Michel Floréal Chasles was a French mathematician. In 1837 he published his Historical view of the origin and development of methods in geometry, a study of the method of reciprocal polars in projective geometry. The work gained him considerable fame and respect and he was appointed Professor at the École Polytechnique.
Background
Chasles was born on November 15, 1793, in Épernon, France, into an upper-middle-class Catholic family. He was given the name Floréal, but it was changed to Michel by court order, 22 November 1809. His father, Charles-Henri, was a lumber merchant and contractor who became president of the chambre de commerce of Chartres.
Education
Chasles received his early education at the Lycée Impérial and entered the École Polytechnique in 1812. In 1814 he was mobilized and took part in the defense of Paris. After the war he returned to the École Polytechnique and was accepted into the engineering corps, but he gave up the appointment in favor of a poor fellow student.
Career
After spending some time at home, Chasles obeyed his father’s wishes and entered a stock brokerage firm in Paris. However, he was not successful and retired to his native region, where he devoted himself to historical and mathematical studies. His first major work, the Aperçu historique, published in 1837, established his reputation both as a geometer and a historian of mathematics.
In 1841 he accepted a position at the École Polytechnique, where he taught geodesy, astronomy, and applied mechanics until 1851. In 1846 a chair of higher geometry was created for him at the Sorbonne, and he remained there until his death. Chasles was elected a corresponding member of the Academy of Sciences in 1839 and a full member in 1851. His international reputation is attested to by the following partial list of his affiliations: member of the Royal Society of London; honorary member of the Royal Academy of Ireland; foreign associate of the royal academies of Brussels, Copenhagen, Naples, and Stockholm; correspondent of the Imperial Academy of Sciences at St. Petersburg; and foreign associate of the National Academy of the United States. In 1865 Chasles was awarded the Copley Medal by the Royal Society of London for his original researches in pure geometry.
Chasles published highly original work until his very last years. His work was marked by its unity of purpose and method. The purpose was to show not only that geometry, by which he meant synthetic geometry, had methods as powerful and fertile for the discovery and demonstration of mathematical truths as those of algebraic analysis, but that these methods had an important advantage, in that they showed more clearly the origin and connections of these truths. The methods were those introduced by Lazare Carnot, Gaspard Monge, and Victor Poncelet and included a systematic use of sensed magnitudes, imaginary elements, the principle of duality, and transformations of figures. The Aperçu historique was inspired by the question posed by the Royal Academy of Brussels in 1829: a philosophical examination of the different methods in modern geometry, particularly the method of reciprocal polars.
Chasles submitted a memoir on the principles of duality and homography. He argued that the principle of duality, like that of homography, is based on the general theory of transformations of figures, particularly transformations in which the cross ratio is preserved, of which the reciprocal polar transformation is an example. The work was crowned in 1830, and the Academy ordered it published. Chasles requested permission to expand the historical introduction and to add a series of mathematical and historical notes, giving the result of recent researches. His books and almost all of his many memoirs are elaborations of points originally discussed in these notes.
Chasles wrote two textbooks for his course at the Sorbonne. The first of these, the Traité de géométrie supérieure (1852), is based on the elementary theories of the cross ratio, homographie ranges and pencils, and involution, all of which were originally defined and discussed in the Aperçu historique; the cross ratio in note 9, involution in note 10. In the case of the cross ratio, which Chasles called the anharmonic ratio, he was anticipated by August Moebius, in his Barycentrische Calcul (1827). However, it was Chasles who developed the theory and showed its scope and power. This book, Chasles felt, showed that the use of sensed magnitudes and imaginary elements gives to geometry the freedom and power of analysis. The second text, the Traité sur des sections coniques (1865), applied these methods to the study of the conic sections. This was a subject in which Chasles was interested throughout his life, and he incorporated many results of his own into the book. The book also contains many of Chasles’s results in what came to be called enumerative geometry. This subject concerns itself with the problem of determining how many figures of a certain type satisfy certain algebraic or geometric conditions.
Chasles considered first the question of systems of conics satisfying four conditions and five conditions (1864). He developed the theory of characteristics and of geometric substitution. The characteristics of a system of conics were defined as the number of conics passing through an arbitrary point and as the number of conics tangent to a given line. He expressed many properties of his system in formulas involving these two numbers and then generalized his results by substituting polynomials in the characteristics for the original values. There are many difficulties in this type of approach, and although Chasles generalized his results to more general curves and to surfaces, and the subject was developed by Hermann Schubert and Hieronymus Zeuthen, it is considered as lacking in any sound foundation.
Chasles did noteworthy work in analysis as well. In particular, his work on the attraction of ellipsoids led him to the introduction and use of level surfaces of partial differential equations in three variables (1837) . He also studied the general theory of attraction (1845), and though many of the results in this paper had been anticipated by George Green and Carl Gauss, it remains worthy of study. Chasles wrote two historical works elaborating points in the Aperçu historique which had given rise to controversy. The Histoire d’arithmétique (1843) argued for a Pythagorean rather than a Hindu origin for our numeral system. Chasles based his claim on the description of a certain type of abacus, which he found in the writings of Boethius and Gerbert. The second work was a reconstruction of the lost book of Porisms of Euclid (1860). Chasles felt that the porisms were essentially the equations of curves and that many of the results utilized the concept of the cross ratio. Neither of these works is accepted by contemporary scholars.
In 1867 Chasles was requested by the minister of public education to prepare a Rapport sur le progrès de la géométrie (1870). Although the work of foreign geometers is treated in less detail than that of the French, the Rapport is still a very valuable source for the history of geometry from 1800 to 1866 and for Chasles’s own work in particular.
Personality
Chasles' few interests outside of his research, teaching, and the Academy, which he served on many commissions, seem to have been in charitable organizations.