*From `A Short Account of the History of Mathematics' (4th edition, 1908)
by W. W. Rouse Ball.*

While Euler, Lagrange, Laplace, and Legendre were perfecting analysis, the members of another group of French mathematicians were extending the range of geometry by methods similar to those previously used by Desargues and Pascal. The revival of the study of synthetic geometry is largely due to Poncelet, but the subject is also associated with the names of Monge and L. Carnot; its great development in more recent times is mainly due to Steiner, von Staudt, and Cremona.

*Gaspard Monge* was born at Beaune on May 10, 1746, and died at
Paris on July 28, 1818. He was the son of a small pedlar, and was
educated in the schools of the Oratorians, in one of which he
subsequently became an usher. A plan of Beaune which he had made fell
into the hands of an officer who recommended the military authorities
to admit him to their training school at Mézières. His birth,
however, precluded his receiving a commission in the army, but his
attendance at an annexe of the school where surveying and drawing were
taught was tolerated, though he was told that he was not sufficiently
well born to be allowed to attempt problems which required calculation.
At last his opportunity came. A plan of a fortress having to be drawn
from the data supplied by certain observations, he did it by a
geometrical construction. At first the officer in charge refused to
receive it, because etiquette required that not less than a certain
time should be used in making such drawings, but the superiority of
the method over that then taught was so obvious that it was accepted;
and in 1768 Monge was made professor, on the understanding that the
results of his descriptive geometry were to be a military secret
confined to officers above a certain rank.

In 1780 he was appointed to a chair in mathematics in Paris, and this with some provincial appointments which he held gave him a comfortable income. The earliest paper of any special importance which he communicated to the French Academy was one in 1781, in which he discussed the lines of curvature drawn on a surface. These had been first considered by Euler in 1760, and defined as those normal sections whose curvature was a maximum or a minimum. Monge treated them as the locus of those points on the surface at which successive normals intersect, and thus obtained the general differential equation. He applied his results to the central quadrics in 1795. In 1786 he published his well-known work on statics.

Monge eagerly embraced the doctrines of the revolution. In 1792 he became minister of the marine, and assisted the committee of public safety in utilizing science for the defence of the republic. When the Terrorists obtained power he was denounced, and escaped the guillotine only by a hasty flight. On his return in 1794 he was made a professor at the short-lived Normal school, where he gave lectures on descriptive geometry; the notes of these were published under the regulation above alluded to. In 1796 he went to Italy on the roving commission which was sent with orders to compel the various Italian towns to offer pictures, sculpture, or other works of art that they might possess, as a present or in lieu of contributions to the French republic for removal to Paris. In 1798 he accepted a mission to Rome, and after executing it joined Napoleon in Egypt. Thence after the naval and military victories of England he escaped to France.

Monge then settled down at Paris, and was made professor at the Polytechnic school, where he gave lectures on descriptive geometry; these were published in 1800 in the form of a textbook entitled Géométrie descriptive. This work contains propositions on the form and relative position of geometrical figures deduced by the use of transversals. The theory of perspective is considered; this includes the art of representing in two dimensions geometrical objects which are of three dimensions, a problem which Monge usually solved by the aid of two diagrams, one being the plan and the other the elevation. Monge also discussed the question as to whether, if in solving a problem certain subsidiary quantities introduced to facilitate the solution become imaginary, the validity of the solution is thereby impaired, and he shewed that the result would not be affected. On the restoration he was deprived of his offices and honours, a degradation which preyed on his mind and which he did not long survive.

Most of his miscellaneous papers are embodied in his works, Application de l'algèbre à la géométrie, published in 1805, and Application de l'analyse à la géométrie, the fourth edition of which, published in 1819, was revised by him just before his death. It contains among other results his solution of a partial differential equation of the second order.

*Lazare Nicholas Marguerite Carnot*, born at Nolay on May 13,
1753, and died at Magdeburg on Aug. 22, 1823, was educated at
Burgundy, and obtained a commission in the engineer corps of
Condé. Although in the army, he continued his mathematical
studies in which he felt great interest. His first work, published in
1784, was on machines; it contains a statement which foreshadows the
principle of energy as applied to a falling weight, and the earliest
proof of the fact that kinetic energy is lost in the collision of
imperfectly elastic bodies. On the outbreak of the revolution in 1789
he threw himself into politics. In 1793 he was elected on the
committee of public safety, and the victories of the French army were
largely due to his powers of organization and enforcing discipline.
He continued to occupy a prominent place in every successive form of
government till 1796 when, having opposed Napoleon's *coup
d'état*, he had to fly from France. He took refuge in Geneva, and
there in 1797 issued his La métaphysique du calcul
infinitésimal. In 1802 he assisted Napoleon, but his sincere
republican convictions were inconsistent with the retention of office.
In 1803 he produced his Géométrie de position. This
work deals with projective rather than descriptive geometry, it also
contains an elaborate discussion of the geometrical meaning of
negative roots of an algebraical equation. In 1814 he offered his
services to fight for France, though not for the empire; and on the
restoration he was exiled.

*Jean Victor Poncelet*, born at Metz on July 1, 1788, and died
at Paris on Dec. 1867, held a commission in the French engineers.
Having been made a prisoner in the French retreat from Moscow in 1812
he occupied his enforced leisure by writing the Traité des
propriétés projectives des figures, published in
1822, which was long one of the best known text-books on modern
geometry. By means of projection, reciprocation, and homologous
figures, he established all the chief properties of conics and
quadrics. He also treated the theory of polygons. His treatise on
practical mechanics in 1826, his memoir on water-mills in 1826, and his
report on the English machinery and tools exhibited at the
International Exhibition held in London in 1851 deserve mention. He
contributed numerous articles to Crelle's journal; the most valuable of
these deal with the explanation, by the aid of the doctrine of
continuity, of imaginary solutions in geometrical problems.

This page is included in a collection of mathematical biographies taken from A Short Account of the History of Mathematics by W. W. Rouse Ball (4th Edition, 1908).

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