The Brachistochrone Problem, a priori a simple game for mathematicians, turns out in the end to be a considerable problem. Indeed, the different approaches tried out in its solution may be considered, in a more or less direct way, as the starting point for new theories. While the true  »mathematical » demonstration involves what we now call the Calculus of Variations, a theory for which Euler and then Lagrange established the foundations, the solution which Jean Bernoulli originally produced, obtained with the help of an analogy with the law of refraction on Optics, was empirical. A similar analogy between Optics and Mechanics reappears when Hamilton applied the principle of least action in Mechanics which Maupertuis justified in the first instance, on the basis of the laws of Optics. This correlation finally suggested to de Broglie and Schroedinger the idea of Wave Mechanics as an analogy of Wave Optics. This article is aimed at teachers of mathematics and other disciplines for use as a means of introducing a historical perspective into the teaching of mathematics. It also contains exercises to be solved according to ancient and modern methods. The chapter ends with a bibliography which contains, in addition to the historical sources that have been used, a certain number of books recommended for further study of the subject. (ZDM/Mathdi)