Sébastien Maronne

I studied the structural method defined by Vuillemin in La Philosophie de l’algèbre and the origins of this method by considering the fourth rule of the Discours de la méthode and its interpretation given by Vuillemin in Mathématiques et métaphysique chez Descartes. To achieve this aim, I analyse Vuillemin’s conception of the history of mathematics and the relationship between method and objects. I also examine the different forms of analysis and abstraction mentioned by Vuillemin and their relationship with Cavaillès’ thematization.

In this article, I examine the respective role of equations and construction in geometrical problem solving according to Descartes and Pascal. I argue that whereas Descartes claims that an equation provides a solution to a geometrical problem even if it leads to an entangled construction, Pascal dismisses this claim and gives priority to construction. To this end, I deal with prototypical problems like Apollonius' problem of the three cir cles or Pappus' problem, both of which were tackled by Descartes and Pascal in their work or correspondence. I also pay close attention to the cor respondence between Pascal and Sluse of October-December 1657 and compare Sluse's conception of algebraic analysis in geometrical problem solving with Descartes'. These letters, not thoroughly studied until now, supply an interesting discussion about what it is to provide a solution to a geometrical problem. Finally, I consider Descartes' algebraization of geometry as a kind of mathematization