Fermat, Newton, Leibniz, and the Bernoullis

Abstract

Fermat wrote nine interesting and important papers on the method of maxima and minima which are grouped together in his collected works. The last two in this set were sent by him in 1662 as attachments to a letter to a colleague, Marin Cureau de la Chambre.¹ As their titles, “The analysis of refractions” and “The synthesis of refractions,” imply, they are companion derivations of the law of refraction, now usually known as Snell’s law. These papers are fundamental for us because Fermat enunciates in them his principle that “nature operates by means and ways that are ‘easiest and fastest.’” He goes on to state that it is not generally true that “nature always acts along shortest paths” (this was the assumption of de la Chambre). Indeed, he cites the example of Galileo that when particles move under the action of gravity they proceed along paths that take the least time to traverse, and not along ones that are of the least length.² This enunciation by Fermat is, as far as I am aware, the first one to appear in correct form and to be used properly.