Circular Error

The pendulum's one great weakness: it only keeps equal time when the swing is small.

Why the Arc Must Stay Small

A pendulum swings in a circular arc, which introduces an inconvenient truth: the period is not quite constant across all amplitudes. This is called circular error, and it is one of the fundamental limits of the pendulum as a timekeeper.

The standard formula — period depends on length and gravity, nothing else — applies only to a pendulum swinging through a vanishingly small angle. In reality, every pendulum swings through a finite arc, and as that arc grows, the period lengthens. The relationship is not linear: a swing of six degrees has a meaningfully different period from a swing of three, and the difference grows faster than intuition suggests. For wide domestic clock swings, the error can amount to several seconds per day, which makes amplitude stability as important as any other factor in the rate.

The underlying geometry is the source of the trouble. The restoring force on a pendulum bob is proportional to the sine of the displacement angle, not the angle itself. For small angles, sine and angle are almost the same — the approximation holds well enough. As the angle grows, sine falls behind, the restoring force is weaker than the idealised formula assumes, and the pendulum takes fractionally longer per beat. There is no design change that eliminates this; it is intrinsic to circular motion.

The consequence for precision work is direct. A regulator — the precision pendulum clock used as a workshop standard — keeps its arc deliberately small, typically no more than one or two degrees either side of centre. The deadbeat escapement was developed partly for this reason: because it delivers its impulse with minimal disturbance, the pendulum can be run at very low amplitude without stalling. A wide swing might seem reassuring, but it is in fact a liability. Every variation in drive force, every draught across the case, every change in the oil at the pallets that alters friction slightly, changes the amplitude — and circular error converts that amplitude change directly into a rate change.

Christiaan Huygens identified the problem in the seventeenth century and invented the cycloidal cheeks, curved guides intended to force the pendulum to trace a cycloidal rather than circular path — theoretically isochronous at any amplitude. In practice, the cure introduced its own errors, and precision clockmakers abandoned it. The clean answer was to keep the swing small enough that the error became negligible, and to hold it there as steadily as possible.