Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of

pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...

, and was a professor at

Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any ...

for 35 years. He postulated what is now known as the Cayley–Hamilton theorem—that every

square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Squ ...

is a root of its own

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to

Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, ...

' concept of

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to ...

s. In group theory,

Cayley table Named after the 19th-century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additi ...

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...

s, and

Cayley's theorem In the mathematical discipline of group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group. More specifically, is isomorphic to a subgroup of the symmetric gro ...

are named in his honour, as well as

Cayley's formula In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n, the number of trees on n labeled vertices is n^. The formula equivalently counts the spanning trees of a ...

in combinatorics.

Early life

Arthur Cayley was born in

Richmond, London Richmond is a town in south-west London,The London Government Act 1963 (c.33) (as amended) categorises the London Borough of Richmond upon Thames as an Outer London borough. Although it is on both sides of the River Thames, the Boundary Commis ...

, England, on 16 August 1821. His father, Henry Cayley, was a distant cousin of

George Cayley Sir George Cayley, 6th Baronet (27 December 1773 – 15 December 1857) was an English engineer, inventor, and aviator. He is one of the most important people in the history of aeronautics. Many consider him to be the first true scientific ...

, the

aeronautics Aeronautics is the science or art involved with the study, design process, design, and manufacturing of air flight-capable machines, and the techniques of operating aircraft and rockets within the atmosphere. While the term originally referred ...

engineer innovator, and descended from an ancient

Yorkshire Yorkshire ( ) is an area of Northern England which was History of Yorkshire, historically a county. Despite no longer being used for administration, Yorkshire retains a strong regional identity. The county was named after its county town, the ...

family. He settled in

Saint Petersburg Saint Petersburg, formerly known as Petrograd and later Leningrad, is the List of cities and towns in Russia by population, second-largest city in Russia after Moscow. It is situated on the Neva, River Neva, at the head of the Gulf of Finland ...

, Russia, as a

merchant A merchant is a person who trades in goods produced by other people, especially one who trades with foreign countries. Merchants have been known for as long as humans have engaged in trade and commerce. Merchants and merchant networks operated i ...

. His mother was Maria Antonia Doughty, daughter of William Doughty. According to some writers she was Russian, but her father's name indicates an English origin. His brother was the linguist Charles Bagot Cayley. Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently at

Blackheath, London Blackheath is an area in Southeast London, straddling the border of the Royal Borough of Greenwich and the London Borough of Lewisham. Historically within the county of Kent, it is located northeast of Lewisham, south of Greenwich, London, G ...

, where Arthur attended a private school. At age 14, he was sent to King's College School. The young Cayley enjoyed complex maths problems, and the school's master observed indications of his mathematical genius. He advised the father to educate his son not for his own business, as he had intended, but at the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

Education

At the age of 17 Cayley began residence at

, where he excelled in Greek, French, German, and Italian, as well as

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. The cause of the Analytical Society had now triumphed, and the '' Cambridge Mathematical Journal'' had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects that had been suggested by reading the ''Mécanique analytique'' of

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace. Cayley's tutor at Cambridge was George Peacock and his private coach was

William Hopkins William Hopkins Fellow of the Royal Society, FRS (2 February 179313 October 1866) was an English mathematician and geologist. He is famous as a private tutor of aspiring undergraduate University of Cambridge, Cambridge mathematicians, earning h ...

. He finished his undergraduate course by winning the place of Senior Wrangler, and the first

Smith's prize Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics and theoretical physics at the University of Cambridge from 1769. Following the reorganization in 1998, they are now awarded under the names ...

. His next step was to take the M.A. degree, and win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; during which time he took some pupils, but his main work was the preparation of 28 memoirs to the '' Mathematical Journal''.

Law career

Because of the limited tenure of his fellowship it was necessary to choose a profession; like De Morgan, Cayley chose law, and was admitted to Lincoln's Inn, London on 20 April 1846 at the age of 24. He made a specialty of

conveyancing In law, conveyancing is the transfer of legal title of real property from one person to another, or the granting of an encumbrance such as a mortgage or a lien. A typical conveyancing transaction has two major phases: the exchange of contract ...

. It was while he was a pupil at the

bar examination A bar examination is an examination administered by the bar association of a jurisdiction that a lawyer must pass in order to be admitted to the bar of that jurisdiction. Australia Administering bar exams is the responsibility of the bar associat ...

that he went to

Dublin Dublin is the capital and largest city of Republic of Ireland, Ireland. Situated on Dublin Bay at the mouth of the River Liffey, it is in the Provinces of Ireland, province of Leinster, and is bordered on the south by the Dublin Mountains, pa ...

to hear

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

's lectures on

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. The algebra of quater ...

s. His friend J. J. Sylvester, his senior by five years at Cambridge, was then an

actuary An actuary is a professional with advanced mathematical skills who deals with the measurement and management of risk and uncertainty. These risks can affect both sides of the balance sheet and require investment management, asset management, ...

, resident in London; they used to walk together round the courts of

Lincoln's Inn The Honourable Society of Lincoln's Inn, commonly known as Lincoln's Inn, is one of the four Inns of Court (professional associations for Barrister, barristers and judges) in London. To be called to the bar in order to practise as a barrister ...

, discussing the theory of invariants and covariants. During these fourteen years, Cayley produced between two and three hundred papers.

Professorship

Around 1860, Cambridge University's

Lucasian Professor of Mathematics The Lucasian Chair of Mathematics () is a mathematics professorship in the University of Cambridge, England; its holder is known as the Lucasian Professor. The post was founded in 1663 by Henry Lucas (politician), Henry Lucas, who was Cambridge U ...

( Newton's chair) was supplemented by the new Sadleirian professorship, using funds bequeathed by Lady Sadleir, with the 42-year-old Cayley as its first holder. His duties were ''"to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."'' He gave up a lucrative legal practice for a modest salary, but never regretted the exchange, since it allowed him to devote his energies to the pursuit that he liked best. He at once married and settled down in Cambridge, and (unlike Hamilton) enjoyed a home life of great happiness. Sylvester, his friend from his bachelor days, once expressed his envy of Cayley's peaceful family life, whereas the unmarried Sylvester had to fight the world all his days. At first the Sadleirian professor was paid to lecture over one of the terms of the academic year, but the university financial reform of 1886 freed funds to extend his lectures to two terms. For many years his courses were attended only by a few students who had finished their examination preparation, but after the reform the attendance numbered about fifteen. He generally lectured on his current research topic. As for his duty to the advancement of mathematical science, he published a long and fruitful series of memoirs ranging over all of pure mathematics. He also became the standing referee on the merits of mathematical papers to many societies both at home and abroad. In 1872, he was made an honorary fellow of Trinity College, and three years later an ordinary fellow, a paid position. About this time his friends subscribed for a presentation portrait.

Maxwell Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell, mathematician and physicist * Justice Maxwell (disambiguation) * Maxwell baronets, in the Baronetage of N ...

wrote an address praising Cayley's principal works, including his Chapters on the Analytical Geometry of

n

dimensions; On the theory of

Determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...

s; Memoir on the theory of Matrices; Memoirs on skew surfaces, otherwise Scrolls; and On the delineation of a Cubic Scroll. In addition to his work on

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, Cayley made fundamental contributions to

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

. Cayley and

Salmon Salmon (; : salmon) are any of several list of commercially important fish species, commercially important species of euryhaline ray-finned fish from the genera ''Salmo'' and ''Oncorhynchus'' of the family (biology), family Salmonidae, native ...

discovered the 27 lines on a

cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than ...

. Cayley constructed the

Chow variety In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow varie ...

of all curves in projective 3-space. He founded the algebro-geometric theory of

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

s. His contributions to

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

include counting the ''nⁿ''⁻²

trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only p ...

on ''n'' labeled vertices by the pioneering use of

generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...

s. In 1876, he published a ''Treatise on

Elliptic Functions In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...

''. He took great interest in the movement for the university education of women. At Cambridge the first women's colleges were Girton and Newnham. In the early days of Girton College he gave direct help in teaching, and for some years he was chairman of the council of Newnham College, in the progress of which he took the keenest interest to the last. In 1881, he received from the

Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...

Baltimore Baltimore is the most populous city in the U.S. state of Maryland. With a population of 585,708 at the 2020 census and estimated at 568,271 in 2024, it is the 30th-most populous U.S. city. The Baltimore metropolitan area is the 20th-large ...

, where Sylvester was then professor of mathematics, an invitation to deliver a course of lectures. He accepted the invitation, and lectured at Baltimore during the first five months of 1882 on the subject of the ''Abelian and Theta Functions''. In 1893, Cayley became a foreign member of the Royal Netherlands Academy of Arts and Sciences.

British Association presidency

In 1883, Cayley was President of the

British Association for the Advancement of Science The British Science Association (BSA) is a Charitable organization, charity and learned society founded in 1831 to aid in the promotion and development of science. Until 2009 it was known as the British Association for the Advancement of Scienc ...

. The meeting was held at Southport, in the north of England. As the President's address is one of the great popular events of the meeting, and brings out an audience of general culture, it is usually made as little technical as possible. took for his subject the Progress of Pure Mathematics.

The ''Collected Papers''

In 1889, the

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

began the publication of his collected papers, which he appreciated very much. He edited seven of the quarto volumes himself, though suffering from a painful internal malady. He died 26 January 1895 at age 73. His funeral at Trinity Chapel was attended by the leading scientists of Britain, with official representatives from as far as Russia and America. The remainder of his papers were edited by Andrew Forsyth, his successor as Sadleirian professor, in total thirteen quarto volumes and 967 papers. His work continues in frequent use, cited in more than 200 mathematical papers in the 21st century alone. Cayley retained to the last his fondness for novel-reading and for travelling. He also took special pleasure in paintings and architecture, and he practiced water-colour painting, which he found useful sometimes in making mathematical diagrams.

Legacy

Cayley is buried in the Mill Road cemetery, Cambridge. An 1874 portrait of Cayley by Lowes Cato Dickinson and an 1884 portrait by William Longmaid are in the collection of

. A number of mathematical terms are named after him: *

* Cayley–Hamilton theorem in

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 matrix (mathemat ...

Cayley–Bacharach theorem In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states: :Assume that two cubics and in the projective plane meet in nine (different) poi ...

* Grassmann–Cayley algebra *

Cayley–Menger determinant In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its ...

* Cayley diagrams – used for finding cognate linkages in mechanical engineering *

Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...

* Cayley algebra (Octonion) *

* Cayley numbers * Cayley's sextic *

* Cayley–Purser algorithm *

* Cayley–Klein metric * Cayley–Klein model of

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

* Cayley's Ω process * Cayley surface *

Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...

* Cayley's nodal cubic surface * Cayley's ruled cubic surface * The crater Cayley on the

Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...

(and consequently the Cayley Formation, a geological unit named after the crater) * Cayley's mousetrap — a card game * Cayleyan * Chasles–Cayley–Brill formula * Hyperdeterminant * Quippian * Tetrahedroid

Bibliography

* * *

References

Sources

* * * *
complete text
at

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)

External links

* * *
Arthur Cayley Letters
to Robert Harley, 1859–1863. Available online through Lehigh University'
I Remain: A Digital Archive of Letters, Manuscripts, and Ephemera
* *

''This article incorporates text from the 1916 ''Lectures on Ten British Mathematicians of the Nineteenth Century'' by Alexander Macfarlane, which is in the

public domain The public domain (PD) consists of all the creative work to which no Exclusive exclusive intellectual property rights apply. Those rights may have expired, been forfeited, expressly Waiver, waived, or may be inapplicable. Because no one holds ...

''. {{DEFAULTSORT:Cayley, Arthur 1821 births 1895 deaths

Arthur Arthur is a masculine given name of uncertain etymology. Its popularity derives from it being the name of the legendary hero King Arthur. A common spelling variant used in many Slavic, Romance, and Germanic languages is Artur. In Spanish and Ital ...

19th-century English mathematicians Group theorists Linear algebraists Algebraic geometers Graph theorists People educated at King's College School, London Newnham College, Cambridge Alumni of Trinity College, Cambridge Fellows of Trinity College, Cambridge Fellows of the Royal Society Fellows of the American Academy of Arts and Sciences Foreign associates of the National Academy of Sciences Members of the Royal Netherlands Academy of Arts and Sciences Members of the Prussian Academy of Sciences Members of the Hungarian Academy of Sciences Presidents of the British Science Association Presidents of the Royal Astronomical Society Recipients of the Copley Medal Royal Medal winners De Morgan Medallists Magic squares Senior Wranglers Sadleirian Professors of Pure Mathematics Presidents of the Cambridge Philosophical Society Historical treatment of octonions