Abstract The ancient Greek astronomical calculating machine, known as the Antikythera Mechanism, predicted eclipses, based on the 223-lunar month Saros cycle. Eclipses are indicated on a four-turn spiral Saros Dial by glyphs, which describe type and time of eclipse and include alphabetical index letters, referring to solar eclipse inscriptions. These include Index Letter Groups, describing shared eclipse characteristics. The grouping and ordering of the index letters, the organization of the inscriptions and the eclipse times have previously been unsolved. A new reading and interpretation of data from the back plate of the Antikythera Mechanism, including the glyphs, the index letters and the eclipse inscriptions, has resulted in substantial changes to previously published work. Based on these new readings, two arithmetical models are presented here that explain the complete eclipse prediction scheme. The first model solves the glyph distribution, the grouping and anomalous ordering of the index letters and the structure of the inscriptions. It also implies the existence of lost lunar eclipse inscriptions. The second model closely matches the glyph times and explains the four-turn spiral of the Saros Dial. Together, these models imply a surprisingly early epoch for the Antikythera Mechanism. The ancient Greeks built a machine that can predict, for many years ahead, not only eclipses but also a remarkable array of their characteristics, such as directions of obscuration, magnitude, colour, angular diameter of the Moon, relationship with the Moon’s node and eclipse time. It was not entirely accurate, but it was an astonishing achievement for its era.

Citation: Freeth T (2014) Eclipse Prediction on the Ancient Greek Astronomical Calculating Machine Known as the Antikythera Mechanism. PLoS ONE 9(7): e103275. https://doi.org/10.1371/journal.pone.0103275 Editor: Luis M. Rocha, Indiana University, United States of America Received: January 30, 2014; Accepted: June 4, 2014; Published: July 30, 2014 Copyright: © 2014 Tony Freeth. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: The author has no support or funding to report. Competing interests: The author is Managing Director of Images First Ltd, a film and television production company, and is an employee of this company. The author’s academic research is carried out entirely independently of the company’s commercial interests. In particular, there are no conflicts of interest in terms of employment, consultancy, patents, products in development or marketed products etc. The author’s role in Images First Ltd in no way alters adherence to PLOS ONE policies on sharing data and materials. Images First Ltd does not own any rights regarding the Antikythera data.

Materials and Methods This study is about the structure of eclipse prediction on the Antikythera Mechanism. Much of the relevant data comes from highly fragmentary inscriptions on the back plate, which are often very hard to decipher. Two techniques were used in the 2005 investigations [1]. PTM [2] combines many digital images, lit from different directions, with computer software (Figure 5 (A)–(G)). This gives the facility to interactively re-light a surface as well as the ability to factor out confusions of colour and texture to reveal essential surface details. A range of filters, such as specular enhancement, diffuse gain and unsharp masking, enable the data to be visualized for maximum character recognition. X-ray CT [3] projects images of the sample from many different angles onto an X-ray detector. These are then combined mathematically into a 3D X-ray volume. X-ray CT viewing software, for example, VGStudio Max (Volume Graphics), enables both 3D volumes as well as single “slices” at any angle through the volume to be isolated and analyzed. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 5. Inscriptions data from the back plate. (A) Fragment A, PTM of back plate with specular enhancement. (B) Fragment A, PTM of back plate with diffuse gain. (C) Fragment A, PTM of back plate with specular enhancement. (D) Fragment A, PTM of impression of back cover with luminance unsharp masking. (E) Fragment A, X-ray CT slice of back plate. (F) Fragment F, X-ray CT slice of back plate. (G) Fragment E, X-ray CT slice of back plate. (H) Fragment E, X-ray CT slice of accretion layer. (I) Fragment E, orthogonal X-ray CT slice of back plate and accretion layer. https://doi.org/10.1371/journal.pone.0103275.g005 To enable the reconstruction of the text shown in Figure 5, more than a hundred X-ray CT slices were exported as image stacks into Photoshop (Adobe) to enable the decipherment of the text. Together with PTMs, these enabled the surviving inscriptions to be traced using a digitizing tablet. The text characters are on average 1.6 mm high, with average line spacing of 2.5 mm. This is tiny text and the small size creates problems reading many of the characters, though it is remarkable how much has been preserved after 2,000 years under water. The quality of the X-ray CT data is variable between fragments. The X-ray technique involves projection of the sample from a microfocus X-ray source onto a 2D detector [3]. To fill the detector, the smaller fragments can be geometrically magnified to a greater degree than the larger fragments: so the resulting 3D X-ray volumes have inherently higher resolution. The resolution for Fragment E was 46 microns; for Fragment F 64 microns; and for Fragment A 101 microns (Scan 5). The highest resolution scan of Fragment A (Scan 6, 54 microns) was seriously compromised by a technical problem during data acquisition, whereby about 27 projections (out of 2,957) failed to record. There is also evidence that the fragment moved during the scan. Attempts to rectify these problems have only been partially successful. The difficulty with lack of resolution of the X-ray CT of Fragment A can be seen in Figure 5 (E). There have been considerable advances in X-ray CT technology since 2005, so it would be of great advantage to gather new X-ray CT data on the Antikythera Mechanism: there is much that cannot be read from the current data and X-ray CT has been developing rapidly in recent years. Some characters are easy to read. For those that are not, many X-ray CT slices, just a few tens of microns apart, are often useful. A character sometimes appears to change as the slices are scrolled through–for example, from Λ to Δ to Α to part of Μ. It is often difficult to get a definitive interpretation, since many random marks often confuse the text. Another aspect, which is sometimes helpful, is that much of the text is overlain with an accretion layer that also includes text information. The text was engraved into bronze: the accretion layer must have built up gradually on the surface, moulding itself to the form of the text letters and finally concreting into a hard deposit over time [1]. This has created a cast of the original engraved surface. The effect of the accretion layer on scrolling through X-ray CT slices is illustrated in Figure 5 (G) and (H). The text characters first appear as black on grey, Figure 5 (G)–black showing where the engraving tool has removed the metal, so there is an absence of X-ray density; then as white or light grey on dark grey, Figure 5 (H), where the X-ray CT slice intersects the cast of the same text characters in the accretion layer. In places the accretion layer has survived better than the original engraving. The advantage can be seen in reading the third character in the top row of the text: in the direct engraving in Figure 5 (G) this character is hard to read; in the accretion layer image in Figure 5 (H) it is evidently Β. In many cases the accretion layer has become detached and slightly displaced from its original position, as seen in Figure 5 (I). In the case of the back cover inscription, Figure 5 (D), most of the original text has been lost and all that is left is the accretion layer, which was deposited onto Fragments A and B and only survives as mirror text on their surfaces [1], [6]. The Antikythera Mechanism is conserved in the National Archaeological Museum in Athens, Greece (http://www.namuseum.gr/collections/bronze/ellinistiki/ellinistiki06-en.html; Accession Number X 15087). Full data from the 2005 investigations can be accessed by application to the Antikythera Mechanism Research Project (http://www.antikythera-mechanism.gr/). All necessary permits for these investigations were obtained from the Central Archaeological Council in Greece.

Conclusions An epoch for the Antikythera Mechanism in 205 BC brings it close to the life of Archimedes, who was killed in the siege of Syracuse in Sicily in 212 BC. It is known from the writings of Cicero that Archimedes built a machine just like the Antikythera Mechanism [6]: “… the famous Sicilian had been endowed with greater genius than one would imagine it possible for a human being to possess… this… globe… on which were delineated the motions of the sun and moon and of those five stars which are called wanderers… (the five planets)… Archimedes… had thought out a way to represent accurately by a single device for turning the globe those various and divergent movements with their different rates of speed…” Cicero, De re publica, 54–51 BC It also brings the Antikythera Mechanism close to Apollonios of Perga, who died in about 190 BC. He initiated the epicyclic theories [15] on which the lunar and (very likely) the planetary mechanisms were based [5]. It would be purely speculative to suggest that the Antikythera Mechanism owed its design to the greatest mathematician and scientist from ancient times, Archimedes, in collaboration with one of the greatest mathematicians and geometers, Apollonios of Perga. The historical record is so fragmentary that it could have been made by an unknown genius, with knowledge of the mathematical astronomy of the era, who made one of the greatest technological advances of all time, yet has left no known trace on history–except the Antikythera Mechanism! The author has found it very productive to view the Antikythera Mechanism from his own academic background as a mathematician. Though subjective, this perspective, emphasizing the idea that the Antikythera Mechanism was essentially a mathematician’s instrument, has proved very successful in discovering its structure and functions. Its Earth-Sun-Moon system has a brilliant design, based on two great arithmetic cycles from ancient Babylon and the beautiful geometric theory of lunar motion from ancient Greece [1]. The mechanism that calculates the lunar phases is an exquisite and economic differential design [17]. The likely incorporation of the planets into the Antikythera Mechanism was almost certainly based on arithmetic period relations from Babylon and virtuoso epicyclic mechanisms to follow variable motions, just like the lunar anomaly mechanism [5]. The design of the upper Metonic calendar dial, with its five-turn spiral of 235 lunar months and 110 excluded days, is a highly ingenious concept [4]. Figure 11 shows a computer reconstruction of the Saros and Exeligmos Dials and associated solar eclipse inscriptions. The mathematical basis of the Antikythera Mechanism is further underlined by this research article, with its eclipse prediction scheme based on the four-turn geometry of the Saros Dial and synchronized with the Full Moon Cycle. It was driven by the Saros cycle–a surprising arithmetic resonance between three orbital periods of the Moon. It was designed using whole number arithmetic, which was highly regarded in ancient Greece [18] as well as the remarkable arithmetic prediction schemes of ancient Babylon [8], [9], [11]. The Antikythera Mechanism was an inspired synthesis of arithmetic and geometry as well as of Babylonian and Greek scientific cultures. It was a brilliant mathematician’s creation. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 11. Computer reconstruction of the Saros and Exeligmos Dials. https://doi.org/10.1371/journal.pone.0103275.g011 The main text is enhanced with notes, some of which include supplementary references. Note S2 includes additional references [19], [20], [21], [22], [23], [24], [25]. Note S3 includes additional references [26], [27], [28]. Note S4 includes additional references [29], [30], [31], [32], [33], [34], [35], [36], [37].

Acknowledgments I am particularly indebted to C. Crowther for his epigraphic input: to R. Gautschy, M. Ossendrijver, J. Evans and C. C. Carman for supplying data and research papers; and to X. Moussas for his support. I would like to thank F. Espenak, who created and maintains the NASA/GSFC eclipse website. Many thanks to the staff of the National Archaeological Museum in Athens, T. Malzbender and his team of imaging experts from Hewlett-Packard, R. Hadland and his team of X-ray specialists from X-Tek Systems (now part of Nikon Metrology), and all in the Antikythera Mechanism Research Project, who were part of the data gathering team in 2005. This article is partly based on data processed, with permission, from the archive of experimental investigations by the Antikythera Mechanism Research Project (Freeth et al. 2006 [1]) in collaboration with the National Archaeological Museum of Athens. The data gathering and subsequent analysis, on which this current research depends, received essential funding from the Leverhulme Trust, the Walter Hudson Bequest, the University of Athens Research Committee, the National Bank of Greece Cultural Foundation, the J. F. Costopoulos Foundation and the A. G. Leventis Foundation.

Author Contributions Conceived and designed the experiments: TF. Performed the experiments: TF. Analyzed the data: TF. Contributed reagents/materials/analysis tools: TF. Wrote the paper: TF. Figures: TF.