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## William Rowan Hamilton
William Rowan Hamilton
Sir
Dr John Brinkley, bishop of Cloyne, is said to have remarked in 1823 of Hamilton at the age of eighteen: “
William Rowan Hamilton's mathematical 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
## Biography## Early LifeHamilton was born in Dublin at 36 Dominick Street. Hamilton showed himself to be a child prodigy. Hamilton was the son of Archibald Hamilton, a solicitor. A branch of the Scottish family to which they belonged had settled in the north of Ireland in the time of James I, and this fact seems to have given rise to the common impression that Hamilton was scottish. Hamilton was educated by James Hamilton (curate of Trim), his uncle and a Anglican priest. Hamilton's genius first displayed itself in the form of a power of acquiring languages. At the age of seven he had already made very considerable progress in Hebrew, and before he was thirteen he had acquired, under the care of his uncle, who was an linguist, almost as many languages as he had years of age. Among these, besides the classical and the modern European languages, were included Persian, Arabic, Hindustani, Sanskrit, and even Malay. But though to the very end of his life he retained much of the singular learning of his childhood and youth, often reading Persian and Arabic in the intervals of sterner pursuits, he had long abandoned them as a study, and employed them merely as a relaxation. Hamilton was part of a small brilliant school of mathematicians associated with Trinity College, Dublin, where he spent his life. He studied both classics and science, and was appointed Professor of Astronomy in 1827, even before he graduated. ## Mathematical studies
Hamilton's mathematicalal studies seem to have been undertaken and carried to their full development without any assistance whatever, and the result is that his writings belong to no particular “
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. In the summer of 1822, in his seventeenth year, he began a systematic study of Laplace’s
It was in the successful effort to open this treasure-house that Hamilton’s mind received its final temper, “ Hamilton’s career at College was perhaps unexampled. Amongst a number of competitors of more than ordinary merit, he was first in every subject and at every examination. He achieved the rare distinction of obtaining an optime for both Greek and for physics. The amount of many more such honours Hamilton might have attained it is impossible to say; 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. This was Hamilton’s appointment to the Andrews professorship of astronomy in the university of Dublin, vacated by Dr Brinkley in 1827. 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 Observatory, Dunsink, 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. And it must be said that Hamilton’s time was better employed in original investigations than it would have been had he spent it in observations made even with the best of instruments. 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. In 1835, being secretary to the meeting of the British Association which was held that year in Dublin, he was knighted by the lord-lieutenant. But far higher honours rapidly succeeded, among which his election in 1837 to the president’s chair in the Royal Irish Academy, and the rare distinction of being made corresponding member of the academy of St Petersburg. 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. ## Optics and Dynamics
He made important contributions to optics and to dynamics. 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 “
Hamilton himself seems not till this period to have fully understood either the nature or importance of optics, as later Hamilton had intentions of applying his method to dynamics. The Royal Irish Academy paper was finally entitled “
The principle of “
The step from optics to dynamics in the application of the method of “ And though differential equations, optics and theoretical dynamics of course are favored in which any such contribution to science can be looked at, the other must not be despised. It is characteristic of most of Hamilton’s, as of nearly all great discoveries, that even their indirect consequences are of high value. ## QuaternionsThe other great contribution made by Hamilton to mathematical science, the invention of Quaternions, is treated under that heading. The following characteristic extract from a letter shows Hamilton’s own opinion of his mathematical work, and also gives a hint of the devices which he employed to render written language as expressive as actual speech. Hamilton discovered quaternions in 1843. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a plane) to higher spatial dimensions. Hamilton could not do so for 3 dimensions, but 4 dimensions produce 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^{2} = j^{2} = k^{2} = ijk = -1 suddenly occurred to him; Hamilton then promptly carved this equation into the side of the nearby Brougham Bridge (now called Broom Bridge). Since 1989, the National University of Ireland, Maynooth has organized a pilgrimage, where mathematicians (including Murray Gell-Mann in 2002 and Andrew Wiles in 2003) take a walk from Dunsink observatory to the bridge where, unfortunately no trace of the carving remains. The quaternion involved abandoning the commutative law, a radical step for the time. 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. His first great work, Lectures on Quaternions (Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals. Hamilton confidently declared that quaternions would be found have a powerful influence as an instrument of research. Peter Guthrie Tait among others, advocated the use of Hamilton's Quaternions. Quaternions is applicable to concise and elegant demonstrations, it is but seldom used by mathematicians today.
There was controversy about the use of 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 [and vector calculus was later applied to four-vectors]), because quaternions provide superior notation. While this is undebatable in four dimensions, quaternions cannot be used with arbitrary dimensionality (though extensions like Octonions and Clifford algebras may be more applicable). Vector notation has replaced the "
Hamilton proceeded to popularize quaternions with several books, the last of which, Hamilton also contributed an alternative formulation of the mathematical theory of classical mechanics. While adding no new physics, this formulation, which builds on that of Joseph Louis Lagrange, provides a more powerful technique for working with the equations of motion. Both the Lagrangian and Hamiltonian approaches were developed to describe the motion of discrete systems, were then extended to continuous system and in this form can be used to define fieldss. In this way, the techniques find use in electromagnetic, quantum and relativity theory. ## Other originality
Hamilton originality matured his ideas before putting pen to paper. The discoveries, papers and treatises previously mentioned might well have formed the whole work of a long and laborious life. 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 developed the variational principle, which was reformulated later by Carl Gustav Jacob Jacobi. He also introduced 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. H. Abel, G. B. Jerrard, and others in their researches on this subject, form another contribution to science. There is next Hamilton's paper on Fluctuating Functions, a subject which, since the time of J. Fourier, has been of immense and ever increasing value in physical applications of mathematics. There is also the extremely ingenious invention of the hodograph. Of his extensive investigations into the solutions (especially by numerical approximation) of certain classes of physical differential equations, only a few items have been published, at intervals, in the Philosophical Magazine. Besides all this, Hamilton was a voluminous correspondent. Often a single letter of Hamilton's occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of Hamilton's mind never to be satisfied with a general understanding of a question; Hamilton pursued the problem until he knew it in all its details. 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. He was excessively precise and hard to please with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little compared with the extent of Hamilton's investigations. ## 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 the 2nd of September 1865, the task of finishing the “ ## Quotes- "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be "time plus space", or "space plus time": and in this sense it has, or at least involves a reference to, four dimensions. And how the One of Time, of Space the Three, Might in the Chain of Symbols girdled be." — William Rowan Hamilton (Quoted in Robert Percival Graves' "
*Life of Sir William Rowan Hamilton*" (3 vols., 1882, 1885, 1889)) - "He used to carry on, long trains of algebraic and arithmetical calculations in his mind, during which he was unconscious of the earthly necessity of eating; we used to bring in a ‘snack’ and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards." — William Edwin Hamilton (his elder son)
## External links, references, and resources
- MacTutor's Sir William Rowan Hamilton. School of Mathematics, University of St Andrews.
- Wilkins, David R.,
*Sir William Rowan Hamilton*. School of Mathematics, Trinity College, Dublin. - Wolfram Research's William Rowan Hamilton
- Cheryl Haefner's Sir William Rowan Hamilton
- 1911
*Britannica Hamilton*
Publications
- Hamilton, William Rowan (Royal Astronomer Of Ireland), "
*Introductory Lecture on Astronomy*". Dublin University Review and Quarterly Magazine Vol. I, Trinity College, January 1833. - Hamilton, William Rowan, "
*Lectures on Quaternions*". Royal Irish Academy, 1853. - David R. Wilkins's collection of Hamilton's Mathematical Papers
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