At 04:07 PM 11/30/2012, you wrote:
Gear wheels have teeth that have to engage, which implies that they have the same pitch.
It is easy to construct gear wheels with 2^n teeth.
Since gear wheels are made from circular blanks, and since the circumference is proportional to the diameter (we don't even have to know the value of pi), we can produce a circular blank whose diameter is any integral ratio k/(2^n) to the diameter of a 2^n circular blank.
All we have to do now is to mark off gear teeth of the same pitch in the new circular blank. If we measured everything carefully, the circumference on the new non-2^k blank should be an integral number of teeth.
Nutz! That should be ^^^^^^^ non-2^n
It's too bad that the antikythera mechanism is so corroded; it would be very interesting to study an uncorroded ancient gear train to see exactly what the shape of the gears was. I'd be willing to bet that the Greeks probably figured out a very close approximation to the correct shape for the gear teeth, as well, so that the gear train would have very little "play" which might throw off the accuracy of the device.
At 11:42 AM 11/30/2012, Henry Baker wrote:
For the non-constructible numbers, how did the Greeks construct them? Which additional operations were allowed to enable the construction of those gear wheels ?