Reference / Paper · 2021

A Model of the Cosmos in the Ancient Greek Antikythera Mechanism — Supplementary Information

Supplementary information to the 2021 UCL model: gear trains, period relations and CT evidence.

Author: T. Freeth, D. Higgon, A. Dacanalis, L. MacDonald, M. Georgakopoulou, A. Wojcik
Year: 2021
Type: Reference / Paper
Credit: Scientific Reports (CC BY 4.0)

antikythera
planets
gear trains

Supplementary Information A Model of the Cosmos in the ancient Greek Antikythera Mechanism Tony Freeth*, David Higgon, Aris Dacanalis, Lindsay MacDonald, Myrto Georgakopoulou, Adam Wojcik* * Corresponding authors: Tony Freeth: [email protected] Adam Wojcik: [email protected] Contents Supplementary Information Supplementary Discussion S1: The Physical Evidence Supplementary Discussion S2: Texts & Inscriptions Supplementary Discussion S3: Planetary Periods Supplementary Discussion S4: Theoretical Mechanisms Supplementary Discussion S5: Matching the Evidence Supplementary Discussion S6: Reconstructing the Cosmos Supplementary Figs. S1–S25. Supplementary Tables S1–S9. Image Credits & Software References 1–26 & Supplementary References 27–61 Supplementary Video S1 Gear-by-gear computer reconstruction of the Cosmos of the Antikythera Mechanism. Supplementary Video S2 A track round all the surviving and reconstructed gears of the Antikythera Mechanism. Supplementary Video S3 An exploded model of the Antikythera Mechanism assembles itself into the complete device. Antikythera Cosmos Supplementary Discussion S1: The Physical Evidence 1. Supplementary Discussion S1: The Physical Evidence 1.1 Review of the Data After its discovery in 1901 by Greek sponge divers, the recovered part of the Antikythera Mechanism was taken to the National Archaeological Museum in Athens, where it has remained ever since2. It is now split into 82 fragments. The major fragments are labelled A – G and the minor fragments 1 – 757. These fragments are the primary evidence for the Mechanism and how it worked. Our project is to reconstruct the front gearing system and dials for the Cosmos. The main fragments that are relevant for this study are shown in Supplementary Fig. S1. The data come from two sources: direct evidence from Fragments A, C, D9 (Supplementary Discussion S5) and indirect evidence from the fragments that reveal the inscriptions: Front Cover—Fragments G, 3, small fragments7,12; Back Cover— Fragments A, B, E,19, small fragments9,13. The evidence consists of high-resolution still photographs, surface imaging with Polynomial Texture Mapping (PTM) and high-resolution 3D X-rays with Microfocus X-ray Computed Tomography (X-ray CT) from new data gathered in 20057,8,22. The X-ray CT showed the surviving internal structure of the device in three dimensions and produced a remarkable revelation: several thousand Greek text characters, which had not been seen since the machine was lost more than 2,000 years ago7,8. These cannot be read in 2D X-rays. Supplementary Fig. S1. Fragments of the Antikythera Mechanism with evidence for this study. The data is PTM, using specular enhancement, from the 2005 investigations7. Fragments A and D are the physical evidence for the Cosmos epicyclic system1,2,3,9. Fragment G is the primary evidence of the Front Cover Inscription but numerous other small fragments contribute scraps of text to the interpretation9,12. The front of Fragment C shows the Zodiac and Calendar scales and has text for the Parapegma; the back includes the lunar phase mechanism10. The back of Fragment A shows the lunar epicyclic system. The Back Cover Inscription is evidenced by an amalgam of Fragments A, B, E and 1913. Fragment B is covered in mirror text with an impression of the original text, which is now lost. —1— Antikythera Cosmos Supplementary Discussion S1: The Physical Evidence These new data have led to a revolution in our understanding of the Antikythera Mechanism7,8,9,23,27. Amongst many discoveries, was an understanding of a remarkable previous observation6 of a pin-and-slot device, which had not been understood. This astonishing discovery revealed a mechanical arrangement for tracking the variable motion of the Moon—the so-called lunar anomaly—using an ancient Greek epicyclic theory of lunar motion. It also explained the surprising identification of epicyclic gearing in the Antikythera Mechanism1,2,6,7, which had previously been misinterpreted. Subsequent work has established beyond doubt that the planets were included in the Antikythera Mechanism1,3,21,9 and very likely used similar mechanical devices to calculate their variable motions. The discovery of epicyclic gearing and the pin-and-slot device in the Antikythera Mechanism has triggered a fundamental reassessment of technology in the ancient world. 1.2 Reconstruction Principles The Antikythera Mechanism’s 82 surviving fragments constitute only about a third of the original device. What principles should we use in reconstructing the original machine? Some parts of the Antikythera Mechanism have now been reconstructed to general agreement, with evidence so persuasive that these aspects are unlikely ever to change: the gears that calculate the sidereal Metonic Cycle2; the synodic Metonic Cycle and its display on the fiveturn spiral Metonic calendar4; the Saros Cycle that predicts eclipses on the four-turn spiral of the Lower Back Dial7; and the gear system that calculates the lunar anomaly and indicates this on the Zodiac Dial7. These involve hypothetical reconstructions of missing gear teeth and missing gears, but the contingent evidence is so strong that these restorations are forced by the data. Other parts of the reconstruction are more conjectural—for example, the Kallippic Dial4,7, a subsidiary dial inside the Metonic calendar, where no physical evidence survives. It is justified because the 76-year Kallippic Cycle is explicitly described in the BCI1,2 and it is the underlying cycle for the calendrical functions on the back of the Mechanism. It is also very easy to mechanize from existing gearing4,7. It is a highly plausible conjecture: not forced by the data but with strong evidence. Our model is rooted in direct evidence of mechanical features in the fragments and indirect evidence from the numerous inscriptions, but we cannot rule out that future discoveries may lead to revisions. A key aspect of our reconstruction has been a very close examination of all the data: all previous models have largely chosen to ignore inconvenient data. With only a third of the original Mechanism surviving, we have followed four Reconstruction Principles in constructing our current model, with a fifth as work-in-progress for our ongoing and future research. 1. 2. 3. 4. 5. Utilise and conform to all the surviving evidence. Reflect the content of the inscriptions in the gears and dials. Match to contemporary astronomy and technology. Prefer simple and economical solutions. Prove the feasibility of the model by making it with ancient techniques. This paper is essentially about our theoretical model of the Antikythera Mechanism, which conforms to Reconstruction Principles 1-4. Experimental archaeology, fulfilling Reconstruction Principle 5, will be a key focus for future research and publications by our team at UCL. We anticipate that this process will tell us much about ancient Greek technology, not just about the Antikythera Mechanism. Some elements of our model are forced by the evidence; some are strongly indicated; and some are hypothetical additions. For each element, we describe the strength of data that validates it. The main structural features of our model are determined by the data: most of the choices are of less consequential aspects, such as the number of teeth of the idler gears. Other choices involve additional hypothetical elements, which fit the evidence and harmonize with its known functions, but which are not prescribed by it. These features respect all the evidence and often explain it. We have created the first model of the Antikythera Mechanism that matches the surviving evidence. —2— Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions 2. Supplementary Discussion S2: Texts & Inscriptions 2.1 Historical Texts Historical texts give strong support for the idea that devices such as the Antikythera Mechanism existed in Hellenistic Greece and that they calculated planetary positions. In Gears from the Greeks2, Derek de Solla Price cites two remarkable quotes from the Roman politician and lawyer, Marcus Tullius Cicero. These appear to describe similar instruments. In De re publica I, xiv (21–22) (54–51 BC), there is the following description of a machine made by Archimedes (c. 287–212 BC): . . .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. . . And when Gallus moved the globe, it was actually true that the moon was always as many revolutions behind the sun on the bronze contrivance as would agree with the number of days it was behind it in the sky. Thus, the same eclipse of the sun happened on the globe as it would actually happen. In De natura deorum II, xxxiv-xxxv (87-88) (45 BC), Cicero writes of a device made by Posidonios of Apamea (c. 135–51 BC), which is almost certainly a first-hand report, since Cicero knew Posidonios and attended his lectures in his Stoic school of philosophy in Rhodes. It describes: ...the orrery recently constructed by our friend Posidonios, which at each revolution reproduces the same motions of the sun, the moon and the five planets that take place in the heavens every day and night... Both descriptions sound very much like the Antikythera Mechanism and offer convincing support to the idea that such machines existed in Hellenistic times. It seems clear from the descriptions that the bronze devices tracked the positions of the planets in the Zodiac, but the descriptions are not specific enough to define their structures. Pappus of Alexandria (c. 290–350 AD) reports that Archimedes wrote a treatise On Sphere Making28, which apparently described astronomical devices, perhaps like the Antikythera Mechanism. This treatise is now lost and there are no known copies. 2.2 Cosmological Order & Textual Analysis Both the Back and Front Cover Inscriptions describe the planets in a particular cosmological order. Expanding this with Moon and Sun, this order is: Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn. This is directly relevant to the inscriptions and the structure of our model. In terms of a modern heliocentric view of the Solar System, the cosmological order of the visible planets is clear: Mercury, Venus, Earth, Mars, Jupiter, Saturn are ordered by their increasing distances from the Sun and their increasing orbital periods. For each planet, this distance varies because of their elliptical orbits, but these variations are small. The situation in a geocentric frame-of-reference is entirely different, since the distances from the Earth to the planets are very variable: for example, Mars is sometimes closer to Earth than Mercury. So, there is no natural order of the planets in terms of their distances from the Earth. In terms of orbital periods in a geocentric frame-of-reference, the mean motions of the superior planets—Mars, Jupiter, Saturn—are progressively slower against the stars, but the inferior planets—Mercury, Venus—have the same mean period of rotation as the Sun. It is not surprising therefore that there was much confusion in antiquity about the cosmological order. Some orders were mentioned in Babylonian and Egyptian astronomy, but they were primarily divinatory/astrological in nature and have little relevance for our study14. The oldest reference to a celestial planetary order in Greek astronomy comes down to us from Theon of Smyrna (1st century AD to 2nd century AD)19, who suggested that the Pythagoreans (6th - 4th century BC) held as true the order: Mercury, Venus, Sun, Mars, Jupiter, Saturn. This is the order in the inscriptions on the Antikythera Mechanism. Plato (c. 426–348 BC) made two references to an order of the planets in his dialogues Timaeus and in The Republic (Book X)29: Sun, Venus, Mercury, Mars, Jupiter, Saturn. A scheme was developed by Aristotle (384 – 322 BC) and his work On the Heavens (350 BC)30, but his ordering of the planets is not the —3— Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions order on the Antikythera Mechanism. Influenced by Aristotle, the order of the superior planets as Mars, Jupiter, Saturn in terms of their progressively longer periods became increasingly universal, but there was still confusion about the inferior planets and the Sun. Geminos (1st century BC), whose astronomical exposition in his Introduction to the Phenomena31 reflects much of the Antikythera Mechanism, asserted the same order as on the Antikythera Mechanism. We shall refer to this order as the “customary cosmological order (CCO)” in ancient Greece. It became the standard order in the medieval and renaissance periods (Supplementary Fig. S2a-c). There is now a very large body of literature related to epigraphic analysis of the inscriptions on the Antikythera Mechanism, which is based on broad scholarship over more than a century of research. Whilst an appreciation of such linguistic analysis has been important to our understanding of the Antikythera Mechanism, detailed investigation extends beyond the scientific and engineering scope of this paper. For the interested reader, the most accessible overview of epigraphic research is given in a series of papers, published in the journal Almagest. The relevant papers for this study are included in the References12,13,24,25. These papers include comprehensive references to the wider body of literature. 2.3 The Back Cover Inscription The Back Cover Inscription (BCI)9,13 (Fig. 1c), shown in Supplementary Table S1, is part of the key evidence that the planets were included in the Antikythera Mechanism. Detailed analyses were published previously9,13. The exact interpretation informs us about the layout of the Cosmos Display. In the light of its crucial importance and the previous decipherment of the text and its translation, we perform our own analysis to confirm the form of the display. Supplementary Table S1. Greek Text and English Translation of the Back Cover Inscription (BCI). Text and translation reproduced from a previous publication 13. Our line-by-line analysis corroborates the previous assessment of this text13, whilst highlighting individual words and phrases. We concentrate on Lines 15 – 30. 15 [ ̣ ̣]οθ̣ε ̣ ̣ τὸ σφαιρίον φερε ̣[ Translation: the little sphere [is carried] —4— Antikythera Cosmos • Supplementary Discussion S2: Texts & Inscriptions σφαιρίον = “tiny sphere” note ending -ιον following ‘sphere’, sphere-ion= little sphere. 16 προέχον αὐτοῦ γνωμόνιον σ[ Translation: Little ruler is protruding from it • gnomon = same word for sundial stick as well as being the word for a geometrical rule. • gnomon-ion = tiny stick / tiny ruler (ending -ion indicating small size). • It is not therefore necessary that it had a pointed end—a sundial stick and a ruler tend not to. There are other words that could have better described a pointy end; this can also imply a relatively thin (as in not wide) rectangular stick protruding outwards, sometimes interpreted as a stick with a truncated end instead of a symmetrical pointer3. 17 φερειῶν ἡ μὲν ἐχομένη τῶι τῆς [ • “φερειων”: this seems to be the last part of the composite word “περιφερειων” which is the plural of “periphery” i.e. what we interpret as the annuli. i.e. "peripheries". • “ ἡ μὲν” = “ and the”; “ἐχομένη” = adjacent to 18 τος, τὸ δὲ δι ̓ αὐτοῦ φερόμεν[ον Translation: …and the [little sphere] being borne through it... • • • The participle “being borne through” seems to agree with the word sphairion (the ending is missing); noun + participle are the subject of a verb which is missing in the gap. In Lines 18, 23, 24 and 25, “τὸ δὲ” was previously translated13 simply as “and”. The development of language from Ancient to Hellenistic Greek32,33 suggests that it might be translated as “the other”. In this case, these lines would describe a planet by its name but refer to another planet (“the other”) in the description of its orbit. This would be a surprising structure for these formulaic inscriptions, but we leave the reader to decide this issue in the context of the cited references. From the point of view of our reconstruction, these considerations make no difference to our reconstruction: the essential role of the inscriptions for our reconstruction is to describe the planets as orbiting in rings, marked by “little spheres”: this is realized in our model in the ring structure of the planetary display. 19 τῆς Ἀφροδίτη<ς> Φωσφόρου ̣ ̣ ̣[ Translation: of the Light-Bearing (in the sense of Light bringer) Aphrodite [Venus] • Φως-φό ρος = light-bearer (φ=phi; phosphorus) - translated as “Lucifer” in Latin. 20 τοῦ [Φω]σφόρου περιφέρειαν ̣[ • The periphery of Phosphorus [Venus]. 21 γνώμω[ ̣] κεῖται χρυσoῦν σφαιρίον ̣ ̣[ Translation: …stick lies a little golden sphere • κείται = it lies from verb κείμαι = to lie, to be found somewhere. • This passage describes a little golden sphere lying on another little stick—again, probably a pointer. It clearly refers to the Sun. • σφαιρίον—see note under Line 15. 22 Ἡλι̣[ου] ἀκτίν ̓, ὑπὲρ δὲ τὸν Ἥλιόν ἐστιν κυ[ Translation: ray of the Sun; beyond the Sun lies a circle • Ἡλι̣[ου] ἀκτίν ̓. Ήλιος=Helios= Sun & ἀκτίν ̓α = ray (the word is the mathematical term for a radius). • The term «ἀκτίν» is in the accusative singular case. The passage may describe a sunray of some sort or it may refer to a pointer, carrying the Sun. • ‘Υπερ’: υπερ = hyper = over, beyond, above. This is indicating that the description of the circle that follows is progressing outwards, beyond the Sun. i.e. —5— Antikythera Cosmos • Supplementary Discussion S2: Texts & Inscriptions there lies a circle beyond where the Sun can be seen. So, up to now we have a series of circles, then the Sun and then more circles (circular paths). This suggests the cosmological order where the Sun is above Mercury and Venus. Κύκλος = cycle = circle. It is now clear that the periphery described above is a circle, strongly suggesting that the description is of concentric circular bands. 23 [ -3- το]ῦ Ἄρεως Πυρόεντος, τὸ δὲ διαπορε[υόμενον Translation: …of Fiery Mars, and travelling through it • For interpretation of “τὸ δὲ”, please see the note under Line 18. • Διαπορευό μενον = δια + πορεύ ομαι. δια = through, πορεύ ομαι = to walk, to move. • This word and the pattern before indicates another little sphere “moving through” a path. “It”, the object of the verb, probably refers to the ‘circle’ in the previous sentence; therefore, 22 combined with 23 seems to say that the circle beyond the Sun is that of the Fiery Mars, and that there is a little sphere moving through that circle. It is now becoming apparent that these circles are not literally geometrical circles; the word is used in the sense of a circular path; perhaps more akin to an annulus rather than a circle of no width. 24 [Διὸς Φα]έθοντος, τὸ δὲ διαπορευόμενον̣ [ Translation: …of Brilliant Zeus, and travelling through it • For interpretation of “τὸ δὲ”, please see the note under Line 18. 25 [νου Φα]ίνοντ̣ος κύκλος, τὸ δὲ σφαιρίον φλ̣[ Translation: …circle [of Saturn], and the little sphere • For interpretation of “τὸ δὲ”, please see the note under Line 18. • The repetitive nature of this passage and its similarity to the previous ones clearly make up a pattern: the description progresses outwards from the centre, identifying the “circles”, or pathways, belonging to each planet, followed by a description of the sphere of that planet—along the lines of: “beyond the Sun lies the circle of Mars, with the little sphere being [colour/material], beyond that lies the circle of Jupiter, with the little sphere being [colour/material], beyond that… 26 [ -7- ]α̣ρα δ̣ὲ τοῦ κόσμου κεῖται ̣ ̣ ̣[ Translation: next to the [world] lies Note that there has been a minor revision of the text here: the uncertain ε̣ has been revised to an uncertain α̣ to match the word Παρά = next to. • Κόσμος = world, universe related to the verb ‘κοσμῶ’ meaning to adorn. Κόσμημα = jewellery ‘item of beauty to be adorned’. • Παρά = next to. • This sentence wraps up the description of what appears to have been nothing other than… the world, the Cosmos. This clearly indicates the intention of whatever is being described to be interpreted as a description of the Cosmos, i.e. the depiction of a Cosmological model with the Earth in the centre, with the five planets and the Sun and the Moon moving in concentric circular paths around it. • Whatever follows this sentence is a description of something which lies beyond the planetarium. It probably refers to the Zodiac annulus or the Egyptian Calendar ring. One would expect the Zodiac to be part of the Cosmos, since the sphere of the fixed stars is the outmost sphere in ancient Greek cosmology30. 27 [ -10- ]μεν[ ̣] στοιχεῖα παρακείμ[ενα Translation: elements lying next to • Στοιχεία = elements; the derivative verb στοιχῶ = to arrange in a straight line, to line up. • Word στοιχεία used in the sense of “elements”, “data”, “information” which is presented for the viewer. Lying next implies there is information lying next to something else – we interpret this as a reference to text lined up next to the Cosmos display, potentially index letters and their descriptions. —6— Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions 28 [ -12- ] α̣ υτα ταῖς ἀσπιδ̣[ίσκαις Translation: …the little shields • ᾿Ασπιδίσκος = small shield. Not simply a disc; a shield implying a curved surface. Δισκος on its own would imply a flat disc; ἀσπιδίσκος implies a curved surface like a shield; the term is in plural form implying a number of such little shields. From the position of this term in the text, it seems likely that these would be outside the planetarium display. • We do not understand the "little shields" in Lines 28 and 30. One possibility is that they refer to the thumb catches34 that held the front dial plate to the Mechanism as clearly seen in the X-ray CT of Fragment C, though it is unclear why these minor mechanical details should be included here. Another possibility is that the phrase "little shields" (ἀσπιδ̣[ίσκαις) refers to a set of stars in the constellation of Argo—the "little shields" protecting the argonauts from arrows on the ship Argo (“little shield” stars are referenced in Ptolemy’s description of Argo35 (Almagest, p.389). Again, it is difficult to see why these stars in a southern constellation would be included here in what could be a description of the Zodiac constellations and major stars that are included in the Parapegma. 29 [ -12- ] π̣ρ̣οειρημένα̣[ Translation: aforementioned 30 [ -16- ]α̣σ̣π̣[ ̣ ̣] ̣[ Translation: … disk(s)(?) … • Ασπ potential element of ἀσπιδίσκος – potential mention of the little shields, possibly the thumb catches, or maybe a removable cover that “shields” the inner workings of the Mechanism. 2.3.1 Structure of the BCI & Implications for the Cosmos Display Supplementary Table S2 shows the structure of the BCI. For the planets, it follows a pattern: “…beyond the Sun lies the circle of Mars, with the little sphere… beyond that lies the circle of Jupiter, with the little sphere… beyond that…” —all in the CCO. The Greek text uses a number of similar words—arcs, periphery, circle. We do not distinguish between them, since they all appear to refer to the ring structure of the Cosmos display. We use the term “indicator” to describe the “little Spheres” that mark the astronomical bodies and the term “pointer” to refer to pointers that indicate ecliptic longitudes on the Zodiac Dial. Supplementary Table S2. Structure of Planetary Inscriptions. Features are highlighted in red where there is evidence—though there are many uncertainties. Two conjectured lines of inscription (in italics) about the Moon and Nodes are included. Evidence in Fragment C shows that the Moon was represented at the front, with its phases shown on a small black-white or semi-silvered sphere10, as in many later astronomical devices20. The other lines show indications of the planets from the BCI. As shown in Supplementary Fig. S2, the same structure is found in geocentric visions of the Cosmos throughout the Medieval and Renaissance periods, which derived so much from ancient Greece. In Supplementary Fig. S2A, note the use of Faeton for Jupiter—the ancient Greek secular name. In Supplementary Fig. S2a-c planets are nested in rings in the CCO. —7— Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions a b c d e f Supplementary Fig. S2. Customary Ancient Greek Cosmos from Mediaeval & Renaissance sources. (a) 10th century copy of a 7th century book On the Nature of Things by Isidore of Seville. (b) Bede, De natura rerum, 11th century AD. (c) The Spheres Between Heaven and Hell, with Fallen Angels Becoming Devils, Neville of Hornby Book of Hours, 14th century AD. (d) Prague, Czech Republic, 1410. (e) Clock by Guillaume Nourrisson, St Jean, Lyon, France, 1661. (f) Clock by Francesco and Giovan Battista Divizioli, 1583-88, in the Torrazzo of Cremona, Italy. For image credits, see information at the end of this document. It is this image that we propose was reproduced in the Cosmos display on the Antikythera Mechanism9. In our model, we have not included “Fallen Angels Becoming Devils” (Supplementary Fig. S2f), because the Antikythera Mechanism is a secular device, which is entirely devoted to scientific astronomy! In Supplementary Fig. S2d-f there are images of astronomical clocks from the 15th-17th centuries20. The dates of the clocks are given but all have been extensively restored and changed over the centuries. All these clocks show the Sun with rays and the phase and position of the Moon. The clock in Supplementary Fig. S2f has a Dragon Hand. The clockfaces are surrounded by a zodiac and a calendar—just as in the Antikythera Mechanism. None of them includes the planets—maybe because the simple ancient Greek epicyclic theories of the planets were known by this era to be inaccurate. A noteworthy clock was the 14th century AD Planetarium of Giovanni de’ Dondi36, which incorporated Ptolemy’s advanced theories of the planets, but this had separate faces for each planet: it would have been very difficult to incorporate Ptolemy’s theories in a coaxial output system. In the period 1905-06, a pioneering researcher, Rehm1, proposed a ring system for the planets (Supplementary Fig. S17), wrongly suggesting that the five rings in Fragment B were for the five planets. We now know that these rings were a fixed part of the Metonic Calendar on the Upper Back Dial4,7. A model of the Antikythera Mechanism (Price & Deroski) in the National Archaeological Museum in Athens has concentric rings for the Sun and Moon. In another publication9, the authors (Freeth & Jones) wanted to create a ring display for the planets because of the description in the BCI but were unable to find a satisfactory mechanical solution for the Moon phase device (Supplementary Discussion S6). It is possible that the Moon and the Nodes were described in the missing lines of text above the surviving inscriptions in the BCI. From the inscriptions we can infer the type of display indicator for each of the astronomical bodies—Supplementary Table S2. The Earth is at the centre. The Moon display is witnessed by Fragment C, which shows that the Moon’s phases were represented by a little black-and-white (or semi-silvered) sphere of about 6 mm diameter10. The Moon’s position in the Zodiac was surely indicated by a pointer attached to —8— Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions the Moon phase cylinder, though this part of the cylinder is missing—possibly torn off by the pointer during the shipwreck. We have no evidence for the Line of Nodes, so this option is included in Supplementary Table S2 in italics and represented by a hypothetical doubleended Dragon Hand, indicating the ascending and descending nodes—often found in mediaeval astronomical clocks20 (Supplementary Fig. S2f). The term Dragon Hand possibly originated in ancient myths that solar eclipses occur when a dragon devours the Sun, such as the Tinnīn of the Arabs, the Djawzahr of the Persians and the demon Rãhu of Hindu mythology37. We interpret the inscriptional data as follows: the Moon and Sun are indicated by “little spheres” with “pointers” attached—the Sun being portrayed with a “ray”, as in many mediaeval astronomical clocks (Supplementary Fig. S2d-f); the Moon’s phase is indicated and the true Sun pointer is attached to a ring; the planets are shown in rings; the Line of Nodes of the Moon by a hypothetical Dragon Hand20; the planets are indicated by “little spheres” attached to concentric rings in the CCO. 2.3.2 Markers for the astronomical bodies An astrological tradition used “magic stones” as markers for the planets, as in a 2nd/3rd century AD papyrus38,9 about an “Astrologer’s Board”, where the astrologer lays out stones: ...a voice comes to you speaking. Let the stars be set upon the board in accordance with [their] nature except for the Sun and Moon. And let the Sun be golden, the Moon silver, Kronos of obsidian, Ares of reddish onyx, Aphrodite lapis lazuli veined with gold, Hermes turquoise; let Zeus be of (whitish?) stone, crystalline (?)... a b c Supplementary Fig. S3. Examples of Hellenistic jewellery, inlaid with semi-precious stones. (a) Bracelets from the Olbia Treasure, 2nd/1st century BC. (b) Jewelled Gold Necklace and Earrings, 1st Century BC. (c) Necklace and Earrings, Late Hellenistic, 1st century BC. For image credits, see information at the end of this document. In Supplementary Fig. S3 there are illustrations of a well-established practice in Hellenistic jewellery of using semi-precious stones with a smooth polished cabochon cut and an oval or circular shape—ideal as markers for the planets. The small-scale workmanship inspires confidence that the skills existed in ancient Greece to make the Antikythera Mechanism. Our model of the Cosmos display, using semi-precious stones as markers, can be seen in Fig. 7, Supplementary Figs. S24, 25 and Supplementary Videos S1, S3. 2.4 The Front Cover Inscription (FCI) 2.4.1 Planetary Motions The inscriptions leave no doubt that the planets were included in the Antikythera Mechanism. The modern term planet derives from the ancient Greek term ‘πλανήτης ἀστέρας’ (planḗtēs astéras) or wandering star. The movements of the planets fascinated the ancient astronomers because of their irregularity. We think of the Solar System as being Sun-centred, with planets rotating in nearly circular orbits. For the ancient astronomers the motions of the planets were Earth-centred, and they were a deep problem because they displayed retrograde motion, whereby they sometimes reverse direction relative to the stars—referred to in astronomy as anomaly. Mostly the planets move along the ecliptic in prograde motion but sometimes they come to a halt, a stationary point, and then reverse direction in retrograde motion before resuming their prograde motion. We think of prograde —9— Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions motion as being represented by anticlockwise rotation: this is how it appears when viewed from north of the ecliptic. However, ancient astronomers tended to view the system from the south celestial pole, as in stereographic projection, which underlies ancient treatises and astrolabes20,42. This is the clockwise convention found on the Antikythera Mechanism. a b c Supplementary Fig. S4. X-ray CT slices showing text in Fragment G. The numbers    (462) in Line 6 and    (442) in Line 33 are highlighted in red. (a) Composite of X-ray CT slices of Fragment G, divided into different regions for each planet, though the exact divisions are uncertain. (b) Close-up X-ray CT slices, spaced at 7 microns, of Line 6 of the FCI. (c) Close-up X-ray CT slices, spaced at 3 microns, of Line 33 of the FCI. — 10 — Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions In Supplementary Fig. S4 there is X-ray CT data of the Front Cover Inscription (FCI), which survives in Fragment G and other small fragments12: it describes parameters of the synodic cycles of the planets. This crucial evidence lists both the main events of the synodic cycles and the time intervals in days between them in the CCO—the same order as the planets on the Back Cover Inscription. Throughout these discussions, blue numbers and gears will incorporate the factors of the number of synodic cycles and red numbers and gears the factors of the number of years. A recent publication12 has identified the surprising numbers 462 in the Venus section and 442 in the Saturn section. The publication that revealed these numbers did not show the evidence for their existence nor did it draw meaningful conclusions about mechanical structure from the newly-discovered period relations: the authors wrote that this was outside the scope of their study. In our view, these numbers are so important that we must show the evidence, so that the numbers cannot be in doubt. In Supplementary Fig. S4b, the characters  can be clearly seen in the X-ray CT slices. In Supplementary Fig. S4c, the characters  can also be seen: the  is harder to decipher than the other characters but it is definite when examined carefully.  does not occur in ancient Greek as part of a longer word39. There are Greek words that include , but the adjacent areas of the plate show that  is not part of a longer word in this context. So, these characters must be numerals:  means 462;  means 442.  is in the section of the Front Cover Inscription that describes Venus and  in the section for Saturn. The newly-discovered periods translate into period relations as follows12: Venus: (289, 462), Saturn: (427, 442). These periods are very surprising in their exquisite accuracy. They clearly have implications for the design of the planetary mechanisms. The authors of the paper that revealed these new periods12 write about Venus: The ratio 462 : 289 factorizes as (2 x 3 x 7 x 11) : (17 x 17), so it can be represented by a gear train with reasonable tooth counts, e.g. (66 : 51) x (63 : 51). They do not describe how this might have worked for Venus in the Antikythera Mechanism. If only four gears are used, then the epicyclic gear would turn in the wrong direction, so an idler gear is necessary. We therefore reject this inappropriate gear train: it does not turn the epicycle in the right direction, and it does not fit the evidence from Fragment A (Supplementary Fig. S9). Gear trains that actually work in the context of the Antikythera Mechanism are discussed in Supplementary Discussion S4. Despite the tooth count 63 being identified for a possible gear, the authors surprisingly make no link with the 63-tooth gear in Fragment D (Supplementary Fig. S13, Supplementary Discussion S5). Accuracy of planetary mechanisms In a pioneering paper3, Michael Wright describes how the gear ratios for the planetary mechanisms were chosen to be accurate to 1° in 500 years. The author used compound gearing to model these highly accurate periods and subsequently40 reinforced his arguments for the inclusion of compound gear trains, incorporating the newly-discovered period relation for Venus into his model16. A previous publication21 showed that simple epicyclic theories of the planets are inherently very inaccurate. For example, Mars will be wrong by as much as 38° within two years. The reason is that the retrogrades of the epicyclic theories are mechanically equal, whereas the actual retrogrades of the planets are of unequal size and are unevenly distributed through the Zodiac, because the planetary orbits are elliptical not circular and are not in the plane of the ecliptic41,42. It seemed unnecessary to pursue the long-term accuracy of compound mechanisms: simple Babylonian Goal Year period relations and the consequent simple gearing should surely be accurate enough. The new periods for Venus and Saturn show that the ancient Greeks did not take this view and used highly accurate period relations. A consequence is that the ancient Greeks were almost certainly unaware of the highly inaccurate nature of their epicyclic theories when the Antikythera Mechanism was made. It has been written14,42 that Hipparchos (c.190 –120 BC) did not propose a theory of planetary motion, maybe because he was aware of the inaccuracies of epicyclic theories but was unable to discover a better theory. Perhaps Hipparchos was influenced by the evident inaccuracies of mechanisms such as the Antikythera Mechanism. — 11 — Antikythera Cosmos Supplementary Discussion S2: Texts & Inscriptions The new period relations must surely mean that they were intended to be used in the gearing of the Mechanism, rather than just being astronomical information in the inscriptions16. Since compound gear trains have already been discovered in the Antikythera Mechanism—for example, for calculating the sidereal Metonic Cycle2—their use for planetary mechanisms is completely justified by our Reconstruction Principles. 2.4.2 Front Cover It has been suggested that the Front and Back Covers might have been hinged to the Mechanism2. The inscriptions on the Front Cover are on the outside, so they could not be viewed at the same time as the Front Dials, if the Front Cover were hinged. Since we propose that the inscriptions in the FCI were indexed to the planetary rings (Fig. 8), the user would need to be able to see both Front Cover and Cosmos Display at the same time. We therefore suggest that the Front Cover was on a separate plate that could be detached from the Mechanism. The Front Cover both protected the front display and gave additional information about the synodic cycles of the planets. There has been a debate about whether the FCI covered the whole front face of the Mechanism or only covered the Cosmos Display12. The Parapegma on the Front Plate above and below the dials8,12 is a detailed description of stellar events, such as the heliacal risings and settings of prominent stars. Together, the BCI, the FCI and the Parapegma comprise a full description of the ancient Greek geocentric Cosmos. It is difficult to see what else might have been included that needed the extra space of a full-length cover: our Front Cover only covers the Front Dials. a b c Supplementary Fig. S5. Fragment 3. (a) Photograph. (b) X-ray CT cross-section. (c) Reconstruction as Front Cover moulding. In Fragment 3, there is a small section of decorative moulding 38.3 mm high, whose function has always been unclear. It is difficult to understand its role as part of the case. We suggest that it might have framed the Front Cover. If the Front Cover only covered the display and was framed by this moulding, then the size of the Front Cover plate would be 128.8 mm square. Fragment G + Fragment 29 (Fig 1b) are together 93.3 mm high for 36 lines of text. This leaves enough space for an additional 12 lines of text and a small margin. To complete the planetary inscriptions, Mercury needs 6 more lines at the beginning and Saturn runs to Line 43. A total of 49 lines would therefore be needed for the synodic cycles of the planets, taking up around 127 mm and fitting neatly on the Front Cover plate. If this is correct, then it does not appear that there was an introductory section in the FCI about the Moon and Sun as previously suggested12: this information was probably given in the BCI. We therefore propose that the FCI was entirely formulaic information about the synodic cycles of the planets and we adopt this hypothesis for our model. — 12 — Antikyth