Kepler in search of the "Anaclastic"

This article was written as a tribute to Enrique Tirapegui who passed away on 10 March 2020. We were working on these issues with him. He will always be remembered.

We describe Kepler's attempt to discover the law of refraction. Its very elegant analysis can be seen as a numerical search for the solution of a differential equation. Kepler's arguments and the reasons for his failure are discussed. This analysis, which precedes Descartes' solution of the inverse tangent problem posed by Florimond de Beaune, is an opportunity to briefly recall some aspects of the history of differential equations. It is also an opportunity to propose a pedagogical activity that consists of introducing the law of refraction to students in the wake of Kepler and Descartes. This article is intended to be pedagogical. It is self-consistent and all the properties of the hyperbola necessary to follow Kepler' arguments are demonstrated using elementary geometry. It will also be the material for the chapter of a book intended to revisit the elementary laws of physics using Euclidean geometry. This project started with Enrique Tirapegui. This work was presented at the XVIII Instabilities and Nonequilibrium Structures Workshop in tribute to Enrique held in December 2021 in Valparaiso, Chile.