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 Ausdehnungslehre (1844, 4th ed. 1969).
..... Click the link for more information. , who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.
These developments continued with the group theory of M. S. LieLie, Marius Sophus
, 1842–99, Norwegian mathematician. He is noted for his contributions to the theories of differential equations and continuous transformation groups.
..... Click the link for more information. in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg CantorCantor, Georg
, 1845–1918, German mathematician, b. St. Petersburg. He studied under Karl Weierstrass and taught (1869–1913) at the Univ. of Halle. He is known for his work on transfinite numbers and on the development of set theory, which is the basis of modern
..... Click the link for more information. , J. W. R. DedekindDedekind, Julius Wilhelm Richard
, 1831–1916, German mathematician. Dedekind studied at Göttingen under the German mathematician Carl Gauss and in 1852 received his doctorate there for a thesis on Eulerian integrals.
..... Click the link for more information. , and K. W. WeierstrassWeierstrass, Karl Wilhelm Theodor
, 1815–97, German mathematician. From 1864 he was professor of mathematics at the Univ. of Berlin. His development of the modern theory of functions is described in his Abhandlungen aus der Funktionenlehre
..... Click the link for more information. . Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. CauchyCauchy, Augustin Louis, Baron
, 1789–1857, French mathematician. He was professor simultaneously (1816–30) at the École polytechnique, the Sorbonne, and the Collège de France in Paris. While a political exile (1830–38) he taught at