1. Applications, examples and notations

According to Heron of Alexandria, the path of a ray of light reflected by a mirror is shorter than any other reflected path. The equality between the angle of incidence and the angle of reflection is immediately deduced from Henon's principle, given that the straight line is the shortest path between two points.

According to Descartes, the denser the medium, the faster light propagates. The law of refraction is a consequence of Descartes' principle. In 1657, Fermat found the same law, assuming that light propagates faster the less dense the medium, and the shorter the travel time.

In 1696, Jean Bernoulli posed the problem of the brachistochrone. As Hadamard notes:

"When Jean Bernoulli posed to the learned world the problem of the curve along which a weighted point would descend as quickly as possible from a given...