William Rowan Hamilton

Irish mathematician

• Born: August 4, 1805
• Birthplace: Dublin, Ireland
• Died: September 2, 1865
• Place of death: Near Dublin, Ireland

While questioning a commonly accepted three-dimensional concept of space on a plane, Hamilton discovered quaternions and, in doing so, drastically altered the study of algebra, forcing the abandonment of the commutative law of multiplication that was dominant in his day and leading the way to new methods of vector analysis.

Early Life

William Rowan Hamilton was born exactly at midnight, a moment poised equally between August 3 and 4, 1805. His father, Archibald Hamilton, was away in the north at the time of his son’s birth, carrying out his duties as agent to Archibald Rowan, a post he had held since 1800. Archibald Rowan, who was William Rowan Hamilton’s godfather, had been in exile for eleven years. His agent, William’s father, worked tirelessly to make possible Rowan’s return to his estate at Killyleagh, an effort that resulted in Rowan’s repatriation in 1806.

To help Rowan meet his expenses, Archibald Hamilton borrowed heavily at high interest rates. When these loans were called, Rowan failed to back Hamilton, who, in a year or two, had no alternative but to declare bankruptcy. By 1808, the family was sufficiently impoverished not to be able to provide for William, then three years old, and his sisters, Grace and Eliza, who had to be sent away to be cared for by relatives. The two girls presumably were sent to live with their father’s sister, Sydney, and young William became the ward of his uncle, James Hamilton, a Church of England clergyman who ran the diocesan school at Trim, some forty miles to the northwest of Dublin in County Meath. William was to remain there until 1823, when he returned to Dublin as a student at Trinity College.

In retrospect, it appears to have been a stroke of good fortune that young William was forced by circumstance to live with his uncle, a man of considerable intellect. Before the boy was four years old, he was able to read English and showed a remarkable understanding of arithmetic. By the time he was five, William was able to translate from Latin, Greek, and Hebrew. He knew Greek and Latin authors well enough to recite from their works, and he was also able to recite passages from works by John Milton and John Dryden. He is said to have mastered fourteen languages by the time he was thirteen. Before he turned twelve, he had compiled a Syriac grammar, and two years later he was sufficiently fluent in Persian to compose a speech of welcome that was delivered to the Persian ambassador when he was a guest in Dublin.

Always advanced in mathematics, Hamilton was enormously exhilarated when he met the American mathematician Zerah Colburn in 1820. Colburn was able to perform complex mathematical computations quickly in his head, a skill that enticed the fifteen-year-old Hamilton. The youth had already read Sir Isaac Newton’s Philosophiae Naturalis Principia Mathematica (1687) and Alexis-Claude Clairaut’s Elémens d’algèbre (1746) by the time he met Colburn. The excitement generated by his meeting with Colburn led Hamilton in the following year to study the completed volumes of Pierre-Simon Laplace’s five-volume Traité de mécanique céleste (1798-1825; A Treatise of Celestial Mechanics, 1829-1839).

Hamilton’s detection of a flaw in Laplace’s reasoning brought him to the attention of John Brinkley, a distinguished professor of astronomy at Trinity College who was then also president of the Royal Irish Academy. The following year, when he was seventeen, Hamilton sent a paper he had written on optics to Brinkley, who, upon reading the paper, declared to the Royal Academy that Hamilton was already the most important mathematician of his time.

Hamilton entered Trinity College in 1823. By 1825, he had completed his paper “On Caustics” and submitted it to the Royal Academy, only to be rebuffed because the members of the Academy could not follow his often convoluted reasoning. Hamilton was awarded the optime in both classics and mathematics, the first Trinity College student to achieve this dual honor. While still an undergraduate, in 1827, he submitted his paper “Theory of Systems of Rays” to the Royal Academy, establishing with that paper a uniform method of solving all problems in the field of geometrical optics. The paper was of sufficient significance that before he had finished his undergraduate studies at Trinity College, the school’s faculty elected William Rowan Hamilton to the Andrews professorship in astronomy, a post that established him as royal astronomer of Ireland and an examiner of graduate students in mathematics at Trinity College. He assumed that post immediately upon graduation.

Life’s Work

The post to which the Trinity College faculty elected Hamilton carried with it a residence at the Dunsink Observatory, some five miles from Trinity College. In October, 1827, Hamilton moved into that residence and remained there for the rest of his life. Although he did not have a distinguished career as an astronomer, Hamilton had a large following of people who attended his lectures on astronomy because the range of his literary as well as his mathematical knowledge was sufficient to enliven his presentations.

Hamilton read encyclopedically and regularly wrote poetry, although his friend, the poet William Wordsworth, advised him that his lasting contributions would lie in mathematics rather than in poetry. In 1832, Hamilton published an important supplement to his paper on the theory of rays. This supplement was purely speculative, postulating a new theory about the refraction of light by biaxial crystals. Augustin Fresnel had already developed the theory of double refraction, but Hamilton took the theory an important step beyond where Fresnel had left it. He contended that in certain circumstances, one ray of incident light could be refracted into an infinite number of rays in a biaxial crystal and would be formed in such a way that a cone would then result. Humphrey Lloyd, following Hamilton’s speculative lead, proved this theory of conical refraction within two months.

In 1833, after six years of living alone in his official residence, Hamilton—a man of average height and ruddy complexion—married Maria Bayley, whose father had been an Anglican rector in County Tipperary. Maria bore three children, two sons and a daughter. Not renowned for her domestic abilities, Maria presided over a somewhat chaotic household. Hamilton considered liquor a more reliable source of nourishment than anything Maria’s cook could provide, and, through the years, he became a heavy drinker.

Hamilton’s “On a General Method in Dynamics,” published in 1835, brought together his work in optics and dynamics. He proposed a theory that showed the duality that exists between the components of momentum in a dynamic system and the coordinates that determine its position. In many ways, this work was some of Hamilton’s most significant, although it took nearly a century for the development of research in quantum mechanics to demonstrate the brilliance and importance of Hamilton’s theory.

Hamilton served as the major local organizer of the British Association for the Advancement of Science meeting in Dublin in 1835, an activity that led to his being knighted in the closing ceremonies of that event. In 1837, he ascended to the presidency of the Royal Irish Academy. In 1843, the Crown awarded him an annual life pension of two hundred pounds. During his final illness, Hamilton received word that he had been ranked first on the list of foreign associates of the National Academy of the United States.

The contribution for which Hamilton is best remembered is his discovery of quaternions . This discovery has fundamentally changed the way in which mathematicians deal with three-dimensional space. Hamilton had begun his extensive investigation into ordered paired numbers more than ten years before he made his monumental discovery of quaternions on October 16, 1843, when, during a walk along Dublin’s Royal Canal, the answer to a question that had been haunting him for nearly a decade flashed almost supernaturally into his mind. So excited was he by this flash of insight that he carved the formula for his discovery, i² = j² = k² = ijk = -1, into the Brougham Bridge.

Hamilton suddenly realized that in three-dimensional space, geometrical operations require not triplets, expressed as i, j, and k and representing space, as had been previously supposed, but rather that, because in three-dimensional space the orientation of the plane is variable, another element, a real term that represents time, must also be considered, resulting in quadruplets rather than triplets. One of the major consequences of this insight was its negation of the previously accepted commutative law of multiplication, which postulates (a x b) = (b x a).

Hamilton’s work with quaternions, to which he devoted the last two decades of his life, was essential to the development of vector analysis. More recently, further important applications of his theory of quaternions have been instrumental in the description of elementary particles. Hamilton published his Lectures on Quaternions in 1853, and his influential The Elements of Quaternions appeared posthumously in 1866. William Rowan Hamilton died of gout on September 2, 1865, after a lingering illness.

Significance

Sir William Rowan Hamilton’s name lives in both the history of mathematics and the histories of physics and optics. His pioneering work in vector analysis forced specialists in that field to abandon the theory of double refraction and to replace it with Hamilton’s expanded theory of conical refraction. The work that led to these changes began while Hamilton was still an undergraduate at Trinity College and reached its culmination in the supplement to his “Theory of Systems of Rays” in 1832.

Hamilton’s next significant achievement posited a duality between the components of momentum in a dynamic system and the coordinates that determine its position, a theory that reduces the field of dynamics to a problem in the calculus of variations. This theory came to have considerable significance as the field of quantum mechanics developed.

Hamilton’s most memorable contribution by far, however, was his discovery of quaternions, which forced mathematicians to break with the commutative law of multiplication. In its simplified form, termed vector analysis and adapted by J. Willard Gibbs from Hamilton’s theory, Hamilton’s theory of quaternions has been of great significance to modern mathematical physicists.

Bibliography

Bell, Eric Temple. Development of Mathematics. New York: McGraw-Hill, 1940. Bell relates Hamilton to some of the salient mathematical developments of his time. The coverage is sketchy and has been superseded by Thomas L. Hankins’s biography (see below).

‗‗‗‗‗‗‗. Men of Mathematics. New York: Simon & Schuster, 1965. Bell puts Hamilton in historical perspective. The chapter “An Irish Tragedy” focuses on Hamilton, but, although interesting, it is not factually dependable in all respects.

Crilly, A. J. Arthur Cayley: Mathematician Laureate of the Victorian Age. Baltimore: Johns Hopkins University Press, 2005. Cayley (1821-1895) was a contemporary of Hamilton; the two men devised a matrix algebra theory that bears their names. Although focusing on Cayley, this biography also describes Hamilton and others who were part of a nineteenth century British mathematical vanguard.

Graves, R. P. Life of Sir William Rowan Hamilton. 3 vols. London: Longmans, Green, 1882. The three enormous volumes of this set include extensive selections from Hamilton’s correspondence, poetry, and miscellaneous writings, as well as extensive commentary. The work, remarkable in its time for its thoroughness, is badly dated and suffers from lack of selectivity.

Hamilton, William Rowan. The Mathematical Papers of Sir William Rowan Hamilton. 4 vols. Cambridge, England: Cambridge University Press, 1931-2000. Volume 1, Geometrical Optics (1931), and volume 2, Dynamics (1940), are edited by A. W. Conway and J. L. Synge; volume 3, Algebra (1967), is edited by H. Halberstam and R. E. Ingram. Volume 4 (2000) is edited by Brendan Scaife and includes Hamilton’s Systems of Rays, two lengthy letters regarding definite integrals and anharmonic coordinates, and reprints of numerous papers about geometry, astronomy, and other topics. Volumes 1 and 3 contain useful introductions. Despite some omissions, these volumes are superbly produced, and the highest standards of scholarship have been observed in their editing.

Hankins, Thomas L. Sir William Rowan Hamilton. Baltimore: Johns Hopkins University Press, 1980. Hankins’s critical biography of Hamilton is the definitive work in the field. Meticulously documented, the book is written in such a lively style that it at times reads like a novel rather than like the eminently scholarly work that it is. The best book to date on Hamilton.

James, Ioan. Remarkable Mathematicians: From Euler to von Neumann. New York: Cambridge University Press, 2002. This collection of brief biographies of prominent mathematicians includes a seven-page biography of Hamilton.

Synge, J. L. Geometrical Optics: An Introduction to Hamilton’s Method. Cambridge, England: Cambridge University Press, 1937. Highly technical in nature, this book contains a brief but valuable preface. This book is for the specialist rather than the beginner.