Girard Desargues

“{{MacTutor Biography|id=Desargues}}”

“An interesting theorem bearing his name and typical of projective geometry is as follows:—If two triangles ABC and A'B'C' are so related that lines joining corresponding vertices meet in a point O, then the intersections of corresponding sides will lie in a straight line A"B"C". It remained for Monge, the inventor of descriptive geometry... and others more than a century later to carry this development forward. Desargues's work was indeed practically lost until Poncelet in 1822 proclaimed him the Monge of his century.”

“Perceiving that the practitioners of these arts ["...among others, the cutting of stones in architecture, that of sun-dials, that of perspective in particular"] had to burden themselves with the laborious acquisition of many special facts in geometry, he sought to relieve them by developing more general methods and printing notes for distribution among his friends.”

“He met Descartes while employed by Cardinal Richelieu at the {{w|siege of Rochelle}}, and they with others met regularly in Paris for the discussion of the new Copernican theory and other scientific problems.”

“Desargues contented himself with enunciating general principles remarking:—"He who shall wish to disentangle this proposition will easily be able to compose a volume."”

“In his chief work Desargues enunciates the propositions:—1. A straight line can be considered as produced to infinity and then the two opposite extremities are united.2. Parallel lines are lines meeting at infinity and conversely.3. A straight line and a circle are two varieties of the same species.On these he bases a general theory of the plane sections of a cone.”

“In general this geometry instead of dealing with definite triangles, polygons, circles, etc., in the Euclidean manner, is based on a consideration of all points of a straight line, of all lines through a common point and of the possible effects of setting up an orderly one-to-one correspondence between them. In particular, Desargues makes a comparative study of the different plane sections of a given cone, deducing from known properties of the circle analogous results for the other conic sections.”

“Hardly less interesting than the new ideas of Descartes and Cavalieri are those of their contemporary Desargues... who made important researches in geometry. But for the still more brilliant geometrical achievements of Descartes, these might have led to the immediate development of projective geometry, the elements of which are contained in Desargues's work.”

“The researches of Kepler and Desargues will serve to remind us that as the geometry of the Greeks was not capable of much further extension, mathematicians were now beginning to seek for new methods of investigation, and were extending the conceptions of geometry. The invention of analytical geometry and of the infinitesimal calculus temporarily diverted attention from pure geometry, but at the beginning of the last century there was a revival of interest in it, and since then it has been a favourite subject of study with many mathematicians.”

“The influence exerted by the lectures of Desargues on Descartes, Pascal and the French geometricians of the seventeenth century was considerable; but the subject of projective geometry soon fell into oblivion, chiefly because the analytical geometry of Descartes was so much more powerful as a method of proof or discovery.”

“In 1636 Desargues issued a work on perspective; but most of his researches were embodied in his Brouillon project on conics, published in 1639, a copy of which was discovered by Chasles in 1845.”

“Girard Desargues... gave some courses of gratuitous lectures in Paris from 1626 to about 1630 which made a great impression upon his contemporaries. Both Descartes and Pascal had a high opionion of his work and abilities, and both made considerable use of the theorems he had enunciated.”

“The book of Desargues (1639), Brouillon project d'une atteinte aux événemens des recontres du cône avec un plan (Schematic Sketch of What Happens When a Cone Meets a Plane), suffered an extreme case of delayed recognition, being completely lost for 200 years.”

“One of the first important steps to be taken in modern times... was due to Desargues. In a work published in 1639 Desargues set forth the foundation of the theory of four harmonic points, not as done today but based on the fact that the product of the distances of two conjugate points from the center is constant. He also treated the theory of poles and polars, although not using these terms.”

“Nous démontrerons aussi la propriété suivante dont le premier inventeur est M. Desargues, Lyonnois, un des grands esprits de ce temps, et des plus versés aux mathématiques, et entre autres aux coniques, dont les écrits sur cette matière, quoiqu'en petit nombre, en ont donné un ample témoignage à ceux qui auront voulu en recevoir l'intelligence. Je veux bien avouer que je dois le peu que j'ai trouvé sur cette matière à ses écrits, et que j'ai tâché d'imiter, autant qu'il m'a été possible, sa méthode...”

“Kepler (and Desargues) regarded the two "ends" of the ["straight"] line as meeting at "infinity" so that the line has the structure of a circle. In fact, Kepler actually thought of a line as a circle with its center at infinity.”

“In 1639, nine years after Kepler's death, there appeared in Paris a remarkably original but little-heeded treatise on the conic sections. ...The work was so generally neglected by other mathematicians that it was soon forgotten and all copies of the publication disappeared. ...in 1845 Chasles happened upon a manuscript copy... made by Desargues' pupil, {{w|Philippe de La Hire}}... and since that time the work has been regarded as one of the classics in the early development of synthetic projective geometry.”

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

“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 {{w|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 "{{w|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.”