Encyclopedia, Difference between revisions of "William Rowan Hamilton" - New World
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Hamilton later atttended [[Westminster School]] with [[Zerah Colburn]]. He was part of a small but well-regarded 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, prior to his graduation. | Hamilton later atttended [[Westminster School]] with [[Zerah Colburn]]. He was part of a small but well-regarded 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, prior to his graduation. | ||
| − | + | At the age of twelve Hamilton engaged [[Zerah Colburn (math prodigy)|Zerah Colburn]], the [[United States|American]] "''[[mental calculator|calculating boy]]''," who was then being exhibited as a curiosity in Dublin, and he had not always the worst of the encounter. | |
| − | |||
| − | About this period Hamilton was also engaged in preparation for entrance at | + | 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 ''[[Celestial mechanics|Mécanique Céleste]]''. Nothing could be better fitted to call forth such mathematical powers as those of Hamilton; for Laplace's great work, rich to profusion in analytical processes alike novel and powerful, demands from the student careful and often laborious study. |
| − | It was in the successful effort to open this treasure-house that Hamilton’s mind received its final temper | + | It was in the successful effort to open this treasure-house that Hamilton’s mind received its final temper. From that time Hamilton appears to have devoted himself almost wholly to the mathematics investigation, though he ever kept himself well acquainted with the [[History of science and technology|progress of science]] both in Britain and abroad. Hamilton detected an important defect in one of Laplace’s demonstrations dealing with vector addition at the beginning of the work, and he was induced by a friend to write out his remarks, that they might be shown to Dr [[John Brinkley (astronomer)|John Brinkley]], then the first [[Astronomer Royal for Ireland]], and an accomplished [[mathematician]]. Brinkley seems at once to have perceived the vast talents of young Hamilton, and to have encouraged him in the kindest manner. |
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 language|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 medal]]s 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 [[Dunsink Observatory]], near Dublin. | 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 language|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 medal]]s 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 [[Dunsink Observatory]], near Dublin. | ||
Revision as of 14:00, 10 August 2007
|
William Hamilton | |
|---|---|
| File:Hamilton.jpg William Rowan Hamilton | |
| Born | |
| Died | September 2, 1865 |
| Residence | Ireland |
| Nationality | Irish, of Scottish descent |
| Field | Mathematician, physicist, and astronomer |
| Institutions | Trinity College Dublin |
| Alma mater | Trinity College Dublin |
| Academic advisor | John Brinkley |
| Known for | Quaternions and Hamiltonians |
| Religious stance | Anglican |
| Note that although Hamilton never had a doctoral advisor, scientific genealogy authorities regard the Reverend John Brinkley as Hamilton's equivalent mentor. | |
Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. His discovery of quaternions is perhaps his best known investigation. Hamilton's work was later significant in the development of quantum mechanics. Hamilton is said to have showed immense talent at a very early age, prompting astronomer Bishop Dr. John Brinkley to remark in 1823 of Hamilton at the age of eighteen: “This young man, I do not say will be, but is, the first mathematician of his age.”
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). Hamilton also invented "Icosian Calculus," which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.
Early life
Hamilton was the fourth of nine children of Archibald Hamilton, a solicitor, and Sarrah Hutton. He was born at 36 Dominick Street, Dublin. When he was a year old, he was placed in the care of his aunt and uncle, James and Sydney Hamilton. Hamilton could read from the bible at age three, and by age four he was able to read some Greek, Latin and Hebrew. At six he was attempting translations of Homer and Virgil. Between the ages of nine and ten he picked up sanscrit, Arabic and Persian, while mastering Italian and French. two years later, he wrote a syriac grammar for publication. 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.
When Hamilton was 12, he lost his mother, and two years later, his father. When he was 15, he began to tackle science and mathematics, beginning with a study of Isaac Newton's Principia. Around the age of 17, he was tackling the infinitesimal calculus, and began a study of Pierre-Simon Laplace's Mechanique Celeste. In this work, Hamilton was able to point out a mistake, an accomplishment that brought him some attention.
Hamilton later atttended Westminster School with Zerah Colburn. He was part of a small but well-regarded 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, prior to his graduation.
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.
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 Mécanique Céleste. Nothing could be better fitted to call forth such mathematical powers as those of Hamilton; for Laplace's great work, rich to profusion in analytical processes alike novel and powerful, demands from the student careful and often laborious study.
It was in the successful effort to open this treasure-house that Hamilton’s mind received its final temper. 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 dealing with vector addition at the beginning of the work, 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. Brinkley seems at once to have perceived the vast talents of young Hamilton, and to have encouraged him in the kindest manner.
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 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. 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. Other 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 “Caustics,” it was presented in 1824 to the Royal Irish Academy. It was referred as usual to a committee. Their report, while acknowledging the novelty and value of its contents recommended that, before being published, it should be still further developed and simplified. During the time between 1825 to 1828 the paper grew to an immense bulk, principally by the additional details which had been inserted at the desire of the committee. But it also assumed a much more intelligible form, and the features of the new method were now easily to be seen. 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. It helped in establishing the wave theory of light. He proposed for it when he first predicted its existence in the third supplement to his "Systems of Rays," read in 1832. The Royal Irish Academy paper was finally entitled “Theory of Systems of Rays,” (April 23, 1827) and the first part was printed in 1828 in the Transactions of the Royal Irish Academy. It is understood that the more important contents of the second and third parts appeared in the three voluminous supplements (to the first part) which were published in the same Transactions, and in the two papers “On a General Method in Dynamics,” which appeared in the Philosophical Transactions in 1834 and 1835.
The principle of “Varying Action“ is the great feature of these papers; 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. This singular result is still known by the name “conical refraction.”
The step from optics to dynamics in the application of the method of “Varying Action” was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on the subject. These display, like the “Systems of Rays,” a mastery over symbols and a flow of mathematical language almost unequalled. But they contain what is far more valuable still, the greatest addition which dynamical science had received since the strides made by Sir Isaac Newton and Joseph Louis Lagrange. C. G. J. Jacobi and other mathematicians have extended Hamilton's processes, and have thus made extensive additions to our knowledge of differential equations.
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.
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
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.
The quaternion involved abandoning commutativity, 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 is Lectures on Quaternions (Dublin, 1852). Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. He popularized quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was published shortly after his death.
Peter Guthrie Tait among others, advocated the use of Hamilton's quaternions. They were made a mandatory examination topic in Dublin, and for a while they were the only advanced mathematics taught in some American universities. However, controversy about the use of quaternions grew in the late 1800s. 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. While this is undeniable for four dimensions, quaternions cannot be used with arbitrary dimensionality (though extensions like Clifford algebras can). Vector notation largely replaced the "space-time" quaternions in science and engineering by the mid-20th century.
Today, the quaternions are in use by computer graphics, control theory, signal processing and orbital mechanics, mainly for representing rotations/orientations. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining many quaternion transformations is more numerically stable than combining many matrix transformations. In pure mathematics, quaternions show up significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry.
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 systems and in this form can be used to define vector fields. In this way, the techniques find use in electromagnetic, quantum relativity theory and field theory.
Other originality
Hamilton originally 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 puzzle which can be solved using the concept of a Hamiltonian path.
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 Joseph 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 his 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 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. The Hamilton Institute is an applied mathematics research institute 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.
Commemorations of Hamilton
- Hamilton's equations are a formulation of classical mechanics.
- Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and a term from graph theory. It can be seen as the Quantum Hamiltonian.
Quotations
- "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)
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.
ReferencesISBN links support NWE through referral fees
- Sir William Rowan Hamilton by Thomas Hankins, 1980 published by The Johns Hopkins University Press, 474 pages. Primarily biographical but covers the math and physics Hamilton worked on in sufficient detail to give a flavor of the work.
- 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
External links
- MacTutor's Sir William Rowan Hamilton. School of Mathematics, University of St Andrews.
- 1911 Britannica Hamilton
- Hamilton Trust
- The Hamilton year 2005 web site
- The Hamilton Mathematics Institute, TCD
- Hamilton Institute
See also
- List of people on stamps of Ireland
- Tarik O'Regan (great, great, great grandson)
- Against the Day
| Persondata | |
|---|---|
| NAME | Hamilton, William |
| ALTERNATIVE NAMES | |
| SHORT DESCRIPTION | Mathematician, physicist, and astronomer |
| DATE OF BIRTH | August 4, 1805 |
| PLACE OF BIRTH | Dublin, Ireland |
| DATE OF DEATH | September 2, 1865 |
| PLACE OF DEATH | Dublin, Ireland |
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