Background
Smith was born in Dublin, Ireland, on the 2nd ol November 1826, the fourth child of John Smith, a barrister, who died when Henry was two. His mother very soon afterwards moved the family to England.
Smith was born in Dublin, Ireland, on the 2nd ol November 1826, the fourth child of John Smith, a barrister, who died when Henry was two. His mother very soon afterwards moved the family to England.
He lived in several places in England as a boy, and had private tutors for his education. His mother did not send him to school but educated him herself until age 11, at which point she hired private tutors. At age 15 Smith was admitted in 1841 to Rugby School in Warwickshire, where Thomas Arnold was the school's headmaster. This came about because his tutor Henry Highton took up a housemaster position there.
At 19 he won an entrance scholarship to Balliol College, Oxford. He graduated in 1849 with high honours in both mathematics and classics. Smith was fluent in French having spent holidays in France, and he took classes in mathematics at the Sorbonne in Paris during the 1846–7 academic year.
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, etc. , 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 Proceedings of that Society 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.
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.
Quotations:
“For each successive class of phenomena, a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. ”
“[Toast:] Pure mathematics; may it never be of any use to anyone. ”
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 was president of the mathematical and physical section of the British Association at Bradford in 1873 and of the London Mathematical Society in 1874 and 1876.