# Differential analyser

The differential analyser is a mechanical analogue computer designed to solve differential  equations by integration, using wheel-and-disc mechanisms to perform the integration. It was one of the first advanced computing devices to be used operationally. The original machines could not add, but then it was noticed that if the two wheels of a rear differential are turned, the drive shaft will compute the average of the left and right wheels. A simple gear ratio of 1:2 then enables multiplication by two, so addition (and subtraction) are achieved. Multiplication is just a special case of integration, namely integrating a constant function.

Differential analyzer, computing device for solving differential equations. Its principal components perform the mathematical operation of integration

A differential analyzer is a complicated arrangement of rods, gears, and spinning discs that can solve differential equations of up to the sixth order. It is like a digital computer in this way, which is also a complicated arrangement of simple parts that somehow adds up to a machine that can do amazing things. But whereas the circuitry of a digital computer implements Boolean logic that is then used to simulate arbitrary problems, the rods, gears, and spinning discs directly simulate the differential equation problem. This is what makes a differential analyzer an analog computer—it is a direct mechanical analogy for the real problem.

## How on earth do gears and spinning discs do calculus?

This is actually the easiest part of the machine to explain. The most important components in a differential analyzer are the six mechanical integrators, one for each order in a sixth-order differential equation. A mechanical integrator is a relatively simple device that can integrate a single input function; mechanical integrators go back to the 19th century. We will want to understand how they work, but, as an aside here, Bush’s big accomplishment was not inventing the mechanical integrator but rather figuring out a practical way to chain integrators together to solve higher-order differential equations.

A mechanical integrator consists of one large spinning disc and one much smaller spinning wheel. The disc is laid flat parallel to the ground like the turntable of a record player. It is driven by a motor and rotates at a constant speed. The small wheel is suspended above the disc so that it rests on the surface of the disc ever so slightly—with enough pressure that the disc drives the wheel but not enough that the wheel cannot freely slide sideways over the surface of the disc. So as the disc turns, the wheel turns too.

The speed at which the wheel turns will depend on how far from the center of the disc the wheel is positioned. The inner parts of the disc, of course, are rotating more slowly than the outer parts. The wheel stays fixed where it is, but the disc is mounted on a carriage that can be moved back and forth in one direction, which repositions the wheel relative to the center of the disc. Now this is the key to how the integrator works: The position of the disc carriage is driven by the input function to the integrator. The output from the integrator is determined by the rotation of the small wheel. So your input function drives the rate of change of your output function and you have just transformed the derivative of some function into the function itself.