The New International Encyclopædia/Riemann, Georg Friedrich Bernhard

RIEMANN, rē'mȧn, (1826-66). One of the foremost German mathematicians of the nineteenth century, particularly in the field of geometry. He was born at Breselenz, near Dannenberg, in Hanover. He studied mathematics at Göttingen and Berlin, and received his doctor's degree at the former university in 1851, his thesis being a well-known contribution to the theory of functions, Grundlagen für allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse. Three years later he was made privat-docent at Göttingen, then (1857) adjunct professor, and finally (1859), on the death of Dirichlet, full professor. His introduction of the notion of geometric order into the theory of Abelian functions, and his invention of the surfaces which bear his name, led to great and rapid advance in the function theory. To him, also, is due (1854) a new system of non-Euclidean geometry, ranking with that of Lobatchevsky and Bolyai (see ), a system which he made known in his thesis, Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (published posthumously, Leipzig, 1867). Riemann's writings, besides those already mentioned, are: Vorlesungen über Schwere, Elektrizität und Magnetismus (1876; 2d ed. 1880, both posthumous); Partielle Differentialgleichungen (1869; 4th ed. 1900-01, both posthumous); Mechanik des Ohres; Elliptische Functionen, Vorlesungen mit Zusätzen (1899); and his Gesammelte mathematische Werke und wissenschaftlicher Nachlass, edited by H. Weber and Dedekind (1876; 2d ed. 1892; French trans., 1898). He also contributed several memoirs on surfaces, which were published in the Annalen and in Crelle's Journal. For the life of Riemann, consult his Gesammelte Werke; Schering, Bernhard Riemann, zum Gedächtniss. For an elementary explanation of Riemann's surfaces, consult: Durège, Theory of Functions (Eng. trans., Philadelphia, 1896); Holzmüller, Einführung in die Theorie der isogonalen Verwandtschaften und der Conformal-Abbildungen (Leipzig, 1882).