He remains arguably the most influential Irish physicist, along withErnest Walton. Since his death, Hamilton has been commemorated throughout the country, with several institutions, streets, monuments, and stamps bearing his name.
William Rowan Hamilton was born on 4 August 1805 inDublin, Ireland, the fourth of nine children of Archibald Hamilton (1778–1819) and Sarah Hutton who lived at 29Dominick Street (later renumbered to 36). Archibald, who was from Dublin, worked as a solicitor. By the age of 3, Hamilton had been sent to live with his uncle James Hamilton, a graduate ofTrinity College Dublin who ran a school in Talbots Castle inTrim, County Meath.[4][3]
Hamilton is said to have shown talent at an early age. His uncle observed that Hamilton, from a young age, had displayed an uncanny ability to acquire languages — a claim which has been disputed by some historians, who claim he had only a basic understanding of them.[5]: 207 At the age of seven, he had already made progress inHebrew, and before he was 13, he had acquired, under the care of his uncle, a dozen languages: classical and modern European languages,Persian,Arabic,Hindustani,Sanskrit,Marathi andMalay.[6] The emphasis of Hamilton's early education on languages is attributed to the wish of his father to see him employed by theBritish East India Company.[7]
An expertmental calculator, the young Hamilton was capable of working out some calculations to many decimal places. In September 1813, the American calculatingprodigyZerah 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 as the clear victor.[5]: 208
In mid-1822, Hamilton began a systematic study ofLaplace'sMécanique Céleste. During this period, he encountered what he believed to be a logical error inMécanique Céleste, an observation which led Hamilton to be introduced toJohn Brinkley, thenRoyal Astronomer of Ireland.[7] In November and December of 1822, he completed his first three original mathematical papers. On his first visit toDunsink Observatory, he showed two of them to Brinkley, who requested that the papers be developed further. Hamilton complied, and early in 1823, Brinkley approved the amended version.[10] In July of 1823, Hamilton earned a place atTrinity College Dublin by examination, at age 17. His tutor there was Charles Boyton, a family friend,[3] who brought to his attention the contemporary mathematics published by the group at theÉcole Polytechnique in Paris.[11] John Brinkley remarked of the precocious Hamilton, "This young man, I do not saywill be, butis, the first mathematician of his age."[12]
The college awarded Hamilton twooptimes, or off-the-chart grades, inGreek and in physics. He was first in every subject and at every examination. He aimed to win a Trinity College fellowship by competitive examination,[3] but this did not happen. Instead, after Brinkley was madeBishop of Cloyne in 1826,[13] Hamilton was appointed to the vacant posts left by Brinkley's departure the following year:Andrews Professor of Astronomy and Royal Astronomer of Ireland.[5]: 209 Despite having his undergraduate career cut short in this way, he earned degrees in both classics and mathematics (BA in 1827, MA in 1837).
Hamilton, now Royal Astronomer of Ireland, took up residence atDunsink Observatory, where he spent the rest of his life.[8] He was there from 1827 until his death in 1865.[14] In his early years at Dunsink, he observed the heavens quite regularly;[15] He left routine observation to his assistant Charles Thompson.[16][17] His sisters also supported the observatory's work.[3]
The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women.[18]Felicia Hemans wrote her poemThe Prayer of the Lonely Student after hearing one of his lectures.[19]
Hamilton invited his four sisters to come and live at the observatory in 1827, and they ran the household until his marriage in 1833. They includedEliza Mary Hamilton (1807–1851), the poet.[3] In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin.[20][21]
Newly appointed to the observatory, Hamilton set off on a tour in Ireland and England withAlexander Nimmo, who was coaching him onlatitude andlongitude.[22] One call was to Sarah Lawrence's school atGateacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes.[23] They visitedWilliam Wordsworth atRydal Mount in September of that year, where the writerCaesar Otway was also present.[24][25]: 410 After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple".[26]
When Wordsworth visited Dublin in the summer of 1829, in a party withJohn Marshall and his family, he stayed at Dunsink with Hamilton.[25]: 411 On a second tour in England with Nimmo in 1831, Hamilton parted from him atBirmingham, to visit the Lawrence sisters and family on his mother's side in the Liverpool area. They met up again in theLake District, where they climbedHelvellyn and had tea with Wordsworth. Hamilton returned to Dublin, via Edinburgh and Glasgow.[27][28]
Hamilton retained his faculties unimpaired to the last, and continued the task of finishing theElements of Quaternions which had occupied the last six years of his life. He died on 2 September 1865 at the age of 60, following a severe attack ofgout.[29] He is buried inMount Jerome Cemetery in Dublin.
His first discovery was in an early paper that he communicated in 1823 to John Brinkley, who presented it under the title ofCaustics in 1824 to theRoyal Irish Academy. It was referred as usual to a committee, which recommended further development and simplification before publication. Between 1825 and 1828 the paper was expanded, and became a clearer exposition of a novel method.[6] Over this period, Hamilton gained an appreciation for the nature and importance of optics.
In 1827, Hamilton presented a theory of a single function, now known asHamilton's principal function, that brings together mechanics and optical theory. It helped to establish the foundations of thewave theory of light inmathematical physics. He proposed it when he first predicted its existence in the third supplement to hisSystems of Rays, read in 1832.
The Royal Irish Academy paper was finally entitledTheory of Systems of Rays (23 April 1827), and the first part was printed in 1828 in theTransactions 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 papersOn a General Method in Dynamics, which appeared in thePhilosophical Transactions in 1834 and 1835. In these papers, Hamilton developed his central principle of "Varying Action".
A result of this work is a prediction for transparent biaxial crystals (i.e.monoclinic,orthorhombic ortriclinic crystals).[30] A ray of light entering such a crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known asconical refraction.[6] Hamilton found it from the geometry of thewave surface introduced byAugustin-Jean Fresnel, which hassingular point.[31] There is a basic mathematical explanation of the phenomenon, namely that the wave surface is not the boundary of a convex body. A fuller understanding awaited themicrolocal analysis of the middle of the 20th century,[32]
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 whosePhilosophical Transactions for 1834 and 1835 there are two papers on the subject.
Hamiltonian mechanics was a powerful new technique for working withequations of motion. Hamilton's advances enlarged the class of mechanical problems that could be solved. His principle of "Varying Action" was based on thecalculus of variations, in the general class of problems included under theprinciple of least action which had been studied earlier byPierre Louis Maupertuis,Euler,Joseph Louis Lagrange and others. Hamilton's analysis uncovered a deeper mathematical structure than had been previously understood, in particular a symmetry between momentum and position. The credit for discovering what are now called theLagrangian andLagrange's equations also belongs to Hamilton.
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 collaboration, and his writings do not belong to any particular school. He 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 restrictions.[6]
Hamilton made his discovery of the algebra ofquaternions in 1843.[5]: 210 Among much prior related work, in 1840Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name.[36]
Hamilton was looking for ways of extendingcomplex numbers (which can be viewed aspoints on a 2-dimensionalArgand diagram) to higher spatial dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along theRoyal Canal in Dublin with his wife when the solution in the form of the equation
i2 =j2 =k2 =ijk = −1
occurred to him; Hamilton then carved this equation using his penknife into the side of the nearbyBroom Bridge (which Hamilton called Brougham Bridge).[5]: 210
The quaternions involved abandoning thecommutative law, a radical step for the time. In the context of this prototypegeometric algebra, Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being thecross product minus thedot product asscalar. 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. He coined theneologisms "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.[37]
In 1824, Hamilton was introduced atEdgeworthstown to the novelistMaria Edgeworth, by The Rev.Richard Butler, vicar ofTrim, County Meath, to whom his uncle James Hamilton was curate.[38][27]: 5, 34 During the same period, his uncle introduced him to the Disney family atSummerhill House, County Meath. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister.[27]: 37 [39]
Hamilton was attracted to Catherine Disney, but her family did not approve and Catherine was required to marry the Rev. William Barlow, a brother of her elder sister's husband. The wedding took place in 1825.[27]: 109, 113 Hamilton wrote in 1826 about his feelings for her in an extended poem, "The Enthusiast". Over twenty years later, in 1847, he confided inJohn Herschel that during this period he might have become a poet.[39]
In 1825, Hamilton metArabella Lawrence, younger sister ofSarah Lawrence, a significant correspondent and frank critic of his poetry. It was a contact he made through Maria Edgeworth's circle.[27]: 26 [40]
Hamilton visitedSamuel Taylor Coleridge atHighgate, 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 ofWilliam Roscoe, who had died in 1831.[41][42]
Hamilton was a Christian, described as "a lover of the Bible, an orthodox and attached member of the Established Church", and as having a "profound conviction of the truth of revealed religion".[43][44][45]
While attending Trinity College, Hamilton proposed to a friend's sister, whose refusal drove the young Hamilton to depression and illness, even to the verge of suicide.[46] He proposed again in 1831 to Ellen de Vere, a sister of the poetAubrey De Vere, who declined as well.[46]
Hamilton married Helen Bayly, the daughter of The Rev. Henry Bayly, rector ofNenagh, County Tipperary, in 1833; she was a sister of neighbours to the observatory.[47][27]: 108 They had three children: the journalistWilliam Edwin Hamilton (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840).[48] 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.[49] Hamilton's married life was reportedly difficult.[5]: 209 In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression.[27]: 125, 126
In 1835, being secretary to the meeting of theBritish Association which was held that year in Dublin, Hamilton wasknighted by theLord Lieutenant. Other honours rapidly succeeded, among which his election in 1837 to the president's chair in theRoyal Irish Academy, and the rare distinction of being made a corresponding member of theSaint Petersburg Academy of Sciences. Later, in 1864, the newly establishedNational Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.[52]
Irish commemorative coin celebrating the 200th anniversary of his birthThe plaque on the birthplace of William Rowan Hamilton on Dominick Street in Dublin
A plaque under theBroom Bridge, associated with the discovery of quaternions, was unveiled by TaoiseachÉamon de Valera on 13 November 1958.[53][54] Since 1989, theNational University of Ireland, Maynooth, has organised a pilgrimage called theHamilton 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.[55]
Twocommemorative stamps valued ½ and 2½ pence were issued by the Irish postal service on 13 November 1943 to mark the centenary of the announcement of quaternions.[57][58] A 10-euro commemorative silverproof coin was issued by theCentral Bank of Ireland in 2005 to commemorate 200 years since his birth.
It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized byCharles Lutwidge Dodgson inAlice in Wonderland. In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert toEuclidean geometry.[60] In September 2022 evidence was presented to counter this suggestion, which appears to have been based on an incorrect understanding of both quaternions and their history.[61]
Hamilton introduced, as a method of analysis, both quaternions andbiquaternions, the extension to eight dimensions by the introduction of complex numbercoefficients. When his work was assembled in 1853, the bookLectures 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 theElements of Quaternions, a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition was prepared byCharles Jasper Joly, when the book was split into two volumes, the first appearing in 1899 and the second in 1901. The subject index and footnotes in this second edition improved theElements accessibility.
^Hamilton was born at midnight. In his younger years, his birthday was celebrated on 3 August, but after the birth of his second son on 4 August 1835 he changed it to 4 August.
^Graves (1882) Vol. I, p. 655: "She was deeply impressed with the picture of astronomical mathematicians in the silence of their closets, living abstracted and apart, and yet in their solitude sympathetic, and able to rule the minds of men."
^De Morgan, Augustus (1866). "Sir W. R. Hamilton".Gentleman's Magazine and Historical Review. Vol. 1. pp. 128–134.In the case of Hamilton there is no occasion to state anything but the simple fact, known to all his intimates, that he was in private profession, as in public, a Christian, a lover of the Bible, an orthodox and attached member of the Established Church, though of the most liberal feelings on all points. He had some disposition towards the life of a clergyman, but preferred to keep himself free to devote all his time to science: he was offered ordination by two bishops.
^Pritchard, Charles (1866). "William Rowan Hamilton".Monthly Notices of the Royal Astronomical Society.26:109–118.This memoir would be incomplete if we did not add, that our deceased member, together with the character of a scholar, a poet, a metaphysician, and a great analyst, combined with that of a kind-hearted, simple-minded Christian gentleman; we say the latter because Sir William Hamilton was too sincere a man ever to disguise, though too diffident to obtrude, his profound conviction of the truth of revealed religion.
^Chase, Gene (1966). "Has Christian theology furthered mathematics". In van der Meer, Jitse M. (ed.).Facets of Faith and Science: The Role of Beliefs in Mathematics and the Natural Sciences: An Augustinian Perspective. Vol. 2. University Press of America; Pascal Centre for Advanced Studies.In Hamilton's Calvinistic[1] theology, as in that of his Scottish friend and pupil Clerk Maxwell, God is the creator both of the universe and of the laws governing it. This means that the lawful relations among material objects are as real as the objects themselves. As a Christian, Hamilton was convinced that the stamp of God is on nature everywhere. He expected a Triune God to leave evidence of the Trinity on everything from three-dimensional space in geometry to an algebra involving triples of numbers. This "metaphysical drive," in the words of Thomas Hankins, his best twentieth-century biographer, "held him to the task" of looking for a generalization of complex numbers to triples."
^abBruno, Leonard C. (2003) [1999].Math and mathematicians: the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 209.ISBN0787638137.OCLC41497065.
