Arthur Cayley (1821-1895) England

Cayley was one of the most prolific mathematicians in history; a list of the branches of mathematics he pioneered will seem like an exaggeration. In addition to being very inventive, he was an excellent algorist; some considered him to be the greatest mathematician of the late 19th century (an era that includes Weierstrass and Poincaré). Cayley was the essential founder of modern group theory, matrix algebra, the theory of higher singularities, and higher-dimensional geometry (anticipating the ideas of Klein), as well as the theory of invariants. Among his many important theorems are the Cayley-Hamilton Theorem, and Cayley's Theorem itself (that any group is isomorphic to a subgroup of a symmetric group). He extended Hamilton's quaternions and developed the octonions, but was still one of the first to realize that these special algebras should be subsumed by general matrix methods. He also did original research in combinatorics (e.g. enumeration of trees), elliptic and Abelian functions, and projective geometry. One of his many famous geometric theorems is a generalization of Pascal's Mystic Hexagram result; another resulted in an elegant proof of the Quadratic Reciprocity law.

Cayley may have been the least eccentric of the great mathematicians: In addition to his life-long love of mathematics, he enjoyed hiking, painting, reading fiction, and had a happy married life. He easily won Smith's Prize and Senior Wrangler at Cambridge, but then worked as a lawyer for many years. He later became professor, and finished his career in the limelight as President of the British Association for the Advancement of Science. He and the great mathematician James Joseph Sylvester (1814-1897) were a source of inspiration to each other. These two, along with Charles Hermite, are considered the founders of the important theory of invariants. Though applied first to algebra, the notion of invariants is useful in many areas of mathematics.

Cayley once wrote: "As for everything else, so for a mathematical theory: beauty can be perceived but not explained."