# Dictionary of National Biography, 1885-1900/Hamilton, William Rowan

**HAMILTON**, Sir WILLIAM ROWAN (1805–1865), mathematician, born in Dublin at midnight, between 3 and 4 Aug. 1805, was the fourth child of Archibald Hamilton, a solicitor there, and his wife Sarah Hutton, a relative of Dr. Hutton the mathematician. Archibald Hamilton was Scottish by birth, and went to Dublin when a boy with his father, William Hamilton, who settled as an apothecary there, and his mother, who was the daughter, of the Rev. James McFerrand, parish minister of Kirkrnichael, Galloway. The Rev. R. P. Graves maintains that William Rowan Hamilton was Irish by descent, while admitting that both the paternal and maternal grandmothers are Scottish; but the express statements of Professor Tait and Dr. Ingleby that the paternal grandfather went to Dublin from Scotland seem conclusive. The apothecary had also brought a second son, James, from Scotland,who studied for the church, became curate of Trim, co. Meath, and earned some reputation as a linguist. To this uncle William Rowan was entrusted by his father, the solicitor, when less than three years old. Hamilton read Hebrew when but seven years of age, at twelve had not only studied Latin, Greek, and the four leading continental languages, but could profess a knowledge of Syriac, Persian, Arabic, Sanskrit, Hindustani, and even Malay, and in 1819 he wrote a letter to the Persian ambassador in his own language. The choice of languages was owing to his father's intention originally to obtain for him a clerkship under the East India Company. The mathematical bent of his mind, however, was presently to assert itself. In his tenth year he was matched in public with Zerah Colburn, the American 'calculating boy,' retiring from the arithmetical duels not without honour. About the same time he fell upon a Latin copy of Euclid, and studied it with' such effect that within two years he read the 'Arithmetica Universalis' of Newton, and soon after began the 'Principia.' In 1822 good evidence shows that he understood much of that work, and had acquired such command of mathematical methods as to speedily master several modern books on analytical geometry and the differential calculus. Hamilton thus appears 'to have been mainly self-taught in mathematical learning. In his seventeenth year, when reading the 'Mécanique Céleste' of Laplace, he found an error in the reasoning on which one of the propositions was based. This discovery led to Hamilton's introduction to Dr. Brinkley, the astronomer royal for Ireland, afterwards bishop of Cloyne, whom he still further surprised by an original paper 'on osculation of certain curves of double curvature. The discipline of Newton and Laplace had already brought into relief the marked features of a mathematical genius of very rare quality and power.

In 1823 Hamilton became a student of Trinity College, Dublin. His achievements in mathematics alone implied great and continuous mental effort, but his success in other departments of thought was scarcely less remarkable. First in all subjects and at all examinations, twice gaining the vice-chancellor's prize for English verse, decorated with the 'double optime' (almost unprecedented), and, but for the appointment to which his special qualifications entitled him, certain to .gain both gold medals (a thing quite unprecedented), he was characterised by a candour and enthusiastic eloquence that well became him as scholar, poet, and metaphysician, not less than as mathematician or natural philosopher.

In 1824, when only a second year's student, Hamilton read before the Royal Irish Academy a 'Memoir on Caustics,' and being invited to develop the subject, he some time after produced a celebrated paper on systems of rays, and predicted 'conical refraction.' Applying the laws of optics he proved that under certain circumstances a ray of light passing through a crystal will emerge not as a single or double ray but as a cone of rays. This theoretical deduction involved the discovery of two laws of light ; and under the mathematical aspect was pronounced by Sir John Herschel to be 'a powerful and elegant piece of analysis,' while Professor Airy, on the physical side, said 'it had made a new science of optics.' This result, that light; refracts as a conical pencil both internally and externally, obtained on purely theoretical grounds, was soon after verified for universal acceptance, when Professor Humphrey Lloyd, at Hamilton's suggestion, put the new law to the test by means of a plate of arragonite (*Transactions of the Royal Irish Academy*, xvii. 145). The ray of light either issues as a cone with its vertex at the surface of emission, or issues as a cylinder after being converted on entering the crystal into a cone whose vertex is at the point of incidence.

Hamilton, when still an undergraduate, was appointed in 1827 Andrews professor of astronomy and superintendent of the observatory, and soon after astronomer royal for Ireland. He was twice honoured with the gold medal of the Royal Society, first for his optical discovery, and secondly, in 1834, for his theory of a general method of dynamics, which resolves an extremely abstruse problem relating to a system of bodies in motion. Next year, on the occasion of the British Association visiting Dublin, Hamilton was knighted by the lord-lieutenant. In 1837 he was chosen president of the Royal Irish Academy, and had the rare distinction of becoming a corresponding member of the academy of St. Petersburg.

About 1843 Hamilton began more or less clearly to shape out the new mathematical method which when perfected was to give him right to rank in originality and insight with Diophantus, Descartes, and La Grange—a method which, as set forth and illustrated in his own writings, can 'only be compared with the "Principia" of Newton and the "Mécanique Céleste " of La Place as a triumph of analytical and geometrical power' (Professor Tait in *North British Review,* September 1866). In 1844, before the Royal Irish Academy, of which he was still president, he formally defined the term 'quaternions,' by which the new calculus was to be known ; but not till 1848 can the method be considered as systematically established, when he began, in Trinity College, Dublin, the 'Lectures on Quaternions,' which were published in 1853. Nearly the whole of this bulky octavo, occupying 808 pages, besides an introduction of 64 pages, can be understood only by advanced mathematicians. But for Professor Tait of Edinburgh, who interpreted the new science for more common-place mathematicians, Hamilton's merits must long have remained unrealised or absolutely unknown. The truth is that this great book of Hamilton's, as well as his so-called 'Elements of Quaternions,' is frequently unpleasant in style, besides being obscure and difficult of interpretation.

Hamilton's method involved a remarkable extension of science. He showed that the 'impossible quantities' which so frequently occur in analysis admit of easy interpretation by a natural extension of the symbol's meaning. The so-called imaginary or unreal factor really denoted an operation to be performed on the line or surface in question, the operation of rotation. If we multiply a line by (-1) the result is the same as if the line were turned through 180° in its plane, and hence if multiplied by (-1)½ the line will be turned through 90°. On that discovery of the operational character of 'imaginary' factors and expressions was based the whole science of quaternions. Warren in 1828, Peacock (see *Algebra*, vol. ii. chap, xxxi.), De Morgan in his 'Double Algebra,' and others had clearly discussed the interpretation of (-1)½. The notion of motion, virtual transference and rotation, was now combined with the application of algebra to geometry, and while the word 'add' represented motion forward and backward, the word 'multiply' was specialised to represent circular motion. Hamilton freed the science from the limitations of ages, and by his new adaptation of symbols dealt with lines in all possible planes, quite irrespective of any such restricting axes of reference as were necessary to the Cartesian system. To bring any line in space to complete coincidence with any other line may be called finding its quaternion : so named from the four numbers or elements occurring in the geometrical question of comparing two lines in space, viz. their mutual angle, the two conditions determining their plane and their relative length.

This new algebra accordingly could express the relations of space directionally as well as quantitatively, and recommended itself as a powerful organ in solid geometry, dynamical questions involving rotation, spherical conics or surfaces of the second order, besides innumerable applications in physical and astronomical problems, crystallography, electrical dynamics, wherever, in short, there occurs motion or implied translation in tridimensional space, or where the notion of polarity is involved.

In spite of the undoubted power of this 'algebra of pure space' and its trenchant disposal of many classes of physical and geometrical problems, the method has not attracted much attention, except among a few advanced mathematicians. Professor Kelland for several years showed the application of the method to elementary geometry, conics, and some central surfaces of the second-order; but at present none of our universities appear to encourage the study, partly from lack of time to deal adequately with the highest physical applications of mathematical work. There are great difficulties from the use of familiar terms in an extended sense which is frequently difficult of interpretation geometrically. As a whole the method is pronounced by most mathematicians to beneither easy nor attractive, the interpretation being hazy or metaphysical and seldom clear and precise.

As a professor of astronomy Hamilton was not successful, especially in the practical part of his duties, partly perhaps from want of previous training in instrumental and technical work. Some of his professorial lectures, however, were admired for their fluent ornate style, frequently rising into eloquence. From the knowledge of languages which he acquired in youth he was able to read Latin, Greek, German, and Arabic for relaxation, and was frequently seen reading Plato and Kant. He had excellent taste in poetical composition, and wrote many sonnets and other poems. He corresponded with Wordsworth, Coleridge, and Southey, and lived on terms of intimacy with Miss Edgeworth and Mrs. Hemans. He had also an extensive correspondence with Professor De Morgan from 1841 till 1865, the year of his death. A mere 'selection' of the letters occupies 390 pages of the concluding volume of the Rev. R. P. Graves's 'Life of Hamilton.' From his genial and candid disposition and the simplicity of his manners, Hamilton was esteemed both by young and old, not only by those in his home circle, but by all with whom he came in contact.

The second great literary work of Hamilton, 'The Elements of Quaternions,' was published posthumously, edited by his son. William Edwin Hamilton, C.E., in 1866. Besides the previous four years spent in accumulating the material of the 'Elements of Quarternions,' the last two years of the author's life were incessantly occupied in the work of revision, selection, and compression. So devoted indeed was his attention that he is supposed to have seriously injured his health, which had already been affected by a gouty illness, and even his brain-power. Latterly there were also epileptic symptoms. He died on 2 Sept. 1865. The pension of 2001. which he had received since he was knighted was afterwards continued to his widow.

A list of Hamilton's papers, memoirs, and posthumous publications is given in the Rev. R. P. Graves's 'Life' (ut supra), iii. 645-54, followed by a bibliography of quaternions.

[Fraser's Mag. January 1842; Dublin Univ. Mag. January 1842; Proc. R.I.A. November 1865, also iii. 47, ix. 67; Gent. Mag. January 1866; North Brit. Rev. September 1866; R.A.S. Monthly Notices, February 1866, also xxvi. 109; Gent. Mag. September 1869, also xxii. 161; Amer. Journ. Sc. 1866; Webb's Comp. Irish Biogr.; the Rev. R. P. Graves's Life of Sir William Rowan Hamilton, 3 vols.]