Mechanical integrator
A wheel resting on a spinning disc computes an integral without knowing it — and it took 55 years to find the force to drive the next wheel.
The construction is almost embarrassingly simple. A disc turns at a steady rate and a small wheel rests against it, free to slide along the radius. Move the wheel toward the rim and it turns faster, toward the centre and it turns slower — so the wheel's total rotation is the integral of its radial position over time. James Thomson described the principle in 1876, and his brother William Thomson, Lord Kelvin, put it straight to work in the tide-predicting machine.
Kelvin also saw that a chain of integrators could solve differential equations, and there it stalled: a friction wheel can barely drive a pointer, let alone the next disc. Only once the torque amplifier was available could Vannevar Bush build the differential analyzer in 1931 on exactly Kelvin's plan. The same wheel-on-disc then sat inside fire-control gear and bombsights well into the 1950s, wherever the answer needed to be roughly right immediately rather than exactly right later.