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**James Joseph Sylvester** FRS HonFRSE (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the *American Journal of Mathematics*. At his death, he was a professor at Oxford University.

James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a Jewish merchant.^{[1]} James later adopted the surname Sylvester when his older brother did so upon emigration to the United States—a country which at that time required all immigrants to have a given name, a middle name, and a surname.^{[citation needed]}

At the age of 14, Sylvester was a student of Augustus de Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student with a knife. Subsequently, he attended the Liverpool Royal Institution.

Sylvester began his study of mathematics at St John's College, Cambridge in 1831,^{[2]} where his tutor was John 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, the tripos, 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 Articles of the Church of England, and Sylvester could not do so because he was Jewish. For the same reason, he was unable to compete for a Fellowship or obtain a Smith's prize.^{[3]} In 1838, Sylvester became professor of natural philosophy at University College London and in 1839 a Fellow of the Royal Society of London. In 1841, he was awarded a BA and an MA by the University of Dublin (Trinity College). In the same year he moved to the United States to become a professor of mathematics at the University of Virginia, but left after less than four months. A student who had been reading a newspaper in one of Sylvester's lectures insulted him and Sylvester struck him with a sword stick. 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]} He moved to New York City and began friendships with the Harvard mathematician Benjamin Peirce (father of Charles Sanders Peirce) and the Princeton physicist Joseph 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 also matrix theory during a long collaboration.^{[5]} 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 Times*.

One of Sylvester's lifelong passions was for poetry; he read and translated works from the original French, German, Italian, Latin and Greek, and many of his mathematical papers contain illustrative quotes from classical poetry. Following his early retirement, Sylvester published a book entitled *The Laws of Verse* in which he attempted to codify a set of laws for prosody in poetry.^{[6]}

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^{[7]} Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, 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.^{[8]}

In 1877, he was elected as a member to the American Philosophical Society.^{[9]}

In 1878 he founded the *American Journal of Mathematics*. The only other mathematical journal in the US at that time was the *Analyst*, which eventually became the *Annals of Mathematics*.

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. He was on the governing body of Abingdon School.^{[10]}

Sylvester died at 5 Hertford Street, London on 15 March 1897. He is buried in Balls Pond Road Cemetery on Kingsbury Road in London.^{[11]}

Sylvester invented a great number of mathematical terms such as "matrix" (in 1850),^{[12]} "graph" (combinatorics)^{[13]} and "discriminant".^{[14]} He coined the term "totient" for Euler's totient function φ(*n*).^{[15]} In discrete geometry he is remembered for Sylvester's problem and a result on the orchard problem, and in matrix theory he discovered
Sylvester's determinant identity,^{[16]} which generalizes the *Desnanot–Jacobi identity*.^{[17]} His collected scientific work fills four volumes. In 1880, the Royal Society of London awarded Sylvester the Copley Medal, its highest award for scientific achievement; in 1901, it instituted the Sylvester Medal in his memory, to encourage mathematical research after his death in Oxford.

Sylvester House, a portion of an undergraduate dormitory at Johns Hopkins University, is named in his honor. Several professorships there are named in his honor 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. - 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.^{[18]} - 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.^{[18]} - 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.^{[19]} - 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.

**^***Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*(PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X.- ^
^{a}^{b}"Sylvester, James Joseph (SLVR831JJ)".*A Cambridge Alumni Database*. University of Cambridge. **^**Bell, Eric Temple (1986).*Men of Mathematics*. Simon Schuster.**^**Biography of Sylvester, MacTutor, University of St. Andrews, accessed 6 October 2021**^**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.**^**Sylvester, J. J. (1870).*The Laws of Verse, or, Principles of Versification Exemplified in Metrical Translations*. London: Longmans, Green and Co.**^**"Preliminary Outline of Instructions for the Session Beginning October 3, 1876".*Johns Hopkins University. Official Circulars*(5). September 1876.**^**Hawkins, Hugh (1960).*Pioneer: A History of the Johns Hopkins University, 1874-1889*. Ithaca, NY: Cornell University Press. pp. 41–43.**^**"APS Member History".*search.amphilsoc.org*. Retrieved 10 May 2021.**^**"School Notes" (PDF). The Abingdonian.**^***Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002*(PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X.**^**Matrices and determinants, The MacTutor History of Mathematics archive**^**See:- J. J. Sylvester (7 February 1878) "Chemistry and algebra,"
*Nature*,**17**: 284. From page 284: "Every invariant and covariant thus becomes expressible by a*graph*precisely identical with a Kekuléan diagram or chemicograph." - J. J. Sylvester (1878) "On an application of the new atomic theory to the graphical representation of the invariants and covariants of binary quantics, — with three appendices,"
*American Journal of Mathematics, Pure and Applied*,**1**(1) : 64-90. The term "graph" first appears in this paper on page 65.

- J. J. Sylvester (7 February 1878) "Chemistry and algebra,"
**^**J. J. Sylvester (1851) "On a remarkable discovery in the theory of canonical forms and of hyperdeterminants,"*Philosophical Magazine*, 4th series,**2**: 391–410; Sylvester coined the term "discriminant" on page 406.**^**J. J. Sylvester (1879) "On certain ternary cubic-form equations,"*American Journal of Mathematics*,**2**: 357–393; Sylvester coins the term "totient" on page 361: "(the so-called*Φ*function of any number I shall here and hereafter designate as its*τ*function and call its Totient)"**^**Sylvester, James Joseph (1851). "On the relation between the minor determinants of linearly equivalent quadratic functions".*Philosophical Magazine*.**1**: 295–305.**^**C.G.J. Jacobi, "De Formatione et Proprietatibus Determinantium", Journal für die reine und angewandte Mathematik, 22, 285-318 (1841)- ^
^{a}^{b}Dickson, L. E. (1909). "Review: Sylvester's*Mathematical Papers*, vols. I & II, ed. by H. F. Baker".*Bull. Amer. Math. Soc*.**15**(5): 232–239. doi:10.1090/S0002-9904-1909-01746-X. **^**Dickson, L. E. (1911). "Review: Sylvester's*Mathematical Papers*, vol. III, ed. by H. F. Baker".*Bull. Amer. Math. Soc*.**17**(5): 254–255. doi:10.1090/S0002-9904-1911-02040-7.

- 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 the Mathematics Genealogy Project
- Collected papers – from the University of Michigan Historical Math Collection
- J. J. Sylvester home page
- Selected Poetry of James Joseph Sylvester
- Works by James Joseph Sylvester at LibriVox (public domain audiobooks)