ume de L'Hôpital, and J. L. LagrangeLagrange, Joseph Louis, Comte
, 1736–1813, French mathematician and astronomer, b. Turin, of French and Italian descent. Before the age of 20 he was professor of geometry at the royal artillery school at Turin.
..... Click the link for more information. . Important advances in geometry began toward the end of the century with the work of Gaspard MongeMonge, Gaspard, comte de Péluse
, 1746–1818, French mathematician, physicist, and public official. He was distinguished for his geometrical research, which laid the foundations of modern descriptive geometry, a field essential to mechanical drawing and architectural
..... Click the link for more information. in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. PonceletPoncelet, Jean Victor
, 1788–1867, French mathematician and army engineer. He taught at the school of mechanics at Metz and at the Faculté des Sciences and the École Polytechnique, both in Paris.
..... Click the link for more information. , who founded projective geometry (1822).
In the Nineteenth Century
The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. GaussGauss, Carl Friedrich
, born Johann Friederich Carl Gauss, 1777–1855, German mathematician, physicist, and astronomer. Gauss was educated at the Caroline College, Brunswick, and the Univ.
..... Click the link for more information. . In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. LobachevskyLobachevsky, Nikolai Ivanovich
, 1793–1856, Russian mathematician. A pioneer in non-Euclidean geometry, he challenged Euclid's fifth postulate that one and only one line parallel to a given line can be drawn through a fixed point external to the line; he developed,
..... Click the link for more information. (1826) and independently by János BolyaiBolyai
, family of Hungarian mathematicians. The father, Farkas, or Wolfgang, Bolyai, 1775–1856, b. Bolya, Transylvania, was educated in Nagyszeben from 1781 to 1796 and studied in Germany during the next three years at Jena
..... Click the link for more information. (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard RiemannRiemann, Bernhard
(Georg Friedrich Bernhard Riemann) , 1826–66, German mathematician. He studied at the universities of Göttingen and Berlin and was professor at Göttingen from 1859.
..... Click the link for more information. (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.
In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. HamiltonHamilton, Sir William Rowan,
1805–65, Irish mathematician and astronomer, b. Dublin. A child prodigy, he had mastered 13 languages by the age of 13 and was still an undergraduate when he became professor of astronomy at the Univ. of Dublin (1827).
..... Click the link for more information. , whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. GrassmannGrassmann, Hermann Günther
, 1809–77, German mathematician and Sanskrit scholar, educated in Berlin. He invented a new algebra of vectors (somewhat similar to quaternions), presented in his book Die Ausdeh