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 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 Transversals, 1806, are important contributions to modern geometry. While Monge revelled mainly in three-dimensional geometry, Carnot confined himself to that of two. By his effort to explain the meaning of the negative sign in geometry he established a "geometry of position," which, however, is different from the "Geometrie der Lage" of to-day. He invented a class of general theorems on projective properties of figures, which have since been pushed to great extent by Poncelet, Chasles, and others.

Jean Victor Poncelet (1788–1867), a native of Metz, took part in the Russian campaign, was abandoned as dead on the bloody field of Krasnoi, and taken prisoner to Saratoff. Deprived there of all books, and reduced to the remembrance of what he had learned at the Lyceum at Metz and the Polytechnic School, where he had studied with predilection the works of Monge, Carnot, and, he began to study mathematics from its elements. He entered upon original researches which afterwards made him illustrious. While in prison he did for mathematics what Bunyan did for literature,—produced a much-read work, which has remained of great value down to the present time. He returned to France in 1814, and in 1822 published the work in question, entitled, Traité des Propriétés projectives des figures. In it he investigated the properties of figures which remain