View source for William Rowan Hamilton

{{Short description|Irish mathematician and physicist (1805–1865)}}
{{Use British English|date=June 2024}}
{{Use dmy dates|date=June 2024}}
{{Infobox scientist
| honorific_prefix  = [[Sir]]
| honorific_suffix  = {{post-nominals|country=GBR|size=100%|FRAS}}
| image             = William Rowan Hamilton portrait oval combined.png
| birth_date        =  4 August 1805
| birth_place       = [[Dublin]], Ireland
| death_date        = {{death date and age|df=y|1865|9|2|1805|8|4}}
| death_place       = Dublin, Ireland
| alma_mater        = [[Trinity College Dublin]]
| known_for         = {{plainlist|
* Introducing [[Hamiltonian mechanics]] (1833)
* Formulating [[Hamilton's principle]] (1834)
*{{nowrap|Introducing [[quaternion]]s (1843)}}
* Coining the term ''[[tensor]]'' (1846)
* Inventing [[icosian calculus]] (1856)
* Inventing the [[icosian game]] (1856)
}}
| spouse            = {{marriage|Helen Marie Bayly|1833}}
| children          = 3, including [[William Edwin Hamilton|William]]
| awards            = {{plainlist|
*[[Cunningham Medal]] (1834, 1848)
*[[Royal Medal]] (1835)
}}
| honours           = [[File:200px ribbon bar of the Knight Bachelor medal (UK).svg|25px]] [[Knight Bachelor]] (1835)
| fields            = {{tree list}}
*[[Mathematics]]
*[[Physics]]
**[[Classical mechanics]]
{{tree list/end}}
| work_institutions = {{tree list}}
*Trinity College Dublin
**[[Dunsink Observatory]]
{{tree list/end}}
| academic_advisors = [[John Brinkley (astronomer)|John Brinkley]]
{{Infobox officeholder
| embed             = yes
| order             = 3rd
| office            = Andrews Professor of Astronomy 
| term_start        = 1827
| term_end          = 1865
| predecessor       = John Brinkley
| successor         = [[Franz Brünnow]]
}}
}}
'''Sir William Rowan Hamilton''' (4 August 1805 – 2 September 1865)<ref>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.</ref><ref>Graves (1882) Vol. I, p. 1</ref> was an Irish [[mathematician]] and [[physicist]] who made major contributions to [[abstract algebra]], [[classical mechanics]], and [[optics]]. His work is considered fundamental to modern [[theoretical physics]], particularly [[Hamiltonian mechanics|his reformulation]] of [[Lagrangian mechanics]]. His career included the analysis of [[geometrical optics]], [[Fourier analysis]], and [[quaternions]], the last of which made him one of the founders of modern [[linear algebra]].<ref name="ODNB"/> 

Hamilton was [[Andrews Professor of Astronomy]] at [[Trinity College Dublin]]. He was also the third director of [[Dunsink Observatory]] from 1827 to 1865. The [[Hamilton Institute]] at [[Maynooth University]] is named after him.

==Early life==
Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819), who lived in Dublin at 29 [[Dominick Street, Dublin|Dominick Street]], later renumbered to 36. 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, a graduate of [[Trinity College Dublin|Trinity College]] who ran a school in Talbots Castle in [[Trim, County Meath|Trim]], County Meath.<ref>Graves (1882) Vol. I, p. 1.</ref><ref name="ODNB">{{cite ODNB|id=12148|title=Hamilton, William Rowan (1805–1865)|year=2004|last=Lewis|first=Albert}}</ref>

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.<ref name="Bruno2003">Bruno (2003)</ref>{{rp|207}} At the age of seven, he had already made progress in [[Hebrew language|Hebrew]], and before he was 13, he had acquired, under the care of his uncle, a dozen languages: classical and modern European languages, [[Persian language|Persian]], [[Arabic]], [[Hindustani language|Hindustani]], [[Sanskrit]], [[Marathi language|Marathi]] and [[Malay language|Malay]].<ref name="EB1911">{{cite EB1911|wstitle=Hamilton, Sir William Rowan}}</ref> The emphasis of Hamilton's early education on languages is attributed to the wish of his father to see him employed by the [[British East India Company]].<ref name="DNB">{{cite DNB|wstitle=Hamilton, William Rowan|volume=24}}</ref>

An expert [[Mental calculation|mental calculator]], the young Hamilton was capable of working out some calculations to many decimal places. In September 1813, the American calculating [[Child prodigy|prodigy]] [[Zerah Colburn (mental calculator)|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 as the clear victor.<ref name="Bruno2003"/>{{rp|208}}

In reaction to his defeat, Hamilton spent less time studying languages, and more on mathematics.<ref name=mactutor>{{MacTutor | id=Hamilton | title=Sir William Rowan Hamilton }}</ref><ref name=Graves1842>{{cite journal | first=Robert Perceval | last=Graves | title=Our portrait gallery – No. XXVI. Sir William R. Hamilton | journal=Dublin University Magazine | volume=19 | date=1842 | pages=94–110 | url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Gallery/Gallery.html | access-date=13 May 2010 | archive-date=17 November 2017 | archive-url=https://web.archive.org/web/20171117041036/https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Gallery/Gallery.html | url-status=live }}</ref> At age ten, he stumbled across a [[Latin]] copy of [[Euclid]]; and at twelve he studied [[Isaac Newton|Newton]]'s ''[[Arithmetica Universalis]]''. By age 16, he had covered much of the ''[[Philosophiae Naturalis Principia Mathematica|Principia]]'', as well as some more recent works on [[analytic geometry]] and [[differential calculus]].<ref name="EB1911"/>

==Student years==
In mid-1822, Hamilton began a systematic study of [[Laplace]]'s ''[[Traité de mécanique céleste|Mécanique Céleste]]''. During this period, he encountered what he believed to be a logical error in ''Mécanique Céleste'', an observation which led Hamilton to be introduced to [[John Brinkley (astronomer)|John Brinkley]], then Royal Astronomer of Ireland.<ref name="DNB"/> In November and December of 1822, he completed his first three original mathematical papers.  On his first visit to Dunsink 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.<ref>Graves (1882) Vol. I, pp. 124, 128</ref> In July of 1823, Hamilton earned a place at [[Trinity College Dublin]] by examination, at age 17. His tutor there was [[Charles Boyton]], a family friend,<ref name="ODNB"/> who brought to his attention the contemporary mathematics published by the group at the [[École Polytechnique]] in Paris.<ref>{{cite web |title=Boyton, Charles, Dictionary of Irish Biography |url=https://www.dib.ie/biography/boyton-charles-a0864 |website=www.dib.ie |language=en}}</ref> John Brinkley remarked of the precocious Hamilton, "This young man, I do not say ''will be'', but ''is'', the first mathematician of his age."<ref>[https://babel.hathitrust.org/cgi/pt?id=mdp.39015027525990;view=1up;seq=155 Sir W. R. Hamilton] {{Webarchive|url=https://web.archive.org/web/20190507113324/https://babel.hathitrust.org/cgi/pt?id=mdp.39015027525990;view=1up;seq=155 |date=7 May 2019 }} The Gentleman's magazine. vol 220, 1866 Jan–Jun, p. 129</ref>

The college awarded Hamilton two ''optime''s, or off-the-chart grades, in [[Ancient Greek language|Greek]] and in physics. He was first in every subject and at every examination. He aimed to win a Trinity College fellowship by competitive examination,<ref name="ODNB"/> but this did not happen. Instead, after Brinkley in 1826 was made [[Bishop of Cloyne]],<ref>{{cite ODNB|id=3438|first=P. A.|last=Wayman|title=Brinkley, John (1766/7–1835)}}</ref> Hamilton was appointed in 1827 to the vacant posts left by Brinkley's departure, Andrews Professor of Astronomy and Royal Astronomer of Ireland.<ref name="Bruno2003"/>{{rp|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).

===Personal life and poetry===
In 1824, Hamilton was introduced at [[Edgeworthstown]] to the novelist [[Maria Edgeworth]], by the Rev. [[Richard Butler (Irish priest)|Richard Butler]], the vicar of [[Trim, County Meath]] to whom his uncle James Hamilton was curate.<ref name="alox">{{alox2|title=Butler, Richard (3)}}</ref><ref name="Hankins">Hankins (1980)</ref>{{rp|5, 34}} During the same period, his uncle introduced him to the Disney family at [[Summerhill House]], County Meath. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister.<ref name="Hankins">Hankins (1980)</ref>{{rp|37}}<ref name="Brown">{{cite journal |last1=Brown |first1=Daniel |title=William Rowan Hamilton and William Wordsworth: the Poetry of Science |journal=Studies in Romanticism |date=2012 |volume=51 |issue=4 |pages=475–501 |jstor=24247229 |url=https://www.jstor.org/stable/24247229 |issn=0039-3762}}</ref>

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.<ref name="Hankins">Hankins (1980)</ref>{{rp|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 in [[John Herschel]] that during this period he might have become a poet.<ref name="Brown"/>

In 1825, Hamilton met [[Arabella Lawrence (educator)|Arabella Lawrence]], younger sister of [[Sarah Lawrence (educator)|Sarah Lawrence]], a significant correspondent and frank critic of his poetry. It was a contact he made through Maria Edgeworth's circle.<ref name="Hankins"/>{{rp|26}}<ref>{{cite journal |last1=Brown |first1=Daniel |title=William Rowan Hamilton and William Wordsworth: the Poetry of Science |journal=Studies in Romanticism |date=2012 |volume=51 |issue=4 |page=490 |jstor=24247229 |url=https://www.jstor.org/stable/24247229 |issn=0039-3762}}</ref>

==At Dunsink Observatory==
Hamilton, now Royal Astronomer of Ireland, took up residence at [[Dunsink Observatory]] where he spent the rest of his life.<ref name=mactutor/> He was there from 1827 until his death in 1865.<ref>Graves (1889) Vol. III, p. 404</ref> In his early years at Dunsink, Hamilton observed the heavens quite regularly;<ref>Graves (1882) Vol. I, p. 326</ref> He left routine observation to his assistant Charles Thompson.<ref>Graves (1882) Vol. I, p. 285</ref><ref>Graves (1882) Vol. I, p. 409</ref> Hamilton's sisters also supported the observatory's work.<ref name="ODNB"/>

The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women.<ref>Graves (1882) Vol. I, p. 655</ref> [[Felicia Hemans]] wrote her poem ''The Prayer of the Lonely Student'' after hearing one of his lectures.<ref>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."</ref>

===Personal life, travel and poetic visits===
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 included Eliza Mary Hamilton (1807–1851), the poet.<ref name="ODNB"/> In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin.<ref>Graves (1882) Vol. I, p. 230</ref><ref>{{cite ODNB|id=61561|first=Virginia H.|last=Blain|title=Hamilton, Eliza Mary (1807–1851)}}</ref>

Newly appointed to the Observatory, Hamilton set off on a tour in Ireland and England with [[Alexander Nimmo]], who was coaching him on [[latitude]] and [[longitude]].<ref>{{cite book |last1=Mollan |first1=Charles |title=It's Part of What We Are – Volumes 1 and 2 – Volume 1: Richard Boyle (1566–1643) to John Tyndall (1820–1893); Volume 2: Samuel Haughton (18210–1897) to John Stewart Bell (1928–1990): Some Irish Contributors to the Development of the Chemical and Physical Sciences |date=2007 |publisher=Charles Mollan |isbn=978-0-86027-055-3 |page=603 |url=https://books.google.com/books?id=cjnJDwAAQBAJ&pg=PA603 |language=en}}</ref> One call was to Sarah Lawrence's school at [[Gateacre]], near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes.<ref>{{cite book |title=The Kaleidoscope: or, Literary and scientific mirror |date=1828 |page=95 |url=https://books.google.com/books?id=ijEFAAAAQAAJ&pg=PA95 |language=en}}</ref> They visited [[William Wordsworth]] at [[Rydal Mount]] in September of that year, where the writer [[Caesar Otway]] was also present.<ref>{{cite book |last1=Gill |first1=Stephen |title=William Wordsworth: A Biography |date=1990 |publisher=Oxford University Press |isbn=978-0-19-282747-0 |page=355 |language=en}}</ref><ref name="Barker 2001">Barker (2001)</ref>{{rp|410}} After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple".<ref>{{cite journal |last1=Brown |first1=Daniel |title=William Rowan Hamilton and William Wordsworth: the Poetry of Science |journal=Studies in Romanticism |date=2012 |volume=51 |issue=4 |page=478 |jstor=24247229 |url=https://www.jstor.org/stable/24247229 |issn=0039-3762}}</ref>
[[File:Master Noakes, a mental calculator Wellcome L0034923.jpg|thumb|Master Noakes, the [[mental calculator]], 1827 lithograph]]

When Wordsworth visited Dublin in the summer of 1829, in a party with [[John Marshall (industrialist)|John Marshall]] and his family, he stayed at Dunsink with Hamilton.<ref name="Barker 2001"/>{{rp|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.<ref name="Hankins"/><ref>{{cite journal |last1=Brown |first1=Daniel |title=William Rowan Hamilton and William Wordsworth: the Poetry of Science |journal=Studies in Romanticism |date=2012 |volume=51 |issue=4 |pages=49–50, 52 |jstor=24247229 |url=https://www.jstor.org/stable/24247229 |issn=0039-3762}}</ref>

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.<ref>{{cite book |last1=Paley |first1=Morton D. |title=Coleridge's Later Poetry |date=1999 |publisher=Clarendon Press |isbn=978-0-19-818685-4 |page=26 |url=https://books.google.com/books?id=QBezJcVeU6UC&pg=PA26 |language=en}}</ref><ref>Graves (1882) Vol. I, p. 191</ref>

He 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".<ref>{{cite magazine |last=De Morgan |first=Augustus |date=1866 |title=Sir W. R. Hamilton |magazine=Gentleman's Magazine and Historical Review |volume=1 |pages=128–134 |quote=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.}}</ref><ref>{{cite journal |last=Pritchard |first=Charles |date=1866 |title=William Rowan Hamilton |journal=Monthly Notices of the Royal Astronomical Society |volume=26 |pages=109–118 |quote=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.}}</ref><ref>{{cite book |last=Chase |first=Gene |date=1966 |chapter=Has Christian theology furthered mathematics |title=Facets of Faith and Science: The Role of Beliefs in Mathematics and the Natural Sciences: An Augustinian Perspective |volume=2 |editor-last=van der Meer |editor-first=Jitse M. |publisher=University Press of America; Pascal Centre for Advanced Studies |quote=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.”}}</ref>

== Family==
While attending Trinity College, Hamilton proposed to his friend's sister, whose refusal drove the young Hamilton to depression and illness, even to the verge of suicide.<ref name="Bruno">{{Cite book|url=https://www.worldcat.org/oclc/41497065|title=Math and mathematicians: the history of math discoveries around the world|last=Bruno|first=Leonard C.|date=2003|orig-date=1999|publisher=U X L|others=Baker, Lawrence W.|isbn=0787638137|location=Detroit, Mich.|page=209|oclc=41497065}}</ref> He proposed again in 1831 to Ellen de Vere, a sister of the poet [[Aubrey Thomas de Vere|Aubrey De Vere]], who declined as well.<ref name="Bruno" /> Hamilton eventually married Helen Marie Bayly in 1833,<ref name="Bruno" /> a country preacher's daughter, and had three children with her: [[William Edwin Hamilton]] (born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840).<ref>Sean O’Donnell (1983) ''William Rowan Hamilton: Portrait of a Prodigy'', Dublin: Boole Press {{ISBN|0-906783-06-2}}</ref> Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, timid, and chronically ill.<ref name="Bruno" />

==Death==
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]].<ref>{{cite news |first=William |last=Reville |title=Ireland's Greatest Mathematician |url=http://undersci.ucc.ie/wp-content/uploads/sites/12/2014/11/William_Rowan_Hamilton.pdf |newspaper=[[The Irish Times]] |date=26 February 2004 |access-date=4 January 2015 |archive-date=4 January 2015 |archive-url=https://web.archive.org/web/20150104195715/http://undersci.ucc.ie/wp-content/uploads/sites/12/2014/11/William_Rowan_Hamilton.pdf |url-status=live }}</ref> He is buried in [[Mount Jerome Cemetery]] in Dublin.

==Physics==
{{Classical mechanics|Topic=Scientists}}

Hamilton made outstanding contributions to [[classical mechanics]] and [[optics]].

His first discovery was in an early paper that he communicated in 1823 to John Brinkley, who presented it under the title of ''Caustics'' in 1824 to the [[Royal 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.<ref name="EB1911" /> Over this period, Hamilton gained an appreciation for the nature and importance of optics.<ref>{{Cite journal|date=2018|url=http://dx.doi.org/10.5040/9781350971424|journal=The BK Bounce|doi=10.5040/9781350971424|title=The BK Bounce|access-date=2 May 2021|archive-date=26 September 2021|archive-url=https://web.archive.org/web/20210926202747/https://www.humankineticslibrary.com/video?docid=HKL_5687412356001&tocid=HKL_file_5687412356001|url-status=live }}</ref>

In 1827, Hamilton presented a theory of a single function, now known as [[Hamilton's principal function]], that brings together mechanics and optical theory. It helped to establish the foundations of the [[wave theory of light]] in [[mathematical physics]]. 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 central principle of "Varying Action".

A result of this work is a prediction for transparent biaxial crystals (i.e. [[monoclinic crystal system|monoclinic]], [[orthorhombic crystal system|orthorhombic]] or [[triclinic crystal system|triclinic]] crystals).<ref>{{cite book |last1=Born |first1=Max |last2=Wolf |first2=Emil |title=Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light |date=28 February 2000 |publisher=CUP Archive |isbn=978-0-521-78449-8 |page=805 |url=https://books.google.com/books?id=oV80AAAAIAAJ&pg=PA805 |language=en}}</ref> A ray of light entering such a crystal at a certain angle would emerge as a hollow cone of rays. This discovery was known as [[conical refraction]].<ref name="EB1911"/> Hamilton found it from the geometry of the [[wave surface]] introduced by [[Augustin-Jean Fresnel]], which has [[Singularity (mathematics)|singular point]].<ref>{{Cite journal |date=2000 |first=Michael |last=Berry |journal=Nature |doi=10.1038/47364|title=Making waves in physics |volume=403 |issue=6765 |page=21 |pmid=10638732 |doi-access=free |bibcode=2000Natur.403...21B }}</ref> 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 the [[microlocal analysis]] of the middle of the 20th century,<ref>{{cite book |last1=Chorin |first1=AlexandreJ |last2=Majda |first2=Andrew J. |title=Wave Motion: Theory, Modelling, and Computation: Proceedings of a Conference in Honor of the 60th Birthday of Peter D. Lax |date=8 March 2013 |publisher=Springer Science & Business Media |isbn=978-1-4613-9583-6 |pages=65–67 |url=https://books.google.com/books?id=mJfbBwAAQBAJ&pg=PA65 |language=en}}</ref>

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.

===Context and importance of the work===
[[Hamiltonian mechanics]] was a powerful new technique for working with [[equations of motion]]. Hamilton's advances enlarged the class of mechanical problems that could be solved. His principle of "Varying Action" was based on the [[calculus of variations]], in the general class of problems included under the [[principle of least action]] which had been studied earlier by [[Maupertuis|Pierre 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 the [[Lagrangian (field theory)|Lagrangian]] and [[Lagrange's equations]] also belongs to Hamilton.

Both the [[Lagrangian mechanics]] and Hamiltonian approaches have proven important in the study of continuous classical systems in physics, and quantum mechanical systems: the techniques find use in [[electromagnetism]], [[quantum mechanics]], [[relativity theory]] and [[quantum field theory]]. In the ''[[Dictionary of Irish Biography]]'' [[David Spearman]] writes:<ref>[https://dib.cambridge.org/home.do Dictionary of Irish Biography: Hamilton, William Rowan] {{Webarchive|url=https://web.archive.org/web/20190406170328/https://dib.cambridge.org/home.do |date=6 April 2019 }} [[Cambridge University Press]]</ref>

{{blockquote|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.}}

Many scientists, including [[Joseph Liouville|Liouville]], [[C. G. J. Jacobi|Jacobi]], [[Jean Gaston Darboux|Darboux]], [[Henri Poincaré|Poincaré]], [[Kolmogorov]], [[Ilya Prigogine|Prigogine]]<ref>{{Cite journal|last1=Petrosky|first1=T|last2=Prigogine|first2=Ilya|title=The extension of classical dynamics for unstable Hamiltonian systems|journal=Computers & Mathematics with Applications|year=1997|volume=34|issue=2–4|pages=1–44|doi=10.1016/S0898-1221(97)00116-8|language=en|doi-access=free}}</ref> and [[V. I. Arnold|Arnold]], have extended Hamilton's work, in [[mechanics]], [[differential equations]] and [[symplectic geometry]].<ref>{{Cite web|last=Hartnett|first=Kevin|title=How Physics Found a Geometric Structure for Math to Play With|url=https://www.quantamagazine.org/how-physics-gifted-math-with-a-new-geometry-20200729/|access-date=2020-07-30|website=Quanta Magazine|date=29 July 2020|language=en|archive-date=29 July 2020|archive-url=https://web.archive.org/web/20200729163345/https://www.quantamagazine.org/how-physics-gifted-math-with-a-new-geometry-20200729/|url-status=live}}</ref>

==Mathematics==

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.<ref name="EB1911"/>

===Quaternions===
{{main|History of quaternions}}
[[File:William Rowan Hamilton Plaque - geograph.org.uk - 347941.jpg|thumb|right|upright=1.4|Quaternion Plaque on [[Broom Bridge]] in Dublin]]
Hamilton made his discovery of the algebra of [[quaternion]]s in 1843.<ref name="Bruno2003"/>{{rp|210}} Among much prior related work, in 1840 [[Olinde Rodrigues|Benjamin Olinde Rodrigues]] had reached a result that amounted to their discovery in all but name.<ref>{{cite journal | author = Simon L. Altmann | title = Hamilton, Rodrigues and the quaternion scandal | journal = Mathematics Magazine | date = 1989 | volume = 62 | pages = 291–308 | doi = 10.2307/2689481 | jstor = 2689481 | issue = 5 }}</ref>

Hamilton was looking for ways of extending [[complex number]]s (which can be viewed as [[Point (geometry)|points]] on a 2-dimensional [[Argand 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 the [[Royal Canal]] in Dublin with his wife when the solution in the form of the equation

:{{bigmath|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = −1}}

occurred to him; Hamilton then carved this equation using his penknife into the side of the nearby [[Broom Bridge]] (which Hamilton called Brougham Bridge).<ref name="Bruno2003"/>{{rp|210}}

The quaternions involved abandoning the [[commutative law]], a radical step for the time. In the context of this prototype [[geometric algebra]], Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the [[cross product]] minus the [[dot product]] as [[Scalar (mathematics)|scalar]]. 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 the [[neologism]]s "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.<ref>{{Cite web |url=http://jeff560.tripod.com/v.html |title=Earliest Known Uses of Some of the Words of Mathematics (V) |access-date=15 June 2019 |archive-date=5 September 2015 |archive-url=https://web.archive.org/web/20150905141835/http://jeff560.tripod.com/v.html |url-status=live }}</ref>

===Other mathematical works===
Hamilton looked into the [[solution of the quintic]] in the [[theory of equations]], examining the results arrived at by [[Niels Henrik Abel]], [[George Jerrard]] and others in their respective research. There is Hamilton's paper on fluctuating functions in [[Fourier analysis]], and the invention of the [[hodograph]]. Of his investigations into the solutions, especially by [[numerical approximation]], of certain classes of physically-important differential equations, only parts were published, at intervals, in the ''[[Philosophical Magazine]]''.<ref name="EB1911"/>

Hamilton also introduced the [[icosian game]] or ''Hamilton's puzzle''. It is based on the concept of a [[Hamiltonian path]] in [[graph theory]].<ref name="ODNB"/>

==Publications==
* Hamilton, Sir W.R. (1853), ''[https://archive.org/details/lecturesonquater00hami Lectures on Quaternions]'' Dublin: Hodges and Smith
* Hamilton, Sir W.R., Hamilton, W.E. (ed) (1866), ''[https://archive.org/details/elementsquaterni00hamirich Elements of Quaternions]'' London: Longmans, Green, & Co.
* Hamilton, W.R. (1833), ''[https://babel.hathitrust.org/cgi/pt?id=uiug.30112084960829&view=1up&seq=84 Introductory Lecture on Astronomy]'' Dublin University Review and Quarterly Magazine Vol. I, Trinity College Dublin
* For Hamilton's mathematical papers see David R. Wilkins, [https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Papers.html Sir William Rowan Hamilton (1805–1865): Mathematical Papers]

Hamilton introduced, as a method of analysis, both quaternions and [[biquaternion]]s, the extension to eight dimensions by the introduction of complex number [[coefficient]]s. 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 in 1899 and the second in 1901. The subject index and footnotes in this second edition improved the ''Elements'' accessibility.

==Honours and awards==
Hamilton was twice awarded the [[Cunningham Medal]] of the [[Royal Irish Academy]].<ref>{{cite web|url= http://www.ria.ie/news-%281%29/cunningham-medal-awarded-to-professor-john-v--mcca.aspx?lang=ga-IE|title= Cunningham Medal Awarded to Professor John V. McCanny, MRIA|publisher= Royal Irish Academy|access-date= 31 October 2014|url-status= dead|archive-url= https://web.archive.org/web/20141031140124/http://www.ria.ie/news-%281%29/cunningham-medal-awarded-to-professor-john-v--mcca.aspx?lang=ga-IE|archive-date= 31 October 2014|df= dmy-all}}</ref> 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.<ref>{{cite web| url = http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Eloge/Eloge.html| title = Memorial Address: Sir William Rowan Hamilton| publisher = Trinity College Dublin| access-date = 31 October 2014| archive-date = 18 February 1999| archive-url = https://web.archive.org/web/19990218064215/http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Eloge/Eloge.html| url-status = live}}</ref> He was to win it again in 1848.

In 1835, being secretary to the meeting of the [[British Association for the Advancement of Science|British Association]] which was held that year in Dublin, Hamilton was [[knight]]ed by the [[Lord Lieutenant of Ireland|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.<ref>Graves (1889) Vol. III, pp. 204–206.</ref>

==Legacy==
[[File:2003 Ireland 10 Euro Sir William Hamilton Reverse.jpg|thumb|200px|[[Euro gold and silver commemorative coins (Ireland)|Irish commemorative coin]] celebrating the 200th anniversary of his birth]]
A plaque under the Broom Bridge, associated with the discovery of quaternions, was unveiled by [[Éamon de Valera]] on 13 November 1958.<ref>[http://www-history.mcs.st-and.ac.uk/history/Biographies/De_Valera.html De Valera] {{Webarchive|url=https://web.archive.org/web/20120401181844/http://www-history.mcs.st-and.ac.uk/history/Biographies/De_Valera.html |date=1 April 2012 }} School of Mathematics and Statistics
University of St Andrews, Scotland</ref><ref>{{Cite web|url=http://www.daviddarling.info/encyclopedia/H/Hamilton.html|title=Hamilton, William Rowan (1805–1865)|first=David|last=Darling|website=www.daviddarling.info|access-date=6 April 2011|archive-date=10 February 2005|archive-url=https://web.archive.org/web/20050210065259/http://www.daviddarling.info/encyclopedia/H/Hamilton.html|url-status=live}}</ref> 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.<ref name="hw20">[http://www.maths.tcd.ie/pub/ims/bull65/M6501.pdf Twenty Years of the Hamilton Walk] {{Webarchive|url=https://web.archive.org/web/20120316003357/http://www.maths.tcd.ie/pub/ims/bull65/M6501.pdf |date=16 March 2012 }} by Fiacre Ó Cairbre, Department of Mathematics, 
National University of Ireland, Maynooth (2005), Irish Math. Soc. Bulletin 65 (2010)</ref>

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. 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]].<ref name=HMI>{{cite web|title=About HMI|url=http://hamilton.tcd.ie/?subsite=main&section=about|website=hamilton.tcd.ie|publisher=Trinity College, Dublin|access-date=1 April 2015|archive-date=17 July 2006|archive-url=https://web.archive.org/web/20060717024533/http://hamilton.tcd.ie/?subsite=main&section=about|url-status=live}}</ref>

Two [[commemorative stamp]]s were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions.<ref name=Colnect.com>{{cite web|title=William Rowan Hamilton|url=https://colnect.com/en/stamps/list/country/2616-Ireland/series/10231-William_Rowan_Hamilton/theme/23-Sculptures/year/1943|website=colnect.com|access-date=8 October 2018}}</ref> A 10-[[euro]] commemorative silver [[proof coin]] was issued by the [[Central Bank of Ireland]] in 2005 to commemorate 200 years since his birth.

=== Commemorations ===

[[File:William Rowan Hamilton Plaque.jpg|thumb|The plaque on the birthplace of William Rowan Hamilton on Dominick Street in Dublin]]

{{Main|List of things named after William Rowan Hamilton}}
* [[Hamilton's equations]] are a formulation of classical mechanics.
* Numerous other concepts and objects in mechanics, such as [[Hamilton's principle]], [[Hamilton's principal function]], the [[Hamilton–Jacobi equation]], [[Cayley–Hamilton theorem|Cayley-Hamilton theorem]] are named after Hamilton. 
* The Hamiltonian is the name of both a [[Hamiltonian mechanics|function]] (classical) and an [[Hamiltonian (quantum mechanics)|operator]] (quantum) in physics, and, in a different sense, a term from [[graph theory]].
* The algebra of [[quaternions]] is usually denoted by {{math|'''H'''}}, or in [[blackboard bold]] by '''<math>\mathbb{H}</math>''', in honour of Hamilton.
* The Hamilton Building at Trinity College Dublin is named after him.<ref>{{Cite web |url=https://www.google.com/maps/place/Hamilton+Building/@53.3431379,-6.2528483,17z/data=!3m1!4b1!4m5!3m4!1s0x48670e90ff2c9f6d:0x4d8738ad2983146b!8m2!3d53.3431379!4d-6.2506596 |title=Hamilton Building TCD |access-date=8 May 2020 |archive-date=26 September 2021 |archive-url=https://web.archive.org/web/20210926202756/https://www.google.com/maps/place/Hamilton+Building/@53.3431379,-6.2528483,17z/data=%213m1%214b1%214m5%213m4%211s0x48670e90ff2c9f6d:0x4d8738ad2983146b%218m2%213d53.3431379%214d-6.2506596 |url-status=live }}</ref>

==In literature==
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]].<ref>{{Cite news|url=https://www.npr.org/templates/story/story.php?storyId=124632317|title=The Mad Hatter's Secret Ingredient: Math|website=NPR.org|access-date=3 July 2018|archive-date=16 March 2010|archive-url=https://web.archive.org/web/20100316053926/https://www.npr.org/templates/story/story.php?storyId=124632317|url-status=live}}</ref> 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.<ref>{{Cite journal |last=Anne van Weerden |date=2022-09-25 |title=Alice without quaternions: another look at the mad tea-party |doi-access=free |journal=British Journal for the History of Mathematics |language=en |volume=37 |issue=3 |pages=230–237 |doi=10.1080/26375451.2022.2085446 |issn=2637-5451}}</ref>

==Family==
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.<ref>Graves (1885) Vol. II, p. 1</ref><ref name="Hankins"/>{{rp|108}} They had three children: the journalist [[William Edwin Hamilton]] (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840).<ref>Graves (1882) Vol. I, p. xix</ref> 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.<ref>{{cite book |last1=Mollan |first1=Charles |title=It's Part of What We Are – Volumes 1 and 2 – Volume 1: Richard Boyle (1566–1643) to John Tyndall (1820–1893); Volume 2: Samuel Haughton (18210–1897) to John Stewart Bell (1928–1990): Some Irish Contributors to the Development of the Chemical and Physical Sciences |date=2007 |publisher=Charles Mollan |isbn=978-0-86027-055-3 |page=610 |url=https://books.google.com/books?id=cjnJDwAAQBAJ&pg=PA610 |language=en}}</ref> Hamilton's married life was reportedly difficult.<ref name="Bruno2003"/>{{rp|209}} In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression.<ref name="Hankins"/>{{rp|125, 126}}

==See also==
* [[List of astronomers]]
* [[List of things named after William Rowan Hamilton]]
* [[Theoretical physics]]

==References==
{{Reflist|30em}}

==Sources==

* {{cite book |last1=Hankins |first1=Thomas L. |title=William Rowan Hamilton |date=1980 |publisher= [[Johns Hopkins University Press]] |location=Baltimore and London|language=en}}
* {{cite web | title=''Life of Sir William Rowan Hamilton, Volume I'' | url=https://archive.org/details/lifeofsirwilliam01gravuoft | author= [[Robert Perceval Graves|Graves, Robert Perceval]] | publisher=Dublin: Hodges, Figgis, & Co. | date=1882}}
* {{cite web | title=''Life of Sir William Rowan Hamilton, Volume II'' | url=https://archive.org/details/lifeofsirwilliam02gravuoft | author=Graves, Robert Perceval | publisher=Dublin: Hodges, Figgis, & Co. | date=1885}}
* {{cite web | title=''Life of Sir William Rowan Hamilton, Volume III'' | url=https://archive.org/details/lifeofsirwilliam03gravuoft | author=Graves, Robert Perceval | publisher=Dublin: Hodges, Figgis, & Co. | date=1889}}
* {{cite book |last1=Barker |first1=Juliet R. V. |title=Wordsworth: A Life |date=2001 |publisher=Penguin |isbn=978-0-14-026162-2 |page=411 |language=en}}
* {{Cite book|url=https://archive.org/details/mathmathematicia00brun|url-access=registration|title=Math and mathematicians: the history of math discoveries around the world|last=Bruno|first=Leonard C.|date=2003|orig-year=1999|publisher=U X L|others=Baker, Lawrence W.|isbn=0787638137|location=Detroit, Mich.|page=|oclc=41497065}}
* Chow, Tai L. (2013). ''[https://www.crcpress.com/Classical-Mechanics/Chow/p/book/9781466569980 Classical Mechanics: Chaper 5: Hamilton Formulation of Mechanics: Description of Motion in Phase Spaces]''. CRC Press, {{ISBN|978-1466569980}}

==External links==
{{sister project links|wikt=no|n=no|v=no|b=no}}
{{EB1911 poster|Hamilton, Sir William Rowan}}
* {{MathGenealogy|id=55187}}
* {{MacTutor|id=Hamilton|title=Sir William Rowan Hamilton}}
* Wilkins, David R., ''[http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/ Sir William Rowan Hamilton]''. School of Mathematics, Trinity College, Dublin.
* [http://scienceworld.wolfram.com/biography/HamiltonWilliamRowan.html Wolfram Research's William Rowan Hamilton]
* Cheryl Haefner's [https://web.archive.org/web/20030814154355/http://www.engr.iupui.edu/~orr/webpages/cpt120/mathbios/hamil.htm Sir William Rowan Hamilton]
* [http://www.hamilton-trust.org.uk/ Hamilton Trust]
* [http://www.hamilton2005.ie/ The Hamilton year 2005 web site]
* [http://www.hamilton.tcd.ie/ The Hamilton Mathematics Institute, TCD]
* [http://www.hamilton.ie Hamilton Institute]
* [http://physicsworld.com/cws/article/print/22749 Hamilton biography]
* [https://www.emis.de/classics/Hamilton/index.html Papers by Hamilton] (transcribed and edited by David R. Wilkins, 2000)

{{Royal Astronomer of Ireland}}
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