In this paper, we try to understand what considerations and possible sources of inspiration Desargues used to formulate his concepts of involution and transversal, and to state the related theorems that are at the basis of his Brouillon project. To this end, we trace some clues which are found scattered throughout his works, we connect them together in the light of his experience and knowledge in the field of perspective, and we investigate what were his motivations within Mersenne’s academy. As (...) a result of our research, we can safely say that were his great geometrical insight and his projective vision of space which, guided by some classical theorems, led him to these completely new concepts in the panorama of the geometry of that time that were destined to remain misunderstood for about two centuries. (shrink)