RIEMANN, GEORG FRIEDRICH BERNHARD (1826-1866), German mathematician, was born on Sept. 17, 1826, at Breselenz, near Dannenberg in Hanover.
In 1840 he went to Hanover, where he attended the lyceum and two years later he entered the Johanneum at Liineburg. The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Euler's works and Legendre s Theory of Numbers). In 1846 Riemann entered the university of Gottingen, where, although supposed to be studying theology, he attended lectures on the numerical solution of equations and on definite integrals by M. A. Stern, on terrestrial magnetism by Goldschmidt, and on the method of least squares by K. F. Gauss. In 1847 he went to Berlin, where P. G. L. Dirichlet, C. G. J. Jacobi, J. Steiner and F. G. M. Eisenstein were professors. Dur ing this period he formed those ideas on the theory of functions of a complex variable which led to his great discoveries.
In 185o he returned to Gottingen and in 1851 obtained his doctorate with his celebrated thesis "Grundlagen fiir eine allge meine Theorie der Functionen einer veranderlichen complexen Grosse." In his Habilitationsschrif t on the "Representation of a Func tion by Means of a Trigonometrical Series," Riemann shows his usual originality and refined style. The subject of his trial lec
ture, chosen by Gauss, was "On the Hypotheses which form the Foundation of Geometry." (See GEOMETRY : Non-Euclidian.) This wonderful work was published in the Gottinger Abhand lungen (1868) and a translation by Clifford in Nature (vol. 8).
Riemann's health had never been strong and now under the strain of work he broke down, and retired to the Harz with his friends Ritter and R. Dedekind, where he gave himself up to excursions and "Naturphilosophie." After his return to Gottingen (Nov. 1857) he was made extraordinary professor, and his salary raised to 30o thalers. Before this he had been in very straitened circumstances, and in 1855 was granted a government stipend of 200 thalers. On Dirichlet's death in 1859, Riemann was ap pointed his successor in Gottingen. He died at Selasca, on Lake Maggiore, on July 20, 1866. Most of his memoirs are master pieces—full of original methods, profound ideas and far-reaching imagination. See RIEMANNIAN GEOMETRY below.
The collected works of Riemann were published by H. Weber, assisted by R. Dedekind (8vo, Leipzig, 1876; 2nd ed., 1892).