Tony Crilly

From Kant to Hilbert: a sourcebook in the foundations of mathematics , William Ewald (ed.). 2 vols. Pp. 1340. 1999. £50 (Paperback). ISBN 0 19 850537 X (Oxford University Press).

Georg Friedrich Bernhard Riemann Hermann von Helmholtz (1821-1894) Julius Wilhelm Richard Dedekind (1831-1916) Georg Cantor (1845-1918) Leopold Kronecker (1823-1891) Christian Felix Klein (1849-1925) Jules Henri Poncare (1854-1912) The French analysts David Hilbert (1862-1943) Luitzen Egbertus Jean Brouwer (1881-1966) Ernst Zermelo (1871-1953) Godfrey Harold Hardy (1877-1947) Nicolas Bourbaki.

The rise of Cayley's invariant theory (1841–1862)

Abstract In his pioneering papers of 1845 and 1846, Arthur Cayley (1821–1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation. This article discusses these papers in the light of Cayleys unpublished correspondence with George Boole, who exercised considerable influence on Cayley at this formative stage of invariant theory. In the 1850s Cayley put forward a new synthesis for invariant theory framed in terms of partial differential equations. In this period he published his memoirs on quantics, the first seven of which appeared in quick succession. This article examines the background of these memoirs and makes use of unpublished correspondence with Cayleys lifelong friend, J. J. Sylvester.

The decline of Cayley's invariant theory (1863–1895)

Abstract From the 1860s the German symbolic approach to invariant theory was in ascendancy. This article discusses the of work Arthur Cayley (1821–1895) and his reaction to this new line of enquiry. The symbolic method is outlined and compared with Cayleys viewpoint in which the calculation and exhibition of invariants and covariants were of primary importance. Cayleys Law and Gordans finiteness theorem, two principal results in the theory, are discussed. Also covered is J. J. Sylvesters Fundamental Postulate, which both reveals the character of the English empirical approach to invariant theory and illustrates its inherent weakness. The article examines the background to Cayleys final three memoirs on quantics, his last work in invariant theory, and it makes use of correspondence with his friend Sylvester.

Cayley's anticipation of a generalised Cayley-Hamilton theorem

Abstract The principal result of Cayleys famouus memoir on matrices of 1858 is his contribution to what is now known as ‘the Cayley-Hamilton theorem’. We discuss this theorem and show that prior to its publication Cayley was aware of a more general theorem, a result that he left unpublished. This theorem is associated with the binary algebraic form det (μP − λQ) analogous to the standard characteristic polynomial det (A − λI).

Arthur Cayley FRS and the four-colour map problem

The four-colour map problem (to prove that on any map only four colours are needed to separate countries) is celebrated in mathematics. It resisted the attempts of able mathematicians for over a century and when it was successfully proved in 1976 the ‘computer proof’ was controversial: it did not allow scrutiny in the conventional way. At the height of his influence in 1878, Arthur Cayley had drawn attention to the problem at a meeting of the London Mathematical Society and it was duly ‘announced’ in print. He made a short contribution himself and he encouraged the young A. B. Kempe to publish a paper on the subject. Though ultimately unsuccessful, the work of Cayley and Kempe in the late 1870s brought valuable insights. Using previously unpublished historical sources, of letters and manuscripts, this article attempts to piece together Cayley’s contribution against the backcloth of his other deliberations. Francis Galton is revealed as the ‘go-between’ in suggesting Cayley publish his observations in Proceedings of the Royal Geographical Society. Of particular interest is that Cayley submitted two manuscripts prior to publication. A detailed comparison of these initial and final manuscripts in this article sheds new light on the early history of this great problem.

The birthday problem for boys and girls

The famous birthday problem is well known to regular Gazette readers. For those readers not familiar with it, the problem is to find the size of the smallest group of people for which there is a probability greater than one-half of two members of the group sharing a birthday. The answer (23 people) is a result which the eminent probability specialist William Feller described as “astounding”.

A gemstone in matrix algebra

The history of mathematics is an abundant field for making surprising discoveries. Some luck is necessary, but whereas the literary tradition has been extensively explored by scholars, finding a nugget in the history of mathematics is not so difficult as turning up a new fragment by Jane Austen, or a lost piece by Mozart. In my case I struck lucky in the theory of matrices. As is well known, the theory of matrices arose during the 1850s – a useful benchmark is the publication of Arthur Cayleys celebrated memoir of 1858, A memoir on the theory of matrices , a paper which generations of mathematicians have taken as signalling the beginning of matrix theory. Even today the opening paragraphs serve as a useful introduction to the subject.

An Improbable Game of Darts

The general utility of the ‘pigeon hole principle’ (that if n + 1 pigeons are placed in n boxes one of the boxes contains more than one pigeon!) is neatly illustrated by the problem of showing that: If seven darts land on a dartboard in any configuration whatsoever, there will always be two darts which are not more than a distance of one radius apart .

What became of Paul Dirac’s classmate?

We examine Paul Dirac’s early life in Bristol and the link with his classmate Herbert Charles Wiltshire. We outline Wiltshire’s subsequent career using archives and the few letters which survive between Dirac and Wiltshire.

BSHM Christmas meeting Birmingham, 5 December 2015

purpose computer, the analytical engine, designed but never built by Charles Babbage. Professor Martin showed how Lovelace’s paper presented the analytical engine as an abstract machine. Also drawing on correspondence between Lovelace and Babbage she described how Lovelace’s speculations on the capabilities and potential of the machine mirror the concerns of modern computer scientists. We also heard what a remarkable person Lovelace was, not only in her background but in her efforts to learn, mathematics in particular, and her ambitions. Drawing on original sources and evaluating secondary material we were presented with a fresh and stimulating evaluation of Lovelace who was born in 1815, dying at age 36. We were all invited to visit the Ada Lovelace mathematical archive to help assess for ourselves this remarkable person. This archive draws on recent archival research and will be released online at www.claymath.org in December 2015. All the lectures were not only informative and stimulating but very enjoyable. If you were unable to be there or simply wish to relive the experience the lectures are available online at: http://www.gresham.ac.uk/women-in-mathematics-the-bicentenary-of-adalovelace Raymond Flood http://dx.doi.org/10.1080/17498430.2016.1215857