L'usage de la combinatoire chez Girard Desargues : le cas du th\'eor\`eme de M\'en\'ela\"us

TL;DR

This paper analyzes Girard Desargues' innovative use of Menelaus' theorem in his work on conics, highlighting his combinatorial approach and its role in key theorems about involution and pencils of conics.

Contribution

It provides a detailed analysis of Desargues' combinatorial methods and their application to classical theorems, revealing new insights into his geometric reasoning.

Findings

Desargues' combinatorial approach is crucial in his proofs.

Invariance of involution configuration under perspective projection.

Application of Menelaus' theorem in Desargues' conic theorems.

Abstract

We show in this article how Girard Desargues, in his well known text on conics, the \textit{Brouillon Project,} manages to use Menelaos' theorem with some awesome virtuosity. To this end, we propose a detailed analysis of his \textit{combinatorial} approach, which was already visible in the development of his notion of involution. We shall study the proofs of two important theorems of the \textit{Brouillon.} The first is the theorem of the "ram\'ee", stating that the configuration of involution is invariant by perspective projection, and the second is the great theorem of Desargues on pencils of conics. We shall also study in the same spirit the first lemma (dealing with the hexagram) of the \textit{Essay pour les coniques} by Pascal and the \textit{Advis charitables} by de Beaugrand.