Hamilton, Sir William Rowan

Life

Like his English contemporaries Thomas Babington Macaulay and John Stuart Mill, Hamilton showed unusual intellect as a child. Before the age of three his parents sent him to live with his father's brother, James, a learned clergyman and schoolmaster at an Anglican school at Trim, a small town near Dublin, where he remained until 1823, when he entered Trinity College, Dublin. Within a few months of his arrival at his uncle's he could read English easily and was advanced in arithmetic; at five he could translate Latin, Greek, and Hebrew and recite Homer, Milton, and Dryden. Before his 12th birthday he had compiled a grammar of Syriac, and by the age of 14 he had sufficient mastery of the Persian language to compose a welcome to the Persian ambassador on his visit to Dublin.

Hamilton became interested in mathematics after a meeting in 1820 with Zerah Colburn, an American who could calculate mentally with astonishing speed. Having read the Eléments d'algèbre of Alexis-Claude Clairaut and Isaac Newton's Principia, Hamilton had immersed himself in the five volumes of Pierre-Simon Laplace's Traité de mécanique céleste (1798-1827; Celestial Mechanics) by the time he was 16. His detection of a flaw in Laplace's reasoning brought him to the attention of John Brinkley, professor of astronomy at Trinity College. When Hamilton was 17, he sent Brinkley, then president of the Royal Irish Academy, an original memoir about geometrical optics. Brinkley, in forwarding the memoir to the Academy, is said to have remarked: "This young man, I do not say will be, but is, the first mathematician of his age."

In 1823 Hamilton entered Trinity College, from which he obtained the highest honours in both classics and mathematics. Meanwhile, he continued his research in optics and in April 1827 submitted his "Theory of Systems of Rays" to the Academy. The paper transformed geometrical optics into a new mathematical science by establishing one uniform method for the solution of all problems in that field. Hamilton started from the principle, originated by the 17th-century French mathematician Pierre de Fermat, that light takes the shortest possible time in going from one point to another, whether the path is straight or is bent by refraction. Hamilton's key idea was to consider the time (or a related quantity called the "action") as a function of the end points between which the light passes and to show that this quantity varied when the coordinates of the end points varied, according to a law that he called the law of varying action. He showed that the entire theory of systems of rays is reducible to the study of this characteristic function.

Shortly after Hamilton submitted his paper and while still an undergraduate, Trinity College elected him to the post of Andrews professor of astronomy and royal astronomer of Ireland, to succeed Brinkley, who had been made a bishop. Thus an undergraduate (not quite 22 years old) became ex officio an examiner of graduates who were candidates for the Bishop Law Prize in mathematics. The electors' object was to provide Hamilton with a research post free from heavy teaching duties. Accordingly, in October 1827 Hamilton took up residence next to Dunsink Observatory, 5 miles (8 km) from Dublin, where he lived for the rest of his life. He proved to be an unsuccessful observer, but large audiences were attracted by the distinctly literary flavour of his lectures on astronomy. Throughout his life Hamilton was attracted to literature and considered the poet William Wordsworth among his friends, although Wordsworth advised him to write mathematics rather than poetry.

Six years after his move to Dunsink, Hamilton married Maria Bayley, daughter of a former rector in County Tipperary, who bore him two sons and a daughter. But his wife was not successful at running the household; as a result, Hamilton never had regular meals and came to rely excessively on alcohol. He would usually work all day in the dining room, and the cook would bring him a mutton chop from time to time. After his death scores of bones were found on plates sandwiched among his papers.

In 1835 Hamilton was the chief local organizer of the British Association for the Advancement of Science meeting in Dublin and at the closing dinner was knighted by the lord lieutenant. Two years later he became president of the Royal Irish Academy. In 1843 he was awarded a Civic List life pension of £200 a year by the British government.

During Hamilton's last illness, an attack of gout, he received with great satisfaction the news that his name had been placed at the head of the first list of Foreign Associates elected by the newly formed National Academy of the United States.

Works

In 1832 a supplement to Hamilton's theory of rays was published. In it he predicted that, as a result of the theory, a wholly unexpected phenomenon would be found in connection with the refraction of light in biaxial crystals, which produce interference figures consisting of two sets of concentric rings when light passes through them. It had been known for some time that certain crystals of this kind, such as topaz, give rise to two refracted rays for each incident ray. The theory of this double refraction had been worked out a few years earlier by Augustin Fresnel. Hamilton found by his general method that under certain conditions a single ray of incident light could actually produce an infinite number of refracted rays in a biaxial crystal and that they would form a cone. Hamilton's prediction of conical refraction, regarded in his lifetime as his most brilliant achievement in optics, was confirmed experimentally within two months by a colleague, Humphrey Lloyd.

Today his unification of optics and dynamics is regarded as far more important than his work on conical refraction. In 1835 his memoir "On a General Method in Dynamics" was published. In it he applied his idea of the characteristic function to the motion of systems of bodies and expressed the equations of motion in a form that revealed the duality between the components of momentum of a dynamical system and the coordinates determining its position. Although Hamilton's canonical equations expressing this duality and his principle that reduces the whole of dynamics to a problem in the calculus of variations have long been familiar to students of dynamics, the deep significance of the duality he discovered was not appreciated for nearly 100 years, until the rise of quantum mechanics.

That same year Hamilton made his famous discovery of quaternions; these ordered sets of four ordinary numbers, satisfying special laws of equality, addition, and multiplication, are useful for studying quantities having magnitude and direction in three-dimensional space. This discovery was a landmark, since it freed algebra from the commutative postulate of multiplication--that the order or sequence of factors does not determine the result. His investigations in algebra had begun 10 years before with a pioneer paper on algebraic couples of numbers in which the basic entity was not a single number but ordered pairs of numbers. Hamilton used this idea to develop a rigorous theory of complex numbers involving the square root of -1. This paper was remarkable as a pioneer attempt to put algebra on an axiomatic basis like geometry. The geometry of complex numbers (i.e., numbers of the form a + bi, in which i is the square root of -1) is that of the two-dimensional vectors in a plane. In attempting to develop analogous technique for three-dimensional space, Hamilton was delayed for many years by a fundamental difficulty that could not be resolved so long as he restricted his attention to "triplets." Suddenly, on Oct. 16, 1843, the solution flashed into his mind as he was walking to Dublin along the Royal Canal: geometrical operations in three-dimensional space require not triplets but "quadruplets." The reason is that, whereas the algebraic couple suffices in the plane because it is equivalent to a multiplier and an angle, in three dimensions the orientation of the plane itself is variable, and this gives rise to two more numbers. Hamilton was so excited by his discovery that as he passed Brougham Bridge he cut the fundamental formulas of quaternions on the stonework: i² = j² = k² = ijk = -1.

Hamilton's discovery was a break with tradition, because it involved the surrender of the commutative law of multiplication--that b times a is the same as a times b. The remaining 22 years of his life were devoted to developing the algebra of quaternions and its applications. This work was published posthumously in 1866 as The Elements of Quaternions. Unfortunately, Hamilton believed that quaternions were ideally suited for the solution of problems in applied mathematics, but it was the simplified version of J. Willard Gibbs, known as vector analysis, that was eventually adopted by mathematical physicists. The value of Hamilton's discovery lay rather in pure mathematics, through its effect on the development of modern abstract algebra.
(G.J.W.)

Bibliography

A full-scale Victorian-style biography, including Hamilton's poems, correspondence, and miscellaneous writings, is Robert Perceval Graves, Life of Sir William Rowan Hamilton, 3 vol. (1882-89, reprinted 1975). Thomas L. Hankins, Sir William Rowan Hamilton (1980), utilizes much unpublished material to relate the mathematician's professional and personal life and includes a useful bibliographic essay.