tautochrone

The tautochrone problem asks: is there a curve along which a frictionless bead, released from any height, always reaches the bottom in exactly the same time? The answer is the cycloid — the curve traced by a point on the rim of a rolling wheel.

The result is counterintuitive. A bead released near the top of a cycloid has farther to travel, but the steepness of the upper portion accelerates it so sharply that it arrives at the bottom at precisely the same instant as a bead released from a point barely above it. The two effects — longer path and greater acceleration — cancel exactly, for any starting point.

Huygens recognised the practical value immediately. An ordinary pendulum is only approximately isochronous; its period stretches as the amplitude grows. But if the bob is constrained to swing along a cycloidal arc — by wrapping the string around cycloidal cheeks at the pivot — the oscillation becomes perfectly isochronous at every amplitude. He built clocks on this principle and published the proof in Horologium Oscillatorium.

Christiaan Huygens proved in Horologium Oscillatorium (1673) that the cycloid is the tautochrone curve. The proof was a landmark in the early calculus of curves and led directly to the theory of evolutes.