Physics World Archive William Rowan Hamilton: mathematical genius David R Wilkins From Physics World August 2005 © IOP Publishing Ltd 2012 ISSN: 0953-8585 Institute of Physics Publishing Bristol and Philadelphia Downloaded on Sun Feb 26 00:21:59 GMT 2012 [37.32.190.226] This year Ireland celebrates the bicentenary of the mathematician William Rowan Hamilton, best remembered for “quaternions” and for his pioneering work on optics and dynamics William Rowan Hamilton: mathematical genius David R Wilkins WILLIAM Rowan Hamilton was born book written by Bartholomew Lloyd, CADEMY in Dublin at midnight between the 3rd A professor of mathematics at Trinity. RISH and 4th of August 1805. His father, I This book awakened Hamilton’s in- OYAL Archibald Hamilton, was a solicitor, R terest in mathematics, and he immedi- and his mother, Sarah (née Hutton), ately set about studying contemporary came from a well-known family of French textbooks and monographs coachbuilders in Dublin. Before his on the subject, including major works third birthday, William was sent to by Lagrange and Laplace. Indeed, he live with his uncle, James Hamilton, found an error in one of Laplace’s who was a clergyman of the Church of proofs, which he rectified with a proof Ireland and curate of Trim, County of his own. He also began to un- Meath. Being in charge of the diocesan dertake his own mathematical re- school there, James was responsible for search. The fruits of his investigations his young nephew’s education. were brought to the attention of John William showed an astonishing ap- Brinkley, then Royal Astronomer of titude for languages, and by the age of Ireland, who encouraged Hamilton five was already making good progress in his studies and gave him a standing in Latin, Greek and Hebrew.Before he invitation to breakfast at the Obser- turned 12, he had broadened his stud- vatory of Trinity College, at Dunsink, ies to include French, Italian, Arabic, whenever he chose. Syriac, Persian and Sanskrit. Of par- ticular interest to him were Semitic Picture of genius – this portrait of William Rowan Optical investigations languages such as Hebrew, Syriac and Hamilton hangs in the Royal Irish Academy. Hamilton achieved first place in the Arabic because of the many early bib- entrance examination for Trinity Col- lical texts written in those languages. Religion was always lege and began his studies there in 1823. His academic pro- important in the lives of Hamilton and his family, although gress as an undergraduate was exceptional. No other student William himself never had any desire to take holy orders. beat him in any examination. He received an “optime” (a dis- As a boy, Hamilton also spent some time studying Indian tinction that was very rarely awarded) in Greek in his first languages because his family thought that there might be a year. Two years later, he received another optime, this time in career opening for him with the East India Company.But as mathematical physics. Nevertheless, he found the intensive he grew older, his uncle ensured that Hamilton concentrated study required to achieve such distinctions increasingly irk- his attention on the classical languages, Latin and Greek, some because it gave him less time to pursue his growing which formed a major portion of the curriculum at Trinity interests in science and mathematics. College, Dublin. At the time, Hamilton was carrying out an intensive study While Hamilton was being prepared for entry to Trinity of Newton’s Principia, and was undertaking highly original College, where he was expected to excel, fellows and professors and sophisticated mathematical investigations in optics. He of the college were setting out to modernize the teaching was employing to the full the methods of calculus and of dif- of mathematics there by introducing French textbooks into ferential geometry that had been developed in France over the curriculum, translating some of these into English, and the previous 50 years. Hamilton initially thought that he was writing their own material. When Hamilton was 16, his uncle the first mathematician to apply such methods to the study of – obviously realizing that William would need to excel in both optical problems. But one day his college tutor took Hamilton mathematics and classics at college – gave him a copy of a text- into the college library and showed him a paper on optics by P HYSICS W ORLD A UGUST 2005 physicsweb.org 33 light takes to travel from one point to another and depends on the co-ordinates of those two points. Hamilton showed that if the form of this characteristic function was known, all signi- ficant optical properties of the system could be expressed in terms of the function and its partial derivatives. Conical refraction When Hamilton presented his third supplement to the Royal Irish Academy in October 1832, he made a surprising pre- diction concerning the way in which certain crystals affect the path of light. Most physical scientists in the early 19th century favoured Isaac Newton’s theory of light as streams of “cor- puscles” travelling in accordance with dynamical laws. They did not accept the rival wave theory of light, which had ori- ginally been advocated by a number of scientists in the 17th century,notably Christiaan Huygens. He had used the wave Hamilton was appointed Royal Astronomer of Ireland at the age of just 22, theory to explain why a ray of unpolarized light entering or which entitled him to live at the Dunsink Observatory near Dublin, where he leaving a “uniaxial” crystal, for example an Iceland spar, is was to remain for the rest of his life. refracted as two rays, each of which is linearly polarized. The wave theory was largely neglected throughout the 18th the French mathematician Etienne-Louis Malus. Dating century because it did not appear to explain phenomena such from 1807, it turned out that Malus had already discovered as polarization. The problem was that it supposed that light these results. waves – like sound waves – are purely longitudinal. However, In 1824, aged just 19, Hamilton submitted a paper on support for the theory was revived at the start of the 19th cen- optics to the Royal Irish Academy, for publication in its tury by Thomas Young, who realized that polarization phe- Transactions. Although the paper was not accepted as it stood, nomena could be best explained if light has both a transverse Hamilton was encouraged to develop and expand on his and a longitudinal component. This inspired a young French ideas and methods. This he did, submitting a substantial scientist and engineer, Augustin Fresnel, to develop an exten- paper to the academy entitled “Theory of systems of rays”. sive theory of light propagation in terms of purely transverse The first part of this paper was published in 1828, the year vibrations of tiny ether particles within each wavefront. after Hamilton graduated. This theory was found to explain the properties of not only Hamilton’s obvious mathematical ability encouraged the uniaxial crystals but also “biaxial” crystals, such as aragonite, college to appoint him to the post of Andrews’ Professor of the optical properties of which had been discovered by Astronomy in 1827, which carried with it the title of Royal David Brewster a few years earlier. These crystals have two Astronomer of Ireland. Hamilton’s home thereafter was at “optic axes”, whereas uniaxial crystals have only one. (Light the observatory at Dunsink, which lies about five miles from travelling down an optic axis may be linearly polarized with the centre of Dublin. Unsuccessful candidates for the post the magnetic field pointing in any direction perpendicular included George Biddell Airy, later to be appointed Astron- to the ray; for light travelling in any other direction, the mag- omer Royal, and a number of experienced fellows of the col- netic field can only point in one of two determined directions, lege. Unfortunately, no astronomical work of any particular which are at right angles to each other and to the ray.) significance was undertaken at the observatory in Hamilton’s In the autumn of 1832 Hamilton undertook a mathemat- time due to antiquated instruments and an inappropriate ob- ical analysis of the wave surface that describes the propaga- serving programme – devised decades earlier – that the Royal tion of light in a biaxial crystal according to Fresnel’s theory. Astronomer was legally required to carry out. This surface represents the position of the light that has pro- Instead, Hamilton focused his talents on his mathematical pagated out from a point source within the crystal in a fixed investigations. In his 1828 paper he had made a detailed study period of time. On the basis of the elegant and general the- of the foci and images that are produced by the reflection of ory of light that he had developed in his third supplement, light off curved mirrors. He studied the aberrations in images Hamilton predicted that the geometrical properties of the produced by reflection, and – using analytical methods that surface should have two surprising physical consequences. he had learnt from his study of Laplace – conducted a thor- First, unpolarized light incident on a biaxial crystal at certain ough investigation into the relative intensity of light along the angles would be refracted to form a hollow cone of rays; this “caustic” surfaces of brightness that are created when light light would then emerge from the crystal in the form of a hol- reflects off a curved mirror. low cylinder. Second, rays of light travelling in certain direc- Hamilton published three supplements to his original tions within the crystal would be refracted on emergence to paper. In the first two, he applied his methods to treat the form a hollow cone of rays.