# Girard Desargues

French mathematician and engineer
Illustration par A. Bosse
du traité de G. Désargues

Girard Desargues (21 February 1591 – September 1661) was a French mathematician, architect and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the Desargues crater are named in his honour.

## Quotes

• Parallel lines have a common end point at an infinite distance.
• When no point of a line is at a finite distance, the line itself is at an infinite distance.
• I freely confess that I never had taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proximate causes... for the good and convenience of life, in maintaining health, in the practice of some art,... having observed that a good part of the arts is based on geometry, among others that cutting of stone in architecture, that of sundials, that of perspective in particular.

Girard Desargues
• Desargues the architect was doubtless influenced by what in his day was surrealism. In any event, he composed more like an artist than a geometer, inventing the most outrageous technical jargon in mathematics for the enlightenment of himself and the mystification of his disciples. Fortunately Desarguesian has long been a dead language.
• After his own fashion, Desargues discussed cross ratio; poles and polars; Kepler's principle (1604) of continuity, in which a straight line is closed at infinity and parallels meet there; involutons; assymptotes at tangents at infinity; his famous theorem on triangles in perspective; and some of the projective properties of quadrilaterals inscribed in conics. Descartes greatly admired Desargue's invention, but happily for the future of geometry did not hesitate on that account to advocate for his own.
• Eric Temple Bell, The Development of Mathematics (1940)
• Pascal made grateful acknowlegement to Desargues for his skill in projective geometry.

Desargues' 6 point Involution
Florian Cajori
A History of Elementary Mathematics:
with Hints on Methods of Teaching
(1897)
• 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 ${\displaystyle AB\cdot AH=AC\cdot AG=AD\cdot AF}$ , 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.
• Florian Cajori, A History of Elementary Mathematics: with Hints on Methods of Teaching (1897)
• 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.
• Florian Cajori, A History of Elementary Mathematics: with Hints on Methods of Teaching (1897)
• 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.
• 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.
• Florian Cajori, A History of Mathematics (1919)
• Pascal greatly admired Desargues' results... Pascal's and Desargues writings contained some of the fundamental ideas of modern synthetic geometry.
• More than two hundred years before Poncelet, the important concept of a point at infinity occurred independently to... Johann Kepler... and the French architect Girard Desargues... Kepler (in his Paralipomena in Vitellionem, 1604) declared that a parabola has two foci, one of which is infinitely distant in two opposite directions, and that any point on the curve is joined to this "blind focus" by a line parallel to the axis. Desargues (in his Brouillion project..., 1639) declared that parallel lines have a common end point at an infinite distance. ...And again ...When no point of a line is at a finite distance, the line itself is at an infinite distance... The groundwork was thus laid for Poncelet to derive projective space from ordinary space by postulating a common "line at infinity" for all the planes parallel to a given plane.
• 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, 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.
• Howard Eves, An Introduction to the History of Mathematics (1964)
• 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.
• Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)
• 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...
• We shall also demonstrate the following property, of which the original inventor is M. Desargues, of Lyon, one of the great minds of our time, and most versed in mathematics, amongst other topics, in conics, and whose writings on this subject, although small in number, have given ample testimony to those who have wished to receive of its knowledge. I am willing to confess that I owe the little I have found on this subject to his writings, and that I have endeavored, as far as possible, to imitate his method...
• Blaise Pascal, "Essais pour les coniques," (Feb, 1640), Œuvres complètes, Vol. 2 (1858) Blaise Pascal, Charles Lahure, Madame Gilberte Perier, pp.354-357, as quoted by Florian Cajori, A History of Elementary Mathematics: with Hints on Methods of Teaching (1905)
• 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.
• The discovery of a method for correct perspective is usually attributed to... Brunelleschi... The first published method appears in... On Painting [Della Pittura] by Alberti. ...Alberti's veil, used a piece of transparent cloth, stretched on a frame... viewing the scene with one eye... fixed... one could trace the scene directly onto the veil. ...The mathematical setting... is the family of lines ("light rays") through the point (the "eye"), together with the plane... (the "veil"). ...In this setting, the problems of perspective and anamorphosis were not very difficult, but the concepts were... a challenge to traditional geometric thought. Contrary to Euclid, one had...
(i) Points at infinity ("vanishing points") where parallels met.
(ii) Transformations that changed lengths and angles (projections).
The first to construct a mathematical theory incorporating these ideas was Desargues,... although the idea of points at infinity had already been used by Kepler...
• The famous geometer Desargues worked on the lines of Kepler and is now commonly credited with the authorship of some of the ideas of his predecessor. ...the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. ...Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity.
• Michel Serres... and more recently, Bernard Cache, have argued that Girard Desargues' mathematics provided a model for Leibniz's monad. ...Desargues was a founder of projective geometry, which offers a mathematical model for the intuitive notions of perspective and horizon by studying what remains invariable in projections. Outlining the concept of the "invariant," he gives his name to the "Desargues theorem," focusing on homological triangles. His disciple was the engraver, Abraham Bosse, author of a Treatise on Projections and Perspective (1665), who later taught linear perspective to stone cutters, carpenters, engravers, manufacturers of instruments and, less successfully, to painters. The perspective that Bosse teaches implicitly introduces the idea of infinity, in that he uses parallel lines with an infinitely extending vanishing point... Moreover, permeated by the knowledge of Desargues, Bosse develops a method for tracing shadows, which was inspired by his master.
• Georges Teyssot, "An Enfolded Membrane,"Architecture in Formation: On the Nature of Information in Digital Architecture (2013) ed. Pablo Lorenzo-Eiroa & Aaron Sprecher, pp.36-37.

### A Short Account of the History of Mathematics (1888)

W. W. Rouse Ball, source (1908)
• 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.
• 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.
• Desrgues commences [in the Brouillon project] with a statement of the doctrine of continuity as laid down by Kepler: thus the points at the opposite ends of a straight line are regarded as coincident, parallel lines are treated as meeting at a point at infinity, and parallel planes on a line at infinity, while a straight line may be considered as a circle whose center is at infinity. The theory of involution of six points, with its special cases, is laid down, and the projective property of pencils in involution is established. The theory of polar lines is expounded and its analogue in space suggested. A tangent is defined as the limiting case of a secant, and an asymptote as a tangent at infinity. Desargues shows that the lines which join four points in a plane determine three pairs of lines in involution on any transversal, and from any conic through the four points another pair of lines can be obtained which are in involution with any two of the former. He proves that the points of intersection of the diagonals and the two pairs of opposite sides of any quadrilateral inscribed in a conic are a conjugate triad with respect to the conic, and when one of the three points is at infinity its polar is a diameter; but he fails to explain the case in which the quadrilateral is a parallelogran, although he had formed the conception of a straight line which was wholly at infinity. The book, therefore, may be fairly said to contain the fundamental theorems on involution, homology, poles and polars, and perspective.
• 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.
• 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.

### A Short History of Science (1917)

William Thompson Sedgwick, Harry Walter Tyler, source
• 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.
• 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.
• 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.
• Desargues contented himself with enunciating general principles remarking:—"He who shall wish to disentangle this proposition will easily be able to compose a volume."

Desargues Theorem of collinearity from Sedgwick &Tyler's A Short History of Science (1917) p. 282.
• 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.
• 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.