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.

Sir William Rowan Hamilton
Sir William Rowan Hamilton (1805–1865)
Born(1805-08-04)4 August 1805
Dublin, Ireland
Died2 September 1865(1865-09-02) (aged 60)
Dublin, Ireland
NationalityIrish
Alma materTrinity College Dublin
Known forHamilton's principle
Hamiltonian mechanics
Hamiltonians
Hamilton–Jacobi equation
Quaternions
Biquaternions
Hamiltonian path
Icosian calculus
Nabla symbol
Versor
Coining the word 'tensor'
Coining the word 'scalar'
cis notation
Hamiltonian vector field
Icosian game
Universal algebra
Hodograph
Hamiltonian group
Cayley–Hamilton theorem
SpouseHelen Maria Bayly
ChildrenWilliam Edwin Hamilton, Archibald Henry Hamilton, Helen Eliza Amelia O'Regan, née Hamilton
AwardsRoyal Medal (1835)
Scientific career
FieldsMathematics, astronomy, physics
InstitutionsTrinity College, Dublin
Academic advisorsJohn Brinkley
InfluencesZerah Colburn
John T. Graves
InfluencedPeter Guthrie Tait

Hamilton's scientific career included the study of geometrical optics, ideas from Fourier analysis, and his work on quaternions which made him one of the founders of modern linear algebra.[3] He made major contributions in 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.

Early life

Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),[4] who lived in Dublin at 29 Dominick Street, later renumbered to 36.[5] 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,[4] a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. Meath.[3] James's daughter Grace, Hamilton's cousin, became the mother of Mary Elizabeth Townsend, philanthropist and co-founder of the Girls' Friendly Society.[6]

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. This has been disputed by some historians, who claim he had only a basic understanding of them.[4]: 207  At the age of seven, he had already made progress in Hebrew, 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 and Malay.[7] The emphasis on languages is attributed to the wish of Hamilton's father to see his son employed by the British East India Company.[8]

An expert mental calculator, the young Hamilton was capable of working out the result of some calculations to many decimal places. In September 1813, an 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.[4]: 208 

In reaction to his defeat, Hamilton spent less time studying languages, and more on mathematics.[9][10][11] At age ten, he stumbled across a Latin copy of Euclid; and at twelve he studied Newton's Arithmetica Universalis. He moved on to read the Principia, and by age 16 he had covered much of it, as well as some more recent works on analytic geometry and the differential calculus.[7] At this period he encountered what he believed to be a logical error in Laplace. It led to an introduction to John Brinkley, then the first Royal Astronomer of Ireland. Hamilton showed him some work on differential geometry of curves.[8]

Student years

In mid-1822 Hamilton began a systematic study of Laplace's Mécanique Céleste. In July 1823, he gained a place at Trinity College Dublin by examination, aged 18. His tutor there was Charles Boyton, a family friend.[3] Boyton brought to his attention contemporary mathematics published by the group at the École Polytechnique in Paris.[12] 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."[13]

The college awarded Hamilton two optimes, or off-the-chart grades, in Greek and in physics. He was in fact first in every subject and at every examination. He was expected to win further student honours, but his undergraduate career was curtailed.[9] He did take degrees in both classics and mathematics (BA in 1827, MA in 1837).

Hamilton was aiming to win a Trinity College fellowship by competitive examination.[3] But that ambition was overtaken by events, after Brinkley in 1826 was made Bishop of Cloyne.[14] Hamilton was still an undergraduate, when he was appointed in 1827 to the vacant posts left by Brinkley's departure, Andrews Professor of Astronomy and Royal Astronomer of Ireland.[4]: 209 

Personal life and poetry

In 1824, Hamilton was introduced at Edgeworthstown to the novelist Maria Edgeworth, by the Rev. Richard Butler, the vicar of Trim, County Meath to whom his uncle James Hamilton was curate.[15][16]: 5, 34  During the same period, his uncle introduced him to the Disney family at Summerhill, County Meath. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister.[16]: 37 [17]

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.[16]: 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.[17]

In 1825, Hamilton met Arabella Lawrence, younger sister of Sarah Lawrence, a significant correspondent and frank critic of his poetry. It was a contact he made through Maria Edgeworth's circle.[16]: 26 [18]

At Dunsink

Hamilton, now Royal Astronomer of Ireland, took up residence at Dunsink Observatory where he spent the rest of his life.[10] He was there from 1827 until his death in 1865.[19] In his early years at Dunsink, Hamilton observed the heavens quite regularly;[20] he left routine observation to his assistant Charles Thompson.[21][22] Hamilton's 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.[23] Felicia Hemans wrote her poem The Prayer of the Lonely Student after hearing one of his lectures.[24]

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.[3] In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin.[25][26]

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.[27] One call was to Sarah Lawrence's school at Gatacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes.[28] They visited William Wordsworth at Rydal Mount in September of that year, where Caesar Otway was also present.[29][30]: 410  After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple".[31]

Master Noakes, the mental calculator, 1827 lithograph

When Wordsworth visited Dublin in summer 1829, in a party with John Marshall and his family, he stayed at Dunsink with Hamilton.[30]: 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.[16][32]

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.[33][34]

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.[9][35] He is buried in Mount Jerome Cemetery in Dublin.

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Hamilton made important contributions to optics and to classical mechanics.

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.[7] Over this period, Hamilton gained appreciation for the nature and importance of optics.[36]

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 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, orthorhombic or triclinic crystals).[37] 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.[7] Hamilton found it from the geometry of the wave surface introduced by Augustin-Jean Fresnel, which has singular points. 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,[38]

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 Pierre Louis Maupertuis, Euler, Joseph Louis Lagrange and others. Hamilton's analysis uncovered 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 and Lagrange's equations belongs also 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:[39]

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 Liouville, Jacobi, Darboux, Poincaré, Kolmogorov, Prigogine[40] and Arnold, have extended Hamilton's work, in mechanics, differential equations and symplectic geometry.[41]

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.[7]

Quaternions
Quaternion Plaque on Broom Bridge

Hamilton made his discovery of the algebra of quaternions in 1843.[4]: 210  Among much prior related work, in 1840 Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name.[42]

Hamilton was looking for ways of extending complex numbers (which can be viewed as 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

i2 = j2 = k2 = 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).[4]: 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. 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 neologisms "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.[43]

Other mathematical works

Hamilton looked into the solution of the quintic in the theory of equations, examining of the results arrived at by Niels Henrik Abel, George Jerrard and others in their researches. 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.[7]

Hamilton also introduced the icosian game or Hamilton's puzzle. It is based on the concept of a Hamiltonian path in graph theory.[3]

Publications

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.

Honours and awards

Hamilton was twice awarded the Cunningham Medal of the Royal Irish Academy.[44] 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.[45] 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, Hamilton 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.[46]

Legacy
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.[47][48] 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.[49]

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.[50]

Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions.[51] 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
  • 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 are named after Hamilton.
  • The Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and, in a different sense, a term from graph theory.
  • The algebra of quaternions is usually denoted by H, or in blackboard bold by , in honour of Hamilton.
  • The Hamilton Building at Trinity College Dublin is named after him.[52]

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.[53] 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.[54]

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.[55][16]: 108  They had three children: William Edwin Hamilton (born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840).[56] 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.[57] Hamilton's married life was reportedly difficult.[4]: 209  In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression.[16]: 125, 6 

See also

References
  1. 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.
  2. Graves (1882) Vol. I, p. 1
  3. Lewis, Albert (2004). "Hamilton, William Rowan (1805–1865)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/12148. (Subscription or UK public library membership required.)
  4. Bruno (2003)
  5. Graves (1882) Vol. I, p. 1.
  6. "Townsend [née Butler], Mary Elizabeth (1841–1918), philanthropist". Oxford Dictionary of National Biography (online ed.). Oxford University Press. 2004. doi:10.1093/ref:odnb/56691. ISBN 978-0-19-861412-8. Retrieved 13 March 2022. (Subscription or UK public library membership required.)
  7. Chisholm, Hugh, ed. (1911). "Hamilton, Sir William Rowan" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
  8. Stephen, Leslie; Lee, Sidney, eds. (1890). "Hamilton, William Rowan" . Dictionary of National Biography. Vol. 24. London: Smith, Elder & Co.
  9. Fountain & Koningsveld (2013)
  10. O'Connor, John J.; Robertson, Edmund F., "Sir William Rowan Hamilton", MacTutor History of Mathematics archive, University of St Andrews
  11. Graves, Robert Perceval (1842). "Our portrait gallery – No. XXVI. Sir William R. Hamilton". Dublin University Magazine. 19: 94–110. Archived from the original on 17 November 2017. Retrieved 13 May 2010.
  12. "Boyton, Charles, Dictionary of Irish Biography". www.dib.ie.
  13. Sir W. R. Hamilton Archived 7 May 2019 at the Wayback Machine The Gentleman's magazine. vol 220, 1866 Jan-Jun, p. 129
  14. Wayman, P. A. "Brinkley, John (1766/7–1835)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/3438. (Subscription or UK public library membership required.)
  15. Foster, Joseph (1888–1892). "Butler, Richard (3)" . Alumni Oxonienses: the Members of the University of Oxford, 1715–1886. Oxford: Parker and Co – via Wikisource.
  16. Hankins (1980)
  17. Brown, Daniel (2012). "William Rowan Hamilton and William Wordsworth: the Poetry of Science". Studies in Romanticism. 51 (4): 475–501. ISSN 0039-3762. JSTOR 24247229.
  18. Brown, Daniel (2012). "William Rowan Hamilton and William Wordsworth: the Poetry of Science". Studies in Romanticism. 51 (4): 490. ISSN 0039-3762. JSTOR 24247229.
  19. Graves (1889) Vol. III, p. 404
  20. Graves (1882) Vol. I, p. 326
  21. Graves (1882) Vol. I, p. 285
  22. Graves (1882) Vol. I, p. 409
  23. Graves (1882) Vol. I, p. 655
  24. 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."
  25. Graves (1882) Vol. I, p. 230
  26. Blain, Virginia H. "Hamilton, Eliza Mary (1807–1851)". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/61561. (Subscription or UK public library membership required.)
  27. Mollan, Charles (15 November 2007). 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. Charles Mollan. p. 603. ISBN 978-0-86027-055-3.
  28. The Kaleidoscope: or, Literary and scientific mirror. 1828. p. 95.
  29. Gill, Stephen (1990). William Wordsworth: A Biography. Oxford University Press. p. 355. ISBN 978-0-19-282747-0.
  30. Barker (2001)
  31. Brown, Daniel (2012). "William Rowan Hamilton and William Wordsworth: the Poetry of Science". Studies in Romanticism. 51 (4): 478. ISSN 0039-3762. JSTOR 24247229.
  32. Brown, Daniel (2012). "William Rowan Hamilton and William Wordsworth: the Poetry of Science". Studies in Romanticism. 51 (4): 49–50, 52. ISSN 0039-3762. JSTOR 24247229.
  33. Paley, Morton D. (1999). Coleridge's Later Poetry. Clarendon Press. p. 26. ISBN 978-0-19-818685-4.
  34. Graves (1882) Vol. I, p. 191
  35. Reville, William (26 February 2004). "Ireland's Greatest Mathematician" (PDF). The Irish Times. Archived (PDF) from the original on 4 January 2015. Retrieved 4 January 2015.
  36. "The BK Bounce". The BK Bounce. 2018. doi:10.5040/9781350971424. Archived from the original on 26 September 2021. Retrieved 2 May 2021.
  37. Born, Max; Wolf, Emil (28 February 2000). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. CUP Archive. p. 805. ISBN 978-0-521-78449-8.
  38. Chorin, AlexandreJ; Majda, Andrew J. (8 March 2013). Wave Motion: Theory, Modelling, and Computation: Proceedings of a Conference in Honor of the 60th Birthday of Peter D. Lax. Springer Science & Business Media. pp. 65–67. ISBN 978-1-4613-9583-6.
  39. Dictionary of Irish Biography: Hamilton, William Rowan Archived 6 April 2019 at the Wayback Machine Cambridge University Press
  40. Petrosky, T; Prigogine, Ilya (1997). "The extension of classical dynamics for unstable Hamiltonian systems". Computers & Mathematics with Applications. 34 (2–4): 1–44. doi:10.1016/S0898-1221(97)00116-8. Archived from the original on 25 June 2021. Retrieved 26 June 2021.
  41. Hartnett, Kevin (29 July 2020). "How Physics Found a Geometric Structure for Math to Play With". Quanta Magazine. Archived from the original on 29 July 2020. Retrieved 30 July 2020.
  42. Simon L. Altmann (1989). "Hamilton, Rodrigues and the quaternion scandal". Mathematics Magazine. 62 (5): 291–308. doi:10.2307/2689481. JSTOR 2689481.
  43. "Earliest Known Uses of Some of the Words of Mathematics (V)". Archived from the original on 5 September 2015. Retrieved 15 June 2019.
  44. "Cunningham Medal Awarded to Professor John V. McCanny, MRIA". Royal Irish Academy. Archived from the original on 31 October 2014. Retrieved 31 October 2014.
  45. "Memorial Address: Sir William Rowan Hamilton". Trinity College Dublin. Archived from the original on 18 February 1999. Retrieved 31 October 2014.
  46. Graves (1889) Vol. III, pp. 204–206.
  47. De Valera Archived 1 April 2012 at the Wayback Machine School of Mathematics and Statistics University of St Andrews, Scotland
  48. Darling, David. "Hamilton, William Rowan (1805-1865)". www.daviddarling.info. Archived from the original on 10 February 2005. Retrieved 6 April 2011.
  49. Twenty Years of the Hamilton Walk Archived 16 March 2012 at the Wayback Machine by Fiacre Ó Cairbre, Department of Mathematics, National University of Ireland, Maynooth (2005), Irish Math. Soc. Bulletin 65 (2010)
  50. "About HMI". hamilton.tcd.ie. Trinity College, Dublin. Archived from the original on 17 July 2006. Retrieved 1 April 2015.
  51. "William Rowan Hamilton". colnect.com. Colnect.com. Retrieved 8 October 2018.
  52. "Hamilton Building TCD". Archived from the original on 26 September 2021. Retrieved 8 May 2020.
  53. "The Mad Hatter's Secret Ingredient: Math". NPR.org. Archived from the original on 16 March 2010. Retrieved 3 July 2018.
  54. Anne van Weerden, 'Alice without quaternions: another look at the mad tea-party', British Journal for the History of Mathematics, published online 25 September 2022.
  55. Graves (1885) Vol. II, p. 1
  56. Graves (1882) Vol. I, p. xix
  57. Mollan, Charles (15 November 2007). 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. Charles Mollan. p. 610. ISBN 978-0-86027-055-3.

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