The hardest part of a tide machine was not the pulleys or the pen — it was making a shaft turn once per 12.4206 hours of simulated time using wheels with whole numbers of teeth. This primer shows how builders squared that circle, starting from the simple 10:1 pair drawn above.
The 10:1 pair, slowly
Look at the sketch: a 100-tooth wheel meshes with a 10-tooth pinion. One turn of the big wheel advances 100 teeth past the contact; the pinion must therefore turn 100 ÷ 10 = 10 times. Speed multiplied by ten; torque divided by ten. Chain two such pairs and you get 100:1. Every machine was a forest of these multiplications and divisions, tuned so that one crank-turn equalled a chosen span of simulated time and each branch shaft ran at its constituent’s speed.
The lunar problem
The moon refuses round numbers. M2’s period, 12.4206 hours, relates awkwardly to the solar day the crank represents. Builders attacked this with compound trains: three or four pairs in series whose product approximates the awkward ratio, e.g. (73/59) × (64/51)… — each fraction a real wheel pair, the product landing within a fraction of a percent of truth. Residual error accumulated over a simulated year, so designers added correction dials and chose factorizations minimizing drift. Doodson’s Bidston team, profiled here, was obsessive on this point.
The cutter’s craft
Teeth had to be numerous (fine pitch for exact ratios), hard (bronze against steel), and true (backlash shows up directly as pen wobble). Surviving wheels show hand-finishing marks; operator notes complain about specific pairs by name. When I handle museum photographs I always zoom to the mesh points first — the machine’s accuracy lived there.
Practical questions
How accurate were the ratios? Good trains held phase within minutes of simulated time over a prediction year — smaller than the uncertainty in the tidal constants themselves. Gears were rarely the limiting factor; friction and wire stretch were.
Why not use chains or belts for ratios? Some machines did use chains for long runs, but toothed wheels alone gave the exact, slip-free ratios prediction demanded. A slipping belt would silently corrupt a whole year.
What should I look at on a museum machine? Find the smallest pinions — the high-speed shallow-water shafts — and the engraved tooth counts. My museums guide tells you which machines let you get close.
A challenge for your next museum visit
Stand before any machine in my museums guide and find the smallest pinion — the high-speed shallow-water shaft. Read its tooth count off the engraving, find its mating wheel, and do the division in your head. You are now doing, on the gallery floor, exactly what the builders did at the bench in 1910. If the ratio bug has bitten you, Kelvin’s prototype shows where the craft began, and Why 37? shows what all that gearing was carrying.