The Original Quaternion

Why not register for our monthly updates?

Portrait of Sir William Rowan Hamilton. Courtesy: Royal Irish Academy. At midnight between the 3-4 August 1805 in Dominick Street, Dublin, the wife of a minor lawyer and land agent, Archibald Hamilton, gave birth to a boy. His mother, Sarah Hutton, was of Huguenot extraction and came from a well-known Dublin family of coach builders. His father was a close friend (and possibly the natural son) of the United Irishman, Archibald Hamilton Rowan, after whom the boy was named William Rowan Hamilton. Two hundred years on, 2005 has been declared ‘Hamilton Year’ to mark the bicentenary of his birth and celebrate Irish Science. But who was Hamilton and why does he merit this commemoration? Hamilton was certainly the greatest mathematician, and arguably the greatest scientist, Ireland has produced to date. Yet his name is little known to the general public and his achievements are unfamiliar even to specialists. One of the aims of the Hamilton Year is to correct this. The more important aim, however, is to raise awareness of Ireland’s rich scientific heritage and of the exciting science being done in Ireland today. While almost everyone is aware of Ireland’s contributions to literature, the many contributions of Irish men and women to the advancement of science have been shamefully neglected. Hamilton first came to fame while still an undergraduate at Trinity College Dublin with a remarkable paper on “Systems of Rays” which he published in the Proceedings of the Royal Irish Academy. He won virtually all available prizes and distinctions and, on the basis of his evident ability and promise, was pushed by his tutor, Charles Boyton, as a possible successor to Brinkley as Andrews’ Professor of Astronomy and Royal Astronomer of Ireland. Brinkley, a very capable astronomer, was (as were most of the academic staff of Trinity in those days) a Church of Ireland clergyman and had just been appointed Bishop of Cloyne in 1826. The resulting vacancy was hotly contested, the strongest candidate being Airy, at that point Lucasian Professor of Mathematics in Cambridge, who subsequently went on to become Astronomer Royal of England. However, Airy demanded a salary of at least £500 a year, which the College was not prepared to pay, and Hamilton was appointed on £300 a year in June 1827 while still an undergraduate! Hamilton then spent the rest of his working life as Professor of Astronomy in Dunsink Observatory until his death in 1865. Although he did his best, at least initially, to fulfill the observational duties attached to the post, Hamilton was an indifferent astronomer and his claim to fame rests solely on his contributions to mathematics and theoretical physics. Here, however, his reputation is secure and rests in particular on two major advances in totally unrelated fields, dynamical systems and pure algebra. The first grew out of his early work on optics. In his “Systems of Rays” Hamilton developed a remarkable mathematical description of all possible optical systems, at least as described by geometrical optics, starting from a variational principle and introducing his “characteristic function”. Basically, what Hamilton showed was the very remarkable result that the entire optical system, and all solutions for paths of light rays through the system, could be completely described by one characteristic function. Actually calculating this function for a real system can be very difficult, but the mere existence of such a function, and the fact that it satisfies certain equations, allows one to state many general results about all optical systems. Hamilton developed this general theory in the context of geometrical optics, but, as he was well aware, it could also be applied to mechanics, where again the classical dynamics of very general systems can be shown to obey a variational principle. In his two great papers on his “General method in dynamics” he develops this idea, first establishing the proper form of the variational principle in terms of what we now call the classical action, then showing that all such systems can be written in a particularly simple canonical (or Hamiltonian) form and finally demonstrating the existence of a principal function from which all the solutions of the dynamical equations can be derived. This work formed the basis for all subsequent work in theoretical mechanics and, remarkably, was found to be precisely the tool needed in the early days of quantum mechanics. The Hamiltonian function survives in modern quantum mechanics as the Hamiltonian operator and the first thing a theoretical physicist usually does when solving a problem is to write down the Hamiltonian. Hamilton’s second great achievement was in the area of abstract algebra. Unlike most of his contemporaries he believed passionately that mathematics, and indeed all science, had to have a philosophical or metaphysical foundation which in his case he found in the works of Kant and Coleridge. In particular he convinced himself that just as geometry, in Kant’s philosophy, derived from our innate perception of space and was the science of pure space, so algebra should be the science of pure time. Whatever the merits of this idea (and even Hamilton was eventually forced to admit that it did not really work) it did enable him to contemplate the possibility of an algebra where multiplication was non-commutative, that is one in which the order in which factors are multiplied matters so that a times b does not equal b times a. This was the crucial element needed for his discovery of quaternions, the inspiration for which famously came to him while walking along the banks of the Royal Canal on his way into Dublin from Dunsink. He immediately scratched the fundamental formulae of quaternion multiplication onto a stone of Broome Bridge and, although his original act of vandalism has long decayed away, a plaque now marks the spot and is the scene of an annual pilgrimage by mathematicians. sensible@screaming.net Hamilton devoted the rest of his life to developing the theory of quaternions, but though beautiful and occasionally useful they have never achieved the importance Hamilton anticipated. (They are used extensively in computer animation software and in the software that controls spacecraft pointing because they give a very efficient and robust method of describing general rotations.) The real importance of his discovery was that it opened peoples’ minds to the possibility of more general algebraic structures and lead within a decade to an explosion of new and important ideas. The modern vector calculus, used by every physicist and engineer, also derives directly from quaternions (the very terms vector and scalar were invented by Hamilton). A mark of the international respect Hamilton enjoyed during his own lifetime was his election as the American National Academy of Sciences’ first foreign associate on 9 January 1865. The honour implied the Academy considered him to be the greatest living non-American scientist. He received the news of this honour shortly before his death in Dunsink on 2 September 1865. Hamilton as a person was deeply romantic and idealistic. As a young man he fell in love with a young girl, Catherine Disney, and continued to worship her all his life even after she was married to another man and Hamilton himself had entered into a not very satisfactory marriage to Helen Bayley. He devoted a lot of time to poetry, and had to be persuaded by no less an authority than Wordsworth that while his poetry was competent, his real strength was in mathematics and he should concentrate on the latter. In many ways this is part of his attraction. He was a genius, but also a human being with very human failings. He had a definite fondness for the bottle, and there is at least one embarrassing episode on record where he got very drunk in public, but descriptions of him as a hopeless alcoholic are clearly exaggerated. He was intensely ambitious and his aim in life was always to make a great original contribution to science, both for his own fame and that of Ireland. In this he succeeded, even if not quite as he anticipated. It is very appropriate, therefore, that we use the occasion of the bicentenary of his birth to reflect on the role of science in Irish society. For further information on events celebrating the bicentenary – www.hamilton2005.ie

© Copyright 2006 by theword.ie

Top of Page