[nograpes]

Can anyone help me understand why gears with prime numbers of teeth would not be mechanizable?

For Venus the original designer faced a dilemma: the known period relation (5, 8) was very inaccurate, whereas the accurate (720, 1151) was not mechanizable because 1151 is a prime number, requiring a gear with 1151 teeth.

I thought that gears with prime numbers of teeth would be advantageous because it would spread the wear evenly across the gear that it contacted.

[vgel]

In addition to the other comments, gears with a prime number of teeth were undesirable because they couldn't be laid out by iterative division of a circle. A gear with 64 teeth can be easily laid out by dividing the circle into fourths, dividing those fourths into fourths, and again to get 64 even divisions. For a gear with a prime number of teeth, the only option is to guess-and-check walk a pair of calipers around the circle, adjusting them iteratively until you make the exact number of steps and wind up at the exact same place. Without vision magnification, this was extremely difficult to do accurately. Clickspring (see top comment) did some experiments with a large dividing plate that makes the process somewhat easier, but it would still be far more difficult than making a non-prime number of teeth.

[CGamesPlay]

The other commenters are all correct, but I wanted to say something I didn't see mentioned: you can totally do it, it's trivial to imagine a working gear with 13 teeth. But you can't make two smaller gears that have a ratio of 1:13, because 13 is prime, so you literally have to make it be 1 gear. That means the 1:1151 ratio literally has to be 1 big gear, which is hard to make.

You do have a misunderstanding about the wear. Unless your gear is oscillating back and forth, it will always move in a complete circle and always wear the teeth evenly. Note that the involute tooth design leads to less wear on an individual tooth, but it doesn't have to do with how many teeth are on the gear.

[biggieshellz]

You can't split them up into several smaller gear pairs, so you actually need two gears with 720 and 1151 teeth. A gear with 1151 teeth is impractical to make, both in terms of size and in terms of the manufacturing capability at the time.

[nwallin]

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.

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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.

[wolfd]

It's tough to make gears with that many teeth, especially if you want them to mesh with smaller gears.

Also, meshing against a 1-tooth gear is problematic, so you would need to probably increase that to >4 teeth to have it work. Then your bigger gear needs to have >4x the teeth to get the desired ratio.

[blt]

1151 is too many teeth for a gear. The teeth would not be deep enough to transmit power effectively. You can't make this exact ratio using gears with fewer teeth because 1151 is prime (more generally, whenever the numerator and denominator are coprime).