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 for its edges.[55] The first property was discovered analytically somewhat earlier in England by Cayley and Salmon, and the second by Sylvester. Steiner's work on this subject was the starting-point of important researches by H. Schröter, F. August, L. Cremona, and R. Sturm. Steiner made investigations by synthetic methods on maxima and minima, and arrived at the solution of problems which at that time altogether surpassed the analytic power of the calculus of variations. He generalised the hexagrammum mysticum and also Malfatti's problem.[59] Malfatti, in 1803, proposed the problem, to cut three cylindrical holes out of a three-sided prism in such a way that the cylinders and the prism have the same altitude and that the volume of the cylinders be a maximum. This problem was reduced to another, now generally known as Malfatti's problem: to inscribe three circles in a triangle that each circle will be 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,