To categorize the innumerable works which are published every year in mathematics is a superhuman task, on account of the close-knit network, almost impossible to unravel, of links which bind together the various theories, and which are the mainsprings of their progress. The list which follows, then, must not be seen as anything else than a very approximate indication of the various directions in which mathematical research might be taken, with respect to problems which often do not seem to point in these directions.

For this ‘centenary issue’, the editor has invited me to write a survey on progress in number theory over the last 100 years. He did not, of course, ask for a proper technical report, but just a short article suitable for Gazette readers. Fortunately number theory is one of those subjects in which one can speak, albeit superficially, on the problems and results without having to explain much of the technical details associated with them.

The evolution of transcendence into a fertile theory with numerous and widespread applications has been an especially exciting development of modern mathematics. The subject was originated by Liouville, Hermite and Lindemann during the last century. Liouville showed that there is a limit to the precision by which an algebraic number, not itself rational, can be approximated by rationals and thereby gave the first examples of transcendental numbers; in fact it suffices to take any non-terminating decimal with sufficiently long blocks of zeros or any continued fraction in which the partial quotients increase sufficiently rapidly.

In my 1982 Presidential Address to the Mathematical Association I tried to explain the general role of Geometry in Mathematics so it seems appropriate, on this centenary occasion, that I should move beyond the confines of Mathematics and discuss the interrelation of Geometry and Physics. There are two very good reasons for doing this. One is historical and arises from the close ties between the two subjects in their early evolution. A second and more topical reason is that, over the past two decades, there has been a remarkable burst of interaction of a quite unexpected kind between Geometry and Physics.

In the development of Mathematics in the past 2500 years we can discern two strands, namely, formal deduction or logic, associated initially with the Stoics and later with Aristotle and Euclid among others, and mathematical analysis, which we see emerging in the same era in the works, for example, of Archimedes and Eudoxus. These strands, for the most part, developed separately until the seventeenth century when Newton and Leibniz invented the calculus. Newton’s presentation, however, was controversial for his arguments deployed infinitesimals and fluxions which some, especially Bishop Berkeley, rightly considered contradictory.

When the Mathematical Gazette was founded, statisticians were mainly concerned with the collection and presentation of data. The mathematical side consisted mainly of developments from the work of Gauss and Laplace in the early part of the 19th century, though some surprisingly modern work was done by Edgeworth towards the close of the century. 1889 saw the publication of Galton’s Natural Inheritance, introducing correlation and regression. These two workers influenced Karl Pearson who, in 1900, began the modern era of mathematical statistics with the concept of the significance test usually known by the symbol he used, chi-squared.

Until the First World War the only effective aids to calculation were the slide-rule, for 1% accuracy, or books of tables of logarithms, roots and trigonometric functions. These were 4 or 5-figure for normal use and 6 or 7- figure for those who needed the extra accuracy. Many ingenious methods were devised for increasing the the length of the slide-rule scale to improve its accuracy: some scales were made spiral and others were wound round a cylinder or printed in multiple parallel lines. But higher accuracy always meant slower calculation.