Babbage's Difference Engine

Do it on paper first. The trick is so simple it looks like a trap.

1. Take the polynomial f(x) = x² + 2x + 3 and tabulate it for x = 0,1,2,3,4,5.
2. Write the differences between consecutive values underneath — that is the first difference row.
3. Write the differences of THOSE — the second difference row.
4. Look at the second difference row.

The second differences are all the same number. For any polynomial of degree n, the nth difference row is constant — and that means you can run the table BACKWARDS: start from the constant, add upward, and generate the next value using only addition.

Extend your table two more rows without ever squaring anything. Add the constant to the last second difference, add that to the last first difference, add that to the last value. Three additions produce the next point of a quadratic.

Make the two parts the machine is built from, and feel the precision problem.

1. Cut a ten-tooth wheel in brass — index the blank at 36° intervals and cut each tooth.
2. Measure the error in each tooth position with a caliper against a dividing plate.
3. Make a second wheel with a single carry cam projecting from one face, positioned to act as that wheel passes from 9 to 0.
4. Mesh them and turn slowly, watching the carry transfer.

The carry cam must act ONLY between 9 and 0 and never at any other position, which means its angular width and timing are as critical as the tooth spacing.