Florian Cajori on Girard Desargues

“Pascal greatly admired Desargues' results... Pascal's and Desargues writings contained some of the fundamental ideas of modern synthetic geometry.”

“Blaise Pascal... was one of the very few contemporaries who appreciated the worth of Desargues. He says in his Essais pour les coniques, "I wish to acknowledge that I owe the little that I have discovered on this subject to his writings."”

“We owe to Desargues the theory of involution and of transversals; also the beautiful conception that the two extremities of a straight line may be considered as meeting at infinity, and that parallels differ from other pairs of lines only in having their points of intersection at infinity. He re-invented the epicycloid and showed its application to the construction of gear teeth, a subject elaborated more fully later by La Hire.”

“Desargues also gives the theory of polar lines. What is called "Desargues' Theorem" in elementary works is as follows: If the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line, and conversely. This theorem has been used since by Brianchon, Sturm, Gergonne, and others. Poncelet made it the basis of his beautiful theory of homological figures.”

“He gives the theory of involution of six points, but his definition of "involution" is not quite the same as the modern definition, first found in Fermat, but really introduced into geometry by Chasles. On a line take the point A as origin (souche), take also the three pairs of points B and H, C and G, D and F; then, says Desargues, if AB cdot AH = AC cdot AG = AD cdot AF, the six points are in "involution." If a point falls on the origin, then its partner must be at an infinite distance from the origin. If from any point P lines be drawn through the six points, these lines cut any transversal MN in six other points, which are also in involution; that is, involution is a projective relation.”

“The beginning of the seventeenth century witnessed also a revival of synthetic geometry. ...it remained for Girard Desargues... and for Pascal to leave the beaten track and cut out fresh paths. They introduced the important method of Perspective. All conics on a cone with circular base appear circular to an eye at the apex. Hence Desargues and Pascal conceived the treatment of the conic sections as projections of circles. Two important and beautiful theorems were given by Desargues: The one is on the "involution of the six points," in which a transversal meets a conic and an inscribed quadrangle; the other is that, if the vertices of two triangles, situated either in space or in a plane, lie on three lines meeting in a point, then their sides meet in three points lying on a line; and conversely. This last theorem has been employed in recent times by Brianchon, C. Sturm; Gergonne, and Poncelet. Poncelet made it the basis of his beautiful theory of homological figures.”