# Mathematician:James Joseph Sylvester

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## Contents

## Mathematician

English mathematician who contributed to matrix theory, invariant theory, number theory, partition theory and combinatorics.

First coined the word matrix.

Contributed notably to the growth of mathematics in the USA.

Tutor of Florence Nightingale.

Also had a passion for poetry.

## Nationality

English

## History

- Born: 3 Sept 1814, London, England
- 1841 - 43: Professorship at the University of Virginia, USA
- 1877 - 83: Professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland, USA
- Died: 15 March 1897, London, England

## Theorems and Definitions

- Chebyshev-Sylvester Constant (with Pafnuty Lvovich Chebyshev)
- Coin Problem
- Sylvester's Sequence
- Sylvester's Formula, or Sylvester's Matrix Theorem
- Sylvester's Determinant Theorem
- Sylvester Matrix
- Sylvester-Gallai Theorem (with Tibor Gallai)
- Sylvester's Law of Inertia
- Sylvester's Criterion

- Rediscovered the Greedy Algorithm for Egyptian Fractions
- First to construct a Hadamard Matrix

Results named for **James Joseph Sylvester** can be found here.

Definitions of concepts named for **James Joseph Sylvester** can be found here.

## Publications

- 1839:
*On rational derivation from equations of coexistence, that is to say, a new and extended theory of elimination, Part I* - 1852:
*On the principle of the calculus of forms* - 1852:
*A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares*(in which he proved Sylvester's Law of Inertia) - 1853:
*On the theory of syzygetic relations and two rational integer functions* - 1857:
*On a Discovery in the Partition of Numbers*(*Quart. J. Pure Appl. Math.***Vol. 1**: 81 – 85) - 1857:
*On a Discovery in the Partition of Numbers -- continued*(*Quart. J. Pure Appl. Math.***Vol. 1**: 141 – 152) - 1869:
*Presidential address to Section A of the British Association* - 1870:
*The Laws of Verse: or Principles of Versification* - 1878: Founded the
*American Journal of Mathematics*which, as of the time of this article, is still going strong. - 1884:
*Question 7382*in*Mathematical Questions from the Educational Times*, in which he provided a partial solution to the Coin Problem - 1897:
*Outlines of seven lectures on the partition of numbers*

## Notable Quotes

*Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined; it is as limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence.*

## Sources

- John J. O'Connor and Edmund F. Robertson: "James Joseph Sylvester": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{XXI}$ - 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Preface - 1992: Ian Stewart:
*The Problems of Mathematics*(2nd ed.): Foreword - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.1$: History - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Sylvester, James Joseph**(1814-97)