**GEORG FRIEDRICH BERNHARD RIEMANN** (1826-1866), mathematician, was born on the 17th September 1826, at Breselenz, near Dannenberg in Hanover. His father Friedrich Bernhard Riemann came from Mecklen-burg, had served in the war of freedom, and had finally settled as pastor in Quickborn. Hero .nth his five brothers and sisters Riemann spent his boyhood and received, chiefly from his father, the elements of his education. He showed at an early age well-marked mathematical powers, and his progress was so rapid in arithmetic and geometry that he was soon beyond the guidance not only of his father but of schoolmaster Schulz, who assisted in the mathematical department of his training.

In 1840 he went to live with his grandmother at Hanover, where he attended the lyceum. After her death, two years later, he entered the Johanneum at Lüneburg, where he finished in four years more his gymnasial educa-tion. Notwithstanding some disadvantages due to defects in his earlier training, and more particularly to shyness arising from his rustic upbringing, he speedily distinguished himself in all the branches of the gymnasial course, and was already known by the school authorities as a mathe-matician of great promise. The director, Schmalfuss, encouraged him in his mathematical studies by lending him books (among them Euler's works and Legendre's Theory of Numbers), and readily understood that he had no ordinary schoolboy to deal with when he found that works of such profoundity were read, mastered, and returned within a few days. In 1846, in his twentieth year, Riemann entered himself as a student of philology and theology in the university of Göttingen. This choice of a university career was dictated more by the natural desire of his father to see his son enter his own profession, and by the poverty of his family, wdiich rendered the i speedy earning of his living a matter of importance, than by his own preference. He sacrificed so far to the bent of his genius as to attend lectures on the numerical solution of equations and on definite integrals by Stern, on terrestrial magnetism by Goldschmidt, and on the method of least squares by Gauss. It soon became evident that his mathematical studies, undertaken at first probably as a relaxation, were destined to be the chief business of his life; and he obtained his father's permis-sion to devote himself entirely to a scientific career. By \ this time he had exhausted the resources of Göttingen in the shape of mathematical lectures; and he proceeded in the beginning of 1847 to Berlin, attracted thither by that brilliant constellation of mathematical genius whose principal stars were Dirichlet, Jacobi, Steiner, and Eisenstein. He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial dif-ferential equations, and Jacobi's on analytical mechanics and higher algebra. It was during this period that he first formed those ideas on the theory of functions of a complex variable which led to most of his great discoveries. One stirring social incident at least marked this part of his life, for, during the revolutionary insurrection in March 1848, the young mathematician, as a member of a company of student volunteers, kept guard in the royal palace from 9 o'clock on the morning, of the 24th March till 1 o'clock on the afternoon of the following day.

In 1850 he returned to Göttingen and began to prepare his doctor's dissertation, busying himself meanwhile with " Naturphilosophie " and experimental physics. In pur-suit of the latter he entered the mathematical and physical seminary, then newly started by Weber, Ulrich, Stern, and Listing. This double cultivation of his scientific powers, doubtless due more to the influence of Göttingen as represented by Gauss than to Berlin, had the happiest effect on his subsequent work; for the greatest achieve-ments of Riemann were effected by the application in pure mathematics generally of a method (theory of potential) which had up to this time been used solely in the solution of certain problems that arise in mathematical physics.

In November 1851 he obtained his doctorate, the thesis being " Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse." This memoir excited the admiration of Gauss, and at once marked its author's rank as a mathematician. The funda-mental method of 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 function as its definition, whereby its value was given for every value of the argument; by our investigation it has been shown that, in consequence of the general 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 rest, and the extent of the deter-mining conditions has been reduced to what js necessary to effect the determination. This essentially simplifies the treatment of such functions. Hitherto, in 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 critical points and at certain boundaries].

The time between his promotion to the doctorate and his habilitation as privat-docent was occupied by re-searches undertaken for his Habilitationsschrift, by "Naturphilosophie," and by experimental work partly as Weber's assistant in the mathematical physical seminary, and partly as collaborates with Weber and Kohlrausch in special researches on electricity. In connexion with the results of Kohlrausch regarding the residual discharge of condensers, Riemann worked out a theory of this pheno-menon which he intended to have published in Poggen-dorfs Anmalen. For some reason not fully explained it was not published at all during his lifetime, and its place in the Annalen was taken by an elegant little paper on Nobili's rings.

The subject he had chosen for his Habilitationsschrift was the " Representation of a Function by means of a Trigono-metrical 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 kind advice and encouragement of Dirichlet himself, who was then on a visit at Göttingen during the preparation of his essay ; but the result was a memoir of such originality and refinement as showed that the pupil was fully the equal of the master. Of the customary three themes which he suggested for his trial lecture, that " On the Hypo-theses which form the Foundation of Geometry" was chosen at the instance of Gauss, who was curious to hear what so young a man had to say on this difficult subject, on which he himself had in private speculated so pro-foundly (see PARALLELS, vol. xviii. p. 254). Dedekind tells us that Eiemann's lecture, which surpassed his utmost expectation, filled him with the greatest astonish-ment, and that on the way back from the faculty meeting he spoke to Wilhelm Weber with the highest apprecia-tion, and with an excitement rare with him, regarding the depth of the thoughts to which Riemanu had given utterance.

In 1855 Gauss died and was succeeded by Dirichlet, who along with others made an effort to obtain Eiemann's nomination as extraordinary professor. In this they were not successful; but a Government stipendium of 200 thalers was given him, and even this miserable pittance was of great importance, so straitened were his circum-stances. But this small beginning of good fortune was embittered by the deaths of his father and his eldest sister, and by the breaking up of the home at Quickborn, where he had so often found solace when ill and dejected. Meantime he was lecturing and writing the great memoir. (Borchardt's Journal, vol. liv., 1857) in which he applied the theory t eveloped in his doctor's dissertation to the Abelian functions. It is amusing to find him speaking jubilantly of the unexpectedly large audience of eight which assembled to hear his first lecture (in 1854) on partial differential equations and their application to physical problems. The rustic shyness which had troubled his schoolboyhood seems still to have haunted him; for he says, speaking of these lectures, " The nervousness which I had at first has pretty well subsided, and I accustom my-self to think more of my hearers than of myself, and to read in their faces whether I may go on, or whether the matter requires further explanation."

Riemann's health had never been strong. Even in his boyhood he had shown symptoms of consumption, the disease that was working such havoc in his family j and now under the strain of work he broke down altogether, and had to retire to the Harz with his friends Eitter and Dedekind, where he gave himself up to excursions and " Naturphilosophie." After his return to Gottingen (November 1857) he was made extraordinary professor, and his salary raised to three hundred thalers. As usual with him, misfortune followed close behind ; for he lost in quick succession his 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. It now seemed for a little as if the world was to go smoothly with him. Well-merited honours began to reach him; and in 1860 he visited Paris, and met with a warm reception there. He married, and married happily, Fraulein Elise Koch in June 1862, but the following month he had an attack of pleurisy which proved the beginning of a long illness that ended only with his death. His physician recommended a sojourn in Italy, for the benefit of his health, and Weber and Sartorius. von Waltershausen obtained from the Government leave of absence and means to defray the cost of the journey. At first it seemed that he would recover; but on his return in June 1863 he caught cold on the Spliigen Pass, and in August of the same year had to go back to Italy. In November 1865 he returned again to Gottingen, but, although he was able to live through the winter, and even to work a few hours every day, it became clear to his friends, and clearest of all to himself, that he was dying. He was very desirous to finish some of the many investi-gations which had presented themselves to him, and eagerly asked his doctors to tell him how long he might reasonably expect to live, so that he might take up what he had most chance of finishing. In order to 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 villa at Selasca near Intra on Lago Maggiore. Here his strength rapidly ebbed away, but his mental faculties remained brilliant to the last. On the 19th of July 1866, attended by his wife, he lay under a fig-tree greatly enjoying the beautiful landscape and working at his last unfinished investigation on the mechanism of the ear. The day following he died.

There are few more pathetic stories than the life of Eiemann, few finer instances of victory gained by inborn genius over a host of adverse circumstances. Few as were the years of work allotted to him, and few as are the printed pages covered by the record of his researches, his name is, and will remain, a household word among mathe-maticians. Most of his memoirs are masterpieces—full of original methods, profound ideas, and far-reaching imagina-tion. Few sources, we imagine, have been fuller of inspira-tion for the younger mathematicians of our day than the octavo volume of five hundred pages or so that contains his works; and many an advance in mathematical science will yet be made, with increase of reputation to the maker, by carrying out his suggestions.

The collected works of Eiemann were published by H. "Weber assisted by R. Dedekind (8vo, Leipsic, 1876). At the end of this volume there is a touching account of his life by the latter, from which the above sketch is almost entirely taken. (G. CH.)

**The above article was written by: **George Chrystal, M.A., Professor of Mathematics, University of Edinburgh.