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Sir William Rowan Hamilton
LL.D Legum Doctor (Latin: “teacher of the laws”) (LL.D.) or, in English, Doctor of Laws, is a doctorate-level academic degree in law or an honorary degree, depending on the jurisdiction. The double “L” in the abbreviation refers to the early ...
, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the
Andrews Professor of Astronomy The Andrews Professor of Astronomy is a chair in astronomy in Trinity College Dublin was established in 1783 in conjunction with the establishment of Dunsink Observatory. Dunsink was founded in 1785 following a bequest by Provost Francis Andrew ...
at
Trinity College Dublin , name_Latin = Collegium Sanctae et Individuae Trinitatis Reginae Elizabethae juxta Dublin , motto = ''Perpetuis futuris temporibus duraturam'' (Latin) , motto_lang = la , motto_English = It will last i ...
, and Royal Astronomer of Ireland, living at Dunsink Observatory. Hamilton's scientific career included the study of geometrical optics, ideas from
Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
, and his work on
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
which made him one of the founders of modern
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
. He made major contributions in optics,
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
. His work was fundamental to modern
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, particularly his reformulation of
Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motio ...
, now called
Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
. It is now central both to
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
and to
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.


Early life

Hamilton was the fourth of nine children born to Sarah Hutton (1780–1817) and Archibald Hamilton (1778–1819),Bruno (2003) who lived in Dublin at 29 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 Trinity College may refer to: Australia * Trinity Anglican College, an Anglican coeducational primary and secondary school in , New South Wales * Trinity Catholic College, Auburn, a coeducational school in the inner-western suburbs of Sydney, New ...
who ran a school in Talbots Castle in
Trim Trim or TRIM may refer to: Cutting * Cutting or trimming small pieces off something to remove them ** Book trimming, a stage of the publishing process ** Pruning, trimming as a form of pruning often used on trees Decoration * Trim (sewing), ...
, Co. Meath. James's daughter Grace, Hamilton's cousin, became the mother of
Mary Elizabeth Townsend Mary Elizabeth Townsend (23 July 1841 - 14 June 1918) was a British philanthropist and co-founder of the Girls' Friendly Society. Early life Mary Elizabeth Butler was born in Kilkenny, then part of the United Kingdom of Great Britain and Irel ...
, philanthropist and co-founder of the
Girls' Friendly Society The Girls' Friendly Society In England And Wales (or just GFS) is a charitable organisation that empowers girls and young women aged 5 to 25, encouraging them to develop their full potential through programs that provide training, confidence bu ...
. 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. At the age of seven, he had already made progress in
Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...
, and before he was 13 he had acquired, under the care of his uncle a dozen languages: classical and modern European languages, Persian,
Arabic Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter ...
, Hindustani,
Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cultural diffusion ...
, Marathi and Malay. The emphasis on languages is attributed to the wish of Hamilton's father to see his son employed by the
British East India Company The East India Company (EIC) was an English, and later British, joint-stock company founded in 1600 and dissolved in 1874. It was formed to trade in the Indian Ocean region, initially with the East Indies (the Indian subcontinent and South ...
. 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 Prodigy, Prodigies or The Prodigy may refer to: * Child prodigy, a child who produces meaningful output to the level of an adult expert performer ** Chess prodigy, a child who can beat experienced adult players at chess Arts, entertainment, and ...
, 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. In reaction to his defeat, Hamilton spent less time studying languages, and more on mathematics.Fountain & Koningsveld (2013) At age ten, he stumbled across a
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
copy of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
; and at twelve he studied Newton's ''
Arithmetica Universalis ''Arithmetica Universalis'' ("Universal Arithmetic") is a mathematics text by Isaac Newton. Written in Latin, it was edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Camb ...
''. 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 In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
and the
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve ...
. At this period he encountered what he believed to be a logical error in
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
. It led to an introduction to John Brinkley, then Royal Astronomer of Ireland. Hamilton showed him some work on
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
of curves.


Student years

In mid-1822 Hamilton began a systematic study of Laplace's '' Mécanique Céleste''. In November and December 1822 he completed his first three original mathematical papers. On his first visit to Dunsink Observatory, he showed two of them to Brinkley, who asked for a more developed form. Hamilton complied, and early in 1823 Brinkley approved the amended version. In July 1823, he gained a place at
Trinity College Dublin , name_Latin = Collegium Sanctae et Individuae Trinitatis Reginae Elizabethae juxta Dublin , motto = ''Perpetuis futuris temporibus duraturam'' (Latin) , motto_lang = la , motto_English = It will last i ...
by examination, aged 18. His tutor there was Charles Boyton, a family friend. Boyton brought to his attention contemporary mathematics published by the group at the
École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
in Paris. 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." The college awarded Hamilton two
optime At the University of Cambridge in England, a "Wrangler" is a student who gains first-class honours in the final year of the university's degree in mathematics. The highest-scoring student is the Senior Wrangler, the second highest is the Se ...
s, or off-the-chart grades, in
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
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. 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. But that ambition was overtaken by events, after Brinkley in 1826 was made Bishop of Cloyne. 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.


Personal life and poetry

In 1824, Hamilton was introduced at
Edgeworthstown Edgeworthstown or Mostrim () is a small town in County Longford, Ireland. The town is in the east of the county, near the border with County Westmeath. Nearby towns are Longford 12 km to the west, Mullingar 26 km to the east, Athlone ...
to the novelist Maria Edgeworth, by the Rev. Richard Butler, the vicar of
Trim, County Meath Trim () is a town in County Meath, Ireland. It is situated on the River Boyne and has a population of 9,194. The town is noted for Trim Castle – the largest Norman castle in Ireland. One of the two cathedrals of the United Dioceses of Meat ...
to whom his uncle James Hamilton was curate.Hankins (1980) During the same period, his uncle introduced him to the Disney family at
Summerhill, County Meath Summerhill () is a heritage village in County Meath, Ireland. It is located in the south of the county, between Trim and Kilcock on the R158 and west of Dunboyne on the R156. It is the site of one of the most important battles in 17th centur ...
. The Disney sons attended Trinity College, and Hamilton had friends among them. At Summerhill, he met Catherine Disney, their sister.Hankins (1980) 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.Hankins (1980) 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 Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor, experimental photographer who invented the blueprint and did botanical wo ...
that during this period he might have become a poet. 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.


At Dunsink

Hamilton, now Royal Astronomer of Ireland, took up residence at Dunsink Observatory where he spent the rest of his life. He was there from 1827 until his death in 1865. In his early years at Dunsink, Hamilton observed the heavens quite regularly; he left routine observation to his assistant Charles Thompson. Hamilton's sisters also supported the observatory's work. The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women. Felicia Hemans wrote her poem ''The Prayer of the Lonely Student'' after hearing one of his lectures.


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. In 1827, Hamilton wrote to his sister Grace about "some of" the Lawrence sisters having met his sister Eliza in Dublin. Newly appointed to the Observatory, Hamilton set off on a tour in Ireland and England with Alexander Nimmo, who was coaching him on
latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...
and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
. One call was to Sarah Lawrence's school at Gatacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes. They visited
William Wordsworth William Wordsworth (7 April 177023 April 1850) was an English Romantic poet who, with Samuel Taylor Coleridge, helped to launch the Romantic Age in English literature with their joint publication '' Lyrical Ballads'' (1798). Wordsworth's ' ...
at
Rydal Mount Rydal Mount is a house in the small village of Rydal, near Ambleside in the English Lake District. It is best known as the home of the poet William Wordsworth from 1813 to his death in 1850. It is currently operated as a writer's home museum ...
in September of that year, where
Caesar Otway Caesar Otway (1780–1842) was born at Castle Otway near Nenagh, Co. Tipperary, Ireland in 1780. He was an Irish author and clergyman who wanted to study and improve the condition of the poor. Life His parents were Cooke and Elizabeth Otway an ...
was also present.Barker (2001) After the visit, Hamilton sent numerous poems to Wordsworth, becoming a "poetic disciple". When Wordsworth visited Dublin in summer 1829, in a party with
John Marshall John Marshall (September 24, 1755July 6, 1835) was an American politician and lawyer who served as the fourth Chief Justice of the United States from 1801 until his death in 1835. He remains the longest-serving chief justice and fourth-longes ...
and his family, he stayed at Dunsink with Hamilton. On a second tour in England with Nimmo in 1831, Hamilton parted from him at
Birmingham Birmingham ( ) is a city and metropolitan borough in the metropolitan county of West Midlands in England. It is the second-largest city in the United Kingdom with a population of 1.145 million in the city proper, 2.92 million in the We ...
, to visit the Lawrence sisters and family on his mother's side in the Liverpool area. They met up again in the
Lake District The Lake District, also known as the Lakes or Lakeland, is a mountainous region in North West England. A popular holiday destination, it is famous for its lakes, forests, and mountains (or '' fells''), and its associations with William Wordswor ...
, where they climbed Helvellyn and had tea with Wordsworth. Hamilton returned to Dublin, via Edinburgh and Glasgow. Hamilton visited
Samuel Taylor Coleridge Samuel Taylor Coleridge (; 21 October 177225 July 1834) was an English poet, literary critic, philosopher, and theologian who, with his friend William Wordsworth, was a founder of the Romantic Movement in England and a member of the Lak ...
at
Highgate Highgate ( ) is a suburban area of north London at the northeastern corner of Hampstead Heath, north-northwest of Charing Cross. Highgate is one of the most expensive London suburbs in which to live. It has two active conservation organisat ...
, 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 William Roscoe (8 March 175330 June 1831) was an English banker, lawyer, and briefly a Member of Parliament. He is best known as one of England's first abolitionists, and as the author of the poem for children ''The Butterfly's Ball, and the G ...
who had died in 1831.


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. He proposed again in 1831 to Ellen de Vere, a sister of the poet Aubrey De Vere (1814-1902), who declined as well. Hamilton eventually married Helen Marie Bayly in 1833, a country preacher's daughter, and had three children with her:
William Edwin Hamilton William Edwin Hamilton (10 May 1834 – 17 March 1902) was the elder son of the Irish mathematician Sir William Rowan Hamilton and Lady Helen Maria Hamilton Bayly. Early life in Ireland William Edwin Hamilton was born at Dunsink Observatory, in ...
(born 1834), Archibald Henry (born 1835), and Helen Elizabeth (born 1840). Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, shy, timid, and chronically ill.


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 Gout ( ) is a form of inflammatory arthritis characterized by recurrent attacks of a red, tender, hot and swollen joint, caused by deposition of monosodium urate monohydrate crystals. Pain typically comes on rapidly, reaching maximal intens ...
. He is buried in
Mount Jerome Cemetery Mount is often used as part of the name of specific mountains, e.g. Mount Everest. Mount or Mounts may also refer to: Places * Mount, Cornwall, a village in Warleggan parish, England * Mount, Perranzabuloe, a hamlet in Perranzabuloe parish, ...
in Dublin.


Physics

Hamilton made important contributions to
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
and to
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
. 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 The Royal Irish Academy (RIA; ga, Acadamh Ríoga na hÉireann), based in Dublin, is an academic body that promotes study in the sciences, humanities and social sciences. It is Ireland's premier learned society and one its leading cultural ...
. 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. Over this period, Hamilton gained appreciation for the nature and importance of optics. In 1827, Hamilton presented a theory of a single function, now known as
Hamilton's principal function Buck Meadows (formerly Hamilton's and Hamilton's Station) is a census-designated place in Mariposa County, California, United States. It is located east-northeast of Smith Peak, at an elevation of . The population was 21 at the 2020 census. Buc ...
, that brings together mechanics and optical theory. It helped to establish foundations of the
wave theory of light In physics, physical optics, or wave optics, is the branch of optics that studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. This usage tends not to include effect ...
in
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
. 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 In crystallography, the monoclinic crystal system is one of the seven crystal systems. A crystal system is described by three vectors. In the monoclinic system, the crystal is described by vectors of unequal lengths, as in the orthorhombic ...
,
orthorhombic In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with ...
or triclinic crystals). 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 Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular the ...
. Hamilton found it from the geometry of the
wave surface In mathematics, Fresnel's wave surface, found by Augustin-Jean Fresnel in 1822, is a quartic surface In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there ...
introduced by
Augustin-Jean Fresnel Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular th ...
, which has
singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
s. 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, 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 ''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journa ...
'' for 1834 and 1835 there are two papers on the subject.


Context and importance of the work

Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
was a powerful new technique for working with
equations of motion In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
. 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 The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, in the general class of problems included under the
principle of least action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
which had been studied earlier by
Pierre Louis Maupertuis Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
,
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
, 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 In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
belongs also to Hamilton. Both the
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
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 In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
. In the ''
Dictionary of Irish Biography The ''Dictionary of Irish Biography'' (DIB) is a biographical dictionary of notable Irish people and people not born in the country who had notable careers in Ireland, including both Northern Ireland and the Republic of Ireland.David Spearman writes: Many scientists, including Liouville,
Jacobi Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, ...
, Darboux, Poincaré,
Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, Prigogine and
Arnold Arnold may refer to: People * Arnold (given name), a masculine given name * Arnold (surname), a German and English surname Places Australia * Arnold, Victoria, a small town in the Australian state of Victoria Canada * Arnold, Nova Scotia U ...
, have extended Hamilton's work, in
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
,
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
and
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
.


Mathematics

Hamilton's
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
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.


Quaternions

Hamilton made his discovery of the algebra of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s in 1843. Among much prior related work, in 1840 Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name. Hamilton was looking for ways of extending
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (which can be viewed as
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * 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 The Royal Canal ( ga, An Chanáil Ríoga) is a canal originally built for freight and passenger transportation from Dublin to Longford in Ireland. It is one of two canals from Dublin to the River Shannon and was built in direct competition ...
in Dublin with his wife when the solution in the form of the equation : occurred to him; Hamilton then carved this equation using his penknife into the side of the nearby
Broom Bridge Broom Bridge ( Irish: ''Droichead Broome''), also called Broome Bridge, and sometimes Brougham Bridge, is a bridge along Broombridge Road which crosses the Royal Canal in Cabra, Dublin, Ireland. Broome Bridge is named after William Broome, on ...
(which Hamilton called Brougham Bridge). The quaternions involved abandoning the commutative law, a radical step for the time. In the context of this prototype
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ...
, Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
minus the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
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
neologism A neologism Ancient_Greek.html"_;"title="_from_Ancient_Greek">Greek_νέο-_''néo''(="new")_and_λόγος_/''lógos''_meaning_"speech,_utterance"is_a_relatively_recent_or_isolated_term,_word,_or_phrase_that_may_be_in_the_process_of_entering_com ...
s "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.


Other mathematical works

Hamilton looked into the
solution of the quintic In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, ...
in the
theory of equations In algebra, the theory of equations is the study of algebraic equations (also called "polynomial equations"), which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an ...
, 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 In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...
, and the invention of the
hodograph A hodograph is a diagram that gives a vectorial visual representation of the movement of a body or a fluid. It is the locus of one end of a variable vector, with the other end fixed. The position of any plotted data on such a diagram is proport ...
. 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 The ''Philosophical Magazine'' is one of the oldest scientific journals published in English. It was established by Alexander Tilloch in 1798;John Burnett"Tilloch, Alexander (1759–1825)" Oxford Dictionary of National Biography, Oxford Univer ...
''. Hamilton also introduced the icosian game or ''Hamilton's puzzle''. It is based on the concept of a Hamiltonian path in
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
.


Publications

* Hamilton, Sir W.R. (1853),
Lectures on Quaternions
' Dublin: Hodges and Smith * Hamilton, Sir W.R., Hamilton, W.E. (ed) (1866),
Elements of Quaternions
' London: Longmans, Green, & Co. * Hamilton, W.R. (1833),
Introductory Lecture on Astronomy
' Dublin University Review and Quarterly Magazine Vol. I, Trinity College Dublin * For Hamilton's mathematical papers see David R. Wilkins

Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by introduction of complex number
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
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 William Edwin Hamilton (10 May 1834 – 17 March 1902) was the elder son of the Irish mathematician Sir William Rowan Hamilton and Lady Helen Maria Hamilton Bayly. Early life in Ireland William Edwin Hamilton was born at Dunsink Observatory, in ...
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 Charles Jasper Joly (27 June 1864 – 4 January 1906) was an Irish mathematician and astronomer who became Royal Astronomer of Ireland.Obituary, New York Times, 5 January 1906 Life He was born at St Catherine's Rectory, Hop Hill, Tullamor ...
, 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 The Cunningham Medal is the premier award of the Royal Irish Academy. It is awarded every three years in recognition of "outstanding contributions to scholarship and the objectives of the Academy". History It was which was established in 1796 at ...
of the
Royal Irish Academy The Royal Irish Academy (RIA; ga, Acadamh Ríoga na hÉireann), based in Dublin, is an academic body that promotes study in the sciences, humanities and social sciences. It is Ireland's premier learned society and one its leading cultural ...
. The first award, in 1834, was for his work on conical refraction, for which he also received the
Royal Medal The Royal Medal, also known as The Queen's Medal and The King's Medal (depending on the gender of the monarch at the time of the award), is a silver-gilt medal, of which three are awarded each year by the Royal Society, two for "the most important ...
of the Royal Society the following year. 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
knight A knight is a person granted an honorary title of knighthood by a head of state (including the Pope) or representative for service to the monarch, the Christian denomination, church or the country, especially in a military capacity. Knighthood ...
ed by the
lord-lieutenant A lord-lieutenant ( ) is the British monarch's personal representative in each lieutenancy area of the United Kingdom. Historically, each lieutenant was responsible for organising the county's militia. In 1871, the lieutenant's responsibilit ...
. Other honours rapidly succeeded, among which his election in 1837 to the president's chair in the
Royal Irish Academy The Royal Irish Academy (RIA; ga, Acadamh Ríoga na hÉireann), based in Dublin, is an academic body that promotes study in the sciences, humanities and social sciences. It is Ireland's premier learned society and one its leading cultural ...
, and the rare distinction of being made a corresponding member of the
Saint Petersburg Academy of Sciences The Russian Academy of Sciences (RAS; russian: Росси́йская акаде́мия нау́к (РАН) ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across t ...
. Later, in 1864, the newly established
United States National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the N ...
elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.


Legacy

A plaque under the Broom Bridge, associated with the discovery of quaternions, was unveiled by
Éamon de Valera Éamon de Valera (, ; first registered as George de Valero; changed some time before 1901 to Edward de Valera; 14 October 1882 – 29 August 1975) was a prominent Irish statesman and political leader. He served several terms as head of govern ...
on 13 November 1958. Since 1989, the
National University of Ireland, Maynooth The National University of Ireland, Maynooth (NUIM; ga, Ollscoil na hÉireann Mhá Nuad), commonly known as Maynooth University (MU), is a constituent university of the National University of Ireland in Maynooth, County Kildare, Ireland. It ...
has organised a pilgrimage called the ''
Hamilton Walk The Hamilton Walk from Dunsink Observatory to Broom Bridge on the Royal Canal in Dublin takes place on 16 October each year. This is the anniversary of the day in 1843 when William Rowan Hamilton discovered the non-commutative algebraic system k ...
'', 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.Twenty Years of the Hamilton Walk
by Fiacre Ó Cairbre, Department of Mathematics, National University of Ireland, Maynooth (2005), Irish Math. Soc. Bulletin 65 (2010)
The
Hamilton Institute The Hamilton Institute is a multi-disciplinary research centre at Maynooth University, named after William Rowan Hamilton, arguably Ireland's most distinguished mathematician. The Hamilton Institute was formally established in November 2001 u ...
is an applied mathematics research institute at
Maynooth University The National University of Ireland, Maynooth (NUIM; ga, Ollscoil na hÉireann Mhá Nuad), commonly known as Maynooth University (MU), is a constituent university of the National University of Ireland in Maynooth, County Kildare, Ireland. I ...
and the
Royal Irish Academy The Royal Irish Academy (RIA; ga, Acadamh Ríoga na hÉireann), based in Dublin, is an academic body that promotes study in the sciences, humanities and social sciences. It is Ireland's premier learned society and one its leading cultural ...
holds an annual public Hamilton lecture at which
Murray Gell-Mann Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...
, 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 , name_Latin = Collegium Sanctae et Individuae Trinitatis Reginae Elizabethae juxta Dublin , motto = ''Perpetuis futuris temporibus duraturam'' (Latin) , motto_lang = la , motto_English = It will last i ...
marked the year by launching the Hamilton Mathematics Institute. Two
commemorative stamp A commemorative stamp is a postage stamp, often issued on a significant date such as an anniversary, to honor or commemorate a place, event, person, or object. The ''subject'' of the commemorative stamp is usually spelled out in print, unlike defi ...
s were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions. A 10-
euro The euro ( symbol: €; code: EUR) is the official currency of 19 out of the member states of the European Union (EU). This group of states is known as the eurozone or, officially, the euro area, and includes about 340 million citizens . ...
commemorative silver proof coin was issued by the
Central Bank of Ireland The Central Bank of Ireland ( ga, Banc Ceannais na hÉireann) is Ireland's central bank, and as such part of the European System of Central Banks (ESCB). It is the country's financial services regulator for most categories of financial firms ...
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 Buck Meadows (formerly Hamilton's and Hamilton's Station) is a census-designated place in Mariposa County, California, United States. It is located east-northeast of Smith Peak, at an elevation of . The population was 21 at the 2020 census. Buc ...
, 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 In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
. * The algebra of
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
is usually denoted by , or in blackboard bold by \mathbb, in honour of Hamilton. * The Hamilton Building at Trinity College Dublin is named after him.


In literature

It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by
Charles Lutwidge Dodgson Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ...
in ''
Alice in Wonderland ''Alice's Adventures in Wonderland'' (commonly ''Alice in Wonderland'') is an 1865 English novel by Lewis Carroll. It details the story of a young girl named Alice who falls through a rabbit hole into a fantasy world of anthropomorphic creatur ...
''. In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
. 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.


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. They had three children:
William Edwin Hamilton William Edwin Hamilton (10 May 1834 – 17 March 1902) was the elder son of the Irish mathematician Sir William Rowan Hamilton and Lady Helen Maria Hamilton Bayly. Early life in Ireland William Edwin Hamilton was born at Dunsink Observatory, in ...
(born 1834), Archibald Henry (born 1835) and Helen Eliza Amelia (born 1840). 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. Hamilton's married life was reportedly difficult. In the troubled period of the early 1840s, his sister Sydney ran his household; when Helen returned, he was happier after some depression.


See also

*
List of astronomers The following is a list of astronomers, astrophysicists and other notable people who have made contributions to the field of astronomy. They may have won major prizes or awards, developed or invented widely used techniques or technologies within as ...
* List of things named after William Rowan Hamilton *
Theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...


References


Sources

* * * * * * * * Chow, Tai L. (2013).
Classical Mechanics: Chaper 5: Hamilton Formulation of Mechanics: Description of Motion in Phase Spaces
'. CRC Press,


External links

* * *Wilkins, David R.,
Sir William Rowan Hamilton
'. School of Mathematics, Trinity College, Dublin.

*Cheryl Haefner'
Hamilton TrustThe Hamilton year 2005 web siteThe Hamilton Mathematics Institute, TCDHamilton InstituteHamilton biography
{{DEFAULTSORT:Hamilton, William Rowan 1805 births 1865 deaths 19th-century Irish people 19th-century British mathematicians Linear algebraists Alumni of Trinity College Dublin Irish people of Scottish descent British physicists Burials at Mount Jerome Cemetery and Crematorium Directors of Dunsink Observatory Members of the Royal Irish Academy Members of the Prussian Academy of Sciences Fellows of the American Academy of Arts and Sciences Foreign associates of the National Academy of Sciences Corresponding members of the Saint Petersburg Academy of Sciences Irish Anglicans Irish astronomers Irish knights Irish mathematicians Irish physicists Mental calculators Optical physicists People from Cabra, Dublin Royal Medal winners Theoretical physicists Mathematical physicists