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One way to do a ratio of 720:1151 is to construct a gear with 720 teeth, and a second gear with 1151 teeth, and mesh them together. This wouldn't be impossible, but it would be a lot of work. If it takes you 5 minutes per tooth, that would be 156 hours sitting there with a file, grinding teeth away. Plus, perhaps most importantly, they'd be enormous- you're talking gears that are like 2 feet across for 1/8" teeth. The Antikythera mechanism was a little over a foot on its long axis, and 8 inches across the other. Let's say instead of a ratio of 720:1151, the ratio happened to be 720:1147. You construct gears with 24, 30, 31, and 37 teeth. (24x30 = 720, 31x37 = 1147) 720 and 1147 are still coprime, so you can't be reduced the way 720:1152 can be reduced to 5:8. (or more likely, 20:32) You connect the teeth of your 24 tooth gear to your 31 tooth gear, connect your 31 tooth gear to the 30 tooth gear via a common shaft, and connect the teeth of your 30 tooth gear to the 37 tooth gear. The final gear ratio of this mechanism will be 720:1147. This only works because both 720 and 1147 can be factorized into manageable primes. Constructing those 122 teeth will take 10 hours at 5 minutes a pop, which is a lot of work but not unmanageable. Furthermore, those four gears can be constructed by four workers, and it will take the 37 tooth worker 3 hours, but a worker working on a 1151 tooth gear will require 96 hours. The gears will be much smaller, easily fitting in your fingers. ---------------------------------------- It's also more difficult to construct gears with prime numbers of teeth. If you need to construct a gear with 32 teeth, you bisect the angles a few times until there are 32 portions, then cut teeth there. If you need to construct a gear with 30 teeth, you split it carefully into thirds, carefully split each third into fifths, and then cut two teeth in each fifth. This would probably be a bit rough, but probably within tolerances. You split into thirds or fifths with a straight edge and compass, the way you can with bisecting, but you can wrap some string around the edges and even split string into thirds pretty easily, and fifths with some effort. If you need to construct a gear with 31 teeth, there's not really a convenient way to do it. There's a lot of pedagogy involved; I think the best way would be to guess about how large a tooth is gonna be, add that to a string you've wrapped around the blank wheel, then repeatedly bisect the string into 32nds, then check if the 32nd tooth overlaps the 1st tooth well enough. If not, start over. But there might be a more convenient way to do it. |