Girard Desargues

Girard Desargues (born February 21, 1591, Lyon, France—died October 1661, France) was a French mathematician who figures prominently in the history of projective geometry. Desargues’s work was well known by his contemporaries, but half a century after his death he was forgotten. His work was rediscovered at the beginning of the 19th century, and one of his results became known as Desargues’s theorem.

Not much is known about Desargues’s early life, which he spent in Lyon where his father worked for the local diocese. In 1626 Desargues proposed a water project to the municipality of Paris, and by 1630 he had become associated with a group of Parisian mathematicians gathered around Father Marin Mersenne. In 1635 Mersenne formed the informal, private Académie Parisienne, whose meetings Desargues attended. Through Mersenne, Desargues had contact with most of the leading French mathematicians of his day; two of the most prominent, René Descartes and Pierre de Fermat, valued his scientific views. It is generally presumed that Desargues worked as an engineer until he took up architecture about 1645. He lived in Lyon again from about 1649 to 1657 before returning to Paris for the remainder of his life.

In 1636 Desargues published Exemple de l’une des manières universelles du S.G.D.L. touchant la pratique de la perspective (“Example of a Universal Method by Sieur Girard Desargues Lyonnais Concerning the Practice of Perspective”), in which he presented a geometric method for constructing perspective images of objects. The painter Laurent de La Hire and the engraver Abraham Bosse found Desargues’s method attractive. Bosse, who taught perspective constructions based on Desargues’s method at the Royal Academy of Painting and Sculpture in Paris, published a more accessible presentation of this method in Manière universelle de Mr. Desargues pour pratiquer la perspective (1648; “Mr. Desargues’s Universal Method of Practising Perspective”). In addition this book contains what is now known as Desargues’s theorem. Desargues also published a primer on music notation, a technique for stonecutting, and a guide for the construction of sundials.

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Desargues’s most important work, Brouillon project d’une atteinte aux événements des rencontres d’un cône avec un plan (1639; “Rough Draft of Attaining the Outcome of Intersecting a Cone with a Plane”), treats the theory of conic sections in a projective manner. In this very theoretical work Desargues revised parts of the Conics by Apollonius of Perga (c. 262–190 bc). Regardless of its theoretical character, Desargues claimed that it was of use for artisans. This statement misled later historians into seeing a strong connection between his perspective method and his treatment of conic sections. Both disciplines deal with central projections but are otherwise rather different. It is likely, however, that one of Desargues’s projective ideas—the concept of points at infinity—came from his theoretical analysis of perspective.

In the 17th century Desargues’s new approach to geometry— studying figures through their projections—was appreciated by a few gifted mathematicians, such as Blaise Pascal and Gottfried Wilhelm Leibniz, but it did not become influential. Descartes’s algebraic way of treating geometrical problems—published in Discours de la méthode (1637; “Discourse on Method”)—came to dominate geometrical thinking and Desargues’s ideas were forgotten. His Brouillon project became known again only after 1822, when Jean-Victor Poncelet drew attention to the fact that in developing projective geometry (which happened while he was a prisoner of war in Russia, 1812–14) he had been preceded—though not inspired—by Desargues in certain aspects.