Babbage's Difference Engine

Before electronic computers existed, Charles Babbage designed a machine that could tabulate polynomials using nothing but repeated addition, propagated through columns of decimal digit wheels connected by carry gearing. This model renders four such registers — the value and its first three forward differences — as real rotating gear wheels. Set the coefficients of a cubic polynomial, turn the crank by hand or let it run automatically, and watch each digit wheel click forward exactly as the method of finite differences dictates: every later value of the polynomial falls out of the machine one addition at a time, with the number of multiplications used staying at zero throughout — the entire reason the design was mechanically realisable with Victorian-era engineering.