Arthur Cayley was born on August 16, 1821, in Cambridge, England.
Arthur was born to Henry Cayley. Although Arthur was born in Cambridge he spent the first eight years of his life in St. Petersburg, Russia, where his dad lived. From the start of Arthur's education at King's College School it was very apparent he had a flare for advanced mathematics. Arthur's math teacher advised Arthur to pursue a career in mathematics against the wishes of his father who wanted Arthur to continue the family business as a merchant.
In 1838, Arthur began his studies at Trinity College located in Cambridge, entering as a pensioner. For 4 years he studied mathematics and graduated in 1842 as a senior wrangler winning the first Smith's prize. While still an undergraduate he had 3 papers published in the Cambridge Mathematical Journal. Arthur went on to teach at Cambridge for four years, publishing twenty-eight more papers in the Cambridge Mathematical Journal.
Arthur then decided to go to law school to become a lawyer. While in training to become a lawyer he continued his interest in mathematics, going to lectures about different fields of math at Dublin College where he met J.J. Sylvester. Sylvester was also in the legal profession and shared the same passion Cayley had for mathematics. During the time Arthur was in training to become a lawyer he published about 250 papers and continued to be a lawyer for fourteen years. Although Arthur was very skilled at being a lawyer he decided to use the money being a lawyer had given him to pursue a career in mathematics he so desperately wanted.
In 1863, Arthur finally got his break as a professor of Pure Mathematics at Cambridge. Arthur's job was explain and teach the principles of pure mathematics and to apply himself to the advancement of Pure Mathematics. Of course going from an experienced lawyer to an inexperienced teacher was quite a decrease in salary. This did not seem to bother Arthur very much because he finally got to do what he wanted to do ever since he was a young boy.
Along with being a professor, Cayley also wrote over 900 papers and noted that explained almost every aspect of modern mathematics. Arthur accomplished many things in the field of mathematics. Some of these accomplishments were he, along with his life time friend Sylvester are considered to be the founder of invariant theory. He was the also solely responsible for matrix theory and proving many of the theory's to be correct. He was one of the first mathematicians to consider geometry of more than three dimensions.
In 1854 Cayley published a paper were he considered a group as a set of symbols, all of them different and such that the product of any two of them, in any order, or the product of any one of them into itself, belongs to a set. This was very different from what earlier mathematicians had formulated. This marked him as the founder of abstract group theory. Arthur also developed the Cayley Table, where he introduced a procedure for defining a finite group by listing its elements in the form of a multiplication table.
In 1876, he published his only book, Treatise on Elliptic Functions. Following that in 1881 he was invited to give a course of lectures at Johns Hopkins University in the United States. He continued to lecture until May of 1882. The following year Cayley became President of the British Association for the Advancement of Science. He remained the President of the British Association for the Advancement of Science and continued teaching at Cambridge until his death on January 26, 1895. He was 74 years old.
RESOURCES
1. http://www.math.ukans.edu/~engheta/bio/cayley.html
2. http://www-groups.dcs.st-and.ac.uk/~histy/Mathematicians/Cayley.html