Girard Desargues and Projective Geometry

Overview

In 1639 Girard Desargues (1591-1661), a military engineer and architect, published the Brouillon project d'une atteinte aux événements des rencontres d'un cône avec un plan (Proposed Draft of an Attempt to Deal with the Events of the Meeting of a Cone with a Plane). In this work Desargues established the principles of projective geometry, an alternative to traditional Euclidean geometry. Projective geometry was concerned with projections, or the extent to which the shape of a figure is changed by the perspective from which that figure is viewed.

Desargues's projective geometry, however, was overshadowed by the work of his contemporary, René Descartes (1596-1650). These two, along with other important scientists and mathematicians of the time, helped to move math and science towards more practical applications. Also, the frequent collaboration between these figures led to the creation of the powerful and important French Académie des Sciences.

Background

Mathematic techniques of the seventeenth century responded rapidly and radically to the conclusions that figures such as Johannes Kepler (1571-1630) and Galileo Galilei (1564-1642) derived from their studies of astronomy. A text such as Galileo's The Assayer (1623), for instance, succeeded in both ridiculing the explanation of comets proposed by Jesuit scholars and establishing new views on scientific reality and the scientific method. Likewise, Kepler's The New Astronomy (1609) overturned the traditional concept of the universe as a series of inter-locking celestial spheres, and necessitated the revision of the laws of planetary motion.

This revision was achieved through a combination of mathematical and philosophical analysis that occurred in the work of Descartes, a French mathematical philosopher intent on establishing a new approach to the universe. Descartes's Discourse on Method, published in 1637, worked towards a new mathematics able to surpass the philosophical and analytical limitations of traditional geometry.

In "Geometry," the third appendix of the Discourse on Method, Descartes explained the fundamentals of algebraic geometry. However, this new geometry did not consist only of the application of algebra to geometry. Instead, it may be better characterized as the translation of algebraic operations into the language of geometry. Indeed, the overall theme of Descartes's "Geometry" is established by its opening sentence:

Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain lines is sufficient for its construction.

Desargues developed a projective geometry, which may be seen as a counterpoint to Descartes's analytic geometry. In fact, Desargues spent many years in Paris with a group of mathematicians that included Descartes and Pascal as well as the Jesuit scientist Marin Mersenne (1588-1648) and Etienne Pascal (1588-1651). Desargues's work on projective geometry was printed principally for this limited readership of friends. Unfortunately, however, his views were very unorthodox and unpopular during his life—Blaise Pascal (1623-1662) was one of his few admirers. Only 50 copies of the Brouillon project were printed, many of them later destroyed by the publisher. Desargues's projective geometry slipped into obscurity for nearly 200 years after the publication of his defining text on the subject.

The group of mathematicians and scientists with whom he associated, however, is notable for more than the academic achievements of its members. The group's informal meetings, which began in Mersenne's "cell" in the Convent of the Annunciation in Paris, later became what is still today the principle scientific organization of France, the Académie des Sciences. As the gatherings of these scientists grew larger, their influence on government increased as well. The author and administrator Charles Perrault suggested the establishment of a scientific academy to Jean-Baptiste Colbert, Minister of Finance to Louis XIV. The academy allowed for and encouraged practical applications of scientific discoveries while providing a forum for intellectual exchange on a previously unprecedented scale. After its establishment in 1666, the academy provided a royal pension and financial assistance for research to its members.

Impact

As an architect and military engineer, Desargues was interested in problems concerning the role of perspective in architecture and geometry. His principal work, Brouillon project, was immensely unsuccessful, even though it laid the foundations of projective geometry. The title was cumbersome (consider the simplicity of Apollonius's [245?-190? b.c.] Conics) and the prose was remarkably unwieldy. He introduced more than 70 new terms in this text, of which only one, involution, has survived. This term, which denotes quite literally the twisted state of young leaves, is used to designate the projective transformation of a line that coincides with its inverse. (Most of the terms that Desargues proposed were based on obscure botanical references.) The deliberate use of these confusing terms resulted in a decidedly negative reception. Indeed, Blaise Pascal was one of the few able to comprehend Desargues's deliberate obfuscation. The work was ignored for roughly two centuries, until the French mathematician Michel Chasles (1793-1880) completed his standard history of geometry.

In addition, Descartes's contemporaneous work further limited interest in Desargues's volume. The crystalline simplicity of Descartes's prose eased readers through difficult concepts. As a result, mathematicians were far more interested in developing the applications made possible by Descartes's powerful contribution to mathematics.

Despite its unwieldy explication, the thought behind Desargues's work is actually quite simple. Projective geometry was indebted to Leon Battista Alberti's (1404-1472) treatment of perspective and Kepler's principle of continuity. Alberti's account of perspective was of key importance for Renaissance painting and architecture. His De pictura seeks to link painting to mathematics and provides criteria for artists interested in creating the illusion of reality in their works. However, in Alberti's treatment of perspective, artists had to use an "eye point" that existed somewhere beyond the edge of the picture. This point was ordinarily positioned at a distance equal to the distance between the picture and the eye of the observer. This was one of the key problems with Alberti's perspective, and a popular problem for mathematicians to attempt to rectify. Desargues's construction allowed this function to be carried out by a point that always lies within the picture, thereby eliminating distortion.

This attention to visual perception characterizes projective geometry. It seeks to understand the extent to which different shapes, or appearances, share the same origin. For example, when a circle is viewed obliquely, it resembles an ellipse. Likewise, the outline of the shadow of a lampshade will appear as either a circle or a hyperbola, depending on whether it is projected on the ceiling or on the wall. In other words, the shape of a figure may change but, despite these changes, the figure maintains many of the same properties. The circle and hyperbola emitted by the lamp appear quantitatively and qualitatively distinct. However, the method of analysis proffered by projective geometry allows one to measure their similarities.

Desargues's projective geometry had little effect on the seventeenth century. It lay dormant until the nineteenth century, when great advances in the subject were made by figures such as Chasles, Charles Dupin (1784-1873), and Victor Poncelet (1788-1867). While Desargues's contemporary, Blaise Pascal, expressed appreciation for the work, Descartes could not stifle his dismay when he heard of the Brouillon project. For Descartes, the notion of treating conic sections without the use of algebra seemed both impossible and implausible. Descartes believed that new achievements in geometry could only be obtained through the use of algebra. Descartes's views echoed those of most of his contemporaries, and this perception dominated mathematics for quite some time.

Prior to the rediscovery of projective geometry, Desargues was known instead for a proposition that does not even appear in the Brouillon project. This theorem, applicable to either two or three dimensions, states:

If two triangles are so situated that lines joining pairs of corresponding vertices are concurrent, then the points of intersection of pairs of corresponding sides are collinear, and conversely.

This theorem was first published in 1648 by Abraham Bosse (1602-1676), an engraver, in La Perspective de M. Desargues (Desargues's Perspective). It became one of the most important propositions of projective geometry.

While Desargues's work was overlooked in the seventeenth century, his involvement with his contemporaries, indicated through his publications, correspondences, and frequent public speeches, was of vital importance. The development of organizations such as the Académie des Sciences in France and the Royal Society in England testify to the growing power of scientists at this time. These groups, which used mathematics as a vehicle for practical applications and philosophical speculations, formed at a time when nation-states were also beginning to develop into their modern incarnations. Mathematic developments of the seventeenth century are emblematic of the transition from the medieval, which focused on exterior causes and the cosmological, to the Age of Reason, which was concerned with interior perception and the individual.

DEAN SWINFORD

Further Reading

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 2nd ed. New York: John Wiley & Sons, 1989.

Coxeter, H.S.M. Projective Geometry. 2nd ed. New York: Springer-Verlag, 1987.

Eves, Howard. An Introduction to the History of Mathematics. 4th ed. New York: Holt, 1976.

Field, J. V., and J. J. Gray. The Geometrical Work of Girard Desargues. New York: Springer-Verlag, 1987.

Rosenfeld, B. A. A History of Non-Euclidean Geometry: Evolution of the Concept of Geometric Space. New York: Springer-Verlag, 1988.

Young, Laurence. Mathematicians and Their Times. Amsterdam: North-Holland Publishing Company, 1981.