studied by Wren and Euler) and discovered between the
theory of surfaces and the integration of partial differential
equations, a hidden relation which threw new light upon both subjects. He gave the differential of curves of curvature, established a general theory of curvature, and applied it to the ellipsoid. He found that the validity of solutions was not impaired when imaginaries are involved among subsidiary quantities. Monge published the following books: Statics, 1786; Applications de l'algèbre à la géométrie, 1805; Application
de l'analyse à la géométrie. The last two contain most of his miscellaneous papers.
Monge was an inspiring teacher, and he gathered around him a large circle of pupils, among which were Dupin, Servois,
Brianchon, Hachette, Biot, and Poncelet.
Charles Dupin (1784–1873), for many years professor of mechanics in the Conservatoire des Arts et Métiers in Paris, published in 1813 an important work on Développements de géométrie, in which is introduced the conception of conjugate tangents of a point of a surface, and of the indicatrix.53 It contains also the theorem known as "Dupin's theorem." Surfaces of the second degree and descriptive geometry were successfully studied by Jean Nicolas Pierre Hachette (1769–1834), who became professor of descriptive geometry at the Polytechnic School after the departure of Monge for Rome and Egypt. In 1822 he published his Traité de géométrie descriptive.
Descriptive geometry, which arose, as we have seen, in technical schools in France, was transferred to Germany at the foundation of technical schools there. G. Schreiber,
professor in Karlsruhe, was the first to spread Monge's
geometry in Germany by the publication of a work thereon in 1828–1829.54 In the United States descriptive geometry was
introduced in 1816 at the Military Academy in West Point by Claude Crozet, once a pupil at the Polytechnic School in
Paris. Crozet wrote the first English work on the subject.2
Lazare Nicholas Marguerite Carnot (1753–1823) was born at Nolay in Burgundy, and educated in his native province. He entered the army, but continued his mathematical studies, and wrote in 1784 a work on machines, containing the earliest proof that kinetic energy is lost in collisions of bodies. With the advent of the Revolution he threw himself into politics, and when coalesced Europe, in 1793, launched against France a million soldiers, the gigantic task of organising fourteen
armies to meet the enemy was achieved by him. He was banished in 1796 for opposing Napoleon's coup d'état. The refugee went to Geneva, where he issued, in 1797, a work still frequently quoted, entitled, Réflexions sur la Métaphysique du Calcul Infinitésimal. He declared himself as an "irreconcilable enemy of kings." After the Russian campaign he offered to fight for France, though not for the empire. On the restoration he was exiled. He died in Magdeburg. His Géométrie de position, 1803, and his Essay on