tangent to two sides of a triangle and to the other two circles. Malfatti gave an analytical solution, but Steiner gave without proof a construction, remarked that there were thirty-two solutions, generalised the problem by replacing the three lines by three circles, and solved the analogous problem for three dimensions. This general problem was solved analytically by C. H. Schellbach (1809–1892)
and Cayley, and by Clebsch with the aid of the addition
theorem of elliptic functions.60
Steiner's researches are confined to synthetic geometry. He hated analysis as thoroughly as Lagrange disliked geometry.
Steiner's Gesammelte Werke were published in Berlin in 1881 and 1882.
Michel Chasles (1793–1880) was born at Epernon, entered
the Polytechnic School of Paris in 1812, engaged afterwards in business, which he later gave up that he might devote all his time to scientific pursuits. In 1841 he became professor of geodesy and mechanics at the Polytechnic School; later,
"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.