# William Rowan Hamilton

Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques. His greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In mathematics, he is perhaps best known for his discovery of quaternions.

## 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.
** Quoted in Robert Percival Graves' "Life of Sir William Rowan Hamilton" (3 volumes, 1882, 1885, 1889))
* The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention... And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning... It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional: extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might (no doubt) be represented by points upon a line, yet I thought that their simple successiveness was better conceived by comparing them with moments of time, divested, however, of all reference to cause and effect; so that the "time" here considered might be said to be abstract, ideal, or pure, like that "space" which is the object of geometry. In this manner I was led, many years ago, to regard Algebra as the Science of Pure Time: and an Essay, containing my views respecting it as such, was published in 1835. ...[I]f the letters A and B were employed as dates, to denote any two moments of time, which might or might not be distinct, the case of the coincidence or identity of these two moments, or of equivalence of these two dates, was denoted by the equation,B = Awhich symbolic assertion was thus interpreted as not involving any original reference to quantity, nor as expressing the result of any comparison between two durations as measured. It corresponded to the conception of simultaneity or synchronism; or, in simpler words, it represented the thought of the present in time. Of all possible answers to the general question, "When," the simplest is the answer, "Now:" and it was the attitude of mind, assumed in the making of this answer, which (in the system here described) might be said to be originally symbolized by the equation above written.
** Preface, Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method of which the Principles were Communicated in 1843 to the Royal Irish Academy... (1853) pp. 1-4. Hamilton makes reference to the article "Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time" (Read November 4th, 1833, and June 1st, 1835) Transactions of the Royal Irish Academy Vol. XVII, Part II (Dublin, 1835) pp 293-422.
* It may not sound very consistent with any such professed humility on my part, if I say to you that, after having served for the Quaternions during fourteen years, and having (as America seems to think) won my Rachel—to be my own by an intellectual marriage—I now wish to wind up several scientific projects, from which those quaternions had for a long time diverted me; and feel as if I were entering, or had already entered, on a new harvest of labour and reputation. As to Fame, if it have not been won or earned already, it is not likely that any future exertion will make it mine. But as to the Labour; that is a thing within everybody's power to judge of, even for himself. I have very long admired Ptolemy's description of his great astronomical Master, Hipparchus... "a labour-loving and truth-loving man."—Be such my epitaph!
** Letter to Mrs. Wilde, (February 11, 1858) as quoted by Robert Perceval Graves, Life of Sir William Rowan Hamilton (1889) Vol.3, p. 230
* … an undercurrent of thought was going on in my mind which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth the herald (as I foresaw immediately) of many long years to come of definitely directed thought and work by myself, if spared, and, at all events, on the part of others if I should even be allowed to live long enough distinctly to communicate the discovery. Nor could I resist the impulse - unphilosophical as it may have been - to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.
** in a letter to his son (dated August 5, 1865), describing his discovery of quaternions on October 16, 1843, in Robert Perceval Graves, Life of Sir William Rowan Hamilton Vol. 2 (1885), pp. 434-435.
* To admire is, to me, questionless, the highest pleasure of life.
** Letter to the Marquess of Northampton (June 17, 1838), in Robert Perceval Graves, Life of Sir William Rowan Hamilton Vol. 2 (1885), pp. 260-261.

## Quotes about Hamilton

* It still remained to be seen whether the laws of motion, as dependent on moving forces, could also be consistently transferred to spherical or pseudospherical space. This investigation has been carried out by Professor Lipschitz of Bonn. It is found that the comprehensive expression for all the laws of dynamics, Hamilton's principle, may be directly transferred to spaces of which the measure of curvature is other than zero. Accordingly, in this respect also, the disparate systems of geometry lead to no contradiction.
** Hermann von Helmholtz, "On the Origin and Significance of Geometrical Axioms" (1870) Lecture delivered in the Docenten Verein in Heidelberg, in Popular Lectures on Scientific Subjects (1881) 2nd series, Tr. E. Atkinson, pp. 53-54.
* To the scientists of 1850, Hamilton's principle was the realization of a dream. ...from the time of Galileo scientists had been striving to deduce as many phenomena of nature as possible from a few fundamental physical principles. ...they made striking progress ...But even before these successes were achieved Descartes had already expressed the hope and expectation that all the laws of science would be derivable from a single basic law of the universe. This hope became a driving force in the late eighteenth century after Maupertuis's and Euler's work showed that optics and mechanics could very likely be unified under one principle. Hamilton's achievement in encompassing the most developed and largest branches of physical science, mechanics, optics, electricity, and magnetism under one principle was therefore regarded as the pinnacle of mathematical physics.
** Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 441.
* The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived... A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.
** Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, p. 442.
* 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)

### "Sir W. R. Hamilton" (January, 1866)

    Augustus De Morgan, Gentleman's Magazine, Vol. 220
* Hamilton was a man who combined different talents to an extent which is often attributed, by exaggeration, to the possessor of one powerful faculty: but in his case there is abundant evidence. He was scholar, poet, metaphysician, mathematician, and natural philosopher. Highly imaginative and fluent of tongue, he was an orator in all that he knew; even in mathematics, to the details of which he could give almost a rhetorical cast in a letter. In metaphysics he was very well read, and could talk in a way which suggested a comparison to Southey, and a difference. Hamilton one day preached to Southey on this subject until the latter remarked, as they passed a ploughman, "If you had been Coleridge, you would have talked to that ploughman just as you have been talking to me."
* Hamilton was not only an Irishman, but Irish: and this with curious oppositions of character. He was a non-combatant: there was too much kindness in his disposition to allow any fight to show itself. Impulsive and enthusiastic, with strong opinions and new views, he was never engaged in a scientific controversy... William Rowan Hamilton's preservative was his dread of wounding the feelings of others. In his youth, "Defender of the Absent" was his nickname. ...He had a morbid fear of being a plagiarist; and the letters which he wrote to those who had treated like subjects with himself sometimes contained curious and far-fetched misgivings about his own priority. But, with all this, there was a touch of the national temperament in him... an Irishman who never gets into a row may give quick but quiet symptoms of opposition of opinion, and of what, were it more than a rudiment, would be called pugnacity.
* Hamilton was apt to work by fits and starts. He has been known several times to work fourteen hours in one day, standing nearly all the while; but there were intervals of comparative inaction... Sometimes a letter was written and copied which was not sent for months, and then only the first sheet, with promise of the rest. It has even happened that the letter was knowingly never forwarded at all, and that when, long after, he found reason to wish to send it, he could not find it and sent the copy instead.

## External links

* Papers by Hamilton (transcribed and edited by David R. Wilkins, 2000)
Mathematics :
| Mathematicians (by country) | Abel • Anaxagoras • Archimedes • Aristarchus of Samos • Averroes • Arnold • Banach • Cantor • Cartan • Chern • Cohen • Descartes • Diophantus • Erdős • Euclid • Euler • Fourier • Gauss • Gödel • Grassmann • Grothendieck • Hamilton • Hilbert • Hypatia • Lagrange • Laplace • Leibniz • Milnor • Newton • von Neumann • Noether • Penrose • Perelman • Poincaré • Pólya • Pythagoras • Riemann • Russell • Schwartz • Serre • Tao • Tarski • Thales • Turing • Weil • Weyl • Wiles • Witten |
| Numbers | 1 • 23 • 360 • e • π • Fibonacci numbers • Irrational number • Negative number • Number • Prime number • Quaternion • Octonion |
| Concepts | Abstraction • Algorithms • Axiomatic system • Completeness • Deductive reasoning • Differential equation • Dimension • Ellipse • Elliptic curve • Exponential growth • Infinity • Integration • Geodesic • Induction • Proof • Partial differential equation • Principle of least action • Prisoner's dilemma • Probability • Randomness • Theorem • Topological space • Wave equation |
| Results | Euler's identity • Fermat's Last Theorem |
| Pure math | Abstract algebra • Algebra • Analysis • Algebraic geometry (Sheaf theory) • Algebraic topology • Arithmetic • Calculus • Category theory • Combinatorics • Commutative algebra • Complex analysis • Differential calculus • Differential geometry • Differential topology • Ergodic theory • Foundations of mathematics • Functional analysis • Game theory • Geometry • Global analysis • Graph theory • Group theory • Harmonic analysis • Homological algebra • Invariant theory • Logic • Non-Euclidean geometry • Nonstandard analysis • Number theory • Numerical analysis • Operations research • Representation theory • Ring theory • Set theory • Sheaf theory • Statistics • Symplectic geometry • Topology |
| Applied math | Computational fluid dynamics • Econometrics • Fluid mechanics • Mathematical physics • Science |
| History of math | Ancient Greek mathematics • Euclid's Elements • History of algebra • History of calculus • History of logarithms • Indian mathematics • Principia Mathematica |
| Other | Mathematics and mysticism • Mathematics education • Mathematics, from the points of view of the Mathematician and of the Physicist • Philosophy of mathematics • Unification in science and mathematics |