550 R I E R I E 1553, affirmed that Mary was illegitimate, and predicted that her accession would be disastrous to the religious interests of England. After the proclamation of Mary he set out for Framlingham to confess his offences against her, but was met with a warrant for his arrest and was committed to the Tower. In March 1554 he was sent down, along with Cranmer and Latimer, to Oxford to be tried before a committee of convocation. He was convicted of heresy, and after refusing to recant was sentenced to death. The trial having been declared irregular, he was, in September 1555, along with Cranmer and Latimer, tried by special com- missioners, and on the 16th October he, in company with Latimer, was burnt at the stake at Oxford. The collected edition of the works of Ridley, published in 1841 with a biographical notice by Rev. Henry Christmas, includes A Treatise concerning Images in Churches ; A brief Declaration of the Lord's Supper ; Certain godly and comfortable Conferences between Bishop Ridley and Mr Hugh Latimer during their imprisonment ; A Comparison between tJie amifortable Doctrine of the Gospel and the Traditions of the Popish Religion; and a variety of other pamphlets. His life by his relative Dr Gloucester Ridley appeared in 1763. See also Foxe's Acts and Monuments ; Strype's Cranmer ; Burnet's History of the Reformation ; Wood's Athense Oxon. ; Cooper's Athenie Cantab. ; and Froude's History of England. EIEMANN, GEORG FRIEDEICH BERNHARD (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. Here with 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 lyceurn. After her death, two years later, he entered the Johanneum at Liineburg, 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 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 permis- sion to devote himself entirely to a scientific career. By this time he had exhausted the resources of Gottingen 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 prin- cipal 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 Gottingen 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 i powers, doubtless due more to the influence of GiJttingen 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 fur eine allgemeine Theorie der Functionen einer veranderlichen complexen Grosse. " 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 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 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 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 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 Gottingen 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
