Tautochrone curve

A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point.

The tautochrone problem, the attempt to identify this curve, was solved by Huygens in 1659. He proved geometrically in his Horologium oscillatorium (1673) that the curve was a cycloid. This solution was later used to attack the problem of the brachistochrone curve.

Later mathematicians such as Lagrange and Euler looked for an analytical solution to the problem.