This article could be subtitled ‘Thoughts on contemplating a model of five tetrahedra inscribed in a dodecahedron’; it is an attempt to communicate the pleasure that ensues when logical reasoning is combined with a visual delight in geometrical figures.

(This article is the result of some work done by a first-year undergraduate, Tat-Sang Fung, at the University of Hong Kong. It went way beyond what his lecturer, Man-Keung Siu, had asked for, and he was so impressed that he submitted it to the Gazette. I have edited it a little, but the ideas are entirely Tat-Sang’s.)

One of the most agreeable fruits of the renaissance which has followed the achievement of national independence in Ireland has been the decision to make accessible, in a form worthy of their importance, the writings of the greatest Irish mathematician and natural philosopher. The responsibility for the work was undertaken by the Royal Irish Academy, with the cooperation of the University of Dublin and the National University of Ireland, and with assistance from the Royal Society and the editorship was entrusted to Professor A. W Conway, F.R.S., of University College, Dublin, and Professor J L. Synge of Trinity College, Dublin. The first volume, devoted to Hamilton’s writings on Geometrical Optics, was published in 1931, and we have now to congratulate the editors (Professor Synge having meanwhile been succeeded by Professor A. J McConnell) on the appearance of the second volume which contains the papers and manuscripts on Dynamics.

I draw your attention to two dates of some importance in the teaching of Geometry. In 1903 Cambridge ruled that “Any proof of a proposition will be accepted that appears to the examiners to form part of a systematic treatment of the subject”; in 1909 the Board of Education issued Circular 711. These two dates determine three periods of time.

Two recent papers published elsewhere [1, 2] set out the original development of logarithms by John Napier: in [2] spreadsheets are used, and the computation of these logarithms is well within the grasp of a student looking for an unusual PC based project. It is an interesting exercise to continue the computer aided investigation of the subsequent development of tables of logarithms in the early 17th century by that other major participant, Henry Briggs (1560–1631): only now can previously unknown flaws in Briggs’ extensive numerical work be found, without the trauma of spending years of arithmetical drudgery. This work is also suitable for a student project using a PC maths package, making use of variable place arithmetic. The story that unfolds is remarkable: some parts, though well-documented, are possibly not so well-known as they deserve - e.g. D. T. Whiteside [3] has shown that Briggs discovered the binomial series expansion for (1 + α)1/2 for small α, while devising a finite difference algorithm for extracting square roots. Florian Cajori [4, pp. 5-14] pieced together the analytical aspects of the development of logarithms in the early years of the 20th century, but largely ignored the numerical aspects of the work, especially that of Briggs (and Mercator). The aspects of Briggs’ work investigated here is guided by his usually good intuition, while of course there are pitfalls into which he occasionally stumbles, as one might expect from someone engaging in numerical work ab initio: with the analytical methods of the calculus still being largely over the mathematical horizon, Briggs presents his methods without formal proof, but uses numerical examples instead.

One of the best known of all classical elementary combinatorial problems is the enumeration of the derangements of n letters A1 A2,…, An, i.e., the permutations of these letters in a row so that, for every i, At is not the ¿th letter. The question is frequently presented in the form of an office boy who puts n differently addressed letters into n envelopes; we then seek the number of ways (w„) in which he can send every letter to a wrong address. Textbooks almost invariably solve the problem by deducing from general reasoning the recurrence relation :

The article which follows is the first of a set of papers sponsored by the Modern Trends sub-committee of the Teaching Committee of the Mathematical Association. It is hoped that future papers will be published either in The Gazette or as a supplement. Each paper will represent the personal view of the author rather than being an official policy statement of the Association. It is hoped that they will lead to correspondence in the form of reports of experiments as well as constructive criticism.

If Pascal’s Triangle is written down, it will be noticed that the number of odd numbers in any row is a power of 2; moreover, if every even number is replaced by 0 and every odd number by 1, the result is an interesting pattern of triangles from which it is possible to deduce a general rule. Again, if divisibility by any prime number p is considered, and if every number in Pascal’s Triangle is replaced by its residue (mod p), an even more intriguing pattern is formed. These considerations led to the following investigation.

I do not know what proportion of the users of mathematical tables go beyond the use of proportional parts in interpolation work. But there are undoubtedly, on the one hand, some tables whose construction would be unnecessarily laborious and expensive were they to be at such an argument interval as to render linear interpolation feasible, and, on the other hand, there are those who find it necessary to use such tables. Given such a table, there are various methods of performing the interpolation. There is the use of the Taylor series, f (a + h) = f(a) + hf′(a) + ½h2f″ (a) + ..., which requires a knowledge of the derivatives of the function as well as the function itself. Then there is the use of the Lagrange formula of interpolation, fx = a0f0 + a1f1 + a2f2 + ..., involving a number of tabular values on both sides of the required value.