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Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific
British British may refer to: Peoples, culture, and language * British people The British people, or Britons, are the citizens of the United Kingdom of Great Britain and Northern Ireland The United Kingdom of Great Britain and Northern Ir ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
. As a child, Cayley enjoyed solving complex maths problems for amusement. He entered
Trinity College, Cambridge Trinity College is a constituent college A collegiate university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) education ...
, where he excelled in
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
,
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officially the French Republic (french: link=no, République française), is a transcontinental country This is a list of co ...

,
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ger ...

, and
Italian Italian may refer to: * Anything of, from, or related to the people of Italy Italy ( it, Italia ), officially the Italian Republic ( it, Repubblica Italiana, links=no ), is a country consisting of a peninsula delimited by the Alps ...

, as well as
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
. He worked as a
lawyer A lawyer or attorney is a person who practices law, as an advocate An advocate is a professional in the field of law. Different countries' legal systems use the term with somewhat differing meanings. The broad equivalent in many English ...

for 14 years. He postulated the
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 a ...
—that every
square matrix In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is a root of its own
characteristic polynomial 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 a ...
, and verified it for matrices of order 2 and 3. He was the first to define the concept of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
in the modern way—as a set with a
binary Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
operation satisfying certain laws. Formerly, when mathematicians spoke of "groups", they had meant
permutation group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s.
Cayley table Named after the 19th century British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity a ...
s and
Cayley graph on two generators ''a'' and ''b'' In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

s as well as
Cayley's theorem In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
are named in honour of Cayley.

# Early years

Arthur Cayley was born in
Richmond, London Richmond is a town in south-west London,The London Government Act 1963 The London Government Act 1963 (c. 33) is an of the , which created and a new structure within it. The Act significantly reduced the number of local government districts ...
,
England England is a Countries of the United Kingdom, country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest of England and the Celtic Sea to the southwest. E ...

, on 16 August 1821. His father, Henry Cayley, was a distant cousin of
Sir George Cayley Sir George Cayley, 6th Baronet (27 December 1773 – 15 December 1857) was an English engineer Engineers, as practitioners of engineering, are Professional, professionals who Invention, invent, design, analyze, build and test Machine, mac ...

, 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. The British Royal Aeronautical So ...

engineer innovator, and descended from an ancient
Yorkshire Yorkshire (; abbreviated Yorks), formally known as the County of York, is a historic county of Northern England Northern England, also known as the North of England or simply the North, is the most northern area of England England ...

family. He settled in

,
Russia Russia ( rus, link=no, Россия, Rossiya, ), or the Russian Federation, is a country spanning Eastern Europe Eastern Europe is the eastern region of . There is no consistent definition of the precise area it covers, partly because th ...

, as a
merchant A merchant is a person who trades in commodities In economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distributi ...

. His
mother A mother is the female Female (symbol: ♀) is the sex of an organism, or a part of an organism, that produces non-mobile ovum, ova (egg cells). Barring rare medical conditions, most female mammals, including female humans, have two X chro ...

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 Charles Bagot Cayley (1823–1883) was an English linguistics, linguist, best known for translating Dante Alighieri, Dante into the metre of the original, with annotations. He also made metrical versions of the ''Iliad'', the ''Prometheia, Promethe ...
. Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently at Blackheath, near London. Arthur was sent to a private school. At age 14 he was sent to
King's College School King's College School, also known as King's College School, Wimbledon, KCS, King's and KCS Wimbledon, is a public school in Wimbledon, southwest London London is the capital Capital most commonly refers to: * Capital letter ...
. The school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but at the
University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Scholars of ...
.

# Education

At the unusually early age of 17 Cayley began residence at
Trinity College, Cambridge Trinity College is a constituent college A collegiate university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) education ...
. The cause of the
Analytical Society The Analytical Society was a group of individuals in early-19th-century Britain Britain usually refers to: * United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,Usag ...
had now triumphed, and the ''Cambridge Mathematical Journal'' had been instituted by Gregory and
Robert Leslie EllisRobert Leslie Ellis (25 August 1817 – 12 May 1859) was an England, English polymath, remembered principally as a mathematician and editor of the works of Francis Bacon. Biography Ellis was the youngest of six children of Francis Ellis (1772–184 ...
. To this journal, at the age of twenty, Cayley contributed three papers, on subjects that had been suggested by reading the ''Mécanique analytique'' of
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
and some of the works of
Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...

. Cayley's tutor at Cambridge was
George Peacock George Peacock Fellow of the Royal Society, FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic. Early life Peacock was born on 9 April 1791 at T ...

and his private coach was
William Hopkins William Hopkins FRS (2 February 179313 October 1866) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as q ...
. He finished his undergraduate course by winning the place of
Senior Wrangler The Senior Wrangler is the top mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...
, and the first
Smith's prize The Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related stru ...
. 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''.

# As a lawyer

Because of the limited tenure of his fellowship it was necessary to choose a profession; like
De Morgan De Morgan or de Morgan is a surname, and may refer to: *Augustus De Morgan Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematic ...

, Cayley chose law, and was admitted to on 20 April 1846 at the age of 24. He made a specialty of
conveyancing In law, conveyancing is the transfer of legal title A title is one or more words used before or after a person's name, in certain contexts. It may signify either generation, an official position, or a professional or academic qualification. I ...
. It was while he was a pupil at the bar examination that he went to
Dublin Dublin (; , or ) is the capital and largest city of Ireland Ireland ( ; ga, Éire ; Ulster-Scots: ) is an island upright=1.15, Great_Britain.html"_;"title="Ireland_(left)_and_Great_Britain">Ireland_(left)_and_Great_Britain_ ...

to hear
Hamilton Hamilton may refer to: People * Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname ** The Duke of Hamilton, the premier peer of Scotland ** Lord Hamilton ...
's lectures on
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s. His friend , his senior by five years at Cambridge, was then an
actuary An actuary is a business professional who deals with the measurement and management of risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty Uncertainty refers to Epistemology, epistemic situat ...
, resident in London; they used to walk together round the courts of
Lincoln's Inn The Honourable Society of Lincoln's Inn is one of the four Inns of Court The Inns of Court in London are the professional associations for barristers A barrister is a type of lawyer A lawyer or attorney is a person who practices ...

, discussing the
theory of invariants Invariant theory is a branch of abstract algebra dealing with Group action (mathematics), actions of group (mathematics), groups on algebraic variety, algebraic varieties, such as vector spaces, from the point of view of their effect on functions. ...
and covariants. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers.

# As a professor

At Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, and is the chair that had been occupied by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

professor Professor (commonly abbreviated as Prof.) is an academic An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary education, secondary or tertiary education, tertiary higher education, hig ...

s, the lectures were extended over two terms. For many years the attendance was small, and came almost entirely from those who had finished their career of preparation for competitive examinations; after the reform the attendance numbered about fifteen. The subject lectured on was generally that of the memoir on which the professor was for the time engaged. The other duty of the chair — the advancement of mathematical science — was discharged in a handsome manner by the long series of memoirs that he published, ranging over every department of pure mathematics. But it was also discharged in a much less obtrusive way; he 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, which meant stipend as well as honour. About this time his friends subscribed for a presentation portrait. wrote an address to the committee of subscribers who had charge of the Cayley portrait fund. The verses refer to the subjects investigated in several of Cayley's most elaborate memoirs; such as, Chapters on the Analytical Geometry of $n$ dimensions; On the theory of
Determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s; Memoir on the theory of Matrices; Memoirs on skew surfaces, otherwise Scrolls; On the delineation of a Cubic Scroll, etc. In addition to his work on
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, Cayley made fundamental contributions to
algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

. Cayley and
Salmon Salmon is the common name for several species of ray-finned fish Actinopterygii ( New Latin ('having rays') + Greek ( 'wing, fins')), members of which are known as ray-finned fishes, is a clade A clade (; from grc, , ''klados'', ...
discovered the 27 lines on a
cubic surface In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

. 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 variet ...
of all curves in projective 3-space. He founded the algebro-geometric theory of
ruled surface In geometry, a surface (mathematics), surface ''S'' is ruled (also called a scroll) if through every point of ''S'' there is a straight line that lies on ''S''. Examples include the plane (mathematics), plane, the lateral surface of a cylinder (ge ...
s. In 1876 he published a ''Treatise on
Elliptic Functions In the mathematical field of complex analysis, elliptic functions are a special kind of Meromorphic function, meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integral ...
''. He took great interest in the movement for the University education of women. At Cambridge the women's colleges are Girton and Newnham. In the early days of
Girton College Girton College is one of the Colleges of the University of Cambridge, 31 constituent colleges of the University of Cambridge. The college was established in 1869 by Emily Davies and Barbara Bodichon as the first women's college in Cambridge. In 1 ...

he gave direct help in teaching, and for some years he was chairman of the council of
Newnham College Newnham College is a women's Colleges of the University of Cambridge, constituent college of the University of Cambridge. The college was founded in 1871 by a group organising Lectures for Ladies, members of which included philosopher Henry Sid ...
, 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 (Johns Hopkins, Hopkins, or JHU) is a private Private or privates may refer to: Music * "In Private "In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty S ...

, Baltimore, 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 The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization An organization, or organisation (Commonwealth English The use of the English langu ...
.

# British Association presidency

In 1883 Cayley was President of the
British Association for the Advancement of Science The British Science Association (BSA) is a charity Charity may refer to: Giving * Charitable organization or charity, a non-profit organization whose primary objectives are philanthropy and social well-being of persons * Charity (practice), ...
. 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 (CUP) is the publishing business of the University of Cambridge , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowled ...
requested him to prepare his mathematical papers for publication in a collected form—a request which he appreciated very much. They are printed in quarto volumes, of which seven appeared under his own editorship. While editing these volumes, he was suffering from a painful internal malady, to which he succumbed on 26 January 1895, in the 74th year of his age. When the funeral took place, a great assemblage met in Trinity Chapel, comprising members of the University, official representatives of Russia and America, and many of the most illustrious philosophers of
Britain Britain usually refers to: * United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain,Usage is mixed. The Guardian' and Telegraph' use Britain as a synonym for the United ...

. The remainder of his papers were edited by
Andrew Forsyth Andrew Russell Forsyth, FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Famil ...
, his successor in the Sadleirian Chair. The Collected Mathematical papers number thirteen quarto volumes, and contain 967 papers. 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 Lowes Cato Dickinson (27 November 1819 – 15 December 1908) was an English portrait painter and Christian socialist. He taught drawing with John Ruskin and Dante Gabriel Rossetti. He was a founder of the Working Men's College in London.
and an 1884 portrait by William Longmaid are in the collection of
Trinity College, Cambridge Trinity College is a constituent college A collegiate university is a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) education ...
. A number of mathematical terms are named after him: *
Cayley's theorem In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
*
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 a ...
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 (mat ...
*
Cayley–Bacharach theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Grassmann–Cayley algebra In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product. It is the most general structure in which projective properties are expressed in a ...
*
Cayley–Menger determinant 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 t ...
* Cayley diagrams – used for finding
cognate linkage In kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an ...
s in mechanical engineering *
Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, each with twice the dimension of a v ...
* Cayley algebra (Octonion) *
Cayley graph on two generators ''a'' and ''b'' In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathemati ...

* Cayley numbers * Cayley's sextic *
Cayley table Named after the 19th century British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity a ...
* Cayley–Purser algorithm * Cayley's formula * Cayley–Klein metric * Klein model, Cayley–Klein model of hyperbolic geometry * Cayley's Ω process * Cayley surface (disambiguation), Cayley surface * Cayley transform * Cayley's nodal cubic surface * Cayley's ruled cubic surface * The crater Cayley (crater), Cayley on the Moon (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

* * *

* List of things named after Arthur Cayley

# Sources

* * * *
complete text
at Project Gutenberg)