######### Card Hero LETTERS #########
Letters to the editors

Vol. 7, NO. 3 / November 2022

To the editors:

It is a pleasure to be invited to respond to this essay by Mike Edmunds, continuing our stimulating discussion almost a decade ago in Leiden. Although, or perhaps because, the Antikythera mechanism is singular, it exists as a signpost to the enormous mass of ancient science and technology that has perished. Edmunds’s essay indeed aims to situate the mechanism in its scientific and technological contexts, and it is that aim that I address here, all too briefly.

When we gaze at the scattered fragments of antiquity, it is as if we are looking across a distant and undulating landscape, with here and there some outstanding landmarks. However, the constrained perspective, perforce, telescopes nearer and farther prominences, so that we join sets of landmarks that were in fact quite distinct and even remote from one another. Our syntheses of the remnants are constructed in a fashion all too collagistic. One of the larger risks is that we retroject things seen or deduced from one era back into times well prior. Some ancient collectors of opinions, for example, attributed a spherical-earth model to Thales of Miletus! What seems obvious or long-established in later generations need not have been prevalent or even present at all in earlier generations.

I agree with Edmunds that the mechanism is “embedded in its era,” both with respect to its mechanical design, and with respect to its underlying astronomical theory. The mechanism fits into its contexts, and is neither too early, nor does it belong to some later context.1 The evolution of both astronomical theory and of technology provide a comprehensible context. As J. B. S. Haldane quipped, finding “rabbits in the Precambrian” would refute an evolutionary account of biology.2 To anticipate, all the rabbits here are in a rabbit-shaped context.

In the section “An Era and Its Design Conventions,” Edmunds identifies four design conventions of the mechanism that manifest its fit into its technological context: the “solar” four-spoke wheel; the spiral calendrical dial; the front panel as an “astrologer’s board”; and “inscribed graduated scales and pointers, following the tradition of markings on sundials.” Those are perceptive insights, and all could fit in the Hellenistic era—as well as much earlier or later. Edmunds’s parallels for the spiral dial are Neolithic—is there a spiral calendar from Greco-Roman antiquity?3 The astrologer’s boards are known primarily from the first through third centuries CE, somewhat later than the Antikythera shipwreck.4

I propose to identify three more design conventions of the device that manifest its fit into its technological context: the design conventions of some automata; the design conventions of epicyclic or eccentric models of planetary motion; and the design conventions of “slide rules” with moving components and scribed lines.

The Antikythera mechanism is not only built with gears, it is a mechanism that automatically produces some result or behavior, starting from a motion that does not transparently manifest that result. That is consonant with the design conventions manifested in some Greek automata. Philo of Byzantium describes two somewhat similar analogues (ca. 200 BCE). There was the repeating ballista—or, polybolos—of Dionysios of Alexandria, where a crank turned a pentagonal gear that drove a link-belt to perform a repeated cycle of operations: load a bolt, draw the bow, fire the bow.5 Likewise, there was the bucket chain for lifting wateror, hálusis (ἅλυσις)6—where chains ran on polygons like those in the repeating catapult, and the buckets automatically filled up at the bottom and poured out at the top. The Roman architect Vitruvius’s account of the hálusis mentions the chains, but does not describe the drive mechanism.7

From the middle of the fourth century BCE through the middle of the second century BCE, the standard model of planetary motion, or at least the only model known to us, was the qualitative model based on concentric spheres, in versions by Eudoxus of Cnidus, Callippus, and Aristotle. This kind of model provided no motivation for making a flat, geared, and predictive device. Even if there had been motivation, the qualitative concentric spheres model could not serve as the basis for designing such a device.

Instead, the mechanical design of the mechanism is consonant with the design conventions of the era of Hipparchus and after. In those 300 years, from the work of Hipparchus in the middle of the second century BCE, through to Ptolemy in the second century CE, the usual model of solar and lunar motion was based on epicycles and eccentrics. The design of the device is based on the structure of an epicyclic mechanism—at least for the surviving parts, the solar and lunar gear trains.8

The Antikythera mechanism is not only built with gears that automatically produce some result or behavior, but it is also a calculating device. Hellenistic scientists knew that calculating could be performed by devices, perhaps a discovery due to Eratosthenes. The problem of computing two means in continuous proportion—i.e., given Α and Δ, the problem of finding Β and Γ such that Α / Β = Β / Γ = Γ / Δ—was not solvable by the canonical methods of Euclidean geometry, and therefore alternative methods were explored. Eratosthenes devised a kind of slide rule (ca. 235 BCE), which when iteratively adjusted could produce the desired values to as great a precision as the machine allowed.9 The operator of this device moved bronze components until certain scribed lines aligned. A second, simpler, device is described much later by Eutokios, and although its attribution has been lost, it is unlikely to have been designed before Eratosthenes.10 A third mechanical method is described by Philon.11

In the section “The Missing Planetary Display,” Edmunds addresses what we can know about the missing pieces of the mechanism. From the inscription on the mechanism, it is clear that the display originally included some indication of the positions of the five planets. Alas, almost no piece of the planetary display survives. There have been many attempts to recreate this display, often as a rotating pointer or ring indicating celestial longitudes.12 The phases for each planet occur in a cyclic order, so the planetary display could instead have been an indicator, rotary or linear, that pointed in sequence to marks which indicated one of the phases of the planet, such as opposition, or station.13

Indeed, what survives of the planetary inscription on the front face describes the motion of the planets solely in terms of their four phases—Mars, Jupiter, and Saturn— or their six phases—Venus and presumably in the lost lines about Mercury—and never mentions planetary longitude per se.14 That by itself does not rule out a set of five distinct rotating pointers. On the one hand, two of the phases—the morning station and the evening station—would be awkward for the user of the device to determine on a pair of the rotating pointers or rings.15 Conjunctions and oppositions,16 on the other hand, could easily be spotted if the planetary display used rotating pointers or rings. In several places, where the text is incomplete, it may originally have specified the longitude difference, or elongation, between the planet and the sun. If so, that would confirm the use of something like rotating pointers or rings.17

The inscription proves there was some planetary display, but we now have no way to know how it worked, or on the basis of what model. The presence of a planetary display means that the engineer who designed the device used some model of planetary motion—one which might not have been very accurate.18 So, the mechanism must postdate the invention of numerical models of planetary motion, and that situates it after the work of Hipparchus, ca. 130 BCE.

As we turn to examining planetary models, let us recall that we are gazing across a remote and uneven conceptual landscape, and beware that we do not retroject landmarks, or otherwise displace them. Among known models of the planets, there are none before Hipparchus that could serve as the basis for a predictive device. Although an ancient engineer could have constructed a representation of the spindle and whorls model at the end of Book X of Plato’s Republic, for example, no such construction could have generated accurate lunar eclipse predictions, let alone even qualitatively correct planetary motion.19 The qualitative model that Plato proposes in the Timaeus is much more complex and more fully three-dimensional. No such construction could have generated accurate eclipse predictions, let alone even qualitatively correct planetary motion. The nested-spheres model of Eudoxus, as modified by Callippus and then by Aristotle, was intended as qualitative.20 Aristotle records the moon occulting Mars, and notes that the Egyptians and Babylonians have closely observed occultations over many years that provide “much reliable data … about each of the stars.”21 But Aristotle himself adhered to a version of the concentric spheres model of Eudoxus, which was qualitative, not predictive.

Two texts from the third-century BCE that refer to the positions of Venus, or those of both Venus and Mercury, are best understood as referring to observed positions, rather than calculated positions. First, in 279/8 BCE, King Ptolemy II exploited the simultaneous appearance of Venus as evening star and Mercury as morning star, with good viewing of the phenomenon for over a month.22 Ptolemy included a crew of the morning star at the head of his all-day procession, and a crew of the evening star at the tail of the same procession. Doing so did not require that astronomers have predicted this configuration; it requires only that once the configuration became visible the astronomers could assure him that the configuration would last long enough to assemble the two crews. Similarly, Apollonius of Rhodes encodes within his epic, Argonautica, the position of Venus as a time marker.23 Apollonius used positions of Venus observed in 238 BCE to arrange the events of the Argonautica 1.1,229–32, 1.1,273–75, 2.40–42, into a temporal frame. His use of those positions does not require that they have been predicted, only that Apollonius could discover after the fact when they had occurred, and what other heavenly events were also observable, e.g., the full moon.

Scholars have suggested that Apollonius of Perga, the mathematician, may have created epicyclic or eccentric models.24 This claim is based on an interpretation of a statement by Ptolemy that Apollonius proved a mathematical equivalence between those two kinds of orbits.25 Yet Ptolemy only cites an equivalence for the sun: “There is a preliminary lemma demonstrated (for a single anomaly, that related to the sun) by a number of mathematicians, notably Apollonius of Perga.” Moreover, Theon of Smyrna (ca. 115 CE) reports that it was his contemporary Adrastus of Aphrodisias who demonstrated the general planetary equivalence:

Adrastus has shown that the hypothesis of the eccentric circle is a consequence of that of the epicycle; but I say further that, the hypothesis of the epicycle is also a consequence of that of the eccentric circle.26

Adrastus was quite willing to retroject the concept of epicycles into the astronomy of Plato and of Aristotle, so the fact that he does not retroject the equivalence of eccentric and epicyclic planetary motions into Apollonius makes it very hard to attribute the equivalence to Apollonius.27 That makes it all the less likely that Apollonius created realizable models of planetary motions.

Not even Hipparchus tried to build mathematical models of planetary motion.28 Indeed, Ptolemy says that, in contrast to Hipparchus’s work on the moon and sun:

He did not even make a beginning in establishing theories for the five planets, not at least in his writings which have come down to us. All that he did was to make a compilation of the planetary observations arranged in a more useful way, and to show by means of these that the phenomena were not in agreement with the hypotheses of the astronomers of that time.29

And indeed, Ptolemy never cites any planetary observation from Hipparchus himself. The planetary observations that Ptolemy does cite are attributed to Aristyllus or Timocharis, which Ptolemy probably extracted from Hipparchus’s compilation.30

The existing astral hypotheses that Hipparchus refuted was evidently an attempt to model, or at least describe, the phenomena, but we do not know how. Perhaps we should translate hypo-theses as sup-positions to reflect the range of the Greek word, and the uncertainty of its reference. The suppositions that Hipparchus claimed to have refuted probably included some concentric sphere models.

In three of his philosophical dialogues, Cicero reports a device apparently similar to the Antikythera mechanism. Scholars often read two of those reports as attributing the mechanism to Archimedes. Cicero’s claims are inconsistent: the mechanism is assigned twice to Archimedes, and once to Posidonius; the reports are found in dialogues whose settings and content are fictional;31 and “up-attribution” is a known trope: admirable or marvelous deeds and sayings are often re-assigned to a better-known figure. We also have evidence of the kind of planetary model Archimedes was using, and it is not consistent with any kind of kinematic model, neither concentric spheres nor epicyclic and eccentric orbits.

Cicero’s reports seem to vary as the literary context shifts from dialogue to dialogue. According to Cicero’s earliest report, in a dialogue set in the time of Hipparchus, one of the participants recounts how General Marcus Claudius Marcellus, sacker of Syracuse, had returned to Rome bearing as booty instruments of Archimedes that included a device showing the sun and moon being eclipsed at the appropriate times.32 That claim is briefly repeated in one of the works that Cicero wrote and set in his last busy year of writing, a work in which Cicero also tells how he rediscovered the tomb of Archimedes.33 In this dialogue, both the attribution of the device and the discovery of the tomb serve symbolic and rhetorical functions. Yet in another work from that same final year, the mechanism was presented as an invention of Posidonius’s time.34

Since all three dialogues have fictional settings and content, there is no reason to treat these attributions as anything other than literary devices by Cicero. Just like Herodotus depicting a meeting between Solon and Croesus (1.29–33), so Cicero also felt free to commit anachronism, and bring together artefacts and people for the sake of the dialogue, and not in order to provide a reliable historical summary.35

The shift in attribution, from Posidonius to Archimedes, undermines any basis for taking either attribution as historical and accurate. Anecdotes and inventions that are attributed to one figure are often reattributed to another better-known exemplary person of the same kind. Thus, both Pythagoras and Thales are credited with having sacrificed a bull in thanksgiving for having proven a theorem about the circle.36 Both men were geometers, but Pythagoras, the better known, would dominate the anecdote—so the earlier version is likely to have concerned the less known figure. A similar transfer or elevation from the known to the well-known occurs in the anecdote about the shipwrecked philosopher and the geometrical diagrams on the beach. A few paragraphs after he attributes the Antikythera-like device to Archimedes, Cicero reports a novelistic tale about Plato, that he was shipwrecked in an unknown land on a deserted shore, and the others were afraid due to their ignorance of the country; then, he noticed geometrical figures traced in the sand, and immediately cried out, “Take heart! For I see human tracks.”37 Vitruvius attributes the same legend to the less famous Aristippus of Cyrene:

The philosopher Aristippus, a follower of Socrates, was cast up from a shipwreck on the Rhodian coast, and when he noticed geometrical diagrams drawn (in the sand), he is said to have exclaimed to his companions: ‘Let us hope for the best! For I see human tracks.’38

A third example of the trope is the riposte by the tutor of geometry to the impatient royal student. When Alexander the Great asked Menaechmus to give him a condensed understanding of geometry, the tutor replied, “O King, for traveling over the country, there are royal roads and roads for common citizens; but in geometry there is one road for all.”39 Almost exactly the same story is told about the first King Ptolemy asking his tutor Euclid whether there was a way to learn geometry in a more condensed fashion than reading the latter’s Elements, to which Euclid replied, “there is no royal shortcut to geometry.”40 In the same fashion, the story that Posidonius had a device like the Antikythera mechanism has been transferred by Cicero to a more famous man, Archimedes—and Cicero performs both that elevation, and the one about the shipwrecked philosopher, in the same few paragraphs.

Perhaps the attribution to Archimedes was supplied by Posidonius. If so, we should beware. Posidonius was not a reliable informant on this topic: neither expert in astronomy,41 nor well-informed about the history of technology.42 His style was tendentious, rhetorical, and exaggerated.43 Very likely, Posidonius indeed had seen, or maybe even owned, something like the Antikythera mechanism, and informed Cicero about it in a letter, filled with colorful exaggeration; Cicero might then have exploited the letter in his fiction about Archimedes. Later attributions can be understood as depending on Cicero’s assignment to Archimedes.

Finally, there is a report of two series of very large distances that Archimedes assigned to the planets: a list of intervals between successive planets, transmitted with a list of total distances from earth to each of the planets.44 So defined, each series should be computable from the other, and ought to reveal that the two series are consistent—but they entirely disagree. Neugebauer calculates that the intervals between successive planets can be generated using a pair of irregular arithmetic series. Such numerical intervals are pure numerology, and there is no hint that the system included any means to represent retrogradations. This does not look at all like Archimedes was working with any kinematic model of planetary motion.

Nevertheless, people have tried to imagine what Archimedes’s device looked like, and even to reconstruct it.45 The effort, although admirably prodigious, is founded on too little data to be convincing.

It was the work of Hipparchus that constituted the initial effort to construct epicyclic and eccentric models of the motions for at least some of the planets. In its full development, the use of epicyclic and eccentric orbits was applied by Ptolemy to all seven stellar vagrants: the Sun, the Moon, and the five known planets—Saturn, Jupiter, Mars, Venus, and Mercury. There is no convincing evidence that any Greek before Hipparchus exploited the Babylonian data to build anything like a predictive model of solar or lunar motion, much less planetary motion.46 Other than Hipparchus, the earliest extant evidence for the translation of Babylonian data into Greek is a papyrus from some time between the late Ptolemaic period and 50 CE that concerns a Babylonian-style model for the moon.47

All evidence for mathematical models of planetary motion is from after the time of Hipparchus. This is consistent with early Greco-Roman astrological practice, which ignored the planets. We have little data on early Greco-Roman astrology, but from the era shortly after Hipparchus, there are four authors or works whose work is well-enough known to make it very likely that they did not use planetary positions, and thus did not make predictions of planetary motion: Epigenes of Byzantium;48 Petosiris;49 the Salmeskhoiniaka;50 and Skulax of Halicarnassus.51

Scraps and hints from the three centuries after Hipparchus and before Ptolemy suggest that astronomers and astrologers developed a wide variety of models. Here are three examples, intended only to display the range of variation:

  • a partially heliocentric concentric-sphere system from ca. 100 BCE;
  • the 248-day arithmetic models of lunar position, known as early as ca. 100 CE; and
  • an unusual epicyclic model of uncertain (but pre-Ptolemaic) date, preserved on papyrus.

There is a modified concentric spheres model that explicitly models the motion of Mercury and Venus as due to their orbiting around the Sun, primarily known from Theon of Smyrna. Four Roman authors also refer to this model.52 Theon says:

For the Sun, Phosphoros [Venus], and Stilbon [Mercury] it may be that … [usual model]. It may also be that there is one hollow sphere common to all, the solid spheres of the three being in its interior, around the same center as one another, with the smallest (and actually solid) sphere being that of the sun, around that one the sphere of Stilbon, then enveloping both and filling the whole interior of the common hollow sphere is that of Phosphoros.53

There were 248-day arithmetic methods, attested by ca. 100 CE, but usually assumed to be older, for the motion of the Moon.54 There were also similar models for planets, known from the first century CE and later.55 Such models might not be immediately usable in a gear-driven calendric device, but they were used in tabular form to predict planetary positions, presumably by astrologers, but probably also by astronomers seeking to develop improved models.

Finally, there is a papyrus that describes an unusual epicyclic model. Scholars have debated the date of the model, although papyrologists attest that the papyrus itself is from ca. 150 CE.56 This model indicates that the three-century long path from Hipparchus to Ptolemy did not consist of only a single step, but rather of many evolutionary increments. No doubt there were many attempts that proved unsatisfactory.57


Things take time to evolve, and counterfactuals are slippery to think with. Some people, as they gaze across the landmarks of a distant and complex conceptual landscape, ask “why didn’t that landmark evolve sooner?” But there is no positive cause. Instead, there is an absence of sufficient, or even necessary, causes, so that things evolve when they do, based on context. Nevertheless, despite the fictional or mistaken nature of the sources claiming for Archimedes a predictive mechanical model of planetary motions, some have argued that a sufficiently talented ancient engineer could have succeeded in creating such a device. The risk of retrojection seems large enough to vitiate any such argument.

Consider the following counterfactuals about the Foucault pendulum and the telescope, deliberately conceived to make the issue very clear. First, let us ask, “could the Foucault pendulum have been invented by Archimedes, or by Galileo?” Both certainly had the interest and the skill. The question, so posed, seems to suggest that if a modern engineer can devise a means to construct a Foucault pendulum, exploiting only the methods known to have been used by Archimedes or Galileo, then the question would be answered, “yes, he could have made a Foucault pendulum.” Such a procedure neglects the crucial conceptual context. Despite the simplicity of the technology of a Foucault pendulum—it is after all little more than a very long pendulum with a heavy bob—the idea that the plane of its swing will remain fixed in inertial space as the earth “rotates out from under” the suspension-point presumes the system of physics published by Laplace.58 Therefore, the only valid answer is “no, it is not conceivable that Archimedes or Galileo, with all their brilliance and skill, could ever have made a Foucault pendulum (any more than Archimedes could have made a bicycle).”59

Second, let us ask, “could the telescope have been invented by Ptolemy, or by Roger Bacon?” Both certainly had the interest and the skill. The question, so posed, seems to suggest that if a modern engineer can devise a means to construct a telescope, exploiting only the methods known to have been used by Ptolemy or Bacon, then the question is answered, “yes, he could have made a telescope.” Such a procedure neglects the crucial conceptual context. Indeed, the technology of a telescope is simple—magnifying mirrors were well-known, and even Seneca knew of magnifying lenses.60 But the idea of the telescope as a device presumes the existence of eye-glasses, which were only invented ca. 1290 CE.61 Therefore, the only valid answer is “no, it is not conceivable that Ptolemy or Bacon, with all their brilliance and skill, could ever have made a telescope (any more than Ptolemy or Bacon could have made a bicycle).”62

As we can see more clearly in the case of the telescope, not only is a conceptual framework necessary for any given discovery or invention to be made, there is also a gap between the state of having all technological and theoretical prerequisites available and the occurrence of the actual impetus for the discovery or invention. That gap can be quite long: for the telescope it was about three centuries, although for the Foucault pendulum it was only about half a century.

In sum, the device found off the coast of Antikythera tells us much that is important and even exciting about ancient technology and astronomy. But it is important to listen to what it does say, and not retroject things that it does not say. It is a calendar computer with triangular-toothed gears, employing differential gearing, that exploited the results of the work of Hipparchus to provide predictions of the positions of the sun and moon, and periods in which lunar and solar eclipses could occur. The display of the phases of the planets played some role, but we do not know precisely how they were displayed. We do not fully know its purpose, but calendrics seem central.

Paul Keyser

Mike Edmunds replies to Paul Keyser, Paul Cartledge, and Kyriakos Efstathiou:

I am very grateful for the kind and informative comments that these three authors have made on my slim essay. Venturing to cross from one discipline to another is a stimulating but dangerous activity, and I have often been conscious of my lack of deep classical and historical background. Nonetheless, we all seem to be in welcome agreement that the Antikythera mechanism is indeed a child of its time.

Paul Keyser provides much additional food for thought—in particular on the idea that there are further design conventions to be recognized in the mechanism. I am sympathetic to Keyser’s feeling that Archimedes has been accorded rather more expertise in the invention of sphaerae than is really justified by the evidence—at least until the contents of his lost book de Sphaerae is discovered. I might be a little more egalitarian than Paul Cartledge in estimating the cost of the mechanism. It was made of bronze not gold; any stones incorporated were semi-precious rather than true gems; and a rough estimate of the mechanical craftsmanship involved might only be a few weeks—i.e., requiring sufficient expertise to make it indeed an expensive object, but not prohibitively so. This suggests that such devices might not have been terribly rare, with the hopeful prospect that another example or variant might yet be found in the future, perhaps preserved at a suddenly-terminated site such as Pompeii or Herculaneum. Kyriakos Efstathiou is of course correct in asserting that no driving knob survives on the side of the mechanism, but the presence of a crown gear that would drive the main Sun gear is excellent circumstantial evidence that a knob or similar drive may have existed. Even if it did not, and the drive was indeed from the moon pointer, the philosophical—or, even theological—nicety of a single driver or Primum Mobile still remains.

We all seem rather stumped by the question of the mechanism’s primary purpose. But perhaps it is appropriate that some uncertainty should remain about this marvelous artefact, especially given that so much valuable de-mystification has taken place in recent years!

  1. Theodosios Tassios, “Prerequisites for the Antikythera Mechanism to be Produced in the 2nd Century BC,” in The Antikythera Shipwreck: The Ship, the Treasures, the Mechanism, ed. Nikos Kaltsas et al., (Athens: Kapon, 2012) 249–55; Alexander Jones, “The Epoch Dates of the Antikythera Mechanism (With an Appendix on its Authenticity),” ISAW Papers 17 (2020), doi:2333.1/ffbg7m07. 
  2. Richard Dawkins, “The Illusion of Design,” Natural History Magazine 9, no. 114 (2005) 35–37. 
  3. Daryn Lehoux, Astronomy, Weather, and Calendars in the Ancient World: Parapegmata and Related Texts in Classical and Near Eastern Societies (Cambridge: Cambridge University Press, 2007), 168–170. In his book, Lehoux records a parapegma with a circular (zodiacal) component, found in the early second-century CE Thermae Traiani of Rome. (I am indebted to Lehoux for supplying this reference.) 
  4. James Evans, “The Astrologer’s Apparatus: A Picture of Professional Practice in Greco-Roman Egypt,” Journal for the History of Astronomy 35, no. 1 (2004): 1–44, doi:10.1177/002182860403500101. In particular, see pp. 4–24. 
  5. Philon of Byzantion, Artillery-Making, in Eric W. Marsden, Greek and Roman Artillery: Technical Treatises, 1971 (UK: Oxford University Press, 1999), 146–153, 177–184. 
  6. The Greek name of the water-lifting device is given by Heron in Dioptra §6. 
  7. Philon, Pneumatics §65. Vitruvius, On Architecture 10.4.4 says:
    a double iron chain is made to revolve on the axle of one and the same wheel, and let down to the lower (water) level, with hanging bronze gallon-buckets. Thus the turning of the wheel makes the chain revolve around the axle, and carries the buckets to the top. When they are carried above the axle, they are made to turn over and pour into the water reservoir what they have raised. (in eiusdem rotae axe inuoluta duplex ferrea catena demissaque ad imum libramentum conlocabitur, habens situlos pendentes aereos congiales. ita uersatio rotae catenam in axem inuoluendo efferet situlos in summum, qui <cum> super axem peruehuntur, cogentur inuerti et infundere in castellum aquae quod extulerint.)
  8. Michael Wright, “A Planetarium Display for the Antikythera Mechanism,” Horological Journal 144, no. 5 (2002): 169–73, and Horological Journal 144, no. 6 (2002): 193; Michael Wright, “Epicyclic Gearing and the Antikythera Mechanism, Part 1,” Antiquarian Horology 27, no. 3 (2003): 267–79; and Michael Wright, “Epicyclic Gearing and the Antikythera Mechanism, Part 2,” Antiquarian Horology 29, no. 2 (2005): 51–63. 
  9. Wilbur Knorr, Textual Studies in Ancient and Medieval Geometry (Boston: Birkhaüser, 1989) 131–53; Klaus Geus, Eratosthenes von Kyrene (Munich: Beck, 2002) 195–205; Carl Huffman, Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King (Cambridge: Cambridge University Press, 2005): 361–401 (in his discussion of A15 of Archytas). Note that the device attributed to Archytas is a fiction or an error by ancient commentators: Huffman, Archytas of Tarentum (2005), 77–83, 342–60. 
  10. Knorr, Textual Studies (1989) 78–80; Reviel Netz, “Plato’s Mathematical Construction,” Classical Quarterly 53, no. 2 (2003): 500–09, doi:10.1093/cq/53.2.500
  11. Philon of Byzantion, Artillery-Making, cited in Eric Marsden, Greek and Roman Artillery, v. 1, Historical Development, 1969 (Oxford: Oxford University Press, 1999) 24–25, 40–41; Knorr, Textual Studies (1989) 41–61. 
  12. Pointers in figure 1 of Wright, “Planetarium Display” (2002): 170; and rings in figures 1a and 7 of Tony Freeth et al., “A Model of the Cosmos in the Ancient Greek Antikythera Mechanism,” Scientific Reports 11 (2021), doi:10.1038/s41598-021-84310-w. 
  13. Discussed by Autolycus of Pitane, On Risings and Settings; see Henry Mendell, “Autolukos of Pitane,” in Paul Keyser and Georgia Irby-Massie, ed., Encyclopedia of Ancient Natural Scientists (London: Routledge, 2008), 183. For the primary importance and role of these phases, see: Otto Neugebauer, A History of Ancient Mathematical Astronomy (Berlin: Springer-Verlag, 1975), 1,090–91; Alexander Jones, “The Adaptation of Babylonian Methods in Greek Numerical Astronomy,” Isis 82, no. 3 (1991): 441–53, see, in particular, p. 443; Alexander Jones, “An Astronomical Ephemeris for A.D. 140: P. Harris I.60,” Zeitschrift für Papyrologie und Epigraphik 100 (1994): 59–63, see pp. 62–63; Magdaleni Anastasiou, “6. The Front Cover Inscription,” in The Inscriptions of the Antikythera Mechanism: Almagest Special Issue 7, no. 1 (2016): 250–97, see pp. 289–90; and Alexander Jones, A Portable Cosmos: Revealing the Antikythera Mechanism, Scientific Wonder of the Ancient World (Oxford: Oxford University Press, 2017), 161–71. 
  14. Anastasiou, et al., “Front Cover Inscription” (2016), 266 Greek, and 276 English. 
  15. The phase “evening station” (ἑσπερινὸς στηριγμός or … στάσις) on lines 9, 19 (twice, once στηριγμός, once στάσις), 22 (στάσις), and 27. The phase “morning station” (ἑώϊος στηριγμός) on lines 21 and 29. On lines 12 and 38, only the word “station” (στηριγμός) is legible. 
  16. The phase “conjunction” (σύνοδον) on lines 7, 20, 28, and 37; the phase “opposition” (κατὰ διάμετρον) on lines 32 and 41. 
  17. The description “distant from the Sun” (ἀπέχων ἀπὸ τοῦ Ἡλίου; with no surviving numeral) on lines 9, 18–19 (Ἡλίου lost), 21, 22–23 (Ἡλίου lost), and 29 (largely restored); the variant “staying away from the Sun” (ἀποστὰς ἀπὸ τοῦ Ἡλίου; also with no surviving numeral) on line 12. See Anastasiou, et al., “Front Cover Inscription” (2016), 287. 
  18. Mike Edmunds, “An Initial Assessment of the Accuracy of the Gear Trains in the Antikythera Mechanism,” Journal for the History of Astronomy 42, no. 3 (2011): 307–20, doi:10.1177/002182861104200302. Edmunds shows that the “lunar position indication would have had angular errors which would at times have reached half a zodiac sign.” That angular error corresponds to a temporal error of ca. 1¼ days. 
  19. If the model in the Republic represented retrograde motions at all, they were performed through the willful intervention of one of the three Fates: Republic 10 (617b9–d2). 
  20. Jones, “Adaptation of Babylonian Methods,” 441:
    The only numerical parameters associated with [Eudoxus’s] concentric spheres in our ancient sources are crude periods of synodic and longitudinal revolution, that is to say, data imposed on the models rather than deduced from them.
  21. Aristotle, On the Heavens, Book 2, Part 12 (292a7–9), ὁμοίως δὲ καὶ περὶ τοὺς ἄλλους ἀστέρας λέγουσιν οἱ πάλαι τετηρηκότες ἐκ πλείστων ἐτῶν Αἰγύπτιοι καὶ Βαβυλώνιοι, παρ᾿ ὧν πολλὰς πίστεις ἔχομεν περὶ ἑκάστου τῶν ἄστρων. The Platonic Epinomis 986e–987a similarly praises Egyptian and “Syrian” observations. 
  22. Paul Keyser, “Venus and Mercury in the Grand Procession of Ptolemy II,” Historia: Zeitschrift für Alte Geschichte 65, no. 1 (2016): 31–52, see pp. 37–40. 
  23. Jackie Murray, “Anchored in Time: The Date in Apollonius’ Argonautica,” in M. Annette Harder et al., eds., Hellenistic Poetry in Context, Hellenestica Groningana 20 (Leuven: Peeters, 2014), 247–83, see pp. 260–67. 
  24. Active ca. 195 BCE; see Reviel Netz, “Apollonios of Perge,” in Keyser and Irby-Massie, Ancient Natural Scientists, 114–15. 
  25. Ptolemy, Almagest 12.1: εἰς δὴ τὴν τοιαύτην διάληψιν προαποδεικνύουσι μὲν καὶ οἵ τε ἄλλοι μαθηματικοὶ καὶ Ἀπολλώνιος ὁ Περγαῖος ὡς ἐπὶ μιᾶς τῆς παρὰ τὸν ἥλιον ἀνωμαλίας. See: Johan Ludvig Heiberg, Claudii Ptolemaei opera quae exstant omnia, vol. 1, part 2 (Leipzig: Teubner, 1903), 450; Neugebauer, Ancient Mathematical Astronomy (1975), 190–201; and Gerald Toomer, Ptolemy’s Almagest (London: Duckworth, 1984) 555–56. 
  26. Bernard Goldstein, “Apollonius of Perga’s Contributions to Astronomy Reconsidered,” Physis 46 (2009): 1–14, see p. 11. Greek in Eduard Hiller, Theonis Smyrnaei philosophi Platonici expositio rerum mathematicarum ad legendum Platonem utilium (Leipzig: Teubner, 1878), 166: δείκνυσι δὲ ὁ Ἄδραστος πρῶτον μὲν πῶς τῇ κατ’ ἐπίκυκλον ἕπεται κατὰ συμβεβηκὸς ἡ κατὰ ἔκκεντρον· ὡς δὲ ἐγώ φημι, καὶ τῇ κατὰ ἔκκεντρον ἡ κατ’ ἐπίκυκλον. 
  27. See: István Bodnár, “Adrastos of Aphrodisias,” in Keyser and Irby-Massie, Ancient Natural Scientists, 31–32; and Alexander Jones, “Theon of Smyrna,” in Keyser and Irby-Massie, Ancient Natural Scientists, 796. 
  28. Neugebauer, Ancient Mathematical Astronomy, 329–32. 
  29. Ptolemy, Almagest 9.2:
    Ἵππαρχον … ὑποθέσεις … ταῖς δὲ τῶν εʹ πλανωμένων διά γε τῶν εἰς ἡμᾶς ἐληλυθότων ὑπομνημάτων μηδὲ τὴν ἀρχὴν ἐπιβάλλειν, μόνον δὲ τὰς τηρήσεις αὐτῶν ἐπὶ τὸ χρησιμώτερον συντάξαι καὶ δεῖξαι δι’ αὐτῶν ἀνομόλογα τὰ φαινόμενα ταῖς τῶν τότε μαθηματικῶν ὑποθέσεσιν.
    See: Heiberg, Ptolemaei, 210; and Toomer, Almagest, 421. 
  30. Toomer, Almagest (1984) 421, see particularly notes 9 and 11; see Alan Bowen, “Aristullos,” in Keyser and Irby-Massie, Ancient Natural Scientists, 155–56, and Alan Bowen, “Timokharis,” in Keyser and Irby-Massie, Ancient Natural Scientists, 812–13. 
  31. Paul Keyser, “Orreries, the Date of [Plato] Letter ii, and Eudoros of Alexandria,” Archiv für Geschichte der Philosophie 80 (1998): 241–67, doi:10.1515/agph.1998.80.3.241, see pp. 246–48. 
  32. On the Republic 1.21–22, written in ca. 54–51 BCE, and set in ca. 130 BCE. This attribution occurs in the almost-immediate context of the tale about Plato and the diagrams on the beach, below. Among the other booty Marcellus took was a sphaera solida, a static depiction of the sky, not a machine with moving parts. Mary Jaeger, Archimedes and the Roman Imagination (Michigan: University of Michigan Press, 2008), 48–72, explores how this solid sphere serves as a foil to the motile machine. 
  33. The Tusculan Disputations was written and set in ca. 45 BCE. For the attribution of the device to Archimedes, see Cicero, Tusculan 1.63. For Cicero’s rediscovery of Archimedes’ tomb, see Cicero, Tusculan 5.64–66, and Mary Jaeger, “Cicero and Archimedes’ Tomb,” Journal of Roman Studies 92 (2002): 49–61, doi:10.2307/3184859, revised in Jaeger, Archimedes (2008), 32–47. 
  34. Cicero, On the Nature of the Gods 2.88, written in ca. 45 BCE, and set in ca. 75 BCE (at a time when Posidonius was “the friend of all of us,” familiaris omnium nostrum). Cicero has the Stoic advocate Balbus say “the thing which our friend Posidonius recently had made” (hanc quam nuper familiaris noster effecit Posidonius). Using effecit rather than fecit conveys the sense of having had it made, rather than creating or building it himself. 
  35. Plato also does this, but some scholars wish to defend Plato’s anachronisms as accurate. On Cicero’s fictions, see Matthew Fox, “Heraclides of Pontus and the Philosophical Dialogue,” in William Fortenbaugh and Elizabeth Pender, eds., Heraclides of Pontus (London: Routledge, 2009), 41–67, see pp. 43, 51–60. 
  36. Diogenes Laertius, Lives and Opinions of Eminent Philosophers 1.24–25. 
  37. Cicero, On the Republic 1.29,
    ut mihi Platonis illud, seu quis dixit alius, perelegans esse uideatur; quem cum ex alto ignotas ad terras tempestas et in desertum litus detulisset, timentibus ceteris propter ignorationem locorum animaduertisse dicunt in arena geometricas formas quasdam esse descriptas; quas ut uidisset, exclamauisse, ut bono essent animo; uidere enim se hominum uestigia.
    The hedge “seu quis dixit alius” suggests that Cicero knew (of) an alternate version. 
  38. Vitruvius, On Architecture 6.pr, Aristippus philosophus Socraticus, naufragio cum eiectus ad Rhodiensium litus animaduertisset geometrica schemata descripta, exclamauisse ad comites ita dicitur: ‘bene speremus! hominum enim uestigia uideo.’ The similarity of vocabulary shows that the tales are related; the greater amount of circumstantial detail (location of wreck, what the philosopher did afterwards) argues for the priority of the Aristippus version; moreover, Diogenes Laertius 2.84 attributes to Aristippus a book “To those who have been shipwrecked” (Πρὸς τοὺς ναυαγούς).  
  39. John of Stoboi (ca. 420 CE), Anthology 2.31.115; Curt Wachsmuth, ed., Anthologium, vol. 2 (Leipzig, 1884), 228: Μέναιχμον τὸν γεωμέτρην Ἀλέξανδρος ἠξίου συντόμως αὐτῷ παραδοῦναι τὴν γεωμετρίαν· ὁ δέ „ὦ βασιλεῦ“, εἶπε, „κατὰ μὲν τὴν χώραν ὁδοί εἰσιν ἰδιωτικαὶ καὶ βασιλικαί, ἐν δὲ τῇ γεωμετρίᾳ πᾶσίν ἐστιν ὁδὸς μία“. The “royal” road is a reference to the Persian “Royal Road” (Herodotus 5.52–54), a circumstantial detail that makes sense in this earlier version. 
  40. Proclus (ca. 460 CE), Procli Diadochi in Primum Euclidis Elementorum Librum Commentarii (Commentary on the First Book of Euclid’s Elements), Gottfried Friedlein, ed. (Leipzig: B. G. Teubneri, 1873; reprinted as Hildesheim: G. Olms, 1961), p. 68.13–17 says about Euclid: οὗτος ὁ ἀνὴρ ἐπὶ τοῦ πρώτου Πτολεμαίου· … μέντοι καί φασιν ὅτι Πτολεμαῖος ἤρετό ποτε αὐτόν, εἴ τίς ἐστιν περὶ γεωμετρίαν ὁδὸς συντομωτέρα τῆς στοιχειώσεως· ὁ δὲ ἀπεκρίνατο, μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπὶ γεωμετρίαν. 
  41. Compare Strabo 2.5.14 where Posidonius imagines that climbing up on a roof will change his effective latitude for observing stars: Ludwig Edelstein and I. G. Kidd, ed., Posidonius (Cambridge: Cambridge University Press, 1972–1999), fragment 204. 
  42. In the only extended fragment on technology, Posidonius attributes technological progress to the practice of philosophy: Seneca, Letters 90.5–13, 20–25, 30–32, see fragment 285 in Edelstein and Kidd, Posidonius (1972–1999). 
  43. As noted by Strabo 2.3.29 and Seneca, Letters 90.20–23, which are testimonia 103 and 106 in Edelstein and Kidd, Posidonius (1972–1999); this sort of color is found in six verbatim fragments: 54, 57, 59, 239, 253, and 257 in Edelstein and Kidd, Posidonius (1972–1999). 
  44. Hippolytus, Refutation 4.8–11; Neugebauer, Ancient Mathematical Astronomy, 647–51. The planets are ordered from the Earth outwards as: Moon, Sun, Venus, Mercury, Mars, Jupiter, and Saturn. 
  45. Jo Marchant, “Archimedes’ Fabled Sphere Brought to Life,” Nature 526 (2015): 19, doi:10.1038/nature.2015.18431. The reconstructor is Michael Wright, probably the most skilled living reconstructor of devices like the Antikythera Mechanism. 
  46. Neugebauer, Ancient Mathematical Astronomy, 271: “there is much in the astronomy of Eudoxus, Aristarchus, and Archimedes … that shows a lack of interest in empirical numerical data in contrast to the emphasis on the purely mathematical structure.” 
  47. Alexander Jones, “A Greek Papyrus Containing Babylonian Lunar Theory,” Zeitschrift für Papyrologie und Epigraphik 119 (1997): 167–72. 
  48. See Francesca Rochberg, “Epigenes of Buzantion,” in Keyser and Irby-Massie, Ancient Natural Scientists, 290, dated to 120–30 BCE. 
  49. See Francesca Rochberg, “Petosiris, or Nekhepso-Petosiris,” in Keyser and Irby-Massie, Ancient Natural Scientists, 637–38, dated to 150–100 BCE. 
  50. See Alexander Jones, “Salmeskhoiniaka,” in Keyser and Irby-Massie, Ancient Natural Scientists, 724, dated to 200–100 BCE. 
  51. See Georgia Irby-Massie, “Skulax of Halikarnassos,” in Keyser and Irby-Massie, Ancient Natural Scientists, 745, dated to 140–90 BCE. 
  52. Vitruvius 9.1.6; Calcidius §109; Macrobius Theodosius 1.19.6; and Martianus Capella 8.857. 
  53. Translation from Georgia Irby-Massie and Paul Keyser, Hellenistic Science Sourcebook (London: Routledge, 2002) 73–74. Theon writes:
    ἐπὶ δὲ τοῦ ἡλίου καὶ Φωσφόρου καὶ Στίλβοντος δυνατὸν μὲν … . δυνατὸν δὲ καὶ μίαν μὲν εἶναι τὴν κοίλην κοινὴν τῶν τριῶν, τὰς δὲ στερεὰς <τῶν> τριῶν ἐν τῷ βάθει ταύτης περὶ τὸ αὐτὸ κέντρον ἀλλήλαις, μικροτάτην μὲν καὶ ὄντως στερεὰν τὴν τοῦ ἡλίου, περὶ δὲ ταύτην τὴν τοῦ Στίλβοντος, εἶτα ἀμφοτέρας περιειληφυῖαν καὶ τὸ πᾶν βάθος τῆς κοίλης καὶ κοινῆς πληροῦσαν τὴν τοῦ Φωσφόρου.
    See: Hiller, Theonis Smyrnaei, 186–87; and Paul Keyser, “Heliocentrism in or out of Heraclides,” in Heraclides of Pontus: Discussion, ed. William Fortenbaugh and Elizabeth Pender (London: Routledge, 2009), 205–35, see pp. 218–33. 
  54. Alexander Jones, “The Development and Transmission of 248-day Schemes for Lunar Motion in Ancient Astronomy,” Archive for History of Exact Sciences 29 (1983): 1–36, doi:10.1007/BF00535977, see pp. 1–2; Jones, “Adaptation of Babylonian Methods,” 449–53; Alexander Jones, “Studies in the Astronomy of the Roman Period. I. The Standard Lunar Scheme,” Centaurus 39, no. 1 (1997): 1–36, doi:10.1111/j.1600-0498.1997.tb00023.x, see pp. 4, 9, and 29–30. 
  55. Neugebauer, Ancient Mathematical Astronomy, 802–05; Alexander Jones, “Pliny on the Planetary Cycles,” Phoenix 45, no. 2 (1991): 148–61, doi:10.2307/1088552; Alexander Jones, “Studies in the Astronomy of the Roman Period. III. Planetary Epoch Tables,” Centaurus 40, no. 1 (1998): 1–41, doi:10.1111/j.1600-0498.1998.tb00416.x, see pp. 33–37. 
  56. The model seems similar to Pliny, Natural History 2.68–76, and Otto Neugebauer, “Planetary Motion in P. Mich. 149,” Bulletin of the American Society of Papyrologists 9, nos. 1–2 (1972): 19–22, initially suggested it could be “perhaps from a time before Apollonius.” Then it was attributed to fourth-century BCE Pythagoreans by Bartel Leendert van der Waerden, “The Earliest Form of the Epicycle Theory,” Journal for the History of Astronomy 5, no. 3 (1974): 175–85, doi:10.1177/002182867400500303 and Bartel Leendert van der Waerden, “The Motion of Venus, Mercury and the Sun in Early Greek Astronomy,” Archive for History of Exact Sciences 26, no. 2 (1982): 99–113, doi:10.1007/BF00348348. Neugebauer, Ancient Mathematical Astronomy, 805–08 did not accept that suggestion, and neither did Alexander Jones in Keyser and Irby-Massie, Ancient Natural Scientists, 619. 
  57. Before turning to the summation, I would like to suggest two small bibliographical corrections. When I mentioned to Edmunds the mechanician Theodoros who remarked on how the cosmos is like a machine, I didn’t have the precise citation handy, and the correction is omitted from his text (n. 10): Georgia Irby-Massie, in Keyser and Irby-Massie, Ancient Natural Scientists, 784–85. Likewise, in his endnote 32, when I mentioned the ancient “mechanical arithmetic calculating devices,” I was dilatory in providing the citation. Such devices, albeit gearless, are attested in the form of Eratosthenes’ means-calculator: Alexander Jones, in Keyser and Irby-Massie Ancient Natural Scientists, 297–300, specifically at 298; a translation of the key texts is in Irby-Massie and Keyser, Sourcebook (2002), ch. 2, §3 (Eratosthenes). 
  58. Pierre-Simon Laplace, Traité de Mecanique Celeste, vol. 4 (Paris: Imprimerie de Crapelet, 1799–1805); the fifth volume, from 1825, is mainly historical. 
  59. Nor could Galileo, but the issue has been rendered controversial by the modern forgery of a drawing of a bicycle in the “Atlantis” codex of Leonardo da Vinci: Hans-Erhard Lessing, “The Evidence against ‘Leonardo’s Bicycle’,” in Nicholas Oddy, ed., Eighth International Cycling History Conference (San Francisco: Van der Plas Publications, 1997) 49–56. 
  60. Seneca, Natural Questions 1.6.5; Jay M. Enoch, “Archeological Optics: The Very First Known Mirrors and Lenses,” Journal of Modern Optics 54, no. 9 (2007): 1,221–39, doi:10.1080/09500340600855106. 
  61. Edward Rosen, “The Invention of Eyeglasses,” Journal of the History of Medicine and Allied Sciences 11, no. 1 (1956): 13–46, doi:10.1093/jhmas/XI.1.13, and 11, no. 2 (1956): 183–218, doi: 10.1093/jhmas/XI.2.183; and Charles Letocha, “The Origin of Spectacles,” Survey of Ophthalmology 31, no. 3 (1986): 185–88, doi:10.1016/0039-6257(86)90038-X. 
  62. Rolf Willach, “The Long Route to the Invention of the Telescope,” Transactions of the American Philosophical Society 98, no. 5 (2008). 

Paul Keyser is an independent scholar.

Mike Edmunds is Emeritus Professor of Astrophysics at Cardiff University and the current President of the Royal Astronomical Society.

More Letters for this Article


Copyright © Inference 2024

ISSN #2576–4403