A water-damaged clockwork solar system has arrived at your bench. 1780. Its gears are lost. Only its numbers remain — and the sky doesn't wait for you to guess.
Real gear ratios. Real continued fractions. No two commissions solved the same way twice.
Every commission gives you a target — a number of days. The Moon's phase repeats every 29.5306 days. The Earth's year is 365.2422. You must build a train of gears whose ratio matches that number.
A train is gears in meshing pairs. Tap a gear to make it the driver, tap a second to make it driven — that's one pair, contributing its tooth ratio (driven ÷ driver) to the train. Chain more pairs and their ratios multiply.
No single pair reaches far. Teeth run 8 to 120 — one pair tops out around 15-to-1. To reach 365, you'll need to chain pairs, the same way a clockmaker really did it: build the ratio in stages, each stage a rational approximation closer than the last. That staged approximation has a name — a continued fraction — and it is the entire craft.
Spin it ten years and watch the drift. A gear train slightly wrong doesn't fail loudly — it just runs a little fast or slow, and after years, the Moon on your dial is full a day before the real one. Get inside the tolerance and the clockmaker bows. Use more gears than you needed, and the work is sound, but not admired.
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