James Joseph Sylvester’s busy life of maths and music

That’s Maths: Sylvester, born on this day 201 years ago, had a prodigious maths talent that became evident early

James Joseph Sylvester was born in London to Jewish parents in 1814 on this day 201 years ago. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic pope from the Roman period – was adopted later.

Sylvester's mathematical talents became evident at an early age. He entered Cambridge in 1831, aged just 17, and came second in the notorious examinations known as the Mathematical Tripos; the man who beat him achieved nothing further in mathematics. Trinity College Dublin awarded Sylvester honours and master's degrees, which Cambridge would not bestow unless he subscribed to the Thirty-Nine Articles of the Church of England, which his Jewish convictions precluded. He was elected a fellow of the Royal Society at the unusually early age of 25.

Sylvester, like many brilliant young people today, had difficulty finding a suitable position. He took a post in Virginia in the US, but he returned within a year or so. He supported himself by taking private pupils, the most distinguished of whom was Florence Nightingale.

Meeting with Cayley

In 1846, aged 32, Sylvester entered the Inner Temple in London to pursue a legal career, and was called to the bar four years later. The greatest benefit of this period was his encounter with Arthur Cayley, another renowned mathematician, which led to a life-long friendship and collaboration. Cayley and Sylvester inspired each other in some of their best work, on the theory of invariants and matrix theory.

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The theory of algebraic invariants had initially been developed by George Boole while he was at Queen's College Cork. At school we learn how to solve quadratic equations such as ax2+bx+c=0. Quadratics have two solutions, called roots. The nature of the solutions depends on a quantity b2-4ac, which is called the discriminant (Sylvester introduced this term). If the discriminant is equal to zero, the two roots are equal. This property remains unchanged under a range of modifications called bilinear transformations: the discriminant is an invariant. Boole had found that such invariants exist for all algebraic equations. Invariance under transformations is fundamental in modern mathematics and also plays a key role in the physical sciences.

In 1854 Sylvester applied for the professorship of mathematics at the Royal Military Academy, Woolwich. He was unsuccessful and the position went to Ennis-born Matthew O'Brien, one of the founders of vector analysis. But, upon O'Brien's death the following year, Sylvester was appointed. After retiring from Woolwich, he took up a professorship in Johns Hopkins University in Baltimore, established in 1876. There he founded the American Journal of Mathematics, among the most celebrated and longest-running mathematical journals. The journal has been in publication ever since.

Vacant chair

In 1883 the brilliant Irish number theorist Henry John Stephen Smith, who held the Savilian Chair of Geometry in Oxford since 1861, died aged 57. Sylvester, aged 70, accepted the offer of the vacant chair and held it until his death in 1897.

Sylvester had a deep knowledge of classical literature, and his mathematical papers are peppered with Latin and Greek quotations. He also wrote poetry and wrote a book, The Laws of Verse.

He had a great interest in music, and took singing lessons from Gounod. Sylvester wrote: “May not music be described as the mathematics of the sense, mathematics as music of the reason? The musician feels mathematics, the mathematician thinks music: music the dream, mathematics the working life.”