by Albert C. Lewis
© Oxford University Press 2004 All rights reserved
Hamilton, Sir William Rowan (1805-1865), mathematician, was born at midnight between 3 and 4 August 1805 in Dominick Street, Dublin, the fourth of nine children of Archibald Hamilton (1778-1819), a solicitor, and his wife, Sarah (1780-1817), from the Dublin family of Hutton. He had four sisters who survived infancy, one older than himself, and no brothers. Before he was three years old his parents sent their son to be raised by a curate uncle, James Hamilton, possibly because of the family's financial difficulties. He attended the Church of Ireland diocesan school in Trim, co. Meath, run by his uncle. Both school and home were in Talbot's Castle overlooking the River Boyne where the young Hamilton enjoyed swimming. James Hamilton, a classics graduate of Trinity College, Dublin, soon recognized his nephew's intellectual capabilities and found him a quick and willing learner of Hebrew, Latin, and Greek; by the age of ten he had also gained a facility in Persian, Arabic, Sanskrit, Chaldee, Syriac, Hindustani, Malay, Marathi, and Bengali, as well as the modern European languages. Against a background of political-religious tensions and the agrarian agitation that grew out of the extreme poverty of farm workers, William Hamilton personally enjoyed a comfortable youth in Trim, upon which he looked back with fondness in later life.
Hamilton appears to have been largely self-taught in mathematics. His interest in the subject lagged somewhat behind that in languages, however, and it was not surprising that he was outmatched in 1818 when put against Zerah Colburn, the 'American Calculating Boy' and a year older than Hamilton, at a performance in Dublin. On Colburn's return to Dublin in 1820 Hamilton was better prepared and, what was more important, was given the chance to discuss techniques with him. He took notes at the time on what he learned of Colburn's methods and appears to have been more interested in why they worked than in duplicating them.
Undergraduate life
His uncle guided Hamilton towards entrance in July 1823 into Trinity College, Dublin, with spectacularly high results on the entrance and prize examinations. Tutored by Charles Boyton, a fellow of Trinity and friend of the family, Hamilton began studying the mathematical texts used at the college. He benefited from recent curriculum reforms that, while keeping Isaac Newton's Principia mathematica (1687) at the core, took advantage of the progress made since Newton, especially in France by Pierre Simon Laplace, Joseph Louis Lagrange, Siméon Denis Poisson, and Sylvestre François Lacroix. Throughout his stay at Trinity he achieved the highest standing at every examination in classics and mathematics that he took, but even with the time spent preparing for term and prize examinations, which involved largely rote memorization, Hamilton kept up his swimming and gymnastics. He is described as being broad-chested and medium-sized with light blue eyes and dark brown hair, and always cheerful and alert. He also found time to pursue his favourite creative activities, original mathematical researches and composition of poetry. Some of his most important later mathematical ideas, for example those on the theory of rays, stem from this time. His poetry has not proven as memorable as his mathematics but he always maintained that the two endeavours share the same creative source.
Hamilton recorded his feelings in poetry when his love for the sister of college friends, Catherine Disney, was rejected. Awkwardness and shyness had prevented him from expressing himself directly to her though there were signs that she reciprocated his feelings. In any case, her family appears to have discouraged any advances from him and she became engaged to someone else in 1825. After their respective marriages, he made efforts to keep within her circle by, for example, acting as tutor to her children. Hamilton himself seemed to wish to make this lost love of his life public knowledge since he published his poem about it in the Dublin Literary Gazette and National Magazine in 1830 and showed it to anyone he thought likely to be sympathetic. There may be some correlation between this episode and Hamilton's periods of creativity and depression, but the evidence is sufficiently indirect that no biographer can do more than speculate on any causal relation.
Astronomer royal
During his college years Hamilton prepared for the competitive examinations leading to a fellowship that would allow him to stay on as a teacher at the college. However, before he had the chance to take them, he was appointed Andrews professor of astronomy in June 1827, which entailed his appointment as astronomer royal of Ireland and director of the Dunsink observatory, 5 miles from Dublin. Boyton was largely responsible for emphasizing Hamilton's work in geometrical optics and organizing the support for his appointment to this prestigious position. Though Hamilton had been a visitor at the observatory, greatly impressing the astronomer royal, John Brinkley, with his mathematical work, Brinkley himself made clear his strong doubts about Hamilton's experience, ability, and willingness to carry out the required duties. The post indeed included the day-to-day management of the observatory at Dunsink and ostensibly required making astronomical observations, an activity for which Hamilton showed neither enthusiasm nor great skill. Nevertheless, it provided him with more time for research than the teaching fellowship would have done. It also meant that his four sisters could live with him at the observatory and this proved to be a key to his making a success of this life for they assisted not only in the management of the household, but in making observations and maintaining the astronomical records.
On 9 April 1833 Hamilton married Helen Maria Bayly (1804-1869) from Nenagh in co. Tipperary. There were none of the poetic signs of passion in this relationship that marked Hamilton's earlier love, and descriptions of Helen are sparse. Her arrival displaced the sisters to some extent and signalled a less well-managed household. Friends, and his first biographer, R. P. Graves, in particular, who were concerned with Hamilton's excessive drinking in the 1840s and with a general change into a less cheerful personality, tended to feel that Helen, if not a direct cause, was at least not taking proper care of him. Nevertheless Hamilton's devotion to her comes across very clearly in letters to her when their travels separated them, and especially in his concern whenever she suffered from one of the incapacitating illnesses that plagued her throughout her life. In spite of doubtful health she bore two sons and a daughter.
Hamilton as astronomer royal made no significant direct contributions to astronomy. Instead his reputation from the beginning rested on his genius for devising general and abstract mathematical theories that his contemporaries recognized as of the highest calibre from an aesthetic, if not from a utilitarian, viewpoint. Hamilton's earliest work in 1824 at the age of eighteen, on the theory of systems of light rays, established him as a major contributor to the mainstream of the mathematics of his time. (Mathematics had not yet experienced the bifurcation into the separate spheres of pure and applied that characterized it in the twentieth century.) Though it has not been of such lasting importance as much of his other work, the experimental verification of his prediction of conical refraction of light was regarded in 1833 as spectacularly brilliant science and earned him the royal medal of the Royal Society of London and a knighthood from the lord lieutenant of Ireland in 1835. Hamilton was elected to the presidency of the Royal Irish Academy in November 1837 and conscientiously performed the burdensome administrative duties of the post until 1846. The academy particularly valued his ability to settle the sometimes contentious disputes between members.
Hamilton's mathematics
Modern historians and philosophers have often and deeply investigated how much philosophy and poetry was essentially involved in the creation, presentation, and justification of Hamilton's mathematics, especially of his contention that algebra was properly the science of pure time. It remains, however, an open and intriguing question. What is clear is that Hamilton claimed that mathematics was akin to poetry, sought advice from his friend William Wordsworth, supported the causes of Samuel Taylor Coleridge, and cited Immanuel Kant in his work. In 1834 he expressed one pervasive, if less poetical, theme of his general methods in dynamics:
In the realm of algebra, Hamilton's invention of quaternions produced one of the most famous moments in mathematical history: a revelatory experience of the key algebraic relationships that came to him on 16 October 1843. Several years after the centenary of this discovery a commemorative plaque was placed at De Valera's suggestion near the spot on the footpath under the Brougham Bridge in Dublin where the solution came to him. Hamilton had long been working on how to represent rotations in space in the same way that rotations in the two-dimensional plane are represented by the multiplication of complex numbers. What he jotted down in his notebook on that day were the multiplication rules for the four base units, i, j, k, and 1, for a new number system, the quaternions: i2 = j2 = k2 = ijk = -1. Hamilton's detailed description of the circumstances and of the justification of his discovery has provided substantial material for sociological and philosophical case studies in mathematical discovery. Mathematically it marked a milestone in the development of modern algebra and he spent most of the rest of his life publicizing its implications. He published 109 papers on quaternions alone and, in 1853, a large compilation, Lectures on Quaternions. On his death he left incomplete another, even larger treatise, Elements of Quaternions, which one of his sons saw through the press in 1866. His quaternionic followers did not regard his writing style as the best for an elementary introduction and thus, to make up for this, Charles Jasper Joly, his successor as astronomer royal, edited a second, enlarged edition, of the Elements, published in two volumes in 1899 and 1901.
Hamilton and his followers had hopes that the quaternion would be the standard notation for representing position and motion in space. Hamilton demonstrated at length the advantages of quaternions over the older notations in mechanics, not least the advantage of compactness that enabled a single-line equation, for example, to replace the three or more lines of equations that were used in the standard eighteenth-century works in mechanics of Lagrange and Laplace. The rise of an alternative, vector notation, especially as expounded by the American mathematical physicist Josiah Willard Gibbs, led to a rivalry about the turn of the twentieth century between advocates of the two systems. For two decades the International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics carried on a vigorous but losing campaign through its publications. Though the more versatile vectors won out over quaternions by the 1920s as the generally preferred notation, this was rather a case of Hamilton's discovery not finding its niche than a sign of its ultimate failure, as some feared at the time. The four quaternion components seemed a natural way to express the four-dimensional space-time continuum of relativity theory in the 1920s, but the main contribution quaternions have made is through their decisive influence on the development of a new branch of mathematics in the late nineteenth century, linear algebra.
After bringing forth quaternions Hamilton returned to an earlier interest in the geometry of polyhedra. At first he regarded it as a largely recreational activity and indeed he created for commercial sale the 'Icosian game' based on his work. In the game a player attempts to trace a one-way path connecting a group of cities on a map without repetition and returning to its starting point. To find such paths, now called Hamiltonian cycles, that connect a collection of points, remains a challenging and important problem in computational mathematics.
Other interests
Hamilton took an interest in Irish politics, usually defending the tory views of his uncle James when his Liberal father provoked him into good-natured argument. He remained a political Conservative, a supporter of the British monarchy, and in favour of Irish and British union throughout his life, but at the same time he insisted on Ireland as a separate culture from England. Throughout the religious contentions of Hamilton's time, as the Catholic church rose in influence, he remained a staunch member of the Church of Ireland and adhered fairly constantly to its evangelical wing, though for a time in the 1840s he was attracted by the transcendentalism of the Oxford Movement.
In the summer of 1865 Hamilton suffered severe attacks of gout and on 2 September died at his home in Dunsink from what doctors described as complications arising from the attacks. The funeral and the procession to the burial at Mount Jerome cemetery, Dublin, on 7 September, were attended by members of Trinity College in academic regalia and members of the Royal Irish Academy. Though Hamilton was often the executor for the estates of other members of the Hamilton family he did not excel at these tasks and never gained any appreciable wealth himself. Friends helped his family overcome financial difficulties in his last years. Hamilton's civil-list pension of £200 was divided equally between his wife and daughter.
Though acknowledging that both Hamilton's grandmothers were Scottish, Graves maintained that Hamilton was of Irish descent. An extensive debate took place in journals in 1891 between Graves and the Scottish physicist P. G. Tait, a close colleague of Hamilton, who maintained that Hamilton's paternal grandfather, William Hamilton, went to Dublin from Scotland. Though Tait insisted on defining Hamilton as Scottish in his article for the Encyclopaedia Britannica, there is no documentary evidence to refute or support the family tradition that their branch of Hamiltons first moved to Ireland from Scotland at the time of James I; a century after Hamilton's most active years, he was unequivocally claimed by Ireland. In 1939 De Valera, a student of mathematics as well as the person chiefly responsible for Ireland's independence from England, invoked Hamilton's name in arguing for the founding of the Institute for Advanced Studies in Dublin as someone who was 'known wherever there is a mathematical physicist or theoretical physicist. This is the country of Hamilton, a country of great mathematicians' (Synge, 645).
ALBERT C. LEWIS
Sources
T. L. Hankins, Sir William Rowan Hamilton (1980)
R. P. Graves, Life of Sir William Rowan Hamilton, 3 vols. (1882-9)
W. R. Hamilton, The mathematical papers, ed. A. W. Conway, J. L. Synge, A. J. McConnell, H. Halberstam, and R. E. Ingram, 3 vols. (1931-67)
M. J. Crowe, A history of vector analysis: the evolution of the idea of a vectorial system, 2nd edn (1985)
S. O'Donnell, William Rowan Hamilton: portrait of a prodigy (1983)
J. L. Synge, Memoirs FRS, 22 (1976), 635-53 [obit. of E. de Valera]
Archives
TCD, corresp. and papers | BL, corresp. with Sir Robert Peel, Add. MSS 40535, 40547, 40599
CUL, letters to Lord Kelvin
RS, letters to Sir John Lubbock
Trinity Cam., letters to William Whewell
University of Limerick Library, corresp. with Lord Dunraven
Likenesses
T. Kirk, bust, before 1830, TCD
T. Kirk, marble bust, 1830, Dunraven-Limerick Estates Co.
T. Farrell, miniature bust, 1833, repro. in Graves, Life
photograph, c.1845 (with his son W. E. Hamilton), TCD [see illus.]
S. Purser, portrait, 1862 (after photograph, 1857), Royal Irish Acad.
J. H. Foley, marble bust, 1867 (after T. Kirk), TCD
C. Grey, etching, repro. in Dublin University Magazine, 19 (1842), facing p. 94
J. Kirkwood, lithograph (after C. Grey), NPG
Wealth at death
£4000: probate, 30 Oct 1865, CGPLA Eng. & Wales
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