An Ancient Greek Astronomical Calculation Machine Reveals New Secrets

In 1900 diver Elias Stadiatis, clad in a copper and brass helmet and a heavy canvas suit, emerged from the sea shaking in fear and mumbling about a “heap of dead naked people.” He was among a group of Greek divers from the eastern Mediterranean island of Symi who were searching for natural sponges. They had sheltered from a violent storm near the tiny island of Antikythera, between Crete and mainland Greece. When the storm subsided, they dived for sponges and chanced on a shipwreck full of Greek treasures—the most significant wreck from the ancient world to have been found up to that point. The “dead naked people” were marble sculptures scattered on the seafloor, along with many other artifacts. Soon after, their discovery prompted the first major underwater archaeological dig in history.

One object recovered from the site, a lump the size of a large dictionary, initially escaped notice amid more exciting finds. Months later, however, at the National Archaeological Museum in Athens, the lump broke apart, revealing bronze precision gearwheels the size of coins. According to historical knowledge at the time, gears like these should not have appeared in ancient Greece, or anywhere else in the world, until many centuries after the shipwreck. The find generated huge controversy.

The lump is known as the Antikythera mechanism, an extraordinary object that has befuddled historians and scientists for more than 120 years. Over the decades the original mass split into 82 fragments, leaving a fiendishly difficult jigsaw puzzle for researchers to put back together. The device appears to be a geared astronomical calculation machine of immense complexity. Today we have a reasonable grasp of some of its workings, but there are still unsolved mysteries. We know it is at least as old as the shipwreck it was found in, which has been dated to between 60 and 70 B.C.E., but other evidence suggests it may have been made around 200 B.C.E.

In March 2021 my group at University College London, known as the UCL Antikythera Research Team, published a new analysis of the machine. The team on that research included me (a mathematician and filmmaker); Adam Wojcik (a materials scientist); Lindsay MacDonald (an imaging scientist); the late Myrto Georgakopoulou (an archaeometallurgist); and two graduate students, David Higgon (a horologist) and Aris Dacanalis (a physicist). Our paper posited a new explanation for the gearing on the front of the mechanism, where the evidence had previously been unresolved. Our findings gave us an even better appreciation for the sophistication of the device—an understanding that challenges many of our preconceptions about the technological capabilities of the ancient Greeks.

Ancient Astronomy

We know the Greeks of that era were accomplished naked-eye astronomers. They viewed the night sky from a geocentric perspective—every night, as Earth turned on its axis, they saw the dome of stars rotating. The stars’ relative positions remained unchanged, so the Greeks called them the “fixed stars.” These early astronomers also saw bodies moving against the background of stars: the moon goes through a rotation against the stars every 27.3 days; the sun takes a year.

The other moving bodies are the planets, named “wanderers” by the Greeks because of their erratic motions. They were the deepest problem for astronomy at the time. Scientists wondered what they were and noticed that sometimes the wanderers move in the same direction as the sun—in “prograde” motion—then come to a stop and reverse direction to move in “retrograde.” After a while they reach another stationary point and resume prograde motion again. These rotations are called the synodic cycles of the planets—their cycles relative to the sun. The seemingly strange reversals happen because, as we know now, the planets orbit the sun—not, as the ancient Greeks believed, Earth.

In modern terms, all the moving astronomical bodies have orbits close to the plane of Earth’s motion around the sun—the so-called ecliptic—meaning that they all follow much the same path through the stars. Predicting the positions of the planets along the ecliptic was very difficult for early astronomers. This task, it turns out, was one of the primary functions of the Antikythera mechanism. Another function was to track the positions of the sun and moon, which also have variable motions against the stars.

Much of the mechanism’s design relies on wisdom from earlier Middle Eastern scientists. Astronomy in particular went through a transformation during the first millennium B.C.E. in Babylon and Uruk (both in modern-day Iraq). The Babylonians recorded the daily positions of the astronomical bodies on clay tablets, which revealed that the sun, moon and planets moved in repeating cycles—a fact that was critical for making predictions. The moon, for instance, goes through 254 cycles against the backdrop of the stars every 19 years—an example of a so-called period relation. The Antikythera mechanism’s design uses several of the Babylonian period relations.

One of the central researchers in the early years of Antikythera research was German philologist Albert Rehm, the first person to understand the mechanism as a calculating machine. Between 1905 and 1906 he made crucial discoveries that he recorded in his unpublished research notes. He found, for instance, the number 19 inscribed on one of the surviving Antikythera fragments. This figure was a reference to the 19-year period relation of the moon known as the Metonic cycle, named after Greek astronomer Meton but discovered much earlier by the Babylonians. On the same fragment, Rehm found the numbers 76, a Greek refinement of the 19-year cycle, and 223, for the number of lunar months in a Babylonian eclipse-prediction cycle called the saros cycle. These repeating astronomical cycles were the driving force behind Babylonian predictive astronomy.

The second key figure in the history of Antikythera research was British physicist turned historian of science Derek J. de Solla Price. In 1974, after 20 years of research, he published an important paper, “Gears from the Greeks.” It referred to remarkable quotations by Roman lawyer, orator and politician Cicero (106–43 B.C.E.). One of these described a machine made by mathematician and inventor Archimedes (circa 287–212 B.C.E.) “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.” This machine sounds just like the Antikythera mechanism. The passage suggests that Archimedes, although he lived before we believe the device was built, might have founded the tradition that led to the Antikythera mechanism. It may well be that the Antikythera mechanism was based on a design by Archimedes.

Fiendishly Complex

For decades researchers were stuck trying to decipher the workings of the device by looking at the surface of its disintegrating fragments. In the early 1970s they finally got to peek inside. Price worked with Greek radiologist Charalambos Karakalos to obtain x-ray scans of the fragments. To their astonishment, the researchers found 30 distinct gears: 27 in the largest fragment and one each in three others. Karakalos, with his wife, Emily, was able to estimate the tooth counts of the gearwheels for the first time, a critical step in understanding what the mechanism calculated. The machine was looking more complicated than anyone had conceived.

The x-ray scans were two-dimensional, meaning that the structure of the gearing appeared flattened, and they revealed only partial pictures of most of the gears. Scientists could only infer the number of teeth on many of the gears. Despite these shortcomings, Price identified a gear train—a set of linked gears—that calculated the average position of the moon on any specific date by using its period relation of 254 sidereal rotations in 19 years. Driven by a prominent feature on the front of the mechanism called the main drive wheel, this gear train starts with a 38-tooth gear (two times 19, as a gear with just 19 teeth would be a bit too small). This 38-tooth gear drives (via some other gears) a 127-tooth gear (half of 254; the full number would require too large a gear).

It seems that the device could be used to predict the positions of the sun, moon and planets on any specific day in the past or future. The maker of the machine would have had to calibrate it with the known positions of these bodies. A user could then simply turn a crank to the desired time frame to see astronomical predictions. The mechanism displayed positions, for instance, on a “zodiac dial” on the front of the mechanism, where the ecliptic was divided into a dozen 30-degree sections representing the constellations of the zodiac. Based on the x-ray data, Price developed a complete model of all the gearing on the device.

Price’s model was my introduction to the Antikythera mechanism. My first paper, in fact, “Challenging the Classic Research,” was a comprehensive demolition of most of Price’s proposed gearing structure for the machine. Nevertheless, Price correctly determined the relative positions of the major fragments and defined the overall architecture of the machine, with date and zodiac dials at the front and two large dial systems at the back. Price’s achievements were a significant step in decoding the Antikythera mystery.

A third key figure in the history of Antikythera research is Michael Wright, a former curator of mechanical engineering at London’s Science Museum. In collaboration with Australian professor of computer science Alan G. Bromley, Wright carried out a second x-ray study of the mechanism in 1990 using an early 3-D x-ray technique called linear tomography. Bromley died before this work bore fruit, but Wright was persistent, making important advances, for example, in identifying the crucial tooth counts of the gears and in understanding the upper dial on the back of the device.

In 2000 I proposed the third x-ray study, which was carried out in 2005 by a team of academics from England and Greece in collaboration with the National Archaeological Museum in Athens. X-Tek Systems (now owned by Nikon) developed a prototype x-ray machine to take high-resolution 3-D x-ray images using microfocus x-ray computed tomography (x-ray CT). Hewlett-Packard used a brilliant digital imaging technique called polynomial texture mapping for enhancing surface details.

The new data surprised us. The first major breakthrough was my discovery that the mechanism predicted eclipses in addition to the motions of the astronomical bodies. This finding was connected to the inscription Rehm had found that mentioned the 223-month saros eclipse cycle. The new x-rays revealed a large, 223-tooth gear at the rear of the mechanism that turns a pointer around a dial that spirals out, making four turns in total that are divided into 223 sections, for 223 months. Named after the customary name of the Babylonian eclipse cycle, the saros dial predicts which months will feature eclipses, along with characteristics of each eclipse as described by inscriptions in the mechanism. The finding revealed an impressive new feature of the device, but it left a massive problem: a group of four gears lying within the circumference of the large gear that appeared to have no function.

It took months to understand these gears. When I did, the results were astonishing. These gears turned out to calculate the variable motion of the moon in a very beautiful way. In modern terms, the moon has variable motion because it has an elliptical orbit: when it is farther from Earth, it moves more slowly against the stars; when it is closer, it moves more quickly. The moon’s orbit, however, is not fixed in space: the whole orbit rotates in a period of just under nine years. The ancient Greeks did not know about elliptical orbits, but they explained the moon’s subtle motion by combining two circular motions in what is called an epicyclic theory.

I figured out how the mechanism calculated the epicyclic theory by building on a remarkable observation by Wright. He had studied two of the four mysterious gears at the back of the mechanism. He saw that one of them has a pin on its face that engages with a slot on the other gear. It might seem to be a useless arrangement because the gears will surely just turn together at the same rate. But Wright noticed that the gears turn on different axes separated by just over a millimeter, meaning that the system generates variable motion. All these details appear in the x-ray CT scan. The axes of the gears are not fixed—they are mounted epicyclically on the large, 223-tooth gear.

Wright discarded the idea that these gears calculated the moon’s variable motion because in his model, the 223-tooth gear turned much too fast for it to make sense. But in my model, the 223-tooth gear rotates very slowly to turn the pointer for the saros dial. Calculating the epicyclic theory of the moon with epicyclic pin-and-slot gears in this subtle and indirect way was an extraordinary conception by the ancient Greeks. This ingenuity reinforces the idea that the machine was designed by Archimedes. This research on the back dials and gearing completed our understanding of the back of the mechanism, reconciling all the evidence up to that point. My colleagues and I published our findings in 2006 in Nature. The other side of the device, however, was still very much a mystery.

The Front of the Mechanism

The most prominent feature of the front of the largest fragment is the main drive wheel, which was designed to rotate once a year. It is not a flat disk like most of the other gears; this one has four spokes and is covered in puzzling features. The spokes show evidence that they held bearings: there are circular holes in them for turning axles. The outer edge of the gear contains a ring of pillars—little fingers that stick up perpendicularly, with shoulders and pierced ends that were clearly intended to carry plates. Four short pillars held a rectangular plate, and four long pillars held a circular plate.

Following Price, Wright proposed that an extensive epicyclic system—the two-circles idea the Greeks used to explain the odd reversing motions of the planets—had been mounted on the main drive wheel. Wright even constructed an actual model gearing system in brass to show how it worked. In 2002 he published a groundbreaking planetarium model for the Antikythera mechanism that displayed all five planets known in the ancient world. (The discovery of Uranus and Neptune in the 18th and 19th centuries, respectively, required the advent of telescopes.) Wright showed that the epicyclic theories could be translated into epicyclic gear trains with pin-and-slot mechanisms to display the planets’ variable motions.

When I first saw Wright’s model, I was shocked by its mechanical complexity. It even featured eight coaxial outputs—tubes all centered on a single axis—that brought information to the front display of the device. Was it really plausible that the ancient Greeks could build such an advanced system? I now believe that Wright’s conception of coaxial outputs must be correct, but his gearing system does not match the economy and ingenuity of the known gear trains. The challenge our UCL team faced was to reconcile Wright’s coaxial outputs with what we knew about the rest of the device.

One crucial clue came from the 2005 x-ray CT study. In addition to showing the gears in three dimensions, these scans made an unexpected revelation—thousands of new text characters hidden inside the fragments and unread for more than 2,000 years. In his research notes from 1905 to 1906, Rehm proposed that the positions of the sun and planets were displayed in a concentric system of rings. The mechanism originally had two covers—front and back—that protected the displays and included extensive inscriptions. The back-cover inscription, revealed in the 2005 scans, was a user manual for the device. In 2016 Alexander Jones, a professor of the history of astronomy at New York University, discovered definitive evidence for Rehm’s idea within this inscription: a detailed description of how the sun and planets were displayed in rings, with marker beads to show their positions.

Any model for the workings of the mechanism should match this description—an explanation literally inscribed onto the back cover of the device describing how the sun and planets were displayed. Yet previous models had failed to incorporate this ring system because of a technical problem that we could not solve. Wright had discovered that the device used a semisilvered ball to show the phase of the moon, which it calculated mechanically by subtracting an input for the sun from an input for the moon. But such a process appeared to be incompatible with a ring system for displaying the planets because the outputs for Mercury and Venus prevented the moon-phase device from accessing the input from the sun gear system. In 2018 Higgon, one of the graduate students on our UCL team, came up with a surprisingly simple idea that neatly fixed this technical problem and explained a mysterious pierced block on one of the spokes of the main drive wheel. This block could transmit the “mean sun” rotation (as opposed to the variable “true sun” rotation) directly to the moon-phase device. This setup enabled a ring system for the front of the Antikythera mechanism that fully reflected the description in the back-cover inscription.

In trying to decipher the front of the device, it was imperative to identify the planetary cycles built into the mechanism because they define how the gear trains calculated planetary positions. Earlier research assumed that they would be based on the planetary period relations derived by the Babylonians. But in 2016 Jones made a discovery that forced us to discard that assumption.

The x-ray CT of the front-cover inscription shows it is divided into sections for each of the five planets. In the Venus section, Jones found the number 462; in the Saturn section, he found the number 442. These numbers were astonishing. No previous research had suggested that ancient astronomers knew them. In fact, they represent more accurate period relations than the ones found by the Babylonians. It seems that the makers of the Antikythera device discovered their own improved period relations for two of the planets: 289 synodic cycles in 462 years for Venus and 427 synodic cycles in 442 years for Saturn.

Jones never figured out how the ancient Greeks derived both these periods. We set out to try ourselves. Dacanalis, our other UCL graduate student, assembled a comprehensive list of the planetary period relations and their estimated errors from Babylonian astronomy. Could combinations of these earlier relations be the key to the more accurate Antikythera period relations? Eventually we found a process, developed by philosopher Parmenides of Elea (sixth to fifth century B.C.E.) and reported by Plato (fifth to fourth century B.C.E.), for combining known period relations to get better ones.

We proposed that any method the Antikythera creators used would have required three criteria: accuracy, factorizability and economy. The method must be accurate to match the known period relations for Venus and Saturn, and it must be factorizable so the planets could be calculated with gears small enough to fit into the mechanism. To make the system economical, different planets could share gears if their period relations shared prime factors, reducing the number of gears needed. Such economy is a key feature of the surviving gear trains. Based on these criteria, our team derived the periods 462 and 442 using the idea from Parmenides and employed the same methods to discover the missing periods for the other planets where the inscriptions were lost or damaged.

Armed with the period relations for the planets, we could now understand how to fit the gear trains for the planets into the tight spaces available. For Mercury and Venus, we theorized economical five-gear mechanisms with pin-and-slot devices, similar to Wright’s mechanisms for these planets. We found strong supporting evidence for our reconstruction in one four-centimeter-diameter fragment. Inside this piece, the x-ray CT shows a disk attached to a 63-tooth gear, which turns in a D-shaped plate. The number 63 shares the prime factors 3 and 7 with 462 (the Venus period). A gear train using the 63-tooth gear could be designed to match a bearing on one of the spokes of the main drive wheel. A similar design for Mercury matches the features on the opposite spoke. These observations gave us great confidence that we were on the right track for Mercury and Venus.

For the other known planets—Mars, Jupiter and Saturn—our team conceived of very compact systems to fit the available space. These designs were a radical departure from Wright’s systems for these planets. Working independently, Christián C. Carman of the National University of Quilmes in Argentina and I had shown that the subtle indirect gearing system for the variable motion of the moon could be adapted for these planets. Our UCL team proved that these gearing systems could be extended to incorporate the new period relations for the planets. This system allowed the Antikythera makers to mount several gears on the same plate and design them to precisely match the period relations.

These economical seven-gear trains could intricately interleave between the plates on the pillars of the main drive wheel so that their outputs conformed to the customary cosmological order of the celestial bodies—moon, Mercury, Venus, sun, Mars, Jupiter and Saturn—that determines the layout of the ring system. The dimensions of the available spaces between the plates were exactly right to accommodate these systems, with some spare capacity and some evidence still unexplained.

We added a mechanism for the variable motion of the sun and an epicyclic mechanism for calculating the “nodes” of the moon—the points at which the moon’s orbit cuts through the plane of the ecliptic, making an eclipse possible. Eclipses happen only when the sun is close to one of these nodes during a full or new moon. Medieval and renaissance astronomers called a double-ended pointer for the nodes of the moon a “dragon hand.” The epicyclic gearing for this dragon hand also exactly explained a prominent bearing on one of the spokes that had previously appeared to have no function. We had finally explained all the features on the main drive wheel; we published our findings in March 2021 in Scientific Reports.

A Beautiful Conception

We now understood how the front display matched the description in the back-cover user’s manual, with the sun and planets shown by marker beads on concentric rings. The front cover also displayed the moon’s phase, position and age (the number of days from a new moon), and the dragon hand that showed eclipse years and seasons.

With the concentric rings for the planets, we realized that we could now make sense of the front-cover inscription as well. This writing is a formulaic list of the synodic events of each planet (such as its conjunctions with the sun and its stationary points) and the intervals in days between them. On the back plate, the eclipse inscriptions are indexed to markings on the saros dial. On the front plate, inscriptions about the risings and settings of stars are indexed to the zodiac dial. Our insight was that the inscriptions on the front could refer to index letters on the planetary rings: if the sun pointer is at one of these letters, then the corresponding inscription entry describes the number of days to the next synodic event. Because the left-hand side of the inscription, where we would expect these index letters to be, is missing, we cannot prove the hypothesis—but it is a compelling explanation.

The device is unique among discoveries from its time. It single-handedly rewrites our knowledge of the technology of the ancient Greeks. We knew they were highly capable—they built the Parthenon and the Lighthouse of Alexandria even earlier than the Antikythera mechanism. They had plumbing and used steam to operate equipment. But before the discovery of the Antikythera mechanism, ancient Greek gears were thought to be restricted to crude wheels in windmills and water mills. Aside from this discovery, the first precision-geared mechanism known is a relatively simple—yet impressive for the time—geared sundial and calendar of Byzantine origin dating to about C.E. 600. It was not until the 14th century that scientists created the first sophisticated astronomical clocks. The Antikythera mechanism, with its precision gears bearing teeth about a millimeter long, is completely unlike anything else from the ancient world.

Why did it take centuries for scientists to reinvent anything as sophisticated as the Antikythera device, and why haven’t archaeologists uncovered more such mechanisms? We have strong reasons to believe this object can’t have been the only model of its kind—there must have been precursors to its development. But bronze was a very valuable metal, and when an object like this stopped working, it probably would have been melted down for its materials. Shipwrecks may be the best prospects for finding more of them. As for why the technology was seemingly lost for so long before being redeveloped, who knows? There are many gaps in the historical record, and future discoveries may well surprise us.

With the Antikythera mechanism, we are clearly not at the end of our story. We believe our work is a significant advance, but there are still mysteries to be solved. The UCL Antikythera Research Team is not certain that our reconstruction is entirely correct because of the huge loss of evidence. It is very hard to match all of the surviving information. Regardless, we can now see more clearly than ever what a towering achievement this object represents.