# Sir William Rowan Hamilton

# Sir William Rowan Hamilton

The Irish mathematical physicist Sir William Rowan Hamilton (1805-1865) reshaped theoretical optics by basing it on his law of varying action. His analysis of motion anticipated several basic notions of relativity and quantum mechanics.

William Rowan Hamilton was born on Aug. 4, 1805, in Dublin, the fourth child of Archibald Hamilton, a solicitor, and Sarah Hutton Hamilton. His father's family was known for a penchant for the gregarious and the romantic, while several members of his mother's family were distinguished scientists. It was the almost clashing union of the scientific and the romantic that became the most marked characteristic of William's personality. At the age of 2 he was entrusted to his uncle, a curate and an accomplished linguist. At the age of 3 William read English fluently; at 5 he was translating Greek and Hebrew; by the time he was 12, he not only had mastered German, French, Italian, and Spanish but also had a working knowledge of Syriac, Persian, Arabic, Sanskrit, and Hindustani. At his father's urging he wrote, at the age of 14, a letter in Persian to the Persian ambassador visiting in Dublin.

In 1820 Hamilton met the American mathematical prodigy Zerah Colburn, then studying in England, and thus became interested in mathematics. Later he began to read Euclid, from which came another powerful boost to Hamilton's single-minded, youthful pursuit of mathematical physics. He was hardly 17 when he mastered Isaac Newton's *Arithmetica universalis* and *Principia,* in addition to Pierre Laplace's *Mécanique céleste.* In the last, the self-taught young genius even discovered an error in connection with Laplace's demonstration of the parallelogram of forces.

## Systems of Rays

In 1822 Hamilton submitted a paper on the osculation of certain curves of double curvature to John Brinkley, professor of astronomy at Trinity College and astronomer royal of Ireland. The following year Hamilton entered Trinity College, earning the highest honors in all examinations and winning practically all the prizes. In 1824 he presented to the Royal Irish Academy a paper, "On Caustics, " the preface of which stated:"The Problems of Optics, considered mathematically, relate for the most part to the intersections of the rays of light proceeding from known surfaces, according to known laws. In the present paper, it is proposed to investigate some general properties common to all such Systems of Rays, and independent of the particular surface or particular law. It is intended in another paper to point out the application of these mathematical principles to the actual laws of Nature." These words prefaced in effect Hamilton's lifelong program in mathematical physics.

The committee appointed to report on the merits of Hamilton's paper requested him to elaborate further on the topic. In 1827 Hamilton, still an undergraduate, presented to the academy the enlarged form of the paper under the title "A Theory of Systems of Rays." His starting point was the well-established principle that light rays travel between two points (extremities, he called them) along the path of least time, or along the path of least action, depending on whether the wave theory or the corpuscular theory of light was considered.

This paper earned Hamilton not only fame but also meteoric rise in the academic world. Although still an undergraduate, he was appointed Andrews professor of astronomy (1827). Connected with the munificently endowed chair were the directorship of the Observatory of Dunsink, the title of astronomer royal of Ireland, and a spacious lodging on the observatory grounds. Furthermore, it was understood that he had no observational duties to perform lest his theoretical investigations be disturbed. The next 7 years in Hamilton's life were bathed in the sunshine of success and glory in every sense. His lectures on astronomy drew crowds that he kept spellbound with his soaring rhetoric. The prodigy was the toast of society, and he made lifelong friendships with Samuel Taylor Coleridge and William Wordsworth. At this time Hamilton's literary ambitions produced an outpouring of sonnets, many inspired by three romances, of which the last ended in his marriage to Helen Maria Bayley in 1833. The marriage was an unhappy one.

## Conical Refraction and Quaternions

In early 1834 Hamilton made the most spectacular discovery of his career—the prediction that under certain circumstances an internal and an external conical refraction would occur. The prediction was soon verified by Humphrey Lloyd, a professor of physics at Trinity. In 1835 Hamilton was knighted. The next year he became president of the Royal Irish Academy.

In 1843 Hamilton announced to the Royal Irish Academy the definition of quaternions; in 1848 he began his "Lectures on Quaternions" (published 1853). The discovery of the quaternions represented for Hamilton the most important event in his life. In his letter of Oct. 15, 1858, to Tait, he described in detail what went on in his mind as he walked on Oct. 16, 1843, toward Broughman Bridge in Dublin:"I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between *i, j, k;* exactly such as I have used them ever
since." From another letter of his we know that on the spur of the moment Hamilton "cut with a knife on a stone of Brougham Bridge … the formula *i ^{2} − j 2 − k 2 − ijk* = − 1." The theory included such points as the principles of noncommutative algebra, the generalized treatment of coordinates and momenta, and the correspondence of multiplication by imaginary numbers to rotation in space. All these topics now form indispensable parts of the mathematics of relativity and quantum mechanics. He also worked feverishly to achieve as much as possible of his great ambition, the detailed formulation of his quaternion theory, in which he saw the geometrical regularity embodied in the physical universe. His

*Elements of Quaternions,*a huge volume comprising the efforts of his last 10 years, was published a year after his death.

Hamilton's last years were marked by alcoholism and by a routine of life that lacked any evidence of orderliness. His study resembled a pigsty; in his cabinets, rows of dry lamb chops alternated with heaps of precious manuscripts. For the most part he was living unmindful of fame, burning with zeal to produce the work that would honor Ireland in the same measure as Newton's *Principia* did glory to England. He felt immense satisfaction on learning shortly before his death, on Sept. 2, 1865, that the recently established National Academy of Sciences in the United States had elected him as its foreign associate. In fact, the academy put Hamilton's name on the top of the list by a majority vote of two-thirds.

## Further Reading

Some of Hamilton's unpublished manuscripts are printed in the Royal Irish Academy's *The Mathematical Papers of Sir William Rowan Hamilton,* edited by A. W. Conway and others (4 vols., 1931-1941). The fullest biography of Hamilton is Robert Perceval Graves, *Life of Sir William Rowan Hamilton* (3 vols., 1882-1889). Shorter biographies of Hamilton are in Eric Temple Bell, *Men of Mathematics* (1937), and Scientific American, *Lives in Science* (1957). Dirk J. Struick, *A Concise History of Mathematics* (1948; 3d rev. ed. 1967), is recommended for general background.

## Additional Sources

Hankins, Thomas L., *Sir William Rowan Hamilton,* Baltimore: Johns Hopkins University Press, 1980. □

# Sir William Rowan Hamilton

# Sir William Rowan Hamilton

**1805-1865**

**Irish Mathematician**

Sir William Hamilton is perhaps best known for developing the theory of quaternions, the first non-cummutative form of algebra. However, he also made significant contributions to the theories of optics and other areas in mathematical physics.

Hamilton was born in Dublin, the son of an attorney. He showed signs of genius early, learning Latin, Greek, and Hebrew by the age of five. At age 13 he was introduced to mathematics and, at age 17, he brought an error in Pierre Laplace's book on celestial mechanics to the attention of John Brinkley, Ireland's Astronomer Royal. After this first meeting, Brinkley commented, "This young man, I do not say will be, but is, the first mathematician of his age."

During his undergraduate years, Hamilton was appointed professor of astronomy at Trinity College. This distinction was followed by his appointment as Astronomer Royal of Ireland. It turned out that both of these appointments were unsatisfying, as Hamilton's primary love was mathematics, not astronomy, and it was in mathematics that he did his most important work.

One of Hamilton's first major accomplishments was a theory of conical refraction, of interest to opticians and verified experimentally within months. He then began work in complex
numbers, systems of numbers in which one term is the square root of -1. Hamilton was among the first to treat complex numbers as points on a plane in which one coordinate is a real number and the other, an imaginary number. (An imaginary number is a number multiplied by *i,* the square root of -1, and is usually expressed as 2*i,* for example.)

He also introduced the concept of the Hamiltonian, a way of describing the total energy of a system. In this system, the total energy *H *of a body is constant over any path it can take, making the dynamical properties of the body more easily and accurately described. In this same area of mathematical physics, Hamilton also described Hamilton's principle, worked on the Hamilton-Jacobi equations, and developed the Hamiltonian operator. All of these are considered developments of fundamental importance to the study of dynamical systems, and all are still taught to physicists today.

Hamilton spent many years working on quaternion theory in one form or another. This work was started by realizing that a description of the motion of an object in three dimensions requires a mathematical description in four dimensions. Such problems could not be handled by the mathematics of the day. After nearly a decade of work, Hamilton finally understood the solution to this problem in the form of quaternions, which could be solved by using the first non-commutative algebra. An algebra is commutative if changing the order of the terms has no effect on the final solution. For example, 2 × 3 gives the same answer as 3 × 2. However, quaternions do not behave in this manner—they produce a different answer if the order of the terms is changed. Quaternions were not of much immediate use and, in fact, were not nearly as important as Hamilton had hoped. William Thomson (1824-1907), Lord Kelvin, wrote, "Quaternions came from Hamilton after his really good work had been done and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way." However, several decades later, Josiah Gibbs (1839-1903) was to find them exceptionally useful in developing vector calculus which, in turn, is widely used in physics today.

In marked contrast to his professional successes, Hamilton's personal life was a disappointment to him. In his early twenties he fell in love with a woman, but lost her to a more financially successful rival. After that, he seemed not to care much for any other women, and eventually married for the sake of marrying. He also had periodic problems with alcohol abuse, alternately controlling and losing control to his cravings. He died of a severe attack of gout at the age of 60, shortly after his election as the first foreign member of the National Academy of Sciences in the United States.

**P. ANDREW KARAM**

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