The Wheel-and-Disc Integrator

Stand a small friction wheel on a flat rotating disc with its axle pointing at the centre and its rim touching at radius x. Turn the disc once and the wheel is dragged along an arc of length two-pi-x, so it turns x over r times. Now move the wheel in and out while the disc keeps turning, and every increment of disc rotation adds whatever x happens to be at that instant. The wheel's total rotation is the integral of x, performed by friction, with no numbers anywhere. William Thomson described this in 1876, together with a scheme for wiring several of them together to solve differential equations, and then could not build it. The reason is in the friction: a wheel pressed onto a disc with a few newtons can deliver perhaps twenty millinewton-metres before it slips, and setting the carriage of the NEXT integrator takes twenty times that. Each unit works; none of them can drive the next. Vannevar Bush's differential analyser of 1931 solved it by putting a torque amplifier after every wheel. That is not a gearbox - gearing up the torque gears down the motion just as far, and the wheel has no motion to spare. It is a power amplifier: a band rides on a continuously driven capstan, the feeble wheel only has to tighten or slacken the band, and a separate motor does the work. Gains of ten thousand, from a mechanism that is itself a feedback loop, built a year before anybody could analyse one. The machine that resulted solved ballistics, power-network transients and atomic-physics problems to about a tenth of a per cent per integrator, which was better than anything else available and is exactly the number that later lost to electronics. The op-amp integrator two blueprints further on does the same calculus by putting charge on a capacitor, and it wins because charge does not slip.