Henry John Stephen Smith

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Henry J. S. Smith,
frontispiece from Biographical Sketches and Recollections (with Early Letters)

Henry John Stephen Smith (2 November 18269 February 1883) was a mathematician remembered for his work in elementary divisors, quadratic forms, matrix theory, and number theory.


Henry John Stephen Smith
  • So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma.
    • As quoted in The Century: A Popular Quarterly (1874) ed. Richard Watson Gilder, Vol. 7, pp. 508-509, "Relations of Mathematics to Physics". Earlier quote without citation in Nature, Volume 8 (1873), page 450.
    • Also quoted partially in Michael Grossman and Robert Katz, Non-Newtonian Calculus (1972) p. iv. ISBN 0912938013.
  • If we except the great name of Newton (and the exception is one that the great Gauss himself would have been delighted to make) it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute vigorousness in demonstration, which the ancient Greeks themselves might have envied. It may be admitted, without any disparagement to the eminence of such great mathematicians as Euler and Cauchy that they were so overwhelmed with the exuberant wealth of their own creations, and so fascinated by the interest attaching to the results at which they arrived, that they did not greatly care to expend their time in arranging their ideas in a strictly logical order, or even in establishing by irrefragable proof propositions which they instinctively felt, and could almost see to be true. With Gauss the case was otherwise. It may seem paradoxical, but it is probably nevertheless true that it is precisely the effort after a logical perfection of form which has rendered the writings of Gauss open to the charge of obscurity and unnecessary difficulty. The fact is that there is neither obscurity nor difficulty in his writings, as long as we read them in the submissive spirit in which an intelligent schoolboy is made to read his Euclid. Every assertion that is made is fully proved, and the assertions succeed one another in a perfectly just analogical order... But when we have finished the perusal, we soon begin to feel that our work is but begun, that we are still standing on the threshold of the temple, and that there is a secret which lies behind the veil and is as yet concealed from us. No vestige appears of the process by which the result itself was obtained, perhaps not even a trace of the considerations which suggested the successive steps of the demonstration. Gauss says more than once that for brevity, he gives only the synthesis, and suppresses the analysis of his propositions. Pauca sed matura—few but well matured... If, on the other hand, we turn to a memoir of Euler's, there is a sort of free and luxuriant gracefulness about the whole performance, which tells of the quiet pleasure which Euler must have taken in each step of his work; but we are conscious nevertheless that we are at an immense distance from the severe grandeur of design which is characteristic of all Gauss's greater efforts.
    • As quoted by Alexander Macfarlane, Lectures on Ten British Physicists of the Nineteenth Century (1916) p. 95, "Henry John Stephen Smith (1826-1883) A Lecture delivered March 15, 1902"

The Collected Mathematical Papers of Henry John Stephen Smith (1894) Vol. 1

ed. James Whitbread Lee Glaisher, source
  • We must confine ourselves to what we may term the great highways of the science; and... we must wholly pass by many outlying researches of great interest and importance, as we propose rather to exhibit in a clear light the most fundamental and indispensable theories, than to embarrass the treatment of a subject, already sufficiently complex, with a multitude of details, which, however important in themselves, are not essential to the comprehension of the whole.
    • Report on the Theory of Numbers (1859) Part I, p. 39.
  • The problem of the direct determination of the primitive roots of a prime number is one of the 'cruces' of the Theory of Numbers. Euler, who first observed the peculiarity of these numbers, has yet left us no rigorous proof of their existence; though assuming their existence, he succeeded in accurately determining their number. The defect in his demonstration was first supplied by Gauss, who has also proposed an indirect method for finding a primitive root.
    • Report on the Theory of Numbers (1859) Part I, p. 49.
  • 'Legendre's Law of Quadratic Reciprocity' ... is ...the most important general truth in the science of integral numbers which has been discovered since the time of Fermat. It has been called by Gauss 'the gem of the higher arithmetic,' and is equally remarkable whether we consider the simplicity of its enunciation, the difficulties which for a long time attended its demonstration, or the number and variety of the results which have been obtained by its means. ...[W]e find in the 'Opuscula Analytica' of Euler... a memoir... which contains a general and very elegant theorem from which the Law of Reciprocity is immediately deducible, and which is, vice versâ, deducible from that law. But Euler... expressly observes that the theorem is undemonstrated; and this would seem to be the only place in which he mentions it in connexion with the theory of the Residues of Powers; though in other researches he has frequently developed results which are consequences of the theorem, and which relate to the linear forms of the divisors of quadratic formulae. But here also his conclusions repose on induction only; though in one memoir he seems to have imagined... that he had obtained a satisfactory demonstration.
    • Report on the Theory of Numbers (1859) Part I, pp. 56-57.
  • The first demonstration (Disq. Arith., Arts. 125-145) which is presented by Gauss in a form very repulsive to any but the most laborious students, has been resumed by Lejeune Dirichlet in a memoir in Crelle's Journal... and has been developed by him with that luminous perspicuity by which his mathematical writings are distinguished.
    • Report on the Theory of Numbers (1859) Part I, p. 59.

Quotes about Smith

H. J. S. Smith
  • I do not know what Henry Smith may be at the subjects of which he professes to know something; but I never go to him about a matter of scholarship, in a line where he professes to know nothing, without learning more from him than I can get from any one else.

Biographical Sketches and Recollections (with Early Letters) (1894)

of Henry John Stephen Smith M.A., F.R.S., Late Savilian Professor of Geometry in the University of Oxford. source
  • Though not a poet or creative genius, he was... possessed of greater natural abilities than any one else whom I have known at Oxford. He had the clearest and most lucid mind, and a natural experience of the world and of human character hardly ever to be found in one so young. He took up all subjects at the right end; he knew whereabouts the truth lay even when he was imperfectly acquainted with the facts. And he was the most amiable and good-natured of young men. I might apply to him the words in which Plato describes the youthful Athenian Mathematician, Theaetetus, where he says: "In all my acquaintance, which is very large, I never knew any one who was his equal in natural gifts. He had a quickness of apprehension which was almost unrivalled, and he was exceedingly gentle. There was a union of qualities in him which I have never seen in any other, and should scarcely have thought possible, for quick wits have generally quick tempers... but he moved surely and smoothly and successfully in the paths of knowledge and enquiry. He flowed on silently like a river of oil. At his age it was wonderful. He was also surprisingly liberal about money, though his fortune was only moderate. (Theætetus 144).
  • In those days he was almost equally a lover of Classics and Mathematics. ...Even in the last years of his life he was in the habit of taking with him Greek books to read during the Vacation.
  • His mathematical speculations could have been shared by a very few, not more than two or three, of his contemporaries at Oxford. Yet he did not withdraw himself from business or society. He was not the silent philosopher who is lost in reverie, or who, while acknowledged to be a mathematical genius, is pointed at by mankind as a poor and eccentric mortal. He was a thorough man of the world and greatly liked by everybody.
  • He was very desirous to promote the interests of Natural Science in Oxford, and was in favour of some measure which would have made the knowledge of a portion of some one of the Natural Sciences the condition of obtaining a degree. The teachers of these sciences had long been fighting a battle against the older traditions of the University; they had now become the study of a few, but he clearly saw that they could never truly flourish until an interest in them was more generally diffused... But he was also the best friend that the older studies... for he could speak with authority, and he was firmly convinced that in education Science should not supersede Literature. He deplored equally the want of literary culture which he observed in many scientific men, and the gross ignorance of the most general facts of Science which prevails in the world at large, especially at English Universities and Public Schools. In a similar spirit he was anxious to encourage at Oxford the study of Medicine and also of Engineering, thinking that they would supply a missing link between the Physical Sciences and the older studies of the University.
  • He was indulgent to the failings of young men, and felt a humane pity for persons who had lost their character. He was one of whom it might be said that 'he would have stood by a friend, not only in adversity, but in disgrace.'
  • He could see through the vanity and folly of a friend, and yet retain a never changing affection for him. Of his own life, he seldom or never spoke; he was not an egotist, and his own sayings or doings did not seem to interest him afterwards.

Lectures on Ten British Mathematicians (1916)

by Alexander MacFarlane, "Henry John Stephen Smith (1826-1883) A Lecture delivered March 15, 1902" source; also see Chapter 7: "Henry John Stephen Smith" .
  • [H]e had a difficulty in deciding between classics and mathematics, and there is a story to the effect that he finally solved the difficulty by tossing up a penny. He certainly used the expression: but the reasons which determined his choice in favor of mathematics were first, his weak sight, which made thinking preferable to reading, and secondly the opportunity...
  • Whewell, the master of Trinity College Cambridge, wrote The Plurality of Worlds... Whewell pointed out what he called law of waste traceable in the Divine economy; and his argument was that the other planets were waste effects, the Earth the only oasis in the desert of our system, the only world inhabited by intelligent beings; Sir David Brewster... wrote a fiery answer entitled "More worlds than one, the creed of the philosopher and the hope of the Christian." In 1855 Smith wrote an essay on this subject... in which the fallibility both of men of science and of theologians was impartially exposed. It was his first and only effort at popular writing.
  • Though addicted to the theory of numbers, he was not in any sense a recluse; on the contrary he entered with zest into every form of social enjoyment in Oxford... He had the rare power of utilizing stray hours of leisure, and it was in such odd times that he accomplished most of his scientific work. After attending a picnic in the afternoon, he could mount to those serene heights in the theory of numbers... Then he could of a sudden come down from these heights to attend a dinner, and could conduct himself there, not as a mathematical genius lost in reverie and pointed out as a poor and eccentric mortal, but on the contrary as a thorough man of the world greatly liked by everybody.
  • In 1858 he was selected by [the British Association] to prepare a report upon the Theory of Numbers. It was prepared in five parts, extending over the years 1859-1865. It is neither a history nor a treatise, but something intermediate. The author analyzes with remarkable clearness and order the works of mathematicians for the preceding century upon the theory of congruences, and upon that of binary quadratic forms. He returns to the original sources, indicates the principle and sketches the course of the demonstrations, and states the result, often adding something of his own. The work has been pronounced to be the most complete and elegant monument ever erected to the theory of numbers, and the model of what a scientific report ought to be.