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Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer. Hamilton's discovery of quaternions is his
best known investigation. Hamilton also contributed to the development of optics,
dynamics, and algebra. Hamilton's research was later significant for the development of quantum mechanics.
Dr. John Brinkley, bishop of Cloyne, is said to have
remarked 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.”
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 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.
Biography
Early Life
Hamilton 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 mathematical 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
"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. But, two years before, he had
accidentally fallen in with a Latin copy of Euclid, which he eagerly devoured; and at twelve Hamilton attacked
Newton’s Arithmetica universalis. This was his introduction
to modern analysis. 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. 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,
"Dês-lors il commença a marcher seul," to use the words of the biographer of another great mathematician. 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, he was
induced by a friend to write out his remarks, that they might be shown to Dr John Brinkley, afterwards bishop of Cloyne, but who was then the first royal
astronomer for Ireland, and a
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 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
“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
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.
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, 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
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 to 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 "space-time" quaternions in science and engineering by the mid-20th century.
Hamilton proceeded to popularize quaternions with several books, the last of which, Elements of Quaternions, had 800
pages and was published shortly after his death. 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.
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 fields. In this way, the techniques find use in electromagnetic, quantum and relativity 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 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 “Elements of Quaternions” which had occupied the last six years of his
life.
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
Publications
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