Sir William Rowan Hamilton

Biography

William Rowan Hamilton's mathematical career included the study of geometrical optics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley-Hamilton Theorem).

Early life

A child prodigy, Hamilton was born the son of Archibald Hamilton, a solicitor, in Dublin at 36 Dominick Street, but was later put up for adoption.

Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College, Dublin, where he spent his life.

Mathematical studies

Hamilton's mathematical studies seem to have been undertaken and carried to their full development without any assistance whatsoever, and the result is that his writings belong to no particular "school", unless indeed we consider them to form, as they are well entitled to do, a school by themselves. As an arithmetical calculator Hamilton was not only an expert, but he seems to have occasionally found a positive experience in working out to an enormous number of places of decimals the result of some irksome calculation. At the age of twelve Hamilton engaged Zerah Colburn, the American "calculating boy", who was then being exhibited as a curiosity in Dublin, and he had not always the worst of the encounter. Hamilton soon commenced to read the Principia, and at sixteen Hamilton had mastered a great part of that work, besides some more modern works on analytical geometry and the differential calculus.

About this period Hamilton was also engaged in preparation for entrance at Trinity College, Dublin, and had therefore to devote a portion of time to classics. Nothing could be better fitted to call forth such mathematical powers as those of Hamilton; From that time Hamilton appears to have devoted himself almost wholly to the mathematics investigation, though he ever kept himself well acquainted with the progress of science both in Britain and abroad. Hamilton detected an important defect in one of Laplace’s demonstrations, and he was induced by a friend to write out his remarks, that they might be shown to Dr John Brinkley, then the first Astronomer Royal for Ireland, and an accomplished mathematician. but Hamilton was expected to win both the gold medals at the degree examination, had his career as a student not been cut short by an unprecedented event. The chair was not exactly offered to him, as has been sometimes asserted, but the electors, having met and talked over the subject, authorized one of their number, who was Hamilton's personal friend, to urge Hamilton to become a candidate, a step which Hamilton's modesty had prevented him from taking. Thus, when barely twenty-two, Hamilton was established at the Dunsink Observatory, near Dublin.

Hamilton was not specially fitted for the post, for although he had a profound acquaintance with theoretical astronomy, he had paid but little attention to the regular work of the practical astronomer. Hamilton was intended by the university authorities who elected him to the professorship of astronomy to spend his time as Hamilton best could for the advancement of science, without being tied down to any particular branch. If Hamilton devoted himself to practical astronomy, the University of Dublin would assuredly have furnished him with instruments and an adequate staff of assistants. These are the few salient points (other, of course, than the epochs of Hamilton's more important discoveries and inventions presently to be considered) in the uneventful life of Hamilton. Hamilton's papers on optics and dynamics demonstrated theoretical dynamics being treated as a branch of pure mathematics. Hamilton's first discovery was contained in one of those early papers which in 1823 Hamilton communicated to Dr Brinkley, by whom, under the title of “Caustics,” it was presented in 1824 to the Royal Irish Academy. Hamilton himself seems not until this period to have fully understood either the nature or importance of optics, as later Hamilton had intentions of applying his method to dynamics.

In 1827, Hamilton presented a theory that provided a single function that brings together mechanics, optics and mathematics. and it is, indeed, that the one particular result of this theory which, perhaps more than anything else that Hamilton has done, something which should have been easily within the reach of Augustin Fresnel and others for many years before, and in no way required Hamilton’s new conceptions or methods, although it was by Hamilton’s new theoretical dynamics that he was led to its discovery. Jacobi and other mathematicians have extended Hamilton's processes, and have thus made extensive additions to our knowledge of differential equations. It is characteristic of most of Hamilton's, as of nearly all great discoveries, that even their indirect consequences are of high value.

Quaternions

The other great contribution made by Hamilton to mathematical science was his discovery of quaternions in 1843.

Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. Hamilton could not do so for 3 dimensions: in fact later mathematicians showed that this would be impossible. Eventually Hamilton tried 4 dimensions and created quaternions. According to the story Hamilton told, on October 16 Hamilton was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

i = k2 = ijk = − 1

suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge.) Since 1989, the National University of Ireland, Maynooth has organized a pilgrimage, where mathematicians take a walk from Dunsink observatory to the bridge where, unfortunately, no trace of the carving remains, though a stone plaque does commemorate the discovery. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the 'scalar' part, and the remaining three as the 'vector' part. In 1852, Hamilton introduced quaternions as a method of analysis. Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. Peter Guthrie Tait among others, advocated the use of Hamilton's quaternions. Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (from developers like Oliver Heaviside and Willard Gibbs), because quaternions provide superior notation. Hamilton also contributed an alternative formulation of the mathematical theory of classical mechanics. Both the Lagrangian and Hamiltonian approaches were developed to describe the motion of discrete systems, were then extended to continuous systems and in this form can be used to define vector fields.

Other originality

Hamilton originally matured his ideas before putting pen to paper. But not to speak of his enormous collection of books, full to overflowing with new and original matter, which have been handed over to Trinity College, Dublin, the previous mentioned works barely form the greater portion of what Hamilton has published.

Hamilton's extraordinary investigations connected with the solution of algebraic equations of the fifth degree, and his examination of the results arrived at by N. There is next Hamilton's paper on Fluctuating Functions, a subject which, since the time of Joseph Fourier, has been of immense and ever increasing value in physical applications of mathematics.

Besides all this, Hamilton was a voluminous correspondent. Hamilton was ever courteous and kind in answering applications for assistance in the study of his works, even when his compliance must have cost him much time.

Death and afterwards

Hamilton retained his faculties unimpaired to the very last, and steadily continued till within a day or two of his death, which occurred on 2 September 1865, the task of finishing the “Elements of Quaternions” which had occupied the last six years of his life.

Hamilton is recognized as one of Ireland's leading scientists and, as Ireland becomes more aware of its scientific heritage, he is increasingly celebrated. There is a research institute named for him at NUI Maynooth and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann, Andrew Wiles and Timothy Gowers have all spoken. 2005 was the 200th anniversary of Hamilton's birth and the Irish government designated that the Hamilton Year, celebrating Irish science. Trinity College Dublin marked the year by launching the Hamilton Mathematics Institute TCD, a mathematics institute modelled on, for example, the Isaac Newton Institute in Cambridge.

Commemorations of Hamilton

External links, references, and resources

Publications