Chapter 4 is devoted to present and comment in depth Girard Desargues’s fundamental work, the Brouillon project d’une atteinte aux evenements des rencontres du Cone avec un Plan, which he published in 1639. He was an architect and passionate amateur mathematician, who participated in the Academy that Marin Mersenne found in Paris and was in touch with the most eminent French mathematicians of his time. In the above mentioned booklet, of only thirty, but very dense, pages, Desargues revolutionized the theory of conic sections, by his systematic use of the notion of points at infinity, which he defined as “the butts at infinite distance where sets of parallel lines converge”; and for the introduction and use of several completely new concepts, as the “involution of six points” which he showed to be invariant under central projection. A large part of this chapter is dedicated to illustrate this notion, and to the proof of the so-called “involution theorem” which revealed itself to be fundamental in his development of the theory of conic sections. This chapter also focus on another important theorem due to Desargues, but that appeared in a work published by his friend engraver Abraham Bosse in 1648, and it is today known as the “theorem of triangles in perspective”, or “homological triangles”. This theorem is at the basis of the modern theory of plane homology. The work of Desargues can be hailed as the “dawn” of Projective geometry, but unfortunately, as it is told in this chapter, it was not much esteemed by his contemporaries except by the young Blaise Pascal. Since the few copies printed of the Brouillon project went lost soon after Desargues’s death, his contribution to the rise of Projective geometry remained virtually unknown, until Chasles’s discovery of a manuscript copy, in de La Hire’s hand, in 1845.