Who is Hamilton?
William Rowan Hamilton was born in 1805 and is universally recognised as the greatest mathematician, and arguably the greatest scientist, that Ireland has produced to date.
He came to early fame for his work on optics, including his remarkable prediction of the phenomenon of conical refraction, but his lasting fame is for his work on the fundamentals of mechanics. Hamilton’s canonical form of the equations of motion, and his emphasis on the importance of variational principles, remain at the heart of theoretical physics and were essential bridges from the world of classical physics to the modern quantum mechanics.
On October 16th 1843 Hamilton’s discovery of quaternions, famously in a moment of inspiration at Broom Bridge, was important in the development of modern abstract algebra and the quaternions remain useful in calculating rotations of solid bodies and are thus important in satellite navigation and video game programming. He states that he pulled out his penknife and scratched on the stone of the bridge the fundamental defining relations for the quaternion algebra. It is annually commemorated by the mathematical community who re-enact the walk and celebrate this flash of creative inspiration at Broom Bridge every 16th October, now called Hamilton Day.
Sir William Rowan Hamilton
by David Spearman, Dictionary of Irish Biography
Sir William Rowan Hamilton (1805–1865), mathematician, was born on the night of 3–4 August 1805 (the time was recorded as midnight) at his father's house in Dominick Street, Dublin. He was the second of five children who survived infancy, and the only son, of Archibald Hamilton (1778–1819) and his wife, Sarah (1780–1817).
Background and education
William's father Archibald was the son of an apothecary; he had served his apprenticeship in an attorney's office and now conducted his own modest legal practice. Both Archibald and his father William before him were freemen of the city of Dublin and his great-uncle Francis Hamilton had been an alderman. William's mother, Sarah, came of a Dublin family called Hutton, who owned a coach-building firm.
At age three William was sent to live under the tutelage of his uncle James, who ran the diocesan school at Trim in Co. Meath. He stayed there, apart from short vacation visits to his family, until he entered Trinity College some fifteen years later. William appears to have been quite happy at Trim. James Hamilton was a classicist with some knowledge of oriental languages; he recognised his nephew's precocious talent and fed him an extraordinary diet of the classics, Hebrew, and a wide range of oriental and modern languages. He was quite a taskmaster, albeit a kindly and supportive one, and his nephew responded positively.
James Hamilton observed William's remarkable computational skill but he was not competent, and probably not inclined, to encourage his mathematical bent. He did, however, produce for him a copy of Analytical geometry by Bartholomew Lloyd, which was to have a decisive effect. William was then sixteen, and his eyes were opened. ‘Ill-omened gift!’ he wrote, ‘It was the commencement of my present course of mathematical reading, which has in so great a degree withdrawn my attention, I may say my affection, from the Classics’ (letter to his cousin Arthur Hamilton, 1822, Graves, i, 112).
Hamilton entered TCD in 1823. He was particularly fortunate that in the decade immediately preceding his entry the teaching of mathematics within the college had been altered almost beyond recognition: new methods from France and Germany were brought into the curriculum, French texts were introduced, and new textbooks were written in English expounding recent continental work. The chief architect of this reform was Bartholomew Lloyd, who was professor of mathematics from 1813 to 1822, then professor of natural philosophy till he was elected provost in 1831. There is no indication that the discipline of regular examinations was harmful to Hamilton, the revised syllabus was reasonably stimulating even to one of such prodigious abilities, and he had every opportunity and encouragement to read outside the course. His tutor, Charles Boyton, was a widely read and intelligent man, a competent mathematician, who, though he did not produce original work himself, could encourage and guide his brilliant student and appears generally to have been an excellent tutor.
Optics and astronomy
Hamilton's earliest attempts at original work were in geometry. His studies of the work of the French geometer Gaspard Monge on families of surfaces and their normals suggested a new approach to optics based on systems of rays treated mathematically as rectilinear congruences. Within a year of entering college Hamilton had submitted his first paper, describing this work, to the RIA. It was referred back with the recommendation that he should develop his ideas further before resubmitting. He did so, adding substantially to what he had already written, and in 1827 his first famous paper ‘Theory of systems of rays’, was read before the academy. His achievement did not pass unrecognised: John Brinkley, the Andrews professor of astronomy, had lately been appointed to the bishopric of Cloyne and the youthful Hamilton, before he had even graduated, was elected to the Andrews chair. This appointment was remarkable, as the new incumbent had almost no practical experience in astronomical observation and the potential field for the job included some able men of established reputation. Hamilton's appointment to this post, which since 1792 had carried the title of ‘royal astronomer of Ireland’, was profoundly significant in the extent to which it shaped his subsequent career.
Fortunately the new royal astronomer did not feel constrained to devote himself unduly to astronomy, though he continued for some time with his important and highly original work in optics. He was, nonetheless, conscious of his responsibility to carry out observations and measurements, and to process and analyse these. Immediately after his appointment he went to stay with Brinkley at Cloyne, receiving valuable advice and guidance; he then accepted the invitation of Thomas Romney Robinson to spend some time with him at the Armagh observatory to gain practical experience. This helped him to assume responsibility for the round of routine observations, which were made with the help of his assistant and his own two sisters who lived with him at the observatory. His position, with a house at Dunsink (a few miles out of Dublin) and a secure if not over-generous salary, was less demanding than a tutorial fellowship, which he would otherwise have held.
The field of optics, to which Hamilton still devoted his main efforts, had recently moved back into the centre of the scientific stage, following the discovery of the phenomenon of interference by Thomas Young, and subsequently the publication of two highly significant memoirs by Augustin Jean Fresnel in the early 1820s. For over a century the conflict between the wave and corpuscular theories of light had remained unresolved. Sir Isaac Newton had been led to reject the wave theory because the type of wave that would be required was transverse rather than longitudinal and he could not conceive of a mechanism whereby transverse light waves could propagate. He concluded that light must be corpuscular in nature. Young's demonstration of the phenomenon of interference for light pointed strongly towards a wave interpretation, as interference effects were already familiar features of wave systems – in sound, for example, or in water waves.
Among the various observed phenomena that any theory of light would have to encompass was that of double refraction, which had been discovered in 1669 by Bartholomeus. Certain crystals, in particular that known as ‘Iceland spar’, gave a double image: a single ray of light entering the crystal produced two refracted rays, and the phenomenon was thus described as ‘double refraction’. Christiaan Huygens, as early as 1690, had invented an elegant and clever procedure for describing wave propagation, which could also allow for the simpler cases of double refraction, though the more complicated phenomenon in so-called ‘biaxial crystals’ could not be described by that method. Fresnel's remarkable achievement was to devise a model for the propagation of transverse light waves in crystals, which led to Huygens's construction where that was applicable but which could also describe the more complicated phenomena involving biaxial crystals. Fresnel's method was mathematically quite involved: it led to wave surfaces that were geometrically rather complicated and whose properties were by no means immediately obvious from their equations.
Hamilton knew and was deeply interested in Fresnel's results. By a stroke of luck or genius or a combination of the two, he noticed a remarkable feature of the wave surface: for a particular direction of the incident ray on a biaxial crystal, instead of double refraction each incident ray gave rise to a complete cone of refracted rays. Hamilton immediately described his discovery to his Trinity colleague, Humphrey Lloyd (son of Bartholomew Lloyd), the recently appointed professor of natural philosophy, and suggested that he should perform the experiment to see if this new and quite unexpected phenomenon did in fact occur. If it did it would strikingly corroborate the Fresnel theory and, by implication, the wave theory of light. Although Lloyd's background and training were in mathematics, he had become interested in experiment and it was as an experimental physicist working in optics and in geomagnetism that he was to achieve distinction. The experiment proposed by Hamilton was not an easy one – the effect was easily obscured and the quality of the crystals available to Lloyd was relatively poor – so the fact that his attempts had a positive outcome was a tribute to Lloyd's considerable skill, as well as to his patience and persistence.
The observation of conical refraction was generally seen as a powerful confirmation of the wave theory, the evidence for which was by now hard to reject, despite reservations about the hypotheses underlying Fresnel's theory and the problems posed by the newly observed absorption phenomena. It was a remarkable discovery, a triumph for Hamilton in particular, and one of the classic vindications of the scientific method. Fresnel's theory had been constructed to accommodate known experimental results; from it, following quite elaborate mathematical argument, a new prediction was extracted, which could not have been anticipated and in fact must have seemed rather improbable. Recognition followed swiftly. At the 1834 meeting of the British Association in Edinburgh, Lloyd was invited to give the main review talk on physical optics, and in 1836 he was elected to the Royal Society. In 1835, the year following the Edinburgh meeting at which Lloyd presented his report, the British Association met in Dublin and on this occasion Hamilton, who acted as local secretary and organiser of the meeting, now at the age of thirty a famous man, was knighted by the lord lieutenant.
Dynamics and algebra
Hamilton's interest now moved from optics to dynamics. In fact this did not represent a total change of direction, for his approach, described in his Essays on a general method in dynamics, closely paralleled his approach to optics. This parallelism between optics and dynamics, emphasised by Hamilton, was to become particularly significant in the twentieth century with the introduction of wave mechanics; Hamilton's work on dynamics is now widely regarded as his most important contribution, though at the time the Essays did not attract a great deal of interest. Only in the twentieth century did the power and generality of the Hamiltonian methods come to be appreciated. Perhaps the most important influence was that on Erwin Schrödinger, who received a thorough grounding in Hamiltonian dynamics from his professor, Friedrich Hasenöhrl, who, in turn, was a student of Arnold Sommerfeld. Schrödinger gave the Hamiltonian formulation a central role in his construction of quantum mechanics.
From the mid-1830s until his death in 1865, Hamilton's preoccupation was with algebraic questions; his culminating achievement, which in his own view outweighed all his other work, was the discovery of quaternions in 1843. Hamilton believed that quaternions would provide a valuable key for the understanding of the physical world. This belief has not been vindicated: quaternions have made little direct contribution to the development of physics, though they have found extremely useful applications in the control of spacecraft and in three-dimensional computer modelling (as, for example, in video games). The profound and lasting influence of Hamilton's work in algebra, which began with his description of complex numbers as number pairs, and was followed by the discovery of quaternions and the recognition of their non-commutativity, was the stimulus it gave to the development of algebra as an abstract axiomatic discipline.
There is some irony in this. Hamilton was led towards quaternions by a deep metaphysical motivation, based particularly on his reading of Kant, to whose work his friend Coleridge directed him. Hamilton grappled with the German original of the Critique of pure reason, and like his literary friends discovered a natural sympathy for Kant's idealism. But he also found in the Critique technical answers to the questions which he regarded as fundamental to his mathematics. Wanting an intuitive interpretation for the objects of mathematics, he could not accept a mathematics in which the basic elements were merely symbols. Following Kant, he gave meaning to the number system through the intuition of pure time. From this starting-point, which he regarded as fundamental, he constructed the real numbers, then the complex numbers as pairs of real numbers, and then quaternions. Hamilton's outlook parallels exactly that of L. E. J. Brouwer and the intuitionists of the twentieth century, whose constructive approach to mathematics contrasts with the emphasis on consistency alone which marks the more widely held formalist viewpoint. From a historical perspective it is interesting that it was Hamilton's intuitionist approach that led to the radical idea of a non-commutative algebra. The formalist outlook of his Cambridge contemporaries could in principle permit non-commutativity without any difficulty, but until Hamilton constructed the algebra of quaternions the old laws, derived from the real number system, were held to be sacrosanct.
Hamilton's own account of his discovery of quaternions, the resolution of the problem that had thwarted and frustrated him for over a decade, is well known. It is an authentic and compelling description of the moment of discovery. It was 16 October 1843 and he was walking from Dunsink into Dublin to attend a meeting at the academy. ‘I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; . . . I felt a problem to have been at that moment solved – an intellectual want relieved – which had haunted me for at least fifteen years before’ (letter to P. G. Tait, 1858, Graves, ii, 435–6). Hamilton straightway recorded the equations that give the multiplication law for his so-called quaternions in his pocket book: ‘nor could I resist’, he wrote, ‘the impulse – unphilosophical as it may have been – to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula’ (letter to his son Archibald, 1865, Graves, ii, 435).
Marriage, friends, and character
Hamilton's personal life did not match the brilliant success of his academic and professional career. He suffered unrequited love for Catherine Disney, the sister of student friends, which was thwarted by her enforced and unhappy marriage to the Rev. William Barlow; as she lay dying, Hamilton, on his knees, offered her his book Lectures on quaternions. In 1833 he married Helen Maria Bayly (1804–69), to whom he seems to have been devoted, though the marriage was difficult and Helen, who was subject to frequent illnesses and was often absent from their home, was unable to provide an orderly domestic regime for her husband. They were plagued by periodic financial anxieties and Hamilton was troubled by excessive drinking during some periods of his life. But he had many close friends in whom he could confide and with whom he maintained a lively correspondence. Apart from his academic colleagues these included the Dunravens, whose son Lord Adare had been his pupil and lived for a time with him in Dunsink, and the de Veres who lived in remote and romantic Curragh Chase. The novelist Maria Edgeworth and her family were also Hamilton's close friends. Another friend was Lady Wilde, or Speranza as she was known, who invited him to be godfather to her son Oscar, describing him as her ‘little pagan’, which may have been what prompted Hamilton, for whom the responsibility of a godparent would not have been taken lightly, to decline.
Hamilton was strongly influenced by the Romantic movement, which accorded with his natural temperament. Wordsworth, whom he visited on several occasions and who came to stay at Dunsink, was a much admired friend, as was Coleridge, who introduced him to the philosophy of Kant. As this influence suggests, Hamilton's science was much affected by his philosophical views and literary connections; at first strongly Berkeleian – not surprising in one connected with Trinity College at that time – his ideas and motives were later influenced by the Kantian perception of algebra as the science of pure time. Poetry was a continuing passion for him, and although Wordsworth gently persuaded him that his vocation was mathematics and not poetry he still wrote verse prolifically, finding an outlet for his deeper sentiments particularly in sonnet form. He saw mathematics as poetic in nature and judged the work of his contemporaries in terms of poetical quality. Among the French mathematicians Fourier commanded his particular respect: He wrote: ‘Fourier was a true poet in mathematics, and in the applications of mathematical science to nature (especially to the theory of heat). So was (though not Laplace) Lagrange, to whose memory I consider as having inscribed those essays on a general method in dynamics’ (letter to John Nichol, 1855, Graves, iii, 48).
Politically Hamilton might be described as a moderate conservative. He was a committed member of the anglican Church of Ireland and a firm supporter of the established order. He certainly did not espouse the radical views that, to some extent, his father shared with his godfather, Archibald Rowan. In 1852 he wrote to the English mathematician Augustus De Morgan: ‘my uncle, the Rev. James Hamilton, was a Tory to the backbone, and doubtless taught me Toryism along with Church of Englandism, Hebrew and Sanskrit. My father used to enjoy provoking me into some political or other argument, in which I always took my uncle's side’ (letter to Augustus de Morgan, 1852, Graves, iii, 392). But although Hamilton did not share his godfather's political outlook, and although he firmly supported the union, he was nonetheless a patriotic Irishman who loved his country and was loyal to, and took pride in, its institutions.
Hamilton was president of the RIA (1837–46). He was deeply committed to the academy, attached considerable significance to his responsibilities as president, and worked assiduously to promote its interests. He received many honours. His knighthood was conferred in 1835. The year before, he was awarded the Cunningham medal of the academy and the royal medal of the Royal Society, both of these in recognition of his work in optics and in particular the discovery of conical refraction. He was an honorary member of many learned societies though (extraordinarily and for some reason not fully understood), he was not a fellow of the Royal Society. Perhaps the best indication of his international standing is the decision of the newly founded National Academy of Sciences in the USA to place his name first on the list of distinguished scientists to be elected foreign associates.
Death and reputation
Hamilton died 2 September 1865 at his home at Dunsink, aged sixty, and was buried at Mount Jerome cemetery. His wife died four years later. There were two sons of their marriage, William Edwin and Archibald, and one daughter, Helen, who married Archdeacon John O'Regan (d. 1898) and whose son, John, was Hamilton's only grandchild. At the time of his death Hamilton's reputation was strongly linked with quaternions. Despite the importance of his contributions to algebra and to optics, posterity accords him greatest fame for his dynamics. The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated. The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this the most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting-point for calculation in almost any area of physics.
Hamilton's manuscript notebooks and miscellaneous papers are in Trinity College Library. A portrait by Sarah Purser hangs in the RIA and there is a portrait bust by J. H. Foley in the Long Room, Trinity College Library.
Sources: Robert Perceval Graves, Life of Sir William Rowan Hamilton (3 vols, 1882–9); Thomas L. Hankins, Sir William Rowan Hamilton (1980); Sean O'Donnell, William Rowan Hamilton: portrait of a prodigy (1983); David R. Wilkins (ed.), Perplexingly easy: selected correspondence between William Rowan Hamilton and Peter Guthrie Tait (2005); William Rowan Hamilton, The mathematical papers (4 vols, 1931–2000); ODNB
Reproduced with thanks to the Dictionary of Irish Biography