Engineering
Antikythera calendar machine
Turn the crank of a bronze reconstruction in a darkened gallery and the centuries pass on its dials: the moon’s pointer makes exactly 254 turns in 19 years, the synodic months come to 235, the Metonic spiral turns five times and the Saros spiral four in 223 months, where the glyphs of the eclipses repeat. Watch the pin-and-slot pair on the side make the moon hurry and dawdle through the anomalistic month, almost exactly as the Hipparchian (e/r) sin M; give the train backlash and reverse the crank, and nothing moves until the slack is taken up. The tooth counts make every ratio an exact fraction.
What happens
A crank turned once a year drives the gear train in the reconstruction of Freeth and colleagues: the moon pointer through 64/38·48/24·127/32 = 254/19, the Metonic spiral through 64/38·53/96·15/53 = 5/19, the Callippic dial a quarter of that, the Saros spiral 5/19·188/223 = 940/4237 (four turns in 223 synodic months) and the Exeligmos dial a twelfth of the Saros. The pin-and-slot pair on the epicyclic platform turns the follower to atan2(r sin M, r cos M − e), the lunar anomaly; the train’s backlash is a dead band that a reversed crank must take up first. The eclipse glyphs come from the true synodic and draconic months against the ecliptic limits.
Try this
Turn the crank for nineteen years and count the moon’s turns; watch the pin-and-slot make the moon hurry and dawdle; add backlash and reverse the crank; follow the eclipse glyphs round the Saros spiral.
The model and its limits
Ideal gears with exact tooth counts, a single lumped backlash, mean motions for the eclipse glyphs; the arrangement of the side gears is illustrative. Dates are those of the mechanism, not of the sky.
Equations used in the module
Equations moon = 64/38·48/24·127/32 = 254/19 turns a year Metonic 5/19, Callippic 5/76, Saros 940/4237, Exeligmos 235/12711 θ_out = atan2(r·sin M, r·cos M − e) ≈ M + (e/r)·sin M 19 years = 254 sidereal = 235 synodic months; 223 months = 4 Saros turns
