**James Joseph Sylvester** was an

EnglishEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

. He made fundamental contributions to matrix theory,

invariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

,

number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, partition theory and

combinatoricsCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the

Johns Hopkins UniversityThe Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...

and as founder of the

American Journal of MathematicsThe American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press.- History :The American Journal of Mathematics is the oldest continuously-published mathematical journal in the United States, established in 1878 at the Johns Hopkins University...

. At his death, he was professor at Oxford.

## Biography

Sylvester was born

**James Joseph** in

LondonLondon is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...

,

EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

. His father, Abraham Joseph, was a merchant. James adopted the surname

**Sylvester** when his older brother did so upon emigration to the

United StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...

—a country which at that time required all immigrants to have a given name, a middle name, and a

surnameA surname is a name added to a given name and is part of a personal name. In many cases, a surname is a family name. Many dictionaries define "surname" as a synonym of "family name"...

. At the age of 14, Sylvester started attending the

University of London-20th century:Shortly after 6 Burlington Gardens was vacated, the University went through a period of rapid expansion. Bedford College, Royal Holloway and the London School of Economics all joined in 1900, Regent's Park College, which had affiliated in 1841 became an official divinity school of the...

, where he was a student of

Augustus De MorganAugustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Following this, he attended the

Liverpool Royal InstitutionThe Liverpool Royal Institution was a learned society set up in 1814 for "the Promotion of Literature, Science and the Arts". William Corrie, William Rathbone IV and William Roscoe were among the founders. A royal charter was granted in 1821. The institute purchased a building on Colquitt Street...

.

Sylvester began his study of

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

at

St John's CollegeSt John's College is a constituent college of the University of Cambridge. The college's alumni include nine Nobel Prize winners, six Prime Ministers, three archbishops, at least two princes, and three Saints....

,

CambridgeThe city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...

in 1831, where his tutor was

John HymersJohn Hymers , mathematician, was born at Ormesby, near Middlesbrough, Yorkshire, the son of Thomas Hymers, a tenant farmer under Sir James Pennymon, and his wife, the daughter of John Parrington, rector of Skelton-in-Cleveland, England.- Education :After attending schools at Witton-le-Wear and...

. Although his studies were interrupted for almost two years due to a prolonged illness, he nevertheless ranked second in Cambridge's famous mathematical examination, the

triposThe University of Cambridge, England, divides the different kinds of honours bachelor's degree by Tripos , plural Triposes. The word has an obscure etymology, but may be traced to the three-legged stool candidates once used to sit on when taking oral examinations...

, for which he sat in 1837. However, Sylvester was not issued a degree, because graduates at that time were required to state their acceptance of the

Thirty-Nine ArticlesThe Thirty-Nine Articles of Religion are the historically defining statements of doctrines of the Anglican church with respect to the controversies of the English Reformation. First established in 1563, the articles served to define the doctrine of the nascent Church of England as it related to...

of the

Church of EnglandThe Church of England is the officially established Christian church in England and the Mother Church of the worldwide Anglican Communion. The church considers itself within the tradition of Western Christianity and dates its formal establishment principally to the mission to England by St...

, and Sylvester - who was of Jewish origin - refused to do so. For the same reason, he was unable to compete for a Fellowship or obtain a

Smith's prizeThe Smith's Prize was the name of each of two prizes awarded annually to two research students in theoretical Physics, mathematics and applied mathematics at the University of Cambridge, Cambridge, England.- History :...

. In 1838 Sylvester became professor of natural philosophy at

University College LondonUniversity College London is a public research university located in London, United Kingdom and the oldest and largest constituent college of the federal University of London...

. In 1841, he was awarded a

BAA Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...

and an

MAA master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...

by

Trinity College, DublinTrinity College, Dublin , formally known as the College of the Holy and Undivided Trinity of Queen Elizabeth near Dublin, was founded in 1592 by letters patent from Queen Elizabeth I as the "mother of a university", Extracts from Letters Patent of Elizabeth I, 1592: "...we...found and...

. In the same year he moved to the

United StatesThe United States of America is a federal constitutional republic comprising fifty states and a federal district...

to become a professor at the

University of VirginiaThe University of Virginia is a public research university located in Charlottesville, Virginia, United States, founded by Thomas Jefferson...

for about six months, and returned to

EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

in November 1843.

On his return to England he studied law, alongside fellow British lawyer/mathematician

Arthur CayleyArthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

, with whom he made significant contributions to matrix theory while working as an

actuaryAn actuary is a business professional who deals with the financial impact of risk and uncertainty. Actuaries provide expert assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms ....

. One of his private pupils was

Florence NightingaleFlorence Nightingale OM, RRC was a celebrated English nurse, writer and statistician. She came to prominence for her pioneering work in nursing during the Crimean War, where she tended to wounded soldiers. She was dubbed "The Lady with the Lamp" after her habit of making rounds at night...

. He did not obtain a position teaching university mathematics until 1855, when he was appointed professor of mathematics at the Royal Military Academy, Woolwich, from which he retired in 1869, because the compulsory retirement age was 55. The Woolwich academy initially refused to pay Sylvester his full pension, and only relented after a prolonged public controversy, during which Sylvester took his case to the letters page of

The TimesThe Times is a British daily national newspaper, first published in London in 1785 under the title The Daily Universal Register . The Times and its sister paper The Sunday Times are published by Times Newspapers Limited, a subsidiary since 1981 of News International...

.

One of Sylvester's lifelong passions was for

poetryPoetry is a form of literary art in which language is used for its aesthetic and evocative qualities in addition to, or in lieu of, its apparent meaning...

; he read and translated works from the original

FrenchFrench is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

,

GermanGerman is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....

,

ItalianItalian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...

,

LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

and

GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

, and many of his mathematical papers contain illustrative quotes from classical poetry. Following his early retirement, published a book entitled The Laws of Verse in which he attempted to codify a set of laws for prosody in poetry.

In 1877 Sylvester again crossed the

Atlantic OceanThe Atlantic Ocean is the second-largest of the world's oceanic divisions. With a total area of about , it covers approximately 20% of the Earth's surface and about 26% of its water surface area...

to become the inaugural professor of mathematics at the new

Johns Hopkins UniversityThe Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...

in Baltimore, Maryland. His salary was $5,000 (quite generous for the time), which he demanded be paid in

goldGold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...

. In 1878 he founded the

American Journal of MathematicsThe American Journal of Mathematics is a bimonthly mathematics journal published by the Johns Hopkins University Press.- History :The American Journal of Mathematics is the oldest continuously-published mathematical journal in the United States, established in 1878 at the Johns Hopkins University...

. The only other mathematical journal in the U.S. at that time was the Analyst, which eventually became the

Annals of MathematicsThe Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...

.

In 1883, he returned to England to take up the Savilian Professor of Geometry at Oxford University. He held this chair until his death, although in 1892 the University appointed a deputy professor to the same chair.

Sylvester invented a great number of mathematical terms such as

discriminantIn algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....

. He has given a name to

Euler's totient functionIn number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...

φ(n). His collected scientific work fills four volumes. In 1880, the

Royal Society of LondonThe Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

awarded Sylvester the

Copley MedalThe Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...

, its highest award for scientific achievement; in 1901, it instituted the

Sylvester MedalThe Sylvester Medal is a bronze medal awarded by the Royal Society for the encouragement of mathematical research, and accompanied by a £1,000 prize...

in his memory, to encourage mathematical research after his death in

OxfordThe city of Oxford is the county town of Oxfordshire, England. The city, made prominent by its medieval university, has a population of just under 165,000, with 153,900 living within the district boundary. It lies about 50 miles north-west of London. The rivers Cherwell and Thames run through...

,

OxfordshireOxfordshire is a county in the South East region of England, bordering on Warwickshire and Northamptonshire , Buckinghamshire , Berkshire , Wiltshire and Gloucestershire ....

,

EnglandEngland is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...

.

Sylvester House, a portion of an undergraduate dormitory at

Mason ClevelandThe Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...

, is named in his honour.

## See also

- Chebyshev–Sylvester constant
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

- Coin problem
The coin problem is a mathematical problem that asks what is the largest monetary amount that cannot be obtained using only coins of specified denominations. For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units...

- Greedy algorithm for Egyptian fractions
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of unit fractions, as e.g. 5/6 = 1/2 + 1/3...

- Hadamard matrix
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal...

- Sylvester's criterion
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester....

- Sylvester domain
In mathematics, a Sylvester domain, named after James Joseph Sylvester by , is a ring in which Sylvester's law of nullity holds. This means that if A is an m by n matrix and B an n by s matrix over R, thenwhere ρ is the inner rank of a matrix...

- Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...

## External links

- Collected papers – from the University of Michigan
The University of Michigan is a public research university located in Ann Arbor, Michigan in the United States. It is the state's oldest university and the flagship campus of the University of Michigan...

Historical Math Collection
- J.J.Sylvester home page
- Selected Poetry of James Joseph Sylvester
- James Joseph Sylvester at Wikiquote
Wikiquote is one of a family of wiki-based projects run by the Wikimedia Foundation, running on MediaWiki software. Based on an idea by Daniel Alston and implemented by Brion Vibber, the goal of the project is to produce collaboratively a vast reference of quotations from prominent people, books,...