Arthur Cayley (1821-1895), dubbed the ‘Mathematician Laureate of the Victorian Age’, was elected an honorary member of the Manchester Literary and Philosophical Society in January 1859. In the world of Victorian mathematics, Cayley was looked on by his contemporaries as a leading figure, and in his lifetime published extensively on all aspects of mathematics, most notably, pure mathematics. His Collected Mathematical Papers were published 1889-1897 by the Cambridge University Press and consists of thirteen large volumes listing 967 works.
The year Cayley was elected in Manchester, he was also given a Royal Society ‘Royal Medal’ in recognition for his important contributions to the ‘advancement of knowledge in the sciences’. Cayley was widely known in London, France and Germany, and with this link to the industrial north of England, established a useful publication outlet for his papers.
Cayley but spent his early childhood in St Petersburg, Russia where his father was an English merchant engaged in the Russian trade. After the family’s return to Britain around 1830, he was educated at a clergyman’s school in Blackheath, a new suburb of London, and then at King’s College London, in the Strand. King’s College was founded in 1829 and was based on the traditions of the Anglican Church. It acted as a counterweight to the nearby ‘Godless’ University College in Gower Street!
King’s College was comprised of a Senior Department, teaching at an A-level standard and a Junior Department, this being more like a lower school. Qualification for the Senior Department normally required entrants to be sixteen, though an exception was made for Cayley who was just fourteen when he was admitted. It was obvious from a young age that he was suited for an academic career, though his father had a commercial profession in mind for him. From King’s, Cayley matriculated for Trinity College Cambridge in October 1838. In the Cambridge Mathematical Tripos competition of 1842, he graduated with a BA, as the ‘Senior Wrangler’, the top student of the university. The Senior Wranglers on either side of his year were George Gabriel Stokes (1841) and John Couch Adams (1843) both honorary members of the Manchester Literary and Philosophical Society. Stokes was a mathematical physicist and Adams, a mathematical astronomer, but within this significant triumvirate, Cayley’s interests centred on pure mathematics.
Following graduation, Cayley became a Fellow of Trinity and set about establishing his academic credentials with a torrent of publications in leading French, German and home-based journals. In this phase of his life and following George Boole’s earlier work on the algebra of forms, he introduced a more general theory which became known as ‘Invariant Theory’. In this branch, he studied algebraic forms which do not change in their essentials when their variables do change. This branch of algebra mushroomed in the nineteenth century and became a theory closely associated with Cayley and his friend, James Joseph Sylvester (an honorary member of the Manchester Literary and Philosophical Society, elected in 1861), so much so that they were known in mathematical circles as the ‘invariant twins’.
Being a Fellow of Trinity entitled Cayley to a share of the ‘College Chest’: a financial provision that would be available for seven years – provided he did not marry. Bearing in mind his background of middle-class stability, he was able to devote himself to scholastic endeavour. It was during this period that he first came in contact with Manchester when adjudicating a paper submitted to the Cambridge Mathematical Journal written by Thomas Kirkman (another mathematician and Manchester-based savant, elected an honorary member of the Manchester Literary and Philosophical Society in October 1852).
In time, Cayley would need a way of earning a living. Taking the route favoured by other Cambridge graduates, he became a trainee conveyancing barrister. Admitted to Lincoln’s Inn in 1846 he was called to the Bar three years later. For fourteen years, Cayley worked in the legal profession but all this time he maintained a vigorous programme of mathematical research. He was elected a member of the Royal Society in 1852 and within a year of election was serving on its Council. This was coupled with active membership of the Royal Astronomical Society and behind the scenes in scientific London, he proved to be an able administrator.
During the 1850s, Cayley contributed to the new algebraic subjects of group theory (where mathematical members of a set are combined under a mathematical ‘binary operation’ [with required properties] to give a result that is part of the original set) and matrix algebra (a study of rectangular arrays of numbers and the operations defined on them). These two lines of enquiry became important during the twentieth century. In the 1850s, Cayley was able to help fellow mathematician Kirkman, further establish himself in the wider scientific community by proposing him for membership of the Royal Society in June 1857 and recommending the publications of his papers in its Philosophical Transactions. In turn, Cayley published a paper in the Memoirs of the Manchester Literary and Philosophical Society, and Kirkman read for him (presented his paper) on this occasion. While Cayley adopted a wide compass for his work, both he and Kirkman were the only two British mathematicians who carried out work on group theory in the 1850s and 1860s which was mainly being researched by French mathematician at the time.
When the British Association for the Advancement of Science met in Manchester in the autumn of 1861, leading mathematicians attended alongside Cayley including the Astronomer Royal, George Biddell Airy and the Irish mathematician, William Rowan Hamilton. Kirkman was a vice president of the mathematics and physics section on the occasion. After this, the harmonious relationship with Manchester was put into jeopardy when Kirkman was said to have held Cayley responsible for the Royal Society’s rejection of a paper he had submitted for inclusion in the Philosophical Transactions and their relationship reportedly suffered. It was unfortunate, and Cayley published only one other paper in Memoirs (on differential equations). It was an opportunity missed, but overall the reputation of the Society had been enhanced by Cayley’s selfless promotion of Kirkman’s work and a recognition that useful mathematical work was being done in Manchester.
Cayley returned to Cambridge when the university changed its rules on marriage and he became the first full-time Sadleirian Professor of Pure Mathematics in 1863. He married Susan Moline the same year and they had two children together (Henry and Mary). This is where he spent the remainder of his career until his death, in Cambridge, on 26 January 1895, aged seventy-three.
To this day, A-level Further Mathematics students still know Cayley’s name, as his eponymously titled ‘Cayley Tables’ in group theory are still taught. This is a square table which displays the results of combining each pair of elements of the group. Cayley’s name appears in a myriad of constructions in modern mathematics, including Cayley graphs and Cayley’s theorem in group theory. Not just honoured in mathematics, his contributions to mathematical astronomy are also recognised. His name lives on in a lunar crater (located to the west of the “Sea of Tranquillity” on the Moon) and a Cayley asteroid, located in the asteroid belt between Mars and Jupiter.
Tony Crilly, member of Manchester Lit & Phil