I should clear up what I was trying to say in my last.
The problem is trying to construct gears. The simplist method uses regular constructible polygons to divide the circle. But there are a limited number of constructible regular polygons. Go here for details:
http://mathworld.wolfram.com/ConstructiblePolygon.html
The series 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24,... are construtible.
If the gear teeth needed are not in this series, it means you have to use measuring device with fractional units. Then you get into a related problem of dividing a line segment.
Any way, the whole device is full of very interesting and `deep' math problems in number theory and geometry...
CG