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**James Joseph Sylvester** FRS HFRSE (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.

**Fellowship of the Royal Society** is an award granted to individuals that the Royal Society of London judges to have made a 'substantial contribution to the improvement of natural knowledge, including mathematics, engineering science, and medical science'.

The **United Kingdom of Great Britain and Northern Ireland**, commonly known as the **United Kingdom** (**UK**) or **Britain**, is a sovereign country located off the north-western coast of the European mainland. The United Kingdom includes the island of Great Britain, the north-eastern part of the island of Ireland, and many smaller islands. Northern Ireland is the only part of the United Kingdom that shares a land border with another sovereign state, the Republic of Ireland. Apart from this land border, the United Kingdom is surrounded by the Atlantic Ocean, with the North Sea to the east, the English Channel to the south and the Celtic Sea to the south-west, giving it the 12th-longest coastline in the world. The Irish Sea separates Great Britain and Ireland. The United Kingdom's 242,500 square kilometres (93,600 sq mi) were home to an estimated 66.0 million inhabitants in 2017.

A **mathematician** is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a 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.

**London** is the capital and largest city of England and the United Kingdom. Standing on the River Thames in the south-east of England, at the head of its 50-mile (80 km) estuary leading to the North Sea, London has been a major settlement for two millennia. *Londinium* was founded by the Romans. The City of London, London's ancient core − an area of just 1.12 square miles (2.9 km^{2}) and colloquially known as the Square Mile − retains boundaries that follow closely its medieval limits. The City of Westminster is also an Inner London borough holding city status. Greater London is governed by the Mayor of London and the London Assembly.

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.

**Augustus 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 **University of London** is a federal research university located in London, England. As of October 2019, the university contains 18 member institutions, central academic bodies and research institutes. The university has over 52,000 distance learning external students and 161,270 campus-based internal students, making it the largest university by number of students in the United Kingdom.

The **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, Thomas Stewart Traill and William Roscoe were among the founders. It was sometimes called the **Royal Society of Liverpool**.

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 following a violent encounter with two students he had disciplined. 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.

**St John's College** is a constituent college of the University of Cambridge founded by the Tudor matriarch Lady Margaret Beaufort. In constitutional terms, the college is a charitable corporation established by a charter dated 9 April 1511. The aims of the college, as specified by its statutes, are the promotion of education, religion, learning and research. It is one of the larger Oxbridge colleges in terms of student numbers. For 2018, St. John’s was ranked 9th of 29 colleges in the Tompkins Table with over 30% of its students earning First-class honours.

**John Hymers** (1803–1887) was an English mathematician and cleric, and, together with his brother Robert, founder of Hymers College, Hull.

At the University of Cambridge, a **Tripos** is any of the undergraduate examinations that qualify an undergraduate for a bachelor's degree or the courses taken by an undergraduate to prepare. For example, an undergraduate studying mathematics is said to be reading for the *Mathematical Tripos*, whilst a student of English literature is reading for the *English Tripos*.

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.^{ [4] }^{[ incomplete short citation ]} 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 *.

**Arthur Cayley** was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics.

In mathematics, a **matrix** is a rectangular *array* of numbers, symbols, or expressions, arranged in *rows* and *columns*. For example, the dimension of the matrix below is 2 × 3, because there are two rows and three columns:

The **Royal Military Academy** (**RMA**) at Woolwich, in south-east London, was a British Army military academy for the training of commissioned officers of the Royal Artillery and Royal Engineers. It later also trained officers of the Royal Corps of Signals and other technical corps. RMA Woolwich was commonly known as "The Shop" because its first building was a converted workshop of the Woolwich Arsenal.

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 (1870) published a book entitled * The Laws of Verse * in which he attempted to codify a set of laws for prosody in poetry.

**Latin** is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets and ultimately from the Phoenician alphabet.

**Greek** is an independent branch of the Indo-European family of languages, native to Greece, Cyprus and other parts of the Eastern Mediterranean and the Black Sea. It has the longest documented history of any living Indo-European language, spanning more than 3000 years of written records. Its writing system has been the Greek alphabet for the major part of its history; other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems.

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^{ [5] } 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.^{ [6] } 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.^{ [7] }

Sylvester died in London on 15 March 1897. He is buried in Balls Pond Road Jewish Cemetery on Kingsbury Road in London.^{ [8] }

Sylvester invented a great number of mathematical terms such as "matrix" (in 1850),^{ [9] } "graph" (combinatorics)^{ [10] } and "discriminant".^{ [11] } He coined the term "totient" for Euler's totient function φ(*n*).^{ [12] }. 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 ^{ [13] }, which generalizes the *Desnanot–Jacobi identity*^{ [14] }. 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*,**I**, New York: AMS Chelsea Publishing, ISBN 978-0-8218-3654-5^{ [15] } - Sylvester, James Joseph (1973) [1908], Baker, Henry Frederick (ed.),
*The collected mathematical papers of James Joseph Sylvester*,**II**, New York: AMS Chelsea Publishing, ISBN 978-0-8218-4719-0^{ [15] } - Sylvester, James Joseph (1973) [1904], Baker, Henry Frederick (ed.),
*The collected mathematical papers of James Joseph Sylvester*,**III**, New York: AMS Chelsea Publishing, ISBN 978-0-8218-4720-6^{ [16] } - Sylvester, James Joseph (1973) [1904], Baker, Henry Frederick (ed.),
*The collected mathematical papers of James Joseph Sylvester*,**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. - 1 2 "Sylvester, James Joseph (SLVR831JJ)".
*A Cambridge Alumni Database*. University of Cambridge. - ↑ Bell, Eric Temple (1986).
*Men of Mathematics*. Simon Schuster. - ↑ Parshall
- ↑ "Preliminary Outline of Instructions for the Session Beginning October 3, 1876".
*Johns Hopkins University. Official Circulars. No. 5*. September 1876. - ↑ Hawkins, Hugh (1960).
*Pioneer: A History of the Johns Hopkins University, 1874-1889*. Ithaca, NY: Cornell University Press. pp. 41–43. - ↑ "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 (February 7, 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 (February 7, 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)
- 1 2 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.

**William Burnside** was an English mathematician. He is known mostly as an early researcher in the theory of finite groups.

**Oscar Zariski** was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.

**Harish-Chandra** FRS was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.

**Kunihiko Kodaira** was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese national to receive this honour.

**Salomon Bochner** was an American mathematician, known for work in mathematical analysis, probability theory and differential geometry.

**Leonard Eugene Dickson** was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory, *History of the Theory of Numbers*.

**Joseph Henry Maclagan Wedderburn** FRSE FRS was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra.

**Markus Rost** is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld.

**Clifford Henry Taubes** is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother, Gary Taubes, is a science writer.

**William Edward Story** was an American mathematician who taught at Johns Hopkins University and Clark University.

**Arthur Byron Coble** was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface. He was President of the American Mathematical Society from 1933 to 1934.

**Douglas Conner Ravenel** is an American mathematician known for work in algebraic topology.

**Jun-Ichi Igusa** was a Japanese mathematician who for over three decades was on the faculty at Johns Hopkins University. He is known for his contributions to algebraic geometry and number theory. The Igusa zeta-function, the Igusa quartic, Igusa subgroups, Igusa curves, and Igusa varieties are named after him.

**William Pitt Durfee** was an American mathematician who introduced Durfee squares. He was a student of James Sylvester, and after obtaining his degree in 1883 he became a professor at Hobart college in 1884 and became dean in 1888. Durfee House and Durfee Hall are named in his honor.

**Karen Hunger Parshall** is an American historian of mathematics.

**Florence Parthenia Lewis** was an American mathematician and astronomer.

**Robert Tucker** (1832–1905) was an English mathematician, who was secretary of the London Mathematical Society for more than 30 years.

**Fabian Franklin** (1853–1939) was a Hungarian-born American engineer, mathematician and journalist, husband of Christine Ladd-Franklin.

**Henry Taber** (1860–1936) was an American mathematician.

- 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 . - Macfarlane, Alexander (2009) [1916],
*Lectures on Ten British Mathematicians of the Nineteenth Century*, Mathematical monographs,**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)

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