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 education. 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 mathematician 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 Gottingen. 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, which rendered the 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 permission 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 differential 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 pursuit 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 fundamental 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 determining conditions has been reduced to what is 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 researches undertaken for his Habilitationsschrift, by “Naturphilosophie,” and by experimental work partly as Weber’s assistant in the mathematical physical seminary, and partly as collaborateur 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 phenomenon which he intended to have published in Poggen-dorff ’ s Annalen. 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 Trigonometrical 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 λvas 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 lie suggested for his trial lecture, that “On the Hypotheses 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 profoundly (see Parallels, vol. xviii. p. 254). Dedekind tells us that Riemann’s lecture, which surpassed his utmost expectation, filled him with the greatest astonishment, and that on the way back from the faculty meeting he spoke to Wilhelm Weber with the highest appreciation, and with an excitement rare with him, regarding the depth of the thoughts to which Riemann had given utterance.

In 1855 Gauss died and was succeeded by Dirichlet, who along with others made an effort to obtain Riemann’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 circumstances. 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 (Borchardtfi Journal, vol. liv., 1857) in which he applied the theory developed 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 myself 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; and now under the strain of work he broke down altogether, and had to retire to the Harz with his friends Ritter and Dedekind, where he gave himself up to excursions and “Naturphilosophie.” After his return to Göttingen (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, Fräulein 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 Splügen Pass, and in August of the same year had to go back to Italy. In November 1865 he returned again to Göttingen, 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 investigations 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 Riemann, 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 mathematicians. Most of his memoirs are masterpieces—full of original methods, profound ideas, and far-reaching imagination. Few sources, we imagine, have been fuller of inspiration 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 Riemann were published by IL 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.

According to Terentius Varro (himself perhaps of Reatino birth) and others who have followed him, the people who founded Cures, and afterwards settled on the Palatine at Rome, were natives of the Reatine territory; but this is of somewhat the same questionable character as the other story about the companion of Hercules who was buried at this point near the Via Salaria. About the Via Salaria itself, there is no doubt it led from the sea to Reate and onwards towards Ancona, and was from a very early date the great route for the conveyance of salt to the Sabine country. While hardly mentioned in connexion with the Punic or Civil Wars, Reate is described by Strabo as exhausted by those long contests. Its inhabitants received the Roman franchise at the same time with the rest of the Sabines (290 b.c.), but it appears as a praefectura and not as a municipium down to the beginning of the empire. It was never made a colonia, though veterans of the Praetorian and of the legions Octava and Décima Augusta were settled there by Vespasian, who belonged to a Reatine family and was born in the neighbourhood. For the contests of the Reatines with the people of Interamna see Terni. About the middle of the 12th century the town was besieged and captured by Roger I. of Sicily. In the struggle between church and empire it always held with the former; and it defied the forces of Frederick II. and Otho IV. Pope Nicholas IV. long resided at Rcati, and it was there he crowned Charles II. of Anjou king of the Two Sicilies. In the 14th century Robert, and afterwards Joanna, of Naples managed to keep possession of Reati for many years, but it returned to the States of the Church under Gregory IX. About the year 1500 tho liberties of the town, long defended against the encroachments of the popes, were entirely abolished. An earthquake in 1785 was in 1799 followed by the much more disastrous pillage of Reati by the papal troops for a space of fourteen days.

See Aldus Manutius, “Dissert, cpistolica de Reati,” in Not. Thes. Aηtiη. Rοrn., i.; Angelotti, Descrizione, 1G35; Sclιenardi, Antiche lapidi, 1820; and Michaeli, Note per la storia di Rieti, 1868, 1870.

For a good biography of Rietschel and account of his works see Appeπnann, Ernst Rietschel, Leipsic, 18G3.