# 1911 Encyclopædia Britannica/Hamilton, Sir William Rowan

**HAMILTON, SIR WILLIAM ROWAN** (1805–1865), Scottish
mathematician, was born in Dublin on the 4th of August 1805.
His father, Archibald Hamilton, who was a solicitor, and his
uncle, James Hamilton (curate of Trim), migrated from Scotland
in youth. 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 an Irishman.

His genius first displayed itself in the form of a wonderful 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 extraordinary 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.

His 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 he was not only wonderfully expert, but he seems to
have occasionally found a positive delight in working out to an
enormous number of places of decimals the result of some irksome
calculation. At the age of twelve he 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 he attacked Newton’s *Arithmetica universalis*.
This was his introduction to modern analysis. He soon commenced
to read the *Principia*, and at sixteen he had mastered
a great part of that work, besides some more modern works on
analytical geometry and the differential calculus.

About this period he was also engaged in preparation for
entrance at Trinity College, Dublin, and had therefore to devote
a portion of his 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 most gifted 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, “Dès-lors il commença à marcher seul,” to use
the words of the biographer of another great mathematician.
From that time he appears to have devoted himself almost
wholly to original investigation (so far at least as regards mathematics),
though he ever kept himself well acquainted with the
progress of science both in Britain and abroad.

Having 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 (1763–1835),
afterwards bishop of Cloyne, but who was then the first
royal astronomer 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. He is said to have remarked in 1823 of this lad
of eighteen: “This young man, I do not say *will be*, but *is*, the
first mathematician of his age.”

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. How many more such honours he might
have attained it is impossible to say; but he 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 his 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 him to become a candidate, a step which
his modesty had prevented him from taking. Thus, when barely
twenty-two, he was established at the Observatory, Dunsink,
near Dublin. He 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 his 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,—infinitely better than if he had spent it on
those of the observatory, which, however good originally, were
then totally unfit for the delicate requirements of modern
astronomy. Indeed there can be little doubt that Hamilton
was intended by the university authorities who elected
him to the professorship of astronomy to spend his time
as he best could for the advancement of science, without being
tied down to any particular branch. Had he devoted himself
to practical astronomy they would assuredly have furnished him
with modern 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 we may merely mention 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 his more important discoveries and
inventions presently to be considered) in the uneventful life of
this great man. He retained his wonderful 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 (his *Elements of Quaternions*) which had occupied the last
six years of his life.

The germ of his first great discovery was contained in one of those
early papers which in 1823 he 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, and the great mathematical skill of its author, recommended
that, before being published, it should be still further developed and
simplified. During the next three years 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 grand 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 the importance
of his discovery, for it is only now that we find him announcing his
intention of applying his method to dynamics. The paper was
finally entitled “Theory of Systems of Rays,” 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 1834–1835. The principle
of “Varying Action” is the great feature of these papers; and it is
strange, indeed, that the one particular result of this theory which,
perhaps more than anything else that Hamilton has done, has
rendered his name known beyond the little world of true philosophers,
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 them that he was
led to its discovery. This singular result is still known by the name
“conical refraction,” which he proposed for it when he first predicted
its existence in the third supplement to his “Systems of
Rays,” read in 1832.

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 grand strides made by Sir Isaac Newton and
Joseph Louis Lagrange. C. G. J. Jacobi and other mathematicians
have developed to a great extent, and as a question of pure mathematics
only, Hamilton’s processes, and have thus made extensive
additions to our knowledge of differential equations. But there can
be little doubt that we have as yet obtained only a mere glimpse
of the vast physical results of which they contain the germ. And
though this is of course by far the more valuable aspect 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.

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. His first great work, *Lectures on Quaternions*
(Dublin, 1852), is almost painful to read in consequence of the
frequent use of italics and capitals.

“I hope that it may not be considered as unpardonable vanity
or presumption on my part, if, as my own taste has always led me
to feel a greater interest in *methods* than in *results*, so it is by
methods, rather than by *any* theorems, which *can* be separately
*quoted*, that I desire and hope to be remembered. Nevertheless it
is only human nature, to derive *some* pleasure from being cited, now
and then, even about a ‘Theorem’; especially where . . . the
quoter can enrich the subject, by combining it with researches of
*his own*.”

The discoveries, papers and treatises we have mentioned might
well have formed the whole work of a long and laborious life. But
not to speak of his enormous collection of MS. books, full to overflowing
with new and original matter, which have been handed over
to Trinity College, Dublin, the works we have already called attention
to barely form the greater portion of what he has published.
His 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 grand contribution to
science. There is next his great 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 solution (especially by
numerical approximation) of certain classes of differential equations
which constantly occur in the treatment of physical questions, 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 his 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 his mind never to be
satisfied with a general understanding of a question; he pursued it
until he knew it in all its details. He 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.

Like most men of great originality, Hamilton generally matured his ideas before putting pen to paper. “He used to carry on,” says his elder son, William Edwin Hamilton, “long trains of algebraical 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.”

For further details about Hamilton (his poetry and his association
with poets, for instance) the reader is referred to the *Dublin University*
*Magazine* (Jan. 1842), the *Gentleman’s Magazine* (Jan. 1866),
and the *Monthly Notices of the Royal Astronomical Society* (Feb. 1866);
and also to an article by the present writer in the *North British*
*Review* (Sept. 1866), from which much of the above sketch has been
taken. His works have been collected and published by R. P.
vols., 1882, 1885, 1889). (P. G. T.)