James Joseph Sylvester was born in London to Jewish parents in 1814, just 201 years ago today. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic Pope from the Roman period – was adopted later. [TM075; or search for “thatsmaths” at irishtimes.com ]

Sylvester’s mathematical talents became evident at an early age. He entered Cambridge in 1831, aged just seventeen and came second in the notorious examinations known as the Mathematical Tripos; the man who beat him achieved nothing further in mathematics!

**Degrees from TCD**

Trinity College Dublin awarded Sylvester the degrees of B.A. and M.A. that 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 twenty five.

Sylvester, like many brilliant young people today, had difficulty in finding a suitable position. He took a post in Virginia, but it turned out badly and he returned from America within a year or so. He supported himself by taking private pupils, the most distinguished of these being Florence Nightingale.

In 1846, aged thirty-two, Sylvester entered the Inner Temple 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.

**Invariants**

The theory of algebraic invariants had initially been developed by George Boole while he was at Queen’s College Cork. At school we learn to solve quadratic equations like *ax*^{2}*+bx+c=0.* Quadratics have two solutions, called roots. The nature of the solutions depends on a quantity *b*^{2}*–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 gave American mathematics a tremendous impetus, and has been in continuous publication ever since.

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

Sylvester had a broad and deep knowledge of classical literature, and his mathematical papers are peppered with Latin and Greek quotations. He also wrote poetry and authored a book, *The Laws of Verse*. He had a great interest in music, once taking 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.”

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