The Machine
Made of Months
A bronze computer, hand-cranked, lost in a shipwreck around 60 BC and pulled from the seabed in 1901. Every pointer it carries is run by a train of gears, and a train of gears is just a product of fractions. Its reconstructed mean gear ratios can be re-derived from tooth counts; day conversions and comparisons need separately supplied astronomical values.
In 1901 sponge divers working a wreck off the islet of Antikythera brought up bronze statues, glassware — and a corroded lump the size of a thick book. It split, drying, into fragments shot through with the green ghosts of gearwheels: thirty of them survive, the largest with 223 teeth, cut by hand into triangular points about a millimetre and a half apart.
Nothing else like it survives from antiquity, and no comparably complex device is known from the next thousand years. Later geared astronomical clocks appear in medieval Europe and the Islamic world. It is a calculator for the sky: turn a crank on the side, and pointers on the front and back faces show the Sun's place in the zodiac, the Moon's place and phase, the cycle that keeps a lunar calendar in step with the year, and a spiral that predicts eclipses. The inscriptions even name games of the four-year Olympiad cycle: Olympia, Nemea, Isthmia, and others.
We can say how it works to the tooth because of X-ray CT scans made in 2005, which read the buried gears Freeth and colleagues published in Nature. What follows uses those reconstructed tooth counts, a supplied year of 365.2425 days, and modern astronomical comparisons. The mean gear ratios and their residuals are computed by a verifier you can run; the browser is a separate illustration with mirrored values.
One turn of the year wheel is one year
The great four-spoked wheel, b1 (223 teeth), turns once a year and carries the Sun pointer around a ring of the twelve zodiac signs. Off that single shaft hangs every other train. The controls here advance days on this annual scale. With the adopted 48-tooth a1 driving 223-tooth b1, one turn of the reconstructed side crank advances 48/223 of a year. Drag the crank, or press play, and watch the Moon chase the Sun: it laps the zodiac about thirteen times a year, and the little ball, half bright and half dark, turns to show the phase, the angle between Moon and Sun.
A schematic pin-and-slot, seen in its carrier's frame. Two equal 50-tooth wheels, k1 and k2, turn on different centres; a pin on one rides a slot in the other. Equal teeth preserve the mean rate, while the offset adds the variable motion of Hipparchos' epicyclic lunar theory. Here the centre offset is exaggerated to 0.34 of the pin radius; the front pointer and degree readout share that schematic exaggeration.
The Moon pointer's mean rate is exactly 254 / 19 turns per year — and its speed is not steady: the pin-and-slot makes it surge and lag, the apse line itself creeping round once every 8.88 years.
Every gear is a fraction
Mesh a wheel of D teeth into one of E teeth and the second turns D/E times for each turn of the first. Chain them and you multiply the fractions. That is the whole secret: each pointer's rate is a single product of small whole numbers, chosen so the product lands on an astronomical ratio. Here are the reconstructed mean gear ratios, with day conversions using the supplied 365.2425-day year and separately supplied modern comparisons; the ledger mirrors those values from the offline verifier.
The Moon train is (38/64)·(24/48)·(32/127) — and that equals 19/254 on the nose. The bronze isn't close to a lunar month. It is one, to fourteen parts in a million.
The headline gear is d2, with 127 teeth, a prime, and an awkward thing to cut by hand. Why 127? Because 127 = 254 ÷ 2, and 254 mean orbits of the Moon take nearly 19 years, approximately the same span as 235 returns to the same phase. In the adopted gear model, 254/19 is the exact mean lunar rate. The maker needed the number 254 in the train, and put half of it into a single wheel. The sky asked for a prime, so they filed a prime.
Counting eclipses on a spiral
The back face carries two spiral dials. The upper one is the Metonic calendar: a five-turn spiral of 235 cells, one per lunar month, that the pointer crawls around once every 19 years, the cycle that lets a Moon-calendar keep step with the Sun. The lower one is the Saros, 223 months of eclipse prediction. Drag the crank below and watch the pointers climb their spirals; the lit cells illustrate a simplified 38-position lunar-prediction scheme. Their positions are synthetic, and the animation uses modern cycle spans.
Why the lunar scheme has 38 predictions
The lunar-prediction scheme uses gaps of six months, sometimes five. That is not decoration; it is forced arithmetic. If a intervals are six months and b are five, then 6a + 5b = 223. This equation has seven nonnegative integer solutions; even spacing alone does not choose the counts. The 2008 analysis adopts 33 six-month and five five-month gaps for lunar possibilities, using astronomical constraints, then distributes them around the cycle:
6·33 + 5·5 = 198 + 25 = 223
→ 33 + 5 = 38 lunar eclipse possibilities in this scheme.
The 2008 reconstruction, Figure 20, has 51 glyphs, combining 38 lunar and 27 solar predictions. Some glyphs contain both. The surviving sequence includes adjacent glyph months 25 and 26, so the five- or six-month spacing does not describe all glyphs together. Our 38-marker illustration shows only a simplified lunar scheme.
And inside the Saros spiral, a small subsidiary dial, the Exeligmos, counts off three Saros cycles at a time. Three, because one Saros is about 6585⅓ days: after a single Saros an eclipse recurs roughly eight hours later and a third of the way round the Earth, so after three it returns to roughly the same longitude and time of day. The gearing for it is a clean ×12: exactly three cycles of the four-turn Saros spiral.
You cannot tile a year with months
Here is the problem the maker actually faced. A year is not a whole number of months — it is about 12.368 of them, and the true ratio never closes. Gears can only have whole teeth. So how do you build a machine whose ratio is, in effect, irrational?
You use the best whole-number approximations there are: the continued-fraction convergents of that ratio. Expand 12.368… as a continued fraction and read off its convergents — each one the best fraction of its size:
Sixth in the list sits 235 / 19 — the Metonic cycle, named for Meton of Athens in 432 BC but known to Babylon before him. It is not an accident the machine uses 19 years: 19 is where the approximation gets suddenly, almost suspiciously good, and the makers cut that convergent straight into the tooth counts. The Antikythera mechanism is a continued fraction you can hold in your hand.
The same incommensurability that this archive's founding layer is about — that no whole number of one thing ever equals a whole number of the other — is the exact problem the bronze solves, with the exact mathematics, twenty centuries early.
Where the bronze runs out
A machine that re-derives the sky from its teeth is only as honest as its account of which teeth we actually have. So, plainly:
Surviving vs. hypothetical gears
28 of the 31 gears listed in the verifier survive; their adopted tooth counts include estimates from partial remains. Three are hypothetical in Freeth's 2008 Figure 14: m3 = 27, n1 = 53, and n3 = 57. The last drives the Olympiad dial: a 57-tooth gear there turns it at exactly ¼ turn a year, and the estimated pitch radii fit the axle spacing. All three are flagged separately. The mechanism's total of 30 surviving gears includes gears outside this list.
The planets, and the missing wheel
The inscriptions describe the motions of all five planets known then, so there were probably planetary pointers on the front, but the planetary trains are largely lost. A 2021 model proposes a Venus role for surviving components in Fragment D. These reconstructions are proposals, and we have left the planetary trains off this page. The surviving r1 with 63 teeth has that proposed Venus role, but no agreed identification.
The machine is no better than its sources — and may not have run
Its cycles are Babylonian period-relations, so its errors are theirs: the Saros gearing lands 4 turns on 6585.24 days where the true value is 6585.32, and the lunar-apse period it builds, 8.8826 years, is about 0.37% off the modern 8.85. We show those residuals rather than hide them. And in 2025 one team argued that the hand-cut triangular teeth carried so much error that the mechanism may never have turned smoothly enough to be usable — while themselves cautioning that the scans may overstate the imperfections. That argument is live; we don't resolve it. What is not in doubt is the design: the ratios are exact, whoever cut them and however well they ran.
Strip away the corrosion and what is left is an argument, in bronze, that the sky is countable. Someone in the Greek world looked at the wandering Moon, the drifting calendar, the returning eclipse, and decided every one of them was a fraction — then cut those fractions into metal so a stranger turning a crank could read the heavens off a dial.
Two thousand years later the metal is green and broken, the maker nameless, the wreck a tourist dive. But the fractions still close. 19 over 254. 235 over 19. Six thirty-threes and five fives make 223. Turn the crank; the months are still there.