Tautochrone Problem

Let a marble roll down a curved slope. In a tautochrone (curve of equal descent), the marble reaches the bottom in the same amount of time no matter where it starts. In a brachistochrone (curve of fastest descent), the marble reaches the bottom in the fastest time. Jakob Bernoulli solved the tautochrone problem in a paper marking the first usage (1690) of an integral. The cycloid is the solution to both problems. His brother Johann posed the brachistochrone problem in 1696. Newton, Leibniz, L'Hôpital, and Jakob Bernoulli sent in solutions.

Huygens found that the end of a pendulum bounded by cycloids traces another cycloid. Mathematically, this gives a perfect pendulum. Realistically, friction causes this pendulum to be worse than an ordinary pendulum whose end traces a circular arc.