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Rh 422 HAMILTON and for physics. How many more such honours he might have attained it is impossible to say ; but he was expected to win both the gold medals at the degree examination, had his career as a student not been cut short by an unprece dented event. This was his appointment to the Andrews professorship of astronomy in the university of Dublin, vacated by Dr Brinkley in 1827. The chair was not exactly offered to him, as has been sometimes asserted, but the electors, having met and talked over the subject, authorized one of their number, who was Hamilton s ! personal friend, to urge him to become a candidate, a step which his modesty had prevented him from taking. Thus, when barely twenty-two, he was established at the Dublin j Observatory. He was not specially fitted for the post, for j although ho had a profound acquaintance with theoretical astronomy, he had paid but little attention to the regular work of the practical astronomer. And it must be said that his time was better employed in grand original investi gations than it would have been had he spent it in meridian observations made even with the best of instruments, infinitely better than if he had spent it on those of the observatory, which, however good originally, were then totally unfit for the delicate requirements of modsrn astro nomy. Indeed there can be little doubt that Hamilton was intended by the university authorities who elected him to the professorship of astronomy to spend his time as he best could for the advancement of science, without being tied down to any particular branch. Had he devoted him self to practical astronomy they would assuredly have furnished him with modern instruments and an adequate staff of assistants. In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, he was | knighted by the lord-lieutenant. Bat far higher honours ; rapidly succeeded, among which we may merely mention his election in 1837 to the president s chair in the Royal Irish Academy, and the rare and coveted distinction of | being made corresponding member of the academy of St Petersburg. These are the few salient points (other, of course, than the epochs of his more important discoveries and inventions presently to be considered) in the uneventful life of this great man. He retained his wonderful faculties unimpaired to the very last, and steadily continued till within a day or two of his death (September 2, 18G5) the task (his Elements of Quaternions) which had occupied the last six years of his life. The genn of his first great discovery was contained in one of those early papers which in 1823 he communicated to Dr Brinkley, by whom, under the title of Caustics, it was presented in 1824 to the Royal Irish Academy. It was referred as usual to a committee. Their report, while acknowledging the novelty and value of its con tents, and the great mathematical skill of its author, recommended that, before being published, it should be still further developed and simplified. During the next three years the paper grew to an immense bulk, principally by the additional details which had been inserted at the desire of the committee. But it also assumed a much more intelligible form, and the grand features of the new method were now easily to be seen. Hamilton himself seems not till this j period to have fully understood either the nature or the importance of his discovery, for it is only now that we find him announcing his ! intention of applying his method to dynamics. The paper was I finally entitled &quot; Theory of Systems of Rays,&quot; and the first part was j printed in 1828 in the Transactions of the Royal Irish Academy. The second and third parts have not yet been printed ; but it is ! understood that their more important contents have appeared in the three voluminous supplements (to the first part) which have j been published in the same Transactions, and in the two papers &quot; Oil a General Method in Dynamics,&quot; which appeared in the Philosophical Transactions in 1834-5. The principle of &quot;Varying Action&quot; is the great feature of these papers ; and it is strange, ! indeed, that the one particular result of this theory which, per- j haps more than anything else that Hamilton has done, has j rendered his name known beyond the little world of true philo- j gophers, should have been easily within the reach of Fresnel and j others for many years before, and in no way required Hamilton s new conceptions or methods, although it was by them that he was ! led to its discovery. This singular result is still known by the name &quot; Conical Refraction,&quot; which he proposed for it when he first pre dicted its existence in the third supplement to his Systems of Rays, read in 1832. The step from optics to dynamics in the application of the method of &quot; Varying Action &quot; was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on the subject. These display, like the &quot;Systems of Rays,&quot; a mastery over symbols and a flow of mathe matical language almost unequalled. But they contain what is far more valuable still, the greatest addition which dynamical science had received since the grand strides made by Newton and Lagrange. Jacobi and other mathematicians have developed to a great extent, and as a question of pure mathematics only, Hamilton s processes, and have thus made extensive additions to our knowledge of differ ential equations. But there can be little doubt that we have as yet obtained only a mere glimpse of the vast physical results of which they contain the germ. And though this is of course by far the more valuable aspect in which any such contribution to science can be looked at, the other must not be despised. It is characteristic of most of Hamilton s, as of nearly all great discoveries, that even their indirect consequences are of high value. The other great contribution made by Hamilton to mathematical science, the invention of QUATERNIONS, is fully treated under that heading. It is not necessary to say here more than this, that quaternions form as great an advance relatively to the Cartesian methods as the latter, when first propounded, formed relatively to Euclidian geometry. The following characteristic extract from a letter shows Hamilton s own opinion of his mathematical work, and also gives a hint of the devices which he employed to render written language as expressive as actual speech. His first great work, Lectures on Quaternions (Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals. &quot; I hope that it may not be considered as unpardonable vanity or presumption on my part, if, as my own taste has always led me to feel a greater interest in methods than in results, so it is by METHODS, rather than by any THEOREMS, which can be separately quoted, that I desire and hope to be remembered. Nevertheless it is only human nature, to derive some pleasure from being cited, now and then, even about a Theorem ; especially where the quoter cau enrich the subject, by combining it with researches of his own.&quot; The discoveries, papers, and treatises we have mentioned might well have formed the whole work of a long and laborious life. But, not to speak of his enormous collection of MS. books, full to over flowing with new and original matter, which have been handed over to Trinity College, Dublin, and of whose contents it is to be hoped a large portion may yet be published, the works we have already called attention to barely form the greater portion of what he has pub lished. His extraordinary investigations connected with the solu tion of algebraic equations of the fifth degree, and his examination of the results arrived at by Abel, Jerrard, and Badano, in their researches on this subject, form another grand contribution to science. There is next his great paper on Fluctuating Functions, a subject which, since the time of Fourier, has been of immense and ever increasing value in physical applications of mathematics. There is also the extremely ingenious invention of the HODOGRAPH (q. v. ). Of his extensive investigations into the solution (especially by numerical approximation) of certain classes of differential equations which constantly occur in the treatment of physical qiicstions, only a few items have been published, at intervals, in the Philosophical Magazine. Besides all this, Hamilton was a voluminous corre spondent. Often a single letter of his occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem : for it was one of the peculiar characteristics of his mind never to be satisfied with a general imderstanding of a question ; he pursued it until he knew it in all its details. He was ever courteous and kind in answering applications for assistance in the study of his works, even when his compliance must have cost him much valuable time. He was excessively precise and hard to please with reference to the final polish of his own works for publication ; and it was probably for this reason that he published so little compared with the extent of his investigations. Like most men of great originality, Hamilton generally matured his ideas before putting pen to paper. &quot; He used to carry on,&quot; says his elder son, &quot;long trains of algebraical and arithmetical calcul ations in his mind, during which he was imconscious of the earthly necessity of eating ; we used to bring in a snack and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards.&quot; For further details about Hamilton (his poetry and his association with poets, for instance), the reader is referred to the Dublin University Magazine (Jan. 18-12), the Gentleman s Magazine (Jan. 18G6), and the Monthly Notices of the Royal Astronomical Society (Feb. I860) ; and also to an article by the present writer in the North British Review (Sept. lCfi), from which much of the above sketch has been taken. (P. &amp;lt;* T.)