Long before electronics, the hardest sums in science were ground out by machines made of shafts and gears. The heart of the greatest of them — Vannevar Bush’s Differential Analyzer, built at MIT around 1931 — is the wheel-and-disk integrator you see here. A flat steel disk spins about its axis; that rotation is the input variable. A small friction wheel rides on the disk’s face, and a carriage slides it in and out along a radius. The further out the wheel sits, the faster it is dragged around — so the wheel’s own turning continuously adds up the area under whatever the carriage position represents. The wheel angle is the integral.
The relationship is exact and mechanical: turn the disk by a small step dx with the wheel at signed radius r, and the wheel turns by dφ = (r ⁄ R) dx, so after a full run φ = (1 ⁄ R) ∫ r dx. Move the carriage past the centre and r goes negative — the integral counts back down. This exhibit runs that kinematics for real: pick cos and the wheel angle traces sin; pick a ramp and it draws a parabola; the live graph lays the mechanical wheel angle over the exact integral and they sit on top of each other to the pixel. The wheel’s grip is feather-light, which is why Bush had to add a torque amplifier — the labelled drum on the output shaft — to make each integrator strong enough to drive the next.
Chain two integrators together, feed each one’s output back as the other’s radius, and put one reversing gear in the loop, and the machine solves a differential equation entirely in hardware: y″ = −y. Set it running and the output pen draws a perfect sine wave — a wave conjured from nothing but spinning brass and steel. Flip that one gear the other way and the same machine runs away into an exponential. The idea goes back to 1876, when James Thomson and his brother Lord Kelvin first described the wheel-and-disk integrator for predicting the tides. This rebuilt exhibit goes further: a differential-gear adder mixes in friction, an input table integrates any curve you draw, and four integrators run in concert to generate sin y on the fly and swing a full nonlinear pendulum — the trick that made Bush’s machine famous.
A differential analyzer is an analog computer: it represents numbers as angles and does calculus by motion. Its digital descendants represent numbers as bits and do arithmetic by logic — the same arithmetic you can wire up gate by gate in DigiSim’s Logic Lab. Stand here and watch a sine appear out of meshing gears, and the line from brass to silicon is suddenly very short.
Specifications
- Principle
- Wheel-and-disk integrator: dφ = (r ⁄ R) dx ⇒ φ = (1 ⁄ R) ∫ r dx
- Mechanism
- Friction wheel on a spinning disk, radius set by a lead-screw carriage
- The loop
- Two integrators + a reversing gear solve y″ = −y → a sine drawn by gears
- Simulated
- Whole state = shaft angles; increments propagate gear by gear — the pen plots only what the shafts compute
- Programs
- sine · exponential · damped swing · pendulum (y″ = −sin y) · area of a drawn curve
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