# William Rowan Hamilton

**Sir William Rowan Hamilton** LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865)^{[1]}^{[2]} was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and a director at Dunsink Observatory. He made major contributions to optics, classical mechanics and abstract algebra. His work was of importance to theoretical physics, particularly his reformulation of Newtonian mechanics, now called Hamiltonian mechanics. It is now central both to electromagnetism and to quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions.

Hamilton's scientific career included the study of geometrical optics, adaptation of dynamic methods in optical systems, 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 was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),^{[3]} who lived in Dublin at 29 Dominick Street, later renumbered to 36.^{[4]} Hamilton's father, who was from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton,^{[5]} a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. Meath.^{[6]} James's daughter Grace, Hamilton's cousin, became the mother of Mary Elizabeth Townsend, philanthropist and co-founder of the Girls' Friendly Society.^{[7]}

Hamilton is said to have shown immense talent at a very early age. Hamilton's predecessor as Royal Astronomer of Ireland and later Bishop of Cloyne Dr. John Brinkley remarked of the 18-year-old Hamilton, 'This young man, I do not say *will be*, but *is*, the first mathematician of his age.'^{[8]}

His uncle observed that Hamilton, from a young age, had displayed an uncanny ability to acquire languages (although this is disputed by some historians, who claim he had only a very basic understanding of them).^{[9]}^{: 207 } At the age of seven, he had already made considerable progress in Hebrew, and before he was thirteen he had acquired, under the care of his uncle (a linguist), almost as many languages as he had years of age. These included the classical and modern European languages, and Persian, Arabic, Hindustani, Sanskrit, and even Marathi and Malay. He retained much of his knowledge of languages to the end of his life, often reading Persian and Arabic in his spare time, although he had long since stopped studying languages, and used them just for relaxation.

In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton. The two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor.^{[10]}^{: 208 } In reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics.^{[11]}^{[12]}^{[13]}

Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College in Dublin, which he entered at age 18. The college awarded him two Optimes, or off-the-chart grades.^{[14]} He studied both classics and mathematics (BA in 1827, MA in 1837). While still an undergraduate, he was appointed Andrews Professor of Astronomy and Royal Astronomer of Ireland in 1827.^{[15]}^{: 209 } He then took up residence at Dunsink Observatory where he spent the rest of his life.^{[12]}

Hamilton retained his faculties unimpaired to the last, and continued the task of finishing the *Elements of Quaternions* which had occupied the last six years of his life. He died on 2 September 1865, following a severe attack of gout.^{[16]}^{[17]} He is buried in Mount Jerome Cemetery in Dublin.

The Hamilton Institute is an applied mathematics research institute at Maynooth University and the Royal Irish Academy holds an annual public Hamilton lecture at which Murray Gell-Mann, Frank Wilczek, Andrew Wiles and Timothy Gowers have all spoken. The year 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.^{[18]}

Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions.^{[19]} A 10-euro commemorative silver proof coin was issued by the Central Bank of Ireland in 2005 to commemorate 200 years since his birth.

In his youth Hamilton owned a telescope,^{[20]} and became an expert at calculating celestial phenomena, for instance the locations of the visibility of eclipses of the moon.^{[21]} On 16 June 1827, just 21 years old and still an undergraduate, he was elected Royal Astronomer of Ireland and went to live at Dunsink Observatory where he remained until his death in 1865.^{[22]}

In his early years at Dunsink, Hamilton observed the heavens quite regularly.^{[23]} Ultimately he left routine observation to his assistant Charles Thompson.^{[24]}^{[25]} His introductory lectures in astronomy were famous; in addition to his students, they attracted scholars and poets, and women;^{[26]} Felicia Hemans wrote her poem *The Prayer of the Lonely Student* after hearing one of his lectures.^{[27]}

Hamilton made important contributions to optics and to classical mechanics.

His first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of *Caustics* in 1824 to the Royal Irish Academy. It was referred as usual to a committee. While their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size, mostly by the additional details that the committee had suggested. But it also became more intelligible, and the features of the new method were now easily seen.^{[28]} Until this period Hamilton himself seems not to have fully understood either the nature or importance of optics,^{[29]} as later he intended to apply his method to dynamics.

In 1827, Hamilton presented a theory of a single function, now known as Hamilton's principal function, that brings together mechanics, optics, and mathematics, and which helped to establish the wave theory of light. He proposed it when he first predicted its existence in the third supplement to his *Systems of Rays*, read in 1832.

The Royal Irish Academy paper was finally entitled *Theory of Systems of Rays* (23 April 1827), and the first part was printed in 1828 in the *Transactions of the Royal Irish Academy*. 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 and 1835. In these papers, Hamilton developed his great principle of "Varying Action". The most remarkable result of this work is the prediction that a single ray of light entering a biaxial crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known as "conical refraction".^{[28]}

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, which, like the *Systems of Rays*, display a mastery over symbols and a flow of mathematical language almost unequaled. The common thread running through all this work is Hamilton's principle of "Varying Action". Although it is based on the calculus of variations and may be said to belong to the general class of problems included under the principle of least action which had been studied earlier by Pierre Louis Maupertuis, Euler, Joseph Louis Lagrange and others (and later by Richard Feynman, and others), Hamilton's analysis revealed much deeper mathematical structure than had been previously understood, in particular the symmetry between momentum and position. Paradoxically, the credit for discovering the quantity now called the Lagrangian and Lagrange's equations belongs to Hamilton. Hamilton's advances enlarged greatly the class of mechanical problems that could be solved, and they represent perhaps the greatest addition which dynamics had received since the work of Isaac Newton and Lagrange. Many scientists, including Liouville, Jacobi, Darboux, Poincaré, Kolmogorov, Prigogine^{[30]} and Arnold, have extended Hamilton's work, thereby expanding our knowledge of mechanics and differential equations, and forming the basis of symplectic geometry.^{[31]}

While Hamiltonian mechanics is based on the same physical principles as the mechanics of Newton and Lagrange, it provides a powerful new technique for working with the equations of motion. More importantly, both the Lagrangian and Hamiltonian approaches, which were initially developed to describe the motion of discrete systems, have proven critical to the study of continuous classical systems in physics, and even quantum mechanical systems. Indeed, the techniques find use in electromagnetism, quantum mechanics, quantum relativity theory and quantum field theory. In the *Dictionary of Irish Biography* David Spearman writes:^{[32]}

Despite the importance of his contributions to algebra and to optics, posterity accords him greatest fame for his dynamics. The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated. The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this the most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting-point for calculation in almost any area of physics.

Hamilton's mathematical studies seem to have been undertaken and carried to their full development without any assistance whatsoever, and the result is that his writings do not belong to any particular "school". Not only was Hamilton an expert as an arithmetic calculator, but he seems to have occasionally had fun in working out the result of some calculation to an enormous number of decimal places. At the age of eight Hamilton engaged Zerah Colburn, the American "calculating boy", who was then being exhibited as a curiosity in Dublin. Two years later, aged ten, Hamilton stumbled across a Latin copy of Euclid, which he eagerly devoured; and at twelve he studied Newton's *Arithmetica Universalis*. This was his introduction to modern analysis. Hamilton soon began to read the *Principia*, and by age sixteen he had mastered a great part of it, as well as some more modern works on analytic geometry and the differential calculus.^{[28]}

Around this time Hamilton was also preparing to enter Trinity College, Dublin, and therefore had to devote some time to classics. In mid-1822 he began a systematic study of Laplace's *Mécanique Céleste*.

From that time Hamilton appears to have devoted himself almost wholly to mathematics, though he always kept himself well acquainted with the progress of science both in Britain and abroad. Hamilton found an important defect in one of Laplace's demonstrations, and he was induced by a friend to write out his remarks, so that they could be shown to Dr. John Brinkley, then the first Royal Astronomer of Ireland, and an accomplished mathematician. Brinkley seems to have immediately perceived Hamilton's talents, and to have encouraged him in the kindest way.

Hamilton's career at College was perhaps unexampled. Amongst a number of extraordinary competitors, he was first in every subject and at every examination. He achieved the rare distinction of obtaining an optime both for Greek and for physics. Hamilton might have attained many more such honours (he was expected to win both the gold medals at the degree examination), if his career as a student had not been cut short by an unprecedented event. This was Hamilton's appointment to the Andrews Professor 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, authorised Hamilton's personal friend (also an elector) 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.^{[28]}

Hamilton was not especially suited for the post, because although he had a profound acquaintance with theoretical astronomy, he had paid little attention to the regular work of the practical astronomer. Hamilton's time was better employed in original investigations than it would have been spent in observations made even with the best of instruments. 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. If Hamilton had devoted himself to practical astronomy, the University of Dublin would assuredly have furnished him with instruments and an adequate staff of assistants.^{[28]}

He was twice awarded the Cunningham Medal of the Royal Irish Academy.^{[33]} The first award, in 1834, was for his work on conical refraction, for which he also received the Royal Medal of the Royal Society the following year.^{[34]} He was to win it again in 1848.

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. Other honours rapidly succeeded, among which his election in 1837 to the president's chair in the Royal Irish Academy, and the rare distinction of being made a corresponding member of the Saint Petersburg Academy of Sciences. Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.^{[35]}

The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843.^{[36]}^{: 210 } However, in 1840, Benjamin Olinde Rodrigues had already reached a result that amounted to their discovery in all but name.^{[37]}

Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He failed to find a useful 3-dimensional system (in modern terminology, he failed to find a real, three-dimensional skew-field), but in working with four dimensions he created quaternions. According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge).^{[38]}^{: 210 } This event marks the discovery of the quaternion group.

A plaque under the bridge was unveiled by the Taoiseach Éamon de Valera, himself a mathematician and student of quaternions,^{[39]} on 13 November 1958.^{[40]} Since 1989, the National University of Ireland, Maynooth has organised a pilgrimage called the *Hamilton Walk*, in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery.^{[41]}

The quaternion involved abandoning commutativity, a radical step for the time. Not only this, but Hamilton also invented the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product. Hamilton also 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 coined the words tensor and scalar, and was the first to use the word vector in the modern sense.^{[42]}

Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by introduction of complex number coefficients. When his work was assembled in 1853, the book *Lectures on Quaternions* had "formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in the Halls of Trinity College, Dublin". Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the *Elements of Quaternions*, a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition was prepared by Charles Jasper Joly, when the book was split into two volumes, the first appearing 1899 and the second in 1901. The subject index and footnotes in this second edition improved the *Elements* accessibility.

One of the features of Hamilton's quaternion system was the differential operator del which could be used to express the gradient of a vector field or to express the curl. These operations were applied by Clerk Maxwell to the electrical and magnetic studies of Michael Faraday in Maxwell's *Treatise on Electricity and Magnetism* (1873). Though the del operator continues to be used, the real quaternions fall short as a representation of spacetime. On the other hand, the biquaternion algebra, in the hands of Arthur W. Conway and Ludwik Silberstein, provided representational tools for Minkowski space and the Lorentz group early in the twentieth century.

Today, the quaternions are used in 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. The rationale is that combining quaternion transformations is more numerically stable than combining many matrix transformations. In control and modelling applications, quaternions do not have a computational singularity (undefined division by zero) that can occur for quarter-turn rotations (90 degrees) that are achievable by many Air, Sea and Space vehicles. In pure mathematics, quaternions show up significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry.

It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in *Alice in Wonderland*. In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry.^{[43]}

Hamilton originally matured his ideas before putting pen to paper. The discoveries, papers, and treatises previously mentioned might well have formed the whole work of a long and laborious life. But not to speak of his enormous collection of books, full to overflowing with new and original matter, which have been handed over to Trinity College Dublin, the previous mentioned works barely form the greater portion of what Hamilton has published.^{[28]} Hamilton developed the variational principle, which was reformulated later by Carl Gustav Jacob Jacobi. He also introduced the icosian game or *Hamilton's puzzle* which can be solved using the concept of a Hamiltonian path.

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*.^{[28]}

Hamilton invited his four sisters to come and live at the observatory in 1827, and they ran the household until his marriage. They included Eliza Mary Hamilton (1807–1851) the poet.^{[6]} Eliza, fostered until about age 15 in a Moravian community near Ballinderry, was strongly influenced by Calvinism. She differed in this from her brother, a devout member of the Church of Ireland.^{[45]}^{[46]} Sydney, another of the sisters, was a Calvinist.^{[47]}^{: 230 }

As a student Hamilton had been attracted to Catherine Disney, sister of one of his Trinity College friends. Her family did not approve, and Catherine was forced to marry the reverend William Barlow, a brother of her elder sister's husband. They married in 1825.^{[48]}^{: 109, 113 }

Hamilton married Helen Bayly, daughter of Rev Henry Bayly, Rector of Nenagh, County Tipperary, in 1833; she was a sister of neighbours to the observatory.^{[49]}^{[50]}^{: 108 } They had three children: William Edwin Hamilton (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840).^{[51]} Helen stayed with her widowed mother at Bayly Farm, Nenagh for extended periods, until her mother's death in 1837. She also was away from Dunsink, staying with sisters, for much of the time from 1840 to 1842.^{[52]} Hamilton's married life was reportedly difficult.^{[53]}^{: 209 } In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression.^{[54]}^{: 125, 6 }

In 1825 Hamilton met Arabella Lawrence, younger sister of Sarah Lawrence, a significant correspondent and frank critic of his poetry. It was a contact made through Maria Edgeworth.^{[55]}^{: 26 }^{[56]} In 1827 Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin.^{[57]}^{[58]} That year, newly appointed to the Observatory, he set off on a tour in Ireland and England with Alexander Nimmo, who was coaching him on latitude and longitude.^{[59]} One call was to Sarah Lawrence's school at Gatacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes.^{[60]} They visited William Wordsworth at Rydal Mount in September of that year, where Caesar Otway was also present.^{[61]}^{[62]}^{: 410 }

When Wordsworth visited Dublin in summer 1829, in a party with John Marshall and his family, he stayed at Dunsink with Hamilton.^{[63]}^{: 411 } On a second tour in England with Nimmo in 1831, Hamilton parted from him at Birmingham, to visit the Lawrence sisters and family on his mother's side in the Liverpool area. They met up again in the Lake District, where they climbed Helvellyn and had tea with Wordsworth. Hamilton returned to Dublin, via Edinburgh and Glasgow.^{[64]}^{[65]}

Hamilton visited Samuel Taylor Coleridge at Highgate, in 1832, helped by an unexpected letter of introduction given to him by Sarah Lawrence on a visit to Liverpool in March of that year. He also paid a call, with Arabella, on the family of William Roscoe who had died in 1831.^{[66]}^{[67]}