MONGE, G.(1746-1818) and CARNOT, L.N.M.(1753-1823)

    Monge was educated at the college of the Oratorians in beaunne, the town of his birth, and at their college in Lyons, where at the age of sixteen he became an instructor in physics.  A skillfully constructed large scale map of his home town led to his acceptance at the military school in Mezieres as a draftsman.   Asked to work out from supplied data the gun emplacements of a proposed fortress, Monge circumvented the long and tedious arithmetic procedure of the time by a rapid geometric one.

  His method, which was one of cleverly represeprojections on the two-dimensional nting three-dimensional objects by appropriate plane, was adopted by the military and classified as top-secrdt.  It later became widely taught as descriptive geometry.  In 1768, Monge became a professor of mathematics, and in 1771, a professor of physics, at Mezieres.  In 1780, he was appointed to a chair of hydraulics at the Lyceum in Paris.
    Monge served as Minister of Marine and engaged in the manufacture of arms and gunpowder for the army.  He was the principal force, under the Directory, in the founding of the Ecole Polytechnique in 1795, and was a professor oof mathematics there.  He gained the close friendship and admiration of Napoleon and accompanied the latter, along with the mathematician Joseph Fourier (1768-1830), on the ill-fated Egyptian expedition of 1798.  Upon returning to Frace, Monge resumed his position at the Ecole Polytechnique, where he proved to be a singularly girted teacher.  His lectures there inspired a large following of able geometers, among whom were Charles Dupin (1784-1873) and jean Victor Poncelet (1788-1867), the former a contributor to the field of differential geometry, and the latter to that of projective geometry.
    In addition to creating descriptive geometry, Monge is considered as the father of differential geometry.  His wirk wntitled Application de l'analyse a la geometrie ran through five editions and was one of the most important of the early treatments of the differential geometry of surfaces.  It is here that Monge introduced, among other things, the concept if lines of curvature of a surface in three-space.  Monge's contributions to differential geometry are devoted principally to the extrinsic geometry of surfaces.

    Lazare Micolas marguerite Carnot (1753-1823), following the custom of many of the sons of will-to-do French families, prepared himself for the army amd was thus led to the military school fo Mezieres, where he studied under Monge, becoming a captain in the engineers in 1783.  In 1784, he wrote his first mathematical work, on mechanics, which contains the earliest proof that kinetic energy is lost in the collision of imperfectly elastic bodies.  With the advent of the French Revolution, he threw himself into politics and embraced the Revolution with enthusiasm and dedication.

  He succeeded to a number of importamt political posts and, in 1793, voted for the execution of Louis XVI as a traitor.  Also in 1793, when a united Europe launched a million-man army against France, Carnot undertook the seemingly impossible task of organizimg fourteem armies to successfully oppose the enemy, winning for himself the name "the Organizer of Victory."  In 1796, he opposed Napoleon's coup d'etat, and had to flee to Geneva, where he wrote a semiphilosophical work on the metaphysics of the calculus.  His two important contributions to geometry, Geometrie de position and Essai sur la theorie des transversals, were published in 1803 and 1806.  As an "irreconcilable enemy of kings" he offered, in 1814, after the Russian campaign, to fight for France but not for the empire.  With the restoration, he was exiled, dying in Madgeburg in straitened circumstances in 1823.