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 water—or, 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

Summation

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!