HENRY JOHN STEPHEN SMITH (1826-1883), English mathematician, was born in Dublin on the 2nd of November 1826, and was the fourth child of his parents. When Henry Smith was just two years old his father died, whereupon his mother left Ireland for England. After being privately educated by his mother and tutors, he entered Rugby school in 1841. Whilst under the first of these tutors, in nine months he read all Thucydides, Sophocles and Sallust, twelve books of Tacitus, the greater part of Horace, Juvenal, Persius, and several plays of Aeschylus and Euripides. He also studied the first six books of Euclid and some algebra, besides reading a considerable quantity of Hebrew and learning the Odes of Horace by heart. On the death of his elder brother in September 1843 Henry Smith left Rugby, and at the end of 1844 gained a scholarship at Balliol College, Oxford. He won the Ireland scholarship in 1848 and obtained a first class in both the classical and the mathematical schools in 1849. He gained the senior mathematical scholarship in 1851. He was elected fellow of Balliol in 1850 and Savilian professor of geometry in 1861, and in 1874 was appointed keeper of the university museum. He was elected F.R.S. in 1861, and was an LL.D. of Cambridge and Dublin. He served on various royal commissions, and from 1877 was the chairman of the managing body of the meteorological office. He died at Oxford on the 9th of February 1883.
After taking his degree he wavered between classics and mathematics, but finally chose the latter. After publishing a few short papers relating to theory of numbers and to geometry, he devoted himself to a thorough examination of the writings of K. F. Gauss, P. G. Lejeune-Dirichlet, E. E. Kummer, &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 publication of Gauss's Disquisitiones arithmeticae. A brief abstract of Smith's methods and results appeared in the Proc. R oy. 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 Academie Frangaise, in ignorance of Smith's work, set the demonstration and completion of Eisenstein's theorems for five squares as the subject 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 Academie 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 problemes 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" made 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 18 741876. 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).
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