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

This article contains a review of the brachistrochrone problem as initiated by Johann Bernoulli in 1696-7. 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. Throughout deliberately an anachronistic style is maintained. Hamilton’s solution recovers the cycloid in a way that is reminiscent of how Newton’s mathematical principles imply Kepler’s laws.