Differential Gear

Draw a corner. Inner wheel radius 4 m, track width 1.4 m, so the outer wheel runs a radius of 5.4 m.

Arc length is proportional to radius, so over a quarter turn the outer wheel travels 5.4/4 = 1.35× as far. The outer wheel must turn 35% faster.

On a rigid axle it cannot. Write down what has to happen instead: one tyre must slide 35% of its rolling distance. That is the scrub the differential removes.

Mount two bevel gears facing each other on a common axis, each fixed to one output shaft, free to rotate.

Around them build a carrier — a frame that can rotate about the same axis, carrying two cross pins at right angles to the shafts.

Fit two smaller bevel pinions on the cross pins so they mesh with both output gears at once.

Each pinion has two jobs simultaneously: it carries drive round with the carrier, and it can spin on its own pin. Those two motions superimpose, and that superposition is the entire mechanism.

Turn the carrier one full revolution with both outputs free.

Expect both outputs to turn once, and the pinions not to rotate on their pins at all. Mark a pinion with tape and watch it — it orbits, it does not spin.

Straight-line driving. The differential is doing nothing, which is correct.

Now clamp one output shaft so it cannot turn, and rotate the carrier one full revolution.

Expect the free output to turn exactly twice.

That is the law in its starkest form: the two outputs average the carrier. One does 0, so the other must do 2 for the average to be 1. Count it out loud — it is exact, not approximate.

Hold one output with your fingers only, lightly, and turn the carrier.

The lightly-held shaft spins fast and the other barely moves. Torque follows the path of least resistance — an open differential sends equal torque to both sides, so the side with less grip sets the limit for both.

This is why a car with one wheel on ice goes nowhere while that wheel spins. Starley's mechanism is exactly right for cornering and exactly wrong for traction, and every limited-slip differential and traction-control system since exists to patch this one behaviour.

The patent. British patent 3388 of 1877, James Starley, for what he called a balance gear. He fitted it to the Salvo tricycle, sold as the Coventry Salvo — a machine with two closely-spaced wheels on one side, which made the cornering problem acute enough to be worth solving.

Starley did not invent the differential. Onésiphore Pecqueur is generally credited with it in 1827 for a steam carriage, and the principle appears far earlier still — the Chinese south-pointing chariot uses differential gearing, though how much its builders understood of the mechanism is genuinely unknown. What Starley did was make it small, cheap, reliable and manufactured in quantity. That is a different achievement, and it should be described as the one it is.

Why a bicycle man mattered to cars. The Coventry cycle trade of the 1870s–80s produced ball bearings, tangent-spoked wheels, chain drive and the differential as a practical package — and then the same workshops and the same people built the first British cars. The car did not invent its own transmission; it inherited the bicycle's.

Where the arithmetic still bites. The averaging law is why you must never jack up one driven wheel of a rear-wheel-drive car and run the engine in gear — the raised wheel spins at double speed while the car sits still. It is also why a limited-slip differential is not a repair but a deliberate compromise: it reintroduces some of the scrub Starley removed, in exchange for traction.