The mathematics of Arthur Cayley with particular reference to linear algebra

This thesis is principally concerned with Arthur Cayley's work on Invariant Theory, but also considers his contribution to matrix algebra and other algebraic systems, drawing on sources including unpublished letters between Cayley and his contemporary, J. J. Sylvester. The history of modern linear algebra and Cayley's part in its development has been extensively researched in the last decade by Thomas Hawkins. However, little has been written on Cayley's contribution to Invariant Theory, a subject to which he constantly reverted over a period of fifty years. In comparison, his work on Matrix Theory was a minor interest. The focal points in Cayley's passage through Invariant theory are investigated with reference being made, inter alia, to his correspondence with J. J. Sylvester which affords special insights into both the development of this Theory and the nature of their collaboration. Where appropriate, particulars of Sylvester's own work are given. Biographical details are included where these are believed to be unpublished or otherwise not generally available. A survey of Cayley's mathematical thought is offered in so far as it may be determined from his scattered remarks. Cayley pursued his algebraic researches on two distinct levels. First, he absorbed himself in calculation which led him to the combinatorial aspects of Invariant Theory and, secondly, he displayed a remarkable proclivity for systemisation, although this expressed itself in the classification of specific forms rather than in the development of an abstract theory as with the German algebraists. The basic text contains four chapters on Cayley's work in approximate chronological order followed by a final chapter on his general mathematical thinking. The Appendices include a statistical survey of his work, a bibliography of manuscripts, including, of course, his letters to Sylvester and a number of, little known photographs associated with Cayley and his times.