Fermat’s method: its status and its diffusion. Algebra and comparison of figures in the history of Fermat’s method.
(La méthode de Fermat: son statut et sa diffusion. Algèbre et comparaison de figures dans l’histoire de Fermat.)

*(French)*Zbl 0819.01013
Cahiers d’Histoire et de Philosophie des Sciences. Nouvelle Série. 33. Paris: Société Française d’Histoire des Sciences de des Techniques, 243 p. (1990).

In 1637 P. de Fermat wrote his “Methodus ad disquirendam maximam et minimam” and described a procedure later known as Fermat’s method. The author undertook an extensive study of it and divides his treatise in six chapters. In the first one she presents four texts, in which Fermat explained his method in different versions. Next she surveys on the historical analyses of Fermat’s work given by Montucla, Duhamel, Itard, Schneider and others. Applications of Fermat’s method are traced out and discussed in the second chapter. The author explains that Fermat used his procedure not only for dealing problems of maxima and minima but also for the determination of tangents and even of centres of gravity. The latter he compares with the procedures of Torricelli, van Schooten and Huygens. In the following chapters she attempts then to demonstrate what Fermat himself does understand by his method and how his contemporaries interpreted it. Various interpretations of Fermat’s method are discussed. The author describes in detail the role of Hérigone in the dissemination of Fermat’s method and van Schooten’s and Huygens’ important treatments of it. Huygens introduced the concept of infinitely small in his connection. After this extensive historical study which makes the book a useful reading the treatise is closed by a comparison between Fermat’s method and the modern theory of synthetic differential geometry. Following the ideas of W. Lawrence, A. Kock and G. E. Reyes the author presents some fundamentals of the synthetic geometry and of the theory of topoi and looks for relations to and common features with Fermat’s method resp. The author tries to make explicit the feelings of Lawrence and others that there is a kind of affinity between symplectic geometry and mathematics of Fermat’s time. This part is quite speculative. There are a name index and a bibliography with 170 items at the end of the book.

Reviewer: K.-H.Schlote (Leipzig)

##### MSC:

01A45 | History of mathematics in the 17th century |