The Differential Analyzer
The differential analyzer was an analog computer invented between 1928 and 1931 by Vannevar Bush, then a professor of engineering at MIT. The machine was built to solve differential equations, which are often used in engineering and physics to model the behavior of dynamic systems. Some kinds of differential equation are trivial to solve, but most others are so complex that it is hard to even express their solutions in concrete mathematical terms.
The differential analyzer solved these complex differential equations by numerically approximating them. The machine could do this even though it contained no electronic circuits, logic gates, or anything else that we might associate with a digital computer. It relied instead on the interplay of various mechanical moving parts to compute results.
The most important component of the differential analyzer was the mechanical integrator. Below is a visualization of how the mechanical integrator worked. The visualization shows just one mechanical integrator, but the differential analyzer built by Bush actually had six—this allowed Bush to solve numerical equations of up to the sixth order. The final output produced by the differential analyzer was a combination of the outputs of the integrators: each individual integrator's output was either fed into the next integrator as input, added to the output the next integrator, or subtracted from the output of the next integrator. Combining the outputs of the integrators was an engineering challenge that Bush was the first to overcome, but the integrators are where the magic happened; they did the real work of solving the equations.
A mechanical integrator is basically a fancy summing machine. Its output is a graph of how many times a particular small wheel has revolved. The rate at which this wheel revolves is determined by a large spinning disk that drives the wheel. The disk can be displaced so that the wheel moves closer or further away from the center of the disk—the further away the wheel is from the center, the faster it spins. The operator of the machine would trace another graph as input to the machine and this input would control the displacement of the wheel.
If you'd like to know more about the differential analyzer, I've written a much longer blog post about how it works that you can find here.