Comment by monknomo

The first thing to keep in mind is that these are not general purpose machines. They only do one thing, which is estimate the trajectory of a particular type of bullet, given several inputs.

This lets the designer do a lot of precalculation, so they might use a series of cams that approximate known curves or differentials some known ratio.

Useful google terms are things like "mechanical fire control computer" or "mechanical tide predictor".

Check out "Old Brass Brains" it might give you an idea of how to approach this kind of problem mechanically; or at least how sophisticated a mechanical approach can be.