Bernoulli's Light Ray Solution of the Brachistochrone Problem Through Hamilton's Eyes

Abstract

This article contains a review of the brachistochrone problem as initiated by Johann Bernoulli in 1696–1697. As is generally known, the cycloid forms the solutions to this problem. We follow Bernoulli's optical solution based on the Fermat principle of least time and later rephrase this in terms of Hamilton's 1828 paper. Deliberately an anachronistic style is maintained throughout. Hamilton's solution recovers the cycloid in a way that is reminiscent of how Newton's mathematical principles imply Kepler's laws.