"Professeur de Géométrie supérieure à la Faculté des Sciences de Paris." He was a voluminous writer on geometrical subjects. In 1837 he published his admirable Aperçu historique sur l'origine et le développement des méthodes en géométrie, containing a history of geometry and, as an appendix, a treatise "sur deux principes généraux de la Science." The Aperçu historique is still a standard historical work; the appendix contains the general theory of Homography (Collineation) and of duality (Reciprocity). The name duality is due to Joseph Diaz Gergonne (1771–1859). Chasles introduced the term anharmonic ratio, corresponding to the German Doppelverhältniss and to Clifford's cross-ratio. Chasles and Steiner elaborated independently the modern synthetic or projective geometry. Numerous original memoirs of Chasles were published later in the Journal de l'École Polytechnique. He gave a reduction of cubics, different from Newton's in this, that the five curves from which all others can be projected are symmetrical with respect to a centre. In 1864 he began the publication, in the Comptes rendus, of articles in which he solves by his "method of characteristics" and the "principle of correspondence" an immense number of problems. He determined, for instance, the number of intersections of two curves in a plane. The method of characteristics contains the basis of enumerative geometry. The application of the principle of correspondence was extended by Cayley, A. Brill, H. G. Zeuthen, H. A. Schwarz, G. H. Halphen (1844–1889), and others. The full value of these principles of Chasles was not brought out until the appearance, in 1879, of the Kalkül der Abzählenden Geometrie by Hermann Schubert of Hamburg. This work contains a masterly discussion of the problem of enumerative geometry, viz. to determine how many geometric figures of given definition satisfy a sufficient number of conditions. Schubert extended his enumerative geometry to -dimensional space.[55]
