Brachistochrone – The Path of Quickest Descent

Abstract

It is now more than three centuries since Johann Bernoulli solved one of the most intriguing problems in the history of the development of mathematics. Adapting Fermat’s principle of least time, applicable for the path followed by a ray of light as it passes through a series of media with decreasing values of refractive index, to the motion of a point mass under the influence of gravity alone, Bernoulli solved the problem of quickest descent of a point mass in a vertical plane from a point to a lower point, but not vertically below it. It is said to be one of the most important problems in mathematics as it paved the way for many branches of modern mathematics, including calculus of variations. In this article, we discuss the historical development of Bernoulli’s challenge problem, its solution, and several anecdotes connected with the story of brachistochrone. We conclude the article with an important property: the ‘tautochronous property’ of the brachistochrone curve, discovered by Huygens and used by himin making clocks. The spirit with which the business of mathematics was transacted in the centuries gone by is highlighted.