λ Ι Ο Ν Ε Σ Τ Ι Ν Κ Υ 22 22 Υ Α Ρ Ε Σ Α Υ Ρ Ο Ε Ν Τ Ο Τ Ο Δ Ε Δ Ι Α Π Ο Ρ Ε 23 23 Ε Θ Ο Ν Ο Σ Τ Ο Δ Ε Δ Ι Α Π Ο Ρ Ε Υ Ο Μ Ε Ν Ο Υ 24 24 Ι Ν Ο Ν Ο Υ Κ Υ Κ Λ Ο Σ Τ Ο Δ Ε Σ Φ Α Ι Ρ Ι Ο Ν Φ 25 25 Μ Ε Τ Ο Υ Κ Ο Σ Μ Ο Υ Κ Ε Ι Τ Α Ι Σ Φ 26 26 Μ Ε Ν Σ Τ Ο Ι Χ Ε Ι Α Π Α Ρ Α Κ Α Ν 27 27 Λ Υ Τ Α Τ Α Ι Σ Α Σ Π Ι Δ 28 28 Α Ο Τ Ω Ν Δ Ι Α Ω Σ Τ Ω Ν Μ Ε Ν 29 - Ν Ο Μ Η Τ Η Ι Ε Λ Ι Κ Ι Τ Μ Η Μ Α Τ Α Σ Λ Ε 30 - Τ Α Ι Σ Κ Α Ι Ε Ξ Α Ι Ρ Ε Σ Ι Μ Ο Ι Η Μ Ε Ρ Α Ι Κ



31 - Χ Ο Ν Σ Τ Η Μ Α Τ Ι Α Δ Υ Ο Π Ε Ρ Ι Τ Υ Μ Π Α Ν 32 - Π Ρ Ο Ε Τ Ρ Η Μ Ε Ν Α Σ Τ Η Μ Α Τ Ι Α Τ Η Μ 33 - Α Τ Ω Ν Τ Ρ Η Μ Α Τ Ω Ν Δ Ι Ε Λ Κ Ε Σ Θ Α Ι 34 30 Ο Μ Ο Ι Α Ω Σ Τ Ο Ι Σ 35 31 Φ Υ Ε Σ Π Ο Ι Η 36 32 Κ Α Ι Σ Υ Μ Φ Υ 37 33 Τ Π Α 38 34 39 35 Ε Ο Υ 40 36 Ε Ν Λ Χ Π Α Ν Ε 41 37 Μ Η Ν Ο Θ Ε Ν Ε Ξ Η Λ 42 38 Τ Η Σ Π Ρ Ω Τ Η Σ Χ Ω Ρ Α Σ 43 39 Μ Ο Ν Ι Α Δ Υ Ο Ω Ν Τ Α Α Κ Ρ Α Φ Ε 44 40 Τ Ε Σ Σ Α Ρ Α Δ Η Λ Ο Ι Δ Ο Μ Ε Ν Τ 45 41 Σ Α Ι Ν Τ Η Σ Ο C L Ι Θ L Τ Ο Υ 46 42 Ο Σ Ε Ι Ε Ι Σ Α Σ Κ Γ Σ Υ Ν Τ Ε Σ 47 43 Ι Ο Ν Τ Ο Σ Δ Ι Α Ι Ρ Ε Θ Η Η Ο Λ Η 48 44 Δ Ο Ι Ε Ε Γ Λ Ε Ι Π Τ Ι Κ Ο Ι Σ 49 45 Ι Ο Μ Ο Τ Ο Ι Σ Ε Π Ι Τ Η Σ Ε 50 46 Χ Ρ Ο Ν Φ Ε Ρ Ε Τ Α 51 47 Π Ι Ν Ε Ν Τ

Greek Text of Back Plate inscription, near the Lower Back Dial Black and blue letters are believed to be good, red and orange are dubious. Black and red are from fragments A and F, blue and orange from the other side of fragment E. 1 Ι Π Ο 2 Ι Κ Ο Λ Ι Τ 3 Ι Ν Ο Ν 4 Α Π Ο Χ Ο Ο 5 Δ Ε Κ Α Τ Ξ Ω Ν Τ Α Ν 6 Λ Ι Β Α Ν Χ Ι Π 7 Λ Μ Α 8 Ν 9 10 Π Ρ Ο 11 Ι Σ Τ Ω Ν Τ Α Σ 12 Κ Α Τ Α Λ Η 13 Π Ρ Ο Σ Α Π Η Λ 14 Ω Τ Η Ν Ω 15 Λ Η Ν Τ Ω 16 Χ Ρ Ω Ν Ι Α 17 Π Υ Ο 18 Ι Ο / / 19 20 Φ 21 Ρ Ι Ι Σ 22 Τ Α Α Δ 23 Ν Ο Τ Ο Ν 24 Κ Α Ι Α Ρ Η Σ 25 Σ Η Φ Α Ρ Ο Σ 26 Λ Ε Ν Τ Η Ν Κ 27 Σ Α Φ Υ Λ Α Ξ Α Σ 28 Ι Λ Α Μ Ε Λ Α Ν 29 Χ 2 Π Κ Ζ Φ 30 Α Π Ο Ν Ο Τ Ο Υ Π Ε Ρ Ι 31 Ι Σ Π Α Ν Ι Α Σ Δ Ε Κ Α 32 Δ Υ Σ Α Ν 33 34



Table 1. Provisional Translation of the Front Door Inscriptions

1. ---

2. ---

3. ---

4. space (or distance) between

5. from the beginning

6. ---

7. --- restore (or which has been restored)

8. ---

9. towards the Sun <370>

10 .equal and brings the Sun upon to the equal

11. brought upon the Sun the minor stationary point <s or 200> then occurs distance

12. brings towards the Sun up to --- and conjunction

13. on to the maximum following within other days

14. [stationa]ry point as the previous one 39

15. day, makes before one to the

16. interval brings upon to the

17. ---

18. <Venus> <approach> the Sun

19. brings upon every <angle> (verb could be coincide)

20. brings upon [days]

21. days and remains until the eastern (eastern = adjective in the sense of dawn)

22. 34<0?> days 270 days ---

23. the stationary point which is at equal distance, is at a distance from the Sun

24. 265 of the Sun, extend the distance

25. 265 of the Sun, has extended four and one seventh

26. 8 days --- of the origin --- dawn

27. interval (or separation, length, distance; greek: diastasin) <2??> large days

28. twelfth part of the circle (greek: dodecatemorios) --- subtract the remaining (genre is feminine)

29. <six> from the evening --- and the remaining

30. in time --- <370> days



31. brings on <139> the Sun the stationary point

32. days 31 is leading <37?> days

33. of the rising is the Sun 40 days

34. day is becoming the <completion>

35. 205 days <equal> days from

36: ---

37: ---

Conventions used:

--- : either unreadable, or non-translatable string

< > : enclosing either dubious characters or one amongst many reading choices (e.g. either number or beginning of word)

[ ] : enclosing restored sections

? : uncertain character

( ) : alternative translation, indication of greek word translated or (if in italics) comment from reader-translator.

Table 2. Provisional Translation of the Back Door Inscriptions

1. this 2. --- 3. and under the 4. --- 5. --- 6. --- 7. --- 8. --- 9. (of the) extremity 10. --- 11. --- 12. --- 13. --- 14. [and is carrying] little [golden] sphere 15. the pointer that protrudes from it 16. carries, of which the next one 17. which is carried through (or the other carried by it) 18. of Venus 19. --- 20. on the [extremity of] the pointer stands a little golden sphere (golden or goldish) 21. the ray [towards the] Sun and above, the Sun is --- 22. --- when it moves through (through its orbit; greek: diaporevomenon) 23. --- and the moving through (same meaning as in line 22) 24. --- circle and the little sphere 25. stands --- the [sphere] of the world (world in greek:cosmos) 26. --- elements --- 27. --- 28. --- 29. the spiral divided in 235 sectors 30. and days to be excluded 2? (twenty to twenty-nine; “excluded” means “taken out of the calendar”) 31. --- two trunnions (greek: stematia) around gear (greek: tympanon) 32. --- perforated trunnions (possibly pre-perforated) 33. through the perforations to be pulled (haul) 34. the same manner as 35. --- 36. --- 37. --- 38. ---



39. --- 40. --- 41. from where it came out of 42. the first position 43. two pointers, whose ends carry 44. four, the one indicates 45. the 76 years, 19 years of the 46. 223 coming together 47. so that the whole will be divided 48. (of the) ecliptic 49. similar to those on the 50. carries 51. --- Conventions used: see Table 1 No translation of the Back Plate Inscriptions, near the Lower Back Dial is attempted, as the text is rather incomplete. Work is in progress.



Supplementary Notes 3 (Gears) The first publication that comprehensively estimated the tooth counts of the gears was Price’s Gears from the Greeks (1974 - reference 1 in the Main Text). The original counts were done for Price by Charalambos and Emily Karakalos (based on conventional film x-radiography). These counts were then adapted by Price to suit his model. M.T. Wright has subsequently re-counted the teeth on digitised X-rays (with some limited use of tomography) undertaken by Bromley and Wright in 1990-1994. Our estimates are based on our CT data.

Gear

Gear Price

Gear Wright

Average outer

radius to gear tips

mm

Inner radius from best-fit circle

0.5 mm

Outer radius from best-fit circle

0.5 mm

Karakalos tooth count

Price tooth count

Wright tooth count

Wright limits

Our best fit tooth count

Our limits

a1 A A 13.60.2 45(48) 48 44-52 48 Definite

b0 B6 20

b1 B1 B1 64.91.1 63.8 65.0 223-226 225 223 216-231

223 223/224

b2 B2 B2 15.50.2 14.9 15.7 64-66 64 64 64 64-66

B3 B3 32 32

b3 B4 B4 8.60.2 8.2 9.3 32 32 32 32 Definite

b4 B6 24

c1 C1 C1 10.30.3 9.4 10.3 38 38 38 38 38/39

c2 C2 C2 11.30.4 10.5 11.0 48 48 48 47-48 47/48 47-49

d1 D1 D1 5.60.3 5.1 5.8 [24] 24 24 24 Definite

d2 D2 D2 31.60.2 30.6 31.7 128 127 127 127 Definite

e1 E1 E6 9.40.3 8.6 9.7 32? 32 32 32 Definite

e2 (E2i)

E7 7.80.2 7.1 7.8 32? 32 32 32 Definite

e3 E4 E4 52.60.3 51.5 52.4 222 222 223 218-228

220-225

217-235

e4 E3 E3 50.20.3 49.1 49.9 192 192 191 188-192

187-191

180-192

e5 (E2ii)

E8 13.40.2 12.2 13.1 (32?) 51 50-52 52 50-52

e6 E5 E5 13.90.2 12.9 13.9 50-52 48 53 51-55 50 49/50

f1 F1 F1 14.00.2 13.6 14.6 54 48 54 53-54 53 53/54

f2 F2 F2 8.30.3 7.4 8.2 30 30 30 30 Definite

g1 G2 G2 14.20.3 13.4 14.4 54/55 60 55 54-55 54 54-56

g2 G1 G1 4.90.1 4.1 4.9 20 20 20 20 20 Definite

h1 H1 H1 14.00.1 13.0 13.7 60-62 60 60 57-64 60-64 60-64

h2 H2 H2 3.90.2 3.0 3.8 16 15 15 15 Definite

i1 I I 13.40.3 12.6 13.2 60 60 60 59-60 60 59-62

k1 (K1) K3 13.50.3 12.6 13.3 (32) 49 48-50 49/50 48-51

k2 K2 K2 14.00.2 13.1 14.0 48 or 51 48 49 48-50 50 48-52

l1 L1 L1 9.10.2 8.3 9.0 36+ 36 38 37-38 38 Definite

l2 L2 L2 13.10.4 12.5 13.3 52 54 53 53 Definite

m1 M1 M1 24.50.5 23.6 24.7 96+ 96 96 95-98 96/97 96-99

m2 M2 M2 4.40.3 3.7 4.0 14 16 15 15 Definite

m3

n1 N1 53

n2 N2 15

o1 O 13.30.1 12.2 12.8 60 57-62 60* 57-61

p1 P1 60

p2 P2 12

q1 Q 5.30.2 24 20 Definite

r1 N Δ1 16.40.2 15.9 16.9 63 64 63 63 Definite

Δ2 65

*Strong preference

(In the table: Price, Karakalos and Wright data taken from Wright, M.T., Bull.Sci.Instr.Soc. 85, 2-7, 2005 – reference 4 in the Main Text) Columns 3-5 give measured radii from CT data. Column 3 is the mean of the radii to the tooth tips from the assumed centres. Columns 4 and 5 gives the radii of “best fit” circles to the pits between the teeth (inner radius) and to the tooth tips (outer radius), with an estimated error of order or less than 0.5 mm. Gears in italics are hypothetical.



Tooth Counting Method

The angular data i of tooth tips from the centre is compared with a model na . The “goodness-of-fit”

is the parameter n

n na2

where n is chosen as the model point closest to the data point.

The shift parameter a is fixed to minimise by requiring that i i

i ia . We then investigate

peaks in /1 as a function of T/360 where T is the implied total number of teeth on the gear, i.e. we seek to minimise . If the errors in the angular data are Gaussian, this should give a “maximum likelihood” estimate of the true total tooth count. The counts are sensitive to the positioning of the assumed centre of the gear. This can be investigated by transformation from a measured set of data (see below), and we endeavour to find the “best” fit – i.e.

strongest peak in /1 - for reasonable variation of the centre position. In some cases a unique tooth count gives a very clear isolated peak, in other cases a range of peaks implies a range of possible total tooth counts, with adopted values implied (subjectively) from the relative peak heights. Gear Count Analysis: Moving the Centre Consider a measured tooth tip T at angle Θ from a given centre O, with a measured distance r from the centre

to the tooth tip. Now move the centre to O by a distance r in direction . The new angle of T from O is

Θ where:

In triangle OOT:

angle OOT = 180 - ( Θ - )

angle OTO = 180 – (Θ - ) – [180 - (Θ- )] = Θ - Θ

By sine rule:

]'sin[)'(180sin

rr

so )'sin(]'sin[

r

r Eqn(1)

Θ

Θ

Δr

r r’

O

O

φ

T



and hence sin'coscos'sinsin'coscos'sin

r

r which re-arranges to:

sinsin'coscoscos'sin

r

r

r

r

Giving finally (or by simple geometry, dropping a perpendicular from T) :

coscos

sinsin

arctan'

r

rr

r

This is the formula used in the analysis of the effects of searching for a “best” centre. A useful approximation

can be made for small r :

From Eqn 1 above: )'sin(]'sin[

r

r

Suppose ' and is small, then

sin)cos(cossin)sin(]'sin[ r

r

r

r

And as 2/1cos;sin

)cos(cos)

21(sin]'sin[

r

r

So to first order if both rr / and are small:

sin]'sin[r

r

So

sin'r

r

By differentiating this, consider the deduced tooth separation with primed and unprimed centres

cos1'r

r

The deduced tooth counts would be /2;'/2' nn , so

Averaging cos over a whole circle of implies ' , but over a half circle (i.e. semi-circular

segment) between 2/ and 2/ (for maximum effect of centre shift)

r

rr

r

2cos1

' 2/

2/

2/

2/

d

d

n

n or

r

r

21

'n

n or

r

r

2nn

As examples, consider the following gears:



e3: radius 52.3 mm; nominal n around 220-230; so a centre shift of 1mm can change the teeth count for a semicircle by 3 h1: radius 14 mm; nominal n of 60-64; so a centre shift of 1mm can change the teeth count for a semicircle by 3. If the segment is less than a semicircle, then the effect can be even stronger, since the average value of

cos( ) will increase a little.

Notes on Individual Gear Measurements On Centres: We estimate that the assumed centre of e3 is within 0.3 mm of the correct centre, both on the x-axis and the y-axis. This implies that the maximum error is about 0.4 mm in terms of distance. Similarly, h1 is within 0.2 mm on both x and y, hence about 0.3 mm in terms of distance. These errors apply to the original hubs. However the teeth are often clearly distorted and moved relative to the hubs. So, for example, there might be a whole sector that has moved by several millimetres relative to the hub even though the stated estimate for the hub itself is within 0.2 mm. However, a best fit circle through the tips may not get a good centre because some of the tips are often corroded to the point of virtual non-existence whereas others look virtually perfect. For nearly all the gears, the assumed centre is based on finding evidence of a hub with some sort of geometry (either a circle or a square). a1 This is the contrate input gear. Since the tooth tips are variable worn, a CT slice has been taken through the tooth pits and the angles of the pits measured rather than the tips. b1 This is the main drive wheel. The teeth are very damaged and mostly non-existent. The centre is easy to identify. The reason for the builder’s choice of 223 or 224 teeth is not known. b2 There is only one sector of teeth. It is a fairly complete gear, though some of the tips are hard to identify. There are two clear central round hubs. They are nearly coaxial and the centre is based on a compromise between the two. b3 It is a complete gear, though some teeth are corroded. c1 The centre is almost impossible to identify accurately. We used a second CT slice that shows it better than the slice for the gear tips. The centre cannot really be identified with a best-fit circle through the tips since they are so damaged. Identification of the tips is also very difficult in some cases. c2 Many of the teeth are hard to identify and the centre is also uncertain. d1 Gear is damaged. Several of the teeth are hard to identify. The centre is clear in one of the CT slices. e1/e2 Both these gears have all their teeth present, though quite damaged in some instances. The angular information should give a fairly good idea of the original variation of angle, but the radial information would not give a good estimate for the original variation of radius. This is because corrosion has removed most of the gear tips in some cases. So the variation of radius mostly represents corrosion differences at the tips, though their angular position has not significantly changed. Even the angular data is not that reliable due to difficulties identifying exactly the right angle (point of the tip) due to corrosion. The hub at e2 (based on our interpretation): This is square with a circle inside it. We believe that the square hub is attached to the pipe that links e2 and e5 and is the way that e2 is attached to the pipe so that it is secure and does not rotate. The circle inside this square hub is the inner shaft that carries the output from e6 to e1 (and thence to b3 and the Front Dial).



e3/e4 The centre has been determined in the same way as e5 and e6 by looking at the pentagon on e5 and the central arbour and pipe on Axis E. Inevitably, it is a bit of a compromise but believed to be quite good. f1 The centre is hard to determine accurately since it looks blurred. We also tried with features in a different CT slice - the bearing in the main plate. This gives a roughly consistent centre. But it could be a few millimetres in any direction. Wright finds "a run of 21 teeth". We find a clear run of 24. Wright also sees "...two points on a projecting tongue of metal on the opposite side...". We think we see these, but that they are not teeth because they are not at the right radius from the centre (wherever the centre actually is). So we have not included them. From images of Axis F: The axis can be seen as a hole just above the centre and just to the right. Gear f2 can be seen as a row of six teeth. We think that a single tooth of f1 can be seen below f2 and slightly to the right (and maybe even three more teeth). It is likely that this is what Wright identified as another couple of teeth. In any case, the CT shows that this part has been displaced from its correct position and should be discounted as data. f2 All the teeth are present, though some are very difficult to see and cannot all be seen in a single CT slice. g1 All the data points are good. The square hub at the centre is fairly clear. There is a part of the gear that has broken off, which contains some extra damaged teeth. None of these have been used as data since they are very clearly in the wrong place. g2 Nearly all the tips are present, but the central square hub is not clear. It was centred using the hub in a parallel CT slice h1 Much of this gear is damaged or missing. The hub is hard to locate and a different CT slice has been used to determine the best guess for the centre. h2 All the teeth are identifiable, though some are a bit faint. The centre is difficult to locate exactly. l1 The data is just one sector. l2 There is a prominent sector of teeth. These teeth are well-formed and regular. A second sector is much more doubtful. The gear is severely cracked and the tips are very worn. There is a clear central hub and the centre is based on this. o1 About one third of the teeth are present and in good condition. The margins of the teeth have the “characteristic” higher density and it looks as if a strip of teeth might be about to tear off as in the gear r1 in Fragment D. The centre was found from a round arbour in a parallel CT slice. q1 This is the small contrate gear in the Moon Phase Mechanism in Fragment C. r1 This is the gear in Fragment D. It is certain that there is only one gear in Fragment D. The outside strip is only present where the gears attached to the main body are rudimentary and conversely, where they are not rudimentary, there is no outside strip. The inside of the torn-off strip of gears matches, tooth for tooth, the shape of the outside of the remaining teeth attached to the body of the gear - where the inside remnant is pointed, the outside one is also; where it is rounded, the outside is also etc. It is clear also in the cross-section that the hub of the gear has split where a pin went through the arbour.



The Gear Trains The existing gear trains appear to be based almost completely (except for the amplitude of the lunar anomaly) on the following Babylonian period relationships: “Metonic” 19 year = 235 synodic months = 254 sidereal months “Saros” 223 synodic months = 239 anomalistic months After a Saros cycle the Moon has returned to its same position relative to the nodes – the points where the Moon’s orbit crosses the plane of the Sun-Earth orbit, and in between has come close enough to a node alignment with the Sun for 38 eclipse possibilities for both sun and moon29. When a gear with p teeth engages with a gear of q teeth, then the second gear rotates at a rate of –p/q times the first. In the following we have suppressed the minus signs for simplicity. The Main Drive and the Date Pointer The Mechanism was driven through crown gear a1, presumably operated by a hand-turned shaft, which drives the large four-spoked gear wheel b1. The outer shaft of b, on which b1 is fixed, probably turned a date marker on the Front Dial, which could also indicate the approximate position of the Sun in the Zodiac. One revolution of b1 represents a year. b2 is fixed to b1 and hence the rest of the Mechanism. The Metonic Gearing The outer shaft of b, indicating the date/mean Sun position rotates once per year. The Metonic dial has 5 full turns in 19 years. So required ratio is:

19

5

The gearing b2-l1+l2-m1+m2-n1 has the ratio:

19

5

53

53

323

53

192

322

53

15

96

53

38

64

as required.

One turn on the “Callippic ” dial of four Metonic cycles of five turns each of the axis n is generated by n2-p1+p2-01 with ratio:

5

1

4

1

60

12

60

15 as required.

The Saros Gearing The Saros dial has four full turns in 223 synodic months. There are 235 synodic months in 19 years, so the required ratio is

19

235

223

4



The gearing b2-l1+l2-m1+m3-e3+e4-f1+f2-g1 has the ratio:

19

235

223

4

1

5

223

1

1

474

19

1

272

532

53

474

223

27

323

53

192

322

54

30

53

188

223

27

96

53

38

64

as required. One turn on the “Exeligmos” dial is three Saros cycles of four turns each of axis g and is generated by g2-h1+h2-i1 with ratio

4

1

3

1

60

15

60

20 as required.

The Sidereal Month The Moon is carried around the Sun with the Earth, so in 19 years there are 19 extra rotations of the Moon relative to the Zodiac in addition to the 235 “synodic” rotations, the origin of the period relation 254 sidereal months = 235 synodic months. Gear e2, and also (we believe) the outer shaft of axis e with e5 attached, rotates once every sidereal month. This requires the ratio:

19

254

235

254

19

235

The gearing b2-c1+c2-d1+d2-e2 has the ratio:

19

254

19

1272

32

127

24

242

192

322

32

127

24

48

38

64

as required.

The Lunar Anomaly Hipparchos developed two equivalent lunar theories based on the idea that the moon exhibits a simple periodic anomaly. In the first eccentric theory, the Moon rotates at the rate of the mean sidereal month about an eccentre that in turn rotates about the Earth at the rate of the anomalistic month. In the second theory, the Moon rotates on an epicycle at the rate of the anomalistic month relative to a deferent circle that rotates at the rate of the sidereal month. Apollonius of Perga (c. 240-190 BC) had already shown30,31 that these are equivalent using (in today’s language) a simple vector diagram and the commutativity of vector addition. The theory introduces a harmonic variation into the Moon’s motion that has the period of the anomalistic month. The lunar display is again driven from b2. The train b2-c1+c2-d1+d2-e2 results in e2 turning with the period of the sidereal month (i.e. position of the Moon relative to the Zodiac). The subsequent gears in the train introduce no further multiplication or division, but introduce a quasi-sinusoidal variation in the Moon’s motion at the period of the anomalistic month – i.e. modelling the “first anomaly”. The sequence starts with an outer shaft, which is free to turn within e3, connecting e2 to e5. The train is then e5-k1+k2-e6+e1-b3 and through to the lunar pointer and phase mechanism on the Front Dial. The link e6+e1 is via the inner shaft of e. A pin-and-slot device on gears k1 and k2, clearly seen in the CT, provides the variation. This device was originally identified by Wright (reference 5 of main text), although he rejected its use as a lunar mechanism. The purpose of mounting the pin-and-slot mechanism on the gear e3 is to change the period of variation from sidereal



month, which would occur if k1 and k2 were on fixed axes, to the anomalistic month by carrying the gears at a rate that is the difference between the rates of the sidereal and anomalistic months – i.e. at the rate of rotation of the Moon’s apogee. We show that this models Hipparchos’ lunar theory. All rotations will be measured in rotations per year with clockwise rotations on the Front Dial being positive. Negative rotations on the Back Dials are clockwise. All the dials of the Antikythera Mechanism run clockwise. Let ωSyn be the rotation of the synodic lunar cycle; ωSi the rotation of the sidereal lunar cycle; ωa the rotation of the Moon’s return to an apse (i.e. the rotational speed corresponding to the anomalistic month); ωn the rotation of the line of apses (apogee and perigee) of the Moon’s orbit. From the Metonic and Saros relationships, we get:

368.1219

235Syn

Si 254

1913.368

a 239

223Syn

239

223235

1913.256

254 239 235 1 56642 56165 477

254 223 239 23519 223 19 19 223 19 223 19 223

0.1126

n Si a

The gearing from the main drive wheel to e3 is b2-l1+l2-m1+m3-e3 and has the ratio:

64

3853

96

27

223 2 32

21953

3 3239

223

539

19223

477

19223

So e3 rotates at the rate of rotation of the line of apses (angular speed ωn ). This is how the prime factor 53 arises in the tooth counts. Here we restore the minus signs to be certain of getting the sense of rotation of the gears correct. The minus sign here means that e3 rotates clockwise if viewed from the back of the Mechanism. In what follows, it is essential to distinguish absolute and relative rotations in the epicyclic system. In order to calculate the rotations on the epicyclic system, we need to look at the rotations relative to e3 since the gears on e3 are on axes that are fixed relative to e3. We can therefore use the basic properties of fixed-axis gearing to calculate the rotations relative to e3. As discussed above, in absolute terms, e5 rotates at the rate of -ωSi and e3 rotates at the rate of -ωn . So, relative to e3, e5 rotates at the rate of -ωSi – (-ωn ) = -ωSi + (ωSi - ωa) = -ωa. Since e5 and k1 both have 50 teeth, relative to e3, k1 rotates at the rate of ωa. This is the critical factor that ensures that the anomaly introduced by the pin-and-slot mechanism has the period of the anomalistic month, as required by Hipparchos’ lunar theory.



Pin-and-Slot Mechanism In the diagram, e3, e5 and e6 rotate about E, k1

rotates about K and k2 rotates about K. The pin-and-slot mechanism on k1/k2 introduces a small quasi-sinusoidal variation in k2’s rotation rate. As k1 rotates, the pin on its face engages

with the slot on k2. k2 rotates about K and is forced to rotate by the pin-and-slot arrangement. The difference between the blue and magenta arrows shows the magnitude of the variation introduced. The period of rotation of k2 relative to e3 is the same as k1—in other words the anomalistic month. Let A(x) be a function (the “anomaly function”) that is the difference between the rotation of k2 and that of k1 after x rotations. This has the correct geometric form for Hipparchos’ eccentric lunar theory and we demonstrate that it acquires the correct period by means of its eccentric placement. We assume that the origin of x is set so that A(0) = A(0.5) = A(1) = 0. Since rotations are no longer constant when the pin-and-slot mechanism takes effect, we need to introduce a time parameter (expressed in years). Recall that rotations are measured in rotations per year. The rotation of k1 relative to e3 at time t is ωat. The rotation of k2 relative to e3 at time t is then given by: ωat + A(ωat). Since k2 and e6 have the same number of teeth, relative to e3,

they rotate at the same rate in opposite directions. So the rotation of e6 relative to e3 at time t is: -ωat - A(ωat) Returning now to absolute rotations, the absolute rotation of e6 is its rotation relative to e3 plus the absolute rotation of e3. In other words: Rotation of e6 at time t = -ωat - A(ωat) + (-ωnt) = -ωat - A(ωat) + (ωa - ωSi)t = - (ωSit + A(ωat)) So e6 rotates at the rate of the mean sidereal month plus an eccentric anomaly that has the period of the anomalistic month. This is Hipparchos’ first lunar theory. In the Mechanism, e6 is linked by a shaft to e1 that engages with b3 (that has the same tooth count as e1) and thence to the lunar indicators on the front dial.



Geometric Proof

We now also give a geometric proof using elementary methods, and so in principle accessible in ancient Greece, establishing that the pin-and-slot mechanism is equivalent to Hipparchos’ epicyclic lunar theory. In the diagram, Q is a point obtained by reflecting the centre of pin P in a rotating mirror defined by the line that is tangent to both pitch circles of e5 and k1. This is referred to as the “e5-k1 mirror” and Q as the “mirror pin”. From Q construct a line segment QR (shown in black) that is parallel to and of the same length as the line segment KK’. The line ER will be our output. First we establish that the mirror pin Q rotates as if fixed to wheel e5. As established previously, e5 rotates at the rate of -ωSi, the epicyclic table e3 rotates at the rate -ωn; and the rate of rotation of e5 relative to e3 is -ωa. In addition, P is fixed to k1 and so rotates at the rate of ωa relative to e3. Therefore its mirror Q rotates at the rate of -ωa relative to e3, which is the same rate as the rotation of e5 relative to e3. Also, EQ = KP, so Q moves on a circle centred at E. Thus Q is fixed to e5 and rotates at the rate ωSi. In order to show that the mechanism satisfies Hipparchos’ lunar theory we want to show that

the rotation of R about Q relative to e5 is ωa and that R appears fixed to e6.

QR is defined to be parallel to KK. So in absolute terms it rotates at the rate of - ωn (the rate of rotation of

e3). But Q rotates about E at the absolute rate ωSi. So, relative to e5, QR rotates at the rate -

ωn Si Si n = ωa. It remains to show that R is fixed to e6. The triangle EQR is congruent to the mirror image of the triangle

PKK. This is because QR is defined to be equal to KK and EQ is equal to KP. Also QR is parallel to KK,

so angle KKP is equal to angle RQE. Hence the angle between the blue and magenta arrows at Axis E is the

same as the angle between the blue and magenta arrows at Axes K and K (in the other direction). In other

words, ER mirrors KP and so R appears to be fixed to e6. In fact it is not difficult to see that ER is the mirror

of KP in the e6-k2 mirror. This establishes that the pin-and-slot mechanism models Hipparchos’ epicyclic lunar theory — subject only to

the correct eccentricity of axis K relative to axis K.



An Alternative Period for the Lunar Anomaly? The period relation 223 synodic months = 239 anomalistic months is not surprising for the period of construction. But if we try to associate the conception of the mechanism with Hipparchos it might be wondered why the Mechanism does not use the relation 251 synodic months = 269 anomalistic months that he is believed to have preferred (e.g. G.J. Toomer’s Note 10 on page 176 of his edition of Ptolemy’s Almagest, Princeton University Press, 1998 – reference 15 in the main text). The period of rotation of the apsides given by the 223/239 relation is 8.8826 years, and by the 251/269 relation is 8.8479 years, which is certainly closer to the modern value 8.8504 years. If Hipparchos was involved, then presumably it was the possibility of the combination of the gearing with the Saros train that appealed and which also avoided a large (and perhaps difficult to accommodate) additional prime 251 gear. Further References

29J. Britton and C. Walker, Astronomy and Astrology in Mesopotamia in Astronomy Before the Telescope, Ed. C. Walker, British Museum Press, 1996 30O. Neugebauer, Apollonius’ Planetary Theory, Communications on Pure and Applied Mathematics Vol 8, pp641-648, 1995 [Reprinted in O. Neugebauer, Astronomy and History, Springer, New York, pp311-318, 1983] 31O. Neugebauer, The Equivalence of Eccentric and Epicyclic Motion According to Apollonius, Scripta Mathematica, Vol 24, pp5-21, 1959


decoding the ancient greek astronomical calculator known as the antikythera mechanism

Documents