James Joseph Sylvester (3 September 1814 – 15 March 1897) was anEnglishmathematician. He made fundamental contributions tomatrix theory,invariant theory,number theory,partition theory, andcombinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at theJohns Hopkins University and as founder of theAmerican Journal of Mathematics. At his death, he was a professor atOxford University.

Professor James Joseph Sylvester FRSHonFRSE
Born	James Joseph (1814-09-03)3 September 1814 London, England
Died	15 March 1897(1897-03-15) (aged 82) London, England
Resting place	Balls Pond Road Cemetery
Alma mater	St. John's College, Cambridge
Known for	Coining the terms 'graph' and 'discriminant' Chebyshev–Sylvester constant Quadruplanar inversor Sylvester's sequence Sylvester's formula Sylvester's determinant theorem Sylvester matrix Sylvester–Gallai theorem Sylvester's law of inertia Sylvester's triangle problem Sylver coinage Sylvester's criterion Sylvester domain
Awards	Royal Medal(1861) Copley Medal(1880) De Morgan Medal(1887)
Scientific career
Fields	Mathematics
Institutions	Johns Hopkins University University College London University of Virginia Royal Military Academy, Woolwich University of Oxford
Academic advisors	John Hymers Augustus De Morgan
Doctoral students	William Durfee George B. Halsted Washington Irving Stringham
Other notable students	Isaac Todhunter William Roberts McDaniel Harry Fielding Reid Christine Ladd-Franklin

James Joseph was born inLondon on 3 September 1814, the son of Abraham Joseph, aJewish merchant.^[1] James later adopted the surnameSylvester when his older brother did so upon emigration to the United States.

At the age of 14, Sylvester was a student ofAugustus De Morgan at the University of London (nowUniversity College London). His family withdrew him from the university after he was accused of stabbing a fellow student with a knife. Subsequently, he attended theLiverpool Royal Institution.

Sylvester began his study of mathematics atSt John's College, Cambridge in 1831,^[2] where his tutor wasJohn Hymers. Although his studies were interrupted for almost two years due to a prolonged illness, he nevertheless ranked second in Cambridge's famous mathematical examination, thetripos, 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 theThirty-nine Articles of theChurch of England, and he could not do so because he was Jewish. For the same reason, he was unable to compete for a Fellowship or obtain aSmith's prize.^[3] In 1838, Sylvester became professor of natural philosophy atUniversity College London and in 1839 a Fellow of the Royal Society of London. In 1841, he was awarded a BA and anMA byTrinity College Dublin.

In the same year he moved to the United States to become a professor of mathematics at theUniversity of Virginia, the first Jewish professor at any American college or university.^[4] He left his appointment after only four months after a classroom incident in which a student he had criticized hit him with a bludgeon and he struck back with a sword-cane. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. Sylvester resigned when he felt that the university authorities had not sufficiently disciplined the student.^[4][5] He moved to New York City and began friendships with the Harvard mathematicianBenjamin Peirce (father ofCharles Sanders Peirce) and the Princeton physicistJoseph Henry. However, he left in November 1843 after being denied appointment as Professor of Mathematics at Columbia College (now University), again for his Judaism, and returned to England.

On his return to England, he was hired in 1844 by the Equity and Law Life Assurance Society for which he developed successful actuarial models and served as de facto CEO, a position that required a law degree. As a result, he studied for the Bar, meeting a fellow British mathematician studying law,Arthur Cayley, with whom he made significant contributions to invariant theory and alsomatrix theory during a long collaboration.^[6] He did not obtain a position teaching university mathematics until 1855, when he was appointed professor of mathematics at theRoyal 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 ofThe Times.

One of Sylvester's lifelong passions was for poetry; he read and translated works from the original French, German, Italian,Latin andGreek, and many of his mathematical papers contain illustrative quotes from classical poetry. Following his early retirement, Sylvester published a book entitledThe Laws of Verse in which he attempted to codify a set of laws forprosody in poetry.^[7]

In 1872, he finally received his B.A. and M.A. from Cambridge, having been denied the degrees due to his being a Jew.^[2]

In 1876^[8] Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the newJohns Hopkins University inBaltimore, Maryland. His salary was $5,000 (quite generous for the time), which he demanded be paid in gold. After negotiation, agreement was reached on a salary that was not paid in gold.^[9]

In 1877, he was elected as a member to theAmerican Philosophical Society.^[10]

In 1878 he founded theAmerican Journal of Mathematics. The only other mathematical journal in the US at that time wasThe Analyst, which eventually became theAnnals of Mathematics.

Also in 1878,Christine Ladd-Franklin was accepted intoJohns Hopkins University with his help. He remembered some of Ladd's earlier works in theEducational Times.^[11] Ladd's application for a fellowship was signed "C. Ladd", and the university offered her the position without realizing she was a woman.^[12] When they did realise her gender, the board tried to revoke the offer, but Sylvester insisted that Ladd should be his student, and so she was.^[12] She held a fellowship at Johns Hopkins University for three years, but the trustees did not allow her name to be printed in circulars with those of other fellows, for fear of setting a precedent.^[12] Furthermore, dissension over her continued presence forced one of the original trustees to resign.^[12]

In 1883, Sylvester returned to England to take up the position ofSavilian Professor of Geometry atOxford University. He held this chair until his death, although in 1892 the university appointed a deputy professor to the same chair. He was on the governing body ofAbingdon School.^[13]

Sylvester died at 5Hertford Street, London on 15 March 1897. He is buried inBalls Pond Road Cemetery on Kingsbury Road in London.^[14]

Sylvester invented a great number of mathematical terms such as "matrix" (in 1850),^[15][16]"graph" in the sense ofnetwork^[17] and "discriminant".^[18] He coined the term "totient" forEuler's totient function φ(n).^[19] Indiscrete geometry he is remembered forSylvester's problem and a result on theorchard problem, and in matrix theory he discoveredSylvester's determinant identity,^[20] which generalizes theDesnanot–Jacobi identity.^[21] His collected scientific work fills four volumes. In 1880, theRoyal Society of London awarded Sylvester theCopley Medal, its highest award for scientific achievement; in 1901, it instituted theSylvester Medal in his memory, to encourage mathematical research after his death inOxford.

Sylvester House, a portion of an undergraduate dormitory atJohns Hopkins University, is named in his honour. Several professorships there are named in his honour also.

• Sylvester, James Joseph (1870).The Laws of Verse, or, Principles of Versification Exemplified in Metrical Translations: Together with an Annotated Reprint of the Inaugural Presidential Address to the Mathematical and Physical Section of the British Association at Exeter. London: Longmans, Green and Co.ISBN 978-1-177-91141-2.{{cite book}}:ISBN / Date incompatibility (help)
• Sylvester, James Joseph (1973) [1904].Baker, Henry Frederick (ed.).The Collected Mathematical Papers of James Joseph Sylvester. Vol. I. New York: AMS Chelsea Publishing.ISBN 978-0-8218-3654-5.^[22]
• Sylvester, James Joseph (1973) [1908]. Baker, Henry Frederick (ed.).The Collected Mathematical Papers of James Joseph Sylvester. Vol. II. New York: AMS Chelsea Publishing.ISBN 978-0-8218-4719-0.^[22]
• Sylvester, James Joseph (1973) [1904]. Baker, Henry Frederick (ed.).The Collected Mathematical Papers of James Joseph Sylvester. Vol. III. New York: AMS Chelsea Publishing.ISBN 978-0-8218-4720-6.^[23]
• Sylvester, James Joseph (1973) [1904]. Baker, Henry Frederick (ed.).The Collected Mathematical Papers of James Joseph Sylvester. Vol. IV. New York: AMS Chelsea Publishing.ISBN 978-0-8218-4238-6.

• Catalecticant
• Clock and shift matrices
• Covariance and contravariance of vectors
• Evectant
• Inclusion–exclusion principle
• Invariant of a binary form
• Sylvester's construction
• Sylvester pentahedron
• Sylvester's problem
• Umbral calculus
• List of things named after James Joseph Sylvester

• Grattan-Guinness, I. (2001), "The contributions of J. J. Sylvester, F.R.S., to mechanics and mathematical physics",Notes and Records of the Royal Society of London,55 (2):253–265,doi:10.1098/rsnr.2001.0142,MR 1840760,S2CID 122748202.
• Macfarlane, Alexander (2009) [1916],Lectures on Ten British Mathematicians of the Nineteenth Century, Mathematical monographs, vol. 17, Cornell University Library,ISBN 978-1-112-28306-2
• Parshall, Karen Hunger (1998),James Joseph Sylvester. Life and work in letters., The Clarendon Press Oxford University Press,ISBN 978-0-19-850391-0,MR 1674190,Review
• Parshall, Karen Hunger (2006),James Joseph Sylvester. Jewish mathematician in a Victorian world, Johns Hopkins University Press,ISBN 978-0-8018-8291-3,MR 2216541

• O'Connor, John J.;Robertson, Edmund F.,"James Joseph Sylvester",MacTutor History of Mathematics Archive,University of St Andrews
• James Joseph Sylvester at theMathematics Genealogy Project
• Collected papers – from theUniversity of Michigan Historical Math Collection
• J. J. Sylvester home page
• Selected Poetry of James Joseph Sylvester
• Works by James Joseph Sylvester atLibriVox (public domain audiobooks)