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Arthur Cayley: Mathematical laureate of the Victorian age. (English) Zbl 1117.01016

Baltimore, MD: The Johns Hopkins University Press (ISBN 0-8018-8011-4/hbk). xxi, 906 p. (2006).
Tony Crilly is without doubt the foremost expert on all matters concerning Arthur Cayley and his biography bearing the subtitle Mathematician Laureate of the Victorian Age documents his insights into the life and the mathematics of this great mathematician. However, as respected Cayley is in some mathematical circles so is he neglected by many as Crilly wearily writes in the introduction. He always was a calm man, quiet and shy in contrast to his friend James Joseph Sylvester who liked to show off in all respects.
Crilly’s biography is divided into five periods, Growing Up (1821-1843), New Vistas (1844-1849), A Rising Star (1850-1826), The High Plateau (1863-1882), and Make One Music as Before (1882-1895). Cayley was born as son of a merchant from Yorkshire who had settled with his family in St Petersburg in Russia. Being part of a small english community Arthur grew up there until the age of eight when his family moved back to England to settle near London where Arthur’s father continued his successfull enterprises. Early on mathematical talent was detected in the boy. When he attended King’s College from his 14th year on his tutor recommended studying mathematics in Cambridge which Arthur did. Not only brilliant in mathematics but also gifted in foreign languages, he became Senior Wrangler in 1842. Already as a freshman had he three mathematical papers published in the Cambridge Mathematical Journal and had won the Smith price. Not surprisingly, he won a fellowship of Trinity College and stayed for some years in Cambridge but then decided to become a barrister - a most remarkable parallel to his friend Sylvester. Cayley went to Dublin to attend lectures of Hamilton on quaternions and started mathematical work with Sylvester. In 1863 he was called to the Sadlerian chair in Cambridge which meant a financial loss to him but his dreams of a life in academe had been fulfilled.
His achievements in mathematics are tremendous: together with Sylvester he dominated the theory of invariants in England. He invented the notion of abstract groups, has left his mark on the theory of matrices and algebraic equations, and is the inventor of the octonions which are called Cayley numbers nowadays. Every mathematician knows the famous Cayley-Hamilton theorem and there is a universe of other important mathematical results connected with his name. Although Crilly has done a lot to keep mathematical details away from the reader who is interested in the biographical details only he goes down to mathematical details from time to time which is unavoidable in a biography of this kind. There is even an appendix containing a glossary of mathematical terms so that the reader who is not a mathematician can manoeuvre through the more mathematically inclined passages of the book.
This biography of Cayley is thoroughly based on hard research and knowledge, written by an expert, and a milestone in the history of the reception of Arthur Cayley’s works. The style of the book is lucid and readable throughout. This is certainly the definitive Cayley biography. As accompanying reading the book on Sylvester by Karen Hunger Parshall [K. H. Parshall, James Joseph Sylvester. Jewish mathematician in a Victorian world. Baltimore, MD: Johns Hopkins University Press. (2006; Zbl 1100.01006)] can be recommended. Reader which may be interested in greater mathematical details of Cayley’s work can look up Cayley’s Collected Works [A. Cayley, The collected mathematical papers. 14 volume set. Reprint of the 1889–1897 original. (2009; Zbl 1194.01173)].

MSC:

01A70 Biographies, obituaries, personalia, bibliographies
01A55 History of mathematics in the 19th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography

Biographic References:

Cayley, Arthur
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