The tautochrone is the curve in which a ball, positioned anywhere on the curve, will take the same time to arrive at its lowest point.

In a physical model where uniform gravity is the only force acting on the ball, the tautochrone curve is a *cycloid *(the curve described by a fixed point in a rolling circle), as correctly proven by Huygens in 1673 [BOYDE] using pure geometric arguments. We are going to present two different ways to arrive at this result.