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. So every astronomical claim the machine makes, you can re-derive from its own teeth.
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 nothing close would be built for more than a thousand years — the next 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 the four-year games — 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 is built entirely from those reconstructed tooth counts. Nothing here is asserted; it is computed — by a verifier you can run, from the integers stamped into the bronze.
One turn of the crank 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. 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, half dark — turns to show the phase, the angle between Moon and Sun.
The lunar anomaly, close up. Two equal 50-tooth wheels, k1 and k2, mounted face-to-face on different centres; a pin on one rides a slot in the other. Equal teeth mean no change on average — but the offset makes the Moon run fast near one side of its orbit and slow near the other, Kepler's law of the ellipse, two millennia early.
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 is the machine's arithmetic, every value below printed by the verifier from the tooth counts alone.
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 orbits of the Moon take exactly 19 years, the same 19 years in which the Moon returns to the same phase 235 times. 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 are the months an eclipse is possible.
Why exactly 38 eclipse glyphs
On the real Saros dial, only some of the 223 cells carry an eclipse glyph, and they are spaced 6 months apart — sometimes 5. That is not decoration; it is forced arithmetic. If a glyphs are 6 months to the next and b are 5 months, then 6a + 5b = 223, and spreading them as evenly as possible around the ring fixes the answer:
6·33 + 5·5 = 198 + 25 = 223
→ 33 + 5 = 38 eclipse-possibility glyphs, exactly as the bronze is engraved.
And below the Saros, a small subsidiary dial — the Exeligmos — counts off three Saros cycles at a time. Three, because one Saros is 6585⅓ days: after a single Saros an eclipse recurs eight hours later and a third of the way round the Earth, so you must wait three for it to come back to the same place and hour. The gearing for it is a clean ×12 — exactly three turns 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:
Attested vs. reconstructed
30 of the 31 gears named on this page have tooth counts read directly off the surviving fragments in the CT scans. One — n3, the 57-tooth wheel that drives the Olympiad dial — is reconstructed: it does not survive, but Freeth's 2008 analysis shows a 57-tooth gear at that spot turns the games dial at exactly ¼ turn a year, and the spacing of the axles fits to the millimetre. We mark it apart because it is an inference, however good.
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 no planetary gearing survives at all. Reconstructions of it (including a 2021 model from the same group) are proposals, not readings, and we have left them off this page entirely. One real surviving gear, r1 with 63 teeth, is still unaccounted for — it fits no train anyone has agreed on.
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.