# James Joseph Sylvester

**James Joseph Sylvester** FRS (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 professor at Oxford.

## Biography

Sylvester was born **James Joseph** in London, England. His father, Abraham Joseph, was a merchant. James 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. 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,^{[1]} 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, being an adherent of Judaism, refused to do so. For the same reason, he was unable to compete for a Fellowship or obtain a Smith's prize.^{[2]} In 1838 Sylvester became professor of natural philosophy at University College London. In 1841, he was awarded a BA and an MA by Trinity College, Dublin. In the same year he moved to the United States to become a professor at the University of Virginia for about six months, and returned to England in November 1843.

On his return to England he studied law, alongside fellow British lawyer/mathematician Arthur Cayley, with whom he made significant contributions to matrix theory while working as an actuary. One of his private pupils was Florence Nightingale. 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 (1870) published a book entitled *The Laws of Verse* in which he attempted to codify a set of laws for prosody in poetry.

In 1872, he finally received his B.A. and M.A. from Cambridge, having been denied the degrees due to him being a Jew.^{[1]}

In 1876^{[3]} 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.^{[4]} 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.

Sylvester invented a great number of mathematical terms such as "matrix",^{[5]} "graph" (combinatorics)^{[6]} and "discriminant".^{[7]} He coined the term "totient" for Euler's totient function φ(*n*).^{[8]} 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. In Discrete geometry he is remembered for Sylvester's Problem and a result on the orchard problem.

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.

## See also

## Publications

- 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>^{[9]} - 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>^{[9]} - 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>^{[10]} - 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## Notes

- ↑
^{1.0}^{1.1}"Sylvester, James Joseph (SLVR831JJ)".*A Cambridge Alumni Database*. University of Cambridge.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Bell, Eric Temple (1986).
*Men of Mathematics*. Simon Schuster.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ "Preliminary Outline of Instructions for the Session Beginning October 3, 1876".
*Johns Hopkins University. Official Circulars. No. 5*. September 1876.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ Hawkins, Hugh (1960).
*Pioneer: A History of the Johns Hopkins University, 1874-1889*. Ithaca, NY: Cornell University Press. pp. 41–43.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - ↑ 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 coins the word "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)" - ↑
^{9.0}^{9.1}**Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).** - ↑
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## References

**Lua error in Module:Citation/CS1/Identifiers at line 47: attempt to index field 'wikibase' (a nil value).**.- 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<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>

## External links

Quotations from Wikiquote | |

Source texts from Wikisource | |

Data from Wikidata |

- O'Connor, John J.; Robertson, Edmund F., "James Joseph Sylvester",
*MacTutor History of Mathematics archive*, University of St Andrews<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>. - 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

- Use dmy dates from July 2012
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- Academics of University College London
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