The mathematical technique in Fermat's deduction of the law of refraction

This article attempts to explain Fermat's not quite obvious calculations connected with his deduction of the law of refraction in Analysis ad refractiones (1662), and to describe the development which led to these calculations. In 1657 Fermat tried to deduce a law of refraction based on the principle that light follows the quickest path between two given points. He did not succeed because he found that the calculations were too long and tedious. The calculations are indeed complicated, but if Fermat, in 1657, had been willing to accept Descartes' law of refraction he would probably also have seen that it solved his problem. However, Fermat was of the opinion that Descartes' law was wrong and, therefore, he did not expect that solution. Only in 1662, when he succeeded in reducing the calculations substantially, did he realize that they led to the sine law of Descartes.