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 P. G. Lejeune-Dirichlet, E. E. Kuromer, &c., on, the theory of numbers. The main results of these researches, which occupied him from 1854 to 1864, are contained in his Report on the Theory of Numbers, which appeared in the British Association volumes from 1859 to 1865. This report contains not only a complete account of all that had been done on this vast and intricate subject but also original contributions of his own. Some of the most important results of his discoveries were communicated to the Royal Society in two memoirs upon “Systems of Linear Indeterminate Equations and Congruences” and upon the “Orders and Genera of Ternary Quadratic Forms” (Phil. Trans., 1861 and 1867). He did not, however, confine himself to the consideration of forms involving only three indeterminates, but succeeded in establishing the principles on which the extension to the general case of n indeterminates depends, and obtained the general formulae, thus effecting what is probably the greatest advance made in the subject since the publican tion of Gauss’s Disquisitiones arithmeticae. A brief abstract of Smith’s methods and results appeared in the ''Proc. Roy. Soc.'' for 1864 and 1868. In the second of these notices he gives the general formulae without demonstrations. As corollaries to the general formulae he adds the formulae relating to the representation of a number as a sum of five squares and also of seven squares. This class of representation ceases when the number of squares exceeds eight. The cases of two, four and six squares had been given by K. G. J. Jacobi and that of three squares by F. G. Eisenstein, who had also given without demonstration some of the results for five squares. Fourteen years later the Académie Française, in ignorance of Smith’s work, set the demonstration and completion of Eisenstein’s theorems for five squares as the object of their “Grand Prix des Sciences Mathematiques.” Smith, at the request of a member of the commission by which the prize was proposed, undertook in 1882 to write out the demonstration of his general theorems so far as was required to prove the results for the special case of five squares. A month after his death, in March 1883, the prize of 3000 francs was awarded to him. The fact that a question of which Smith had given the solution in 1867, as a corollary from general formulae governing the whole class of investigations to which it belonged, should have been set by the Académie as the subject of their great prize shows how far in advance of his contemporaries his early researches had carried him. Many of the propositions contained in his dissertation are general; but the demonstrations are not supplied for the case of seven squares. He was also the author of important papers in which he extended to complex quadratic forms, many of Gauss’s investigations relating to real quadratic forms. After 1864 he devoted himself chiefly to elliptic functions, and numerous papers on this subject were published by him in the ''Proc. Lond. Math. Soc.'' and elsewhere. At the time of his death he was engaged upon a memoir on the Theta and Omega Functions, which he left nearly complete. In 1868 he was awarded the Steiner prize of the Berlin Academy for a geometrical memoir, Sur quelques problèmes cubiques et biquadratiques. He also wrote the introduction to the collected edition of Clifford’s Mathematical Papers (1882). The three subjects to which Smith’s writings relate are theory of numbers, elliptic functions and modern geometry; but in all that he wrote an “arithmetical” mode of thought is apparent, his methods and processes being arithmetical as distinguished from algebraic. He had the most intense admiration of Gauss. He Was president of the mathematical and physical section of the British Association at Bradford in 1873 and of the London Mathematical Society in 1874–1876. His Collected Papers were edited by J. W. L. Glaisher and published in 1894.

An article in the Spectator of the 17th of February 1883, by Lord Justice Bowen, gives perhaps the best idea of Smith’s extraordinary personal qualities and influence. See also J. W. L. Glaisher’s memoir in the ''Monthly Notices of the Roy. Ast. Soc.'' (vol. xliv., 1884).

SMITH, HENRY PRESERVED (1847–), American Biblical scholar, was born in Troy, Ohio, on the 23rd of October 1847. He graduated at Amherst College in 1869 and studied theology in Lane Theological Seminary in 1869–1872, in Berlin in 1872–1874 and in Leipzig in 1876–1877. He was instructor in church history in 1874–1875, and in Hebrew in 1875–1876, and was assistant-professor in 1877–1879 and professor in 1879–1893 of Hebrew and Old Testament exegesis in Lane Theological Seminary. In 1892 he was tried for heresy by the Presbytery of Cincinnati, was found guilty of teaching (in a pamphlet entitled Biblical Scholarship and Inspiration, 1891) that there were “errors of historic fact,” suppressions of “historic truths,” &c., in the books of Chronicles, and that the “inspiration of the Holy Scriptures is consistent with the unprofitableness of portions of the sacred writings,”—in other words, that inspiration does not imply inerrancy,—and he was suspended from the ministry. Dr Smith retired from the denomination, and in 1893, upon becoming a professor at Andover Theological Seminary, entered the ministry of the Congregational Church. From 1897 to 1906 he was a professor in Amherst College, and in 1907 became a professor in the Meadville (Pennsylvania) Theological School.

SMITH, JAMES (1775–1839), and HORACE (1779–1849), authors of the Rejected Addresses, sons of a London solicitor, were born, the former on 10th February 1775 and the latter on 31st December 1779, both in London. The occasion of their happy jeu d’esprit was the rebuilding of Drury Lane theatre in 1812, after a fire in which it had been burnt down. The managers had offered a prize of £50 for an address to be recited at the re-opening in October. Six weeks before that date the happy thought occurred to the brothers Smith of feigning that the most popular poets of the time had been among the competitors and issuing a volume of unsuccessful addresses in parody of their Various styles. They divided the task between them, James taking Wordsworth, Southey, Coleridge and Crabbe, while Byron, Moore, Scott and Bowles were assigned to Horace. Seven editions were called for within three months. The Rejected Addresses are the most widely popular parodies ever published in England, and take classical rank in literature. The brothers fairly divided the honours: the elder brother’s Wordsworth is evenly balanced by the younger’s Scott, and both had a hand in Byron. A striking feature is the absence of malice; none of the poets caricatured took offence, while the imitation is so clever that both Byron and Scott are recorded to have said that they could hardly believe they had not written the addresses ascribed to them. The only other undertaking of the two brothers was Horace in London (1813). James Smith made another hit in writing Country Cousins, A Trip to Paris, A Trip to America, and other lively skits for Charles Mathews who said he was “the only man who can write clever nonsense.” His social reputation as a wit stood high. He was reputed one of the best of talkers in an age when the art was studied, and it was remarked that he held his own without falling into the great error of wits—sarcasm. But in his old age the irreverent Fraser’s put him in its gallery of living portraits as a gouty and elderly but painstaking joker. He died in London on the 24th of December 1839. After making a fortune as a stockbroker, Horace Smith followed in the wake of Scott and wrote about a score of historical novels—Brambletye House (1826), Tor Hill (1826), Reuben Apsley (1827), Zillah (1828), The New Forest (1829), Walter Colyton (1830), &c. His sketches of eccentric character are brilliant and amusing; but he was more of an essayist than a story-teller. Three volumes of Gaieties and Gravities, published by him in 1826, contain many witty essays both in prose and in verse, but the only single piece that has taken a permanent place is the “Address to the Mummy in Belzoni’s Exhibition.” In private life Horace Smith was not less popular than his brother, though less ambitious as a talker. It was of him that Shelley said: “Is it not odd that the only truly generous person I ever knew who had money enough to be generous with should be a stock-broker? He writes poetry and pastoral dramas and yet knows how to make money, and does make it, and is still generous.” Horace Smith died at Tunbridge Wells on 12th July 1849.

SMITH, JOHN (1579–1631), usually distinguished as Captain John Smith, sometime president of the English colony in Virginia, was the elder son of George Smith, a well-to-do tenant-farmer on the estate of Lord Willoughby d’Eresby at Willoughby, near Alford in Lincolnshire. The life of this Virginian hero falls conveniently into five periods. The first of these, up to 1596, that of his early youth, is thus described by himself in his Travels: “He was born (1579) in Willoughby in Lincolnshire, and was a scholar in the two free schools of Alford and Louth.