# William Rowan Hamilton

William Hamilton

William Rowan Hamilton
Born

August 4, 1805
Dublin, Ireland

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
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 mathematical physicist who recast the the laws governing the motion of bodies in a simplified and elegant form called Hamilton's equations. He also made contributions in optics through a similar formulation of its laws. Hamilton invented quaternions, a four-dimensional extension of complex numbers.

## Early life

Hamilton was the fourth of nine children of Archibald Hamilton, a solicitor, and Sarah Hutton. He was born at 36 Dominick Street, Dublin.

When he was a year-old, he was placed in the care of an uncle and aunt, 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 Sanskrit, Arabic, and Persian, while mastering Italian and French. Two years later, he wrote a Syriac grammar for publication. In his later years, he would take up these pursuits as a form of relaxation.

Around this time, Hamilton encountered math prodigy Zerah Colburn, who could do elaborate calculations in his head. Hamilton competed with Colburn but was never able to match his acuity in the sphere of calculation, although it opened the budding mathematician's eyes to new possibilities.

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 was 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, he began a systematic study of Pierre-Simon Laplace's Mécanique Céleste.

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, Hamilton detected an important defect in one of Laplace’s demonstrations dealing with the composition of forces 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.

## University years

Hamilton was 18 when he entered Trinity College, and during his time there achieved great honors. Amongst a number of competitors of more than ordinary merit, he was first in every subject and at every examination. In 1824, he submitted his first paper for publication touching on themes of optics that would later win him an important place in the history of physics. In the same year, he met and hoped to marry Catherine Disney, the daughter of family friends, but lost out to a clergyman who Catherine's mother thought was better situated socially. This created a deep disturbance in Hamilton's emotional life, but he managed to press through in his studies and research. He achieved the rare distinction of obtaining an optime for both Greek and for physics in 1826. He also won awards for his poetry, which would later be commented on, with reserved praise, by the famous bard William Wordsworth. Wordsworth became one of Hamilton's life-long friends.

## Scientific career

### Early research

Hamilton's first discovery, a method of mathematical investigation equally applicable to optics and dynamics, was contained in an early paper which in 1823 Hamilton communicated to Dr. Brinkley, by whom, under the title of “Caustics,” it was presented to the Royal Irish Academy in 1824. 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 was renamed The theory of systems of rays, and 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.

### Hamilton as astronomer

Hamilton's career as a student was cut short by his appointment to the Andrews Professorship of Astronomy in the University of Dublin, vacated by 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 22, 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.

### Least action and conical refraction

In 1828, Hamilton published the first part of his Theory of Systems of Rays in the Transactions of the Royal Irish Academy. Supplements were published later.

The third portion of "Systems of Rays" appeared in 1832. It was in this paper that he predicted conical refraction, based on his formulation of optics using the principle of what Hamilton called varying action. but which is often referred to as least action. Conical refraction results in a conical formation of light rays from a single ray passing through certain crystal formations classified as biaxial. There was some challenge mounted to Hamilton's being the first to uncover this phenomenon, but it was later admitted that Hamilton had taken the final and necessary step toward this discovery. Humphrey Lloyd was able to confirm Hamilton's prediction experimentally by observing a conical shaft of light projected from a crystal. While information on conical refraction was included in the supplement to "System of Rays," it had been announced earlier in communications to the Royal Irish Academy.

A year after his discovery of conical refraction, Hamilton married Helen Maria Bayly. The couple had three children: a daughter and two sons.

### Hamiltonian dynamics

In what is probably his greatest claim to popular fame, Hamilton published a paper entitled "On a General Method in Dynamics" in the Philosophical Transactions of 1834 and 1835. There, he utilizes the principle of least action, long known by physicists, to produce an elegant and revealing formulation of the dynamical principles of moving bodies. This opened up a new area of mathematical physics known as Hamiltonian dynamics. In this formulation, the Hamiltonian, a function that Hamilton represented by the letter "V" (later changed to "H" in his honor) can, when employed in conjunction with his equations of motion, predict the motion of a body or collection of bodies under idealized circumstances. Hamilton's equations are a more elegant formulation of those produced earlier by Isaac Newton and Joseph Louis Lagrange. They were later expanded upon and improved by the mathematician C. G. J. Jacobi.

In 1835, Hamilton was knighted by the lord-lieutenant. Other honors rapidly followed, including his election in 1837 to the president’s chair of the Royal Irish Academy, and the rare distinction of being made corresponding member of the Academy of St. Petersburg.

### Quaternions

The other great contribution made by Hamilton to mathematical science was the introduction of quaternions, beginning in 1843.

Hamilton was looking for ways of extending complex numbers, which are expressed as the sum of a real and an imaginary number (an imaginary number is one whose square yields a negative number). Hamilton could not extend this branch of mathematics to three dimensions: in fact later mathematicians showed that this would be impossible. Eventually Hamilton tried four dimensions and, by doing so, created quaternions. According to the story Hamilton told, on October 16, he was out walking along the Royal Canal in Dublin with his wife, when the solution in the form of the equation

${\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}$

suddenly occurred to him. Hamilton then promptly carved this equation into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). He developed the rules for multiplication and division of quaternions, which had alluded him for many years.

The quaternion involved abandoning commutativity, a radical step for the time. Hamilton 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.

Hamilton introduced his first great work on the subject, Lectures on Quaternions, in 1852, and confidently declared that quaternions would become a powerful instrument of research. He popularized quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was just short of being completed at the time of his death.

Peter Guthrie Tait, among others, advocated the use of Hamilton's quaternions. However, controversy about their use 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 Josiah Willard Gibbs). Vector notation largely replaced the "space-time" quaternions in science and engineering by the mid-twentieth century.

### Contemporary uses for quaternions

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. .

### Last years

Toward the end of his life, Hamilton was completing "Elements of Quaternions"—what he hoped would be an improvement over his "Lectures on Quaternions" published earlier. But during these years he drove himself exceedingly hard. He was in the habit of putting in a 12-hour work day, often laboring through the night and into the morning.

In early June of 1865, Hamilton was besieged by an intense attack of the gout, followed by convulsions. He seemed to recover, and resumed his arduous work schedule. Around this time, the National Academy of Science, an American institution, was forming, and admitted him into its roster of founding members. In August, he wrote his acceptance letter, while continuing to work on Elements, the publication costs for which were an increased source of worry for him. On September 2, 1865, Hamilton called his friend, Charles Graves, to his observatory, and confided to him that his death was imminent. After recommending the sentiments expressed in the 145th psalm of the bible, he died that very afternoon. He was buried September 7 at Mount Jerome Cemetery.

## Other accomplishments

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.

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.

Hamilton had designed an entertainment called the "Icosian Game," which he sold for 25 pounds to some entrepreneurs who hoped to make a profit by mass-marketing it. The game, however, did not catch on, and the venture proved to be a financial failure.

## Honors

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 Institue[1] 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.

## Character and legacy

### Character

Although his researches consumed much of his time, Hamilton was a great socializer, and a voluminous correspondent. Often a single letter of Hamilton's occupied from 50 to 100 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.

He was obviously swept up by the romantic notions of his era, not only writing poetry of his own, but also spending time with the poets William Wordsworth and Samuel Taylor Coleridge. He corresponded with a great number of women, who provided inspiration for his work and a psychological release, outside of his voluminous correspondence with his scientific colleagues.

Of his marriage, Hamilton said that he was as happy as he expected, and happier than he deserved (Ball 1895, 217). Yet, his wife once took the children and lived away from her husband for almost a year. It is said that his wife was not the best housekeeper, and the chaotic state of his study was legendary. During his wife's absence, Hamilton began drinking heavily, which undoubtedly took a toll on both his physical and psychological health. He tried to quit for a time in the mid-1840s, but was shamed for doing so by a fellow scientist and fell back into the habit. Alcoholism is likely to have been a contributing cause to his early death at age 60. Be that as it may, he was a family man, and his children were devoted to him, often commenting (after his death) about what life with their father was like.

Interestingly, Hamilton had two rather distinct anomalies in his physical condition that, while not in any way connected, bear an interesting similarity. He is said to have had two distinct pitches to his voice, one high, another, low, and he used these interchangeably in conversation as well as when musing to himself.

Hamilton also appears to have had a problem with his eyesight, because of which he saw individual images with each eye, rather than a single image from both eyes. The use of a stereoscope demonstrated to him for the first time the manner in which most people see images. This led him to remark that depth perception was not due to stereoscopic vision, and could equally be perceived by vision in a single eye.

### Legacy

Any scientist contributing a new formulation to a science as influential and important as the dynamics of moving bodies will naturally be memorialized by future generations. While adding no new physics, his 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.

Hamilton's quaternions, while not generally celebrated any longer as a subject taught in undergraduate work, are still used in specialized fields.

## 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.