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RIEMANN, GEORG FRIEDRICH BERNHARD (18...

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Originally appearing in Volume V23, Page 323 of the 1911 Encyclopedia Britannica.
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RIEMANN, GEORG See also:FRIEDRICH BERNHARD (1826–1866) , See also:German mathematician, was See also:born on the 17th of See also:September II 1826, at Breselenz, near Dannenberg in See also:Hanover. His See also:father, Friedrich Bernhard Riemann, came from See also:Mecklenburg, had served in the See also:war of freedom, and had finally settled as pastor in Quickborn. Here with his five See also:brothers and sisters Riemann spent his boyhood and received, chiefly from his father, the elements of his See also:education. He showed at an See also:early See also:age well-marked mathematical See also:powers, and his progress was so rapid in See also:arithmetic and See also:geometry that he was soon beyond the guidance not only of his father but of schoolmaster Schulz, who assisted in the mathematical See also:department of his training. In 184o he went to Hanover, where he attended the See also:lyceum, and two years later he entered the Johanneum at See also:Luneburg. The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Leonhard See also:Euler's See also:works and Adrien See also:Marie See also:Legendre's Theory of See also:Numbers), which Riemann read, mastered and returned within a few days. In 1846 Riemann entered himself as a student of See also:philology and See also:theology in the university of See also:Gottingen. This choice of a university career was dictated more by the natural See also:desire of his father to see his son enter his own profession, and by the poverty of his See also:family, than by his own preference. He attended lectures on the numerical See also:solution of equations and on definite integrals by M. A. Stern, on terrestrial See also:magnetism by See also:Goldschmidt, and on the method of least squares by K. F.

See also:

Gauss. It soon became evident that his mathematical studies, undertaken at first probably as a relaxation, were destined to be the See also:chief business of his See also:life. He proceeded in the beginning of 1847 to See also:Berlin, attracted thither by that brilliant See also:constellation of mathematical See also:genius whose See also:principal stars were P. G. L. Dirichlet, C. G. J. See also:Jacobi, J. See also:Steiner and F. G. M.

Eisenstein. He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial See also:

differential equations, and Jacobi's on See also:analytical See also:mechanics and higher See also:algebra. It was during this See also:period that he first formed those ideas on the theory of functions of a complex variable which led to most of his See also:great discoveries. One stirring social incident at least marked this See also:part of his life, for, during the revolutionary insurrection in See also:March 1848, the See also:young mathematician, as a member of a See also:company of student See also:volunteers, kept guard in the royal See also:palace from 9 o'See also:clock on the See also:morning of the 24th of March till 1 o'clock on the afternoon of the following See also:day. In 185o he returned to Gottingen and began to prepare,his See also:doctor's dissertation, busying himself meanwhile with " Naturphilosophie " and experimental physics. This See also:double cultivation of his scientific powers had the happiest effect on his subsequent See also:work; for the greatest achievements of Riemann were effected by the application in pure See also:mathematics generally of a method (theory of potential) which had up to this See also:time been used solely in the solution of certain problems that arise in mathematical physics. In See also:November 1851 he obtained his doctorate, the thesis being " Grundlagen See also:fur eine allgemeine Theorie der Functionen einer veranderlichen complexen See also:Grosse." This memoir excited the admiration of Gauss, and at once marked its author's See also:rank as a mathematician. The fundamental method of See also:research which Riemann employed has just been alluded to; the results will be best indicated in his own words: " The methods in use hitherto for treating functions of a complex variable always started from an expression for the See also:function as its See also:definition, whereby its value was given for every value of the See also:argument ; by our investigation it has been shown that, in consequence of the See also:general See also:character of a function of a complex variable, in a definition of this sort one part of the determining conditions is a consequence of the See also:rest, and the extent of the determining conditions has been reduced to what is necessary to effect the determination. This essentially simplifies the treatment of such functions. Hitherto, in See also:order to prove the equality of two expressions for the same function, it was necessary to transform the one into the other, i.e. to show that both expressions agreed for every value of the variable; now it is sufficient to prove their agreement to a far less extent " [merely in certain See also:critical points and at certain boundaries]. The time between his promotion to the doctorate and his habilitation as Privatdozent was occupied by researches undertaken for his Habilitationsschrift, by " Naturphilosophie,"and by experimental work. The subject he had chosen for his Habilitationsschrift was the "See also:Representation of a Function by Means of a Trigonometrical See also:Series," a subject which Dirichlet had made his own by a now well-known series of researches.

It was fortunate, no doubt, for Riemann that he had the See also:

kind See also:advice and encouragement of Dirichlet himself, who was then on a visit at Gottingen during the preparation of his See also:essay; but the result was a memoir of such originality and refinement as showed that the See also:pupil was fully the equal of the See also:master. Of the customary three themes which he suggested for his trial lecture, that " On the Hypotheses which See also:form the See also:Foundation of Geometry" was chosen at the instance of Gauss, who was curious to hear what so young a See also:man had to say on this difficult subject, on which he himself had in private speculated so profoundly (see GEOMETRY, NON-EUCLIDIAN). In 1855 Gauss died and was succeeded by Dirichlet, who along with others made an effort to obtain Riemann's nomination as extraordinary See also:professor. In this they were not successful; but a See also:government See also:stipend of 200 thalers was given him, and even this miserable See also:pittance was of great importance, so straitened were his circumstances. But this small beginning of See also:good See also:fortune was embittered by the deaths of his father and his eldest See also:sister, and by the breaking up of the See also:home at See also:Quick-born. Meantime he was lecturing and See also:writing the great memoir (Borchardt's See also:Journal, vol. liv., 1857) in which he applied the theory See also:developed in his doctor's dissertation to the Abelian functions. It is amusing to find him speaking jubilantly of the unexpectedly large See also:audience of eight which assembled to hear his first lecture (in 1854) on partial differential equations and their application to See also:physical problems. Riemann's See also:health had never been strong. Even in his boyhood he had shown symptoms of See also:consumption, the disease that was working such havoc in his family; and now under the See also:strain of work he See also:broke down altogether, and had to retire to the Harz with his See also:friends See also:Ritter and R. Dedekind, where he gave himself up to excursions and " Naturphilosophie." After his return to Gottingen (November 1857) he was made extra-See also:ordinary professor, and his See also:salary raised to 300 thalers. As usual with him, misfortune followed See also:close behind; for he lost in quick See also:succession his See also:brother Wilhelm and another sister. In 1859 he lost his friend Dirichlet; but his reputation was now so well established that he was at once appointed to succeed him.

Well-merited honours began to reach him; and in 186o he visited See also:

Paris, and met with a warm reception there. He married Elise See also:Koch in See also:June 1862, but the following See also:month he had an attack of See also:pleurisy which proved the beginning of a See also:long illness that ended only with his See also:death. His physician recommended a sojourn in See also:Italy, for the benefit of his health, and See also:Weber and Sartorius von See also:Waltershausen obtained from the government leave of See also:absence and means to defray the cost of the See also:journey. At first it seemed that he would recover; but on his return in June 1863 he caught See also:cold on the Splugen Pass, and in See also:August of the same See also:year had to go back to Italy. In November 1865 he returned again to Gottingen, but, although he was able to live through the See also:winter, and even to work a few See also:hours every day, it became clear to his friends, and clearest of all to himself, that he was dying. In order to See also:husband his few remaining days he resolved in June 1866 to return once more to Italy. Thither he journeyed through the confusion of the first days of the Austro-Prussian War, and settled in a See also:villa at Selasca near See also:Intra on Lago See also:Maggiore. Here his strength rapidly ebbed away, but his See also:mental faculties remained brilliant to the last. On the 19th of See also:July 1866 he was working at his last unfinished investigation on the mechanism of the See also:ear. The day following he died. Few as were the years of work allotted to him, and few as are the printed pages covered by the See also:record of his researches, his name is, and will remain, a See also:household word among mathematicians. Most of his See also:memoirs are masterpieces—full of See also:original methods, profound ideas and far-reaching See also:imagination.

The collected works of Riemann were published by H. Weber, assisted by R. Dedekind (8vo, See also:

Leipzig, 1876; 2nd ed., 1892). At the end of this See also:volume there is a touching See also:account of his life disparagingly of him. But his See also:power was already beginning by the latter. - (G. CH.) to wane.

End of Article: RIEMANN, GEORG FRIEDRICH BERNHARD (1826–1866)

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