Machining Nerd Time

 As I have previously discussed, the core of all machine tools is a 20 threads/inch leadscrew. One turn of the screw is 0.50". If you can narrow that down to 100 divisions, you can easily move pieces on that lead screw .01". This is beginning to get accurate for mildly precise work. But how do divide a circles into 100 equal divisions. Think back to the suffering of high school geometry. Dividing a circle in half is just drawing a line across the center point. Another diameter that crosses the center point gets you 4 divisions of 90 degrees. How do you get to 100 divisions? Even more useful, 360 divisions? I often think about having to rebuild civilization even without a 20 threads/inch screw thread. (Okay, not on your list of concerns.)

How do you get this capability if you lack a starting reference for 360 degrees. This has been bugging me for years. This video explains how it was done.

Short answer is that you use a worm gear to turn a wheel with a 40:1 gear ratio. Forty turns of the worm gear did one full rotation. One rotation was 9 degrees. Keep increasing the worm gear ratio to get finer divisions. The rotating table that I use with my Sherline mill uses this technique to get 0.1 degree divisions. Actually with some interpolation better than that!

When I was young, my father taught me how to use a caliper with a Vernier scale. I learned and forgot. Digital calipers are too easy to use and cheap, too! But how do they work? I have a slight inkling from using a Foucault tester for making a telescope mirror. It used a nomograph for allowing measurement of hundredths of an inch of movement. You make a nomograph by drawing a series of lines .1" apart in the X direction. Then you draw diagonal lines from left side of 0" to right side of .1". Now measure 1" wide line with .1" divisions on 0" line. Now draw lines from each .1" division to the last line along Y. 

This is an oversimplified version, not done very well using OpenOffice Draw:

Where the objects sits as you reach the point where the mirror edge exactly shows a proper paraboloid image, you look at where the knife edge intersects the diagonal. The magic of the Foucault tester is that a very simple set of parts lets you measure optical precision to millionths of an inch.

This video explains the math and history of the vernier scale:

I find myself wondering about use of vernier scales for angles (for which Vernier first developed the idea). Equatorial telescope mounts have an adjustment for the polar axis, which needs to be aimed at your latitude above the horizon in order track across the sky.

As you can sort of see, the degree divisions are necessarily not very fine. On smaller, cheap mounts, you are lucky to get 2 degree divisions. A vernier scale should make this much more precise, more precise than it really needs to be.