On the analytic and synthetic demonstrations in Fermat’s work on the law of refraction

The history of refraction is extensive and quite complex, it can be dated back to Ptolemy and Ibn Sahl, and the law of refraction itself is usually called Snell’s law. We do not wish however, to give a complete account of this history, our aim in this paper is rather to study Fermat’s contributions which are divided into two periods. In the first period, commencing with the publication of Descartes’ Discours de la Méthode, Fermat rejects the law of refraction as proposed by Descartes claiming that the method used to obtain it is flawed and that the law must therefore be wrong. During the second period, dating from 1657 to 1662, Fermat presents two proofs of the law, one which is analytic and one that is synthetic, and concludes (to his own surprise) that Descartes’ law was in fact correct. Our goal is to study in detail in what ways these proofs are analytic and synthetic and then account for Fermat’s results given he had set out to disprove them.