### Cycloid

The cycloid is the curve traced out by a point on the rim of a wheel as the wheel rolls along a flat surface. The name was first used by Galileo, who approximately measured the area under a cycloid by cutting one out of a sheet of metal, and comparing its weight with that of a circular disk the same size as the wheel. He concluded it was a little over 3, but in fact it's exactly 3, see the fantastic visual proof here (easy to understand!). The upside-down cycloid (an arch) has the property that a smooth small object placed on it takes the same time to slide to the bottom from any point. If a pendulum is constrained so it swings in a cycloidal path, the time of a swing is independent of amplitude, unlike an ordinary pendulum. This was used long ago in constructing accurate clocks.