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

We show in this article how Girard Desargues, in his well known text on conics, the $\mathit{\text{Brouillon}}\mathit{\text{Project,}}$ manages to use Menelaos’ theorem with some awesome virtuosity. To this end, we propose a detailed analysis of his $\mathit{\text{combinatorial}}$ approach, which was already visible in the development of his notion of involution. We shall study the proofs of two important theorems of the $\mathit{\text{Brouillon.}}$ 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 $\mathit{\text{Essay}}\mathit{\text{pour}}\mathit{\text{les}}\mathit{\text{coniques}}$ by Pascal and the $\mathit{\text{Advis}}\mathit{\text{charitables}}$ by de Beaugrand.