Girard Desargues and Projective Geometry

Born to one of the richest men in France, Girard Desargues (1591 – 1661) is credited with unifying the theory of conics.  A highly competent and original mathematician, he conceived problems in three-dimensional terms, just like Benedetti did fifty years earlier.  The difference was that unlike Benedetti, Desargues fully appreciated the power of this method, and used it to give a projective treatment of conics.

Desargues wrote his most important work, the treatise on projective geometry, when he was 48.  It was entitled “Rough draft for an essay on the results of taking plane sections of a cone” (Brouillon proiect d’une atteinte aux evenemens des rencontres du cone avec un plan).  In the course of his work, he became recognized as the first mathematician to get the idea of infinity properly under control, in a precise, mathematical way.  For example, Desargues defined a line as one to be produced to infinity in both directions, while a plane was similarly taken to extend to infinity in all directions.  A set of parallel lines were defined as a set of concurrent lines whose meeting point lied at an infinite distance.

One of the most important parts of his treatise can be viewed as a generalization of a theorem proved by Piero della Francesca (Fig. 20).  Desargues’ theorem allowed a way of defining the pattern of division even when the second line was not parallel to the first.  In a sense, Desargues was generalizing perspective into becoming a technique of use to mathematicians. 

Fig. 29 – Diagrams for Desargues’ proof of the theorem that if we have six points in involution, BDCGFH, then their images under projection from a point K onto another line, bdcgfh, will also be six points in involution.

Desargues is not looking at what is changed by perspective, as artists were, but is instead looking for what is unchanged.  Perspective is not being seen as a procedure that “degrades” but merely as one that transforms, leaving certain mathematical relationships unchanged (invariance).  These are now what we term the “projective properties” of the figure concerned.  Specifically, Desargues’ theorem states that six points in involution are projected onto another six points in involution.