Here's a series of 1950s-vintage videos illustrating the workings of mechanical Navy ballistics computers — enormous arrays of switches and gears that are used to quickly and reliably do complex mathematics in response to changing wind, targets and other factors. It's a really good, easy-to-follow guide to the underlying logic in a computer, and it's excitingly mechanical.

u.s navy vintage fire control computers (part 1)

(Thanks, Harold D and everyone else who suggested this!)