It has now been possible to study the many comments received since the publication of the “interim statement” in Gazette No. 407 (March 1975). The recommendations given below are based on the original suggestions, taking these comments into account. Sidelining indicates a change or modification of the original suggestion.

It is proposed also to publish these recommendations as a pamphlet, so that they may be widely available to authors and others.

The following paper is an attempt to explain the action of a class of three-dimensional mechanisms, first from a theoretical standpoint and afterwards to suggest some practical applications. One or two rather more general mechanisms are investigated than have previously been studied and well-known mechanisms are deduced as special cases of these. There is no modern textbook or recent publication dealing with these mechanisms and their properties are not generally known.

This note describes a possible approach to the direct calculation of both derivatives and integrals of power functions using rather simple algebra and limits that could be easily accessible to students beginning a calculus course. The calculations provide a powerful example of the fundamental theorem, and could be used to allow students to discover the fundamental theorem from the example. The main calculation, which is not found in many calculus textbooks and is due to Descartes, is described by Maor [2]. While this approach may seem like the popular American ‘reform’ calculus, one may find a somewhat similar approach (presenting integral calculus from first principles) in Tom Apostol’s classic and lovely text [1].

Some three weeks into the academic year I presented, off the cuff, the following morsel to engage some lower-sixth formers who had polished off some routine derivatives all too quickly:

Sketch the curve whose equation is given by x3 + y3 + 3xy – 1 = 0 , and find the gradient at the point (–1 , –1).

Since this was near the end of a double period I thought no more of it. At the end of the next double period one student remarked that he had been having difficulty in sketching it. “It seems to be a straight line”, he said. I rashly suggested that he could well be mistaken, but promised to look at it. If my faith in the student had been greater I could have saved myself some labour searching for simple solutions of the cubic. I had noticed the solution (½ , ½) by the way. I began to have delusions of grandeur, imagining myself to be the natural successor of Diophantus, as I found the solutions (5,–4) and (–4,5).

I believe many people have been inspired by Grunbaum and Shephard's book on tilings and patterns. On reading the book, as a computer programmer, I thought it would be a challenge to display the tiling patterns in colour on the screen of my home computer. Not having a hard disc, I had found the storage for fractals too demanding, notwithstanding the time needed to compute the intricate patterns. Hence the idea of producing a variety of interesting patterns without such overhead was attractive.

Take any positive integer written in the usual base 10 notation, reverse the digits and add to the original number. If the result is a palindrome, stop. If not, continue. Will a palindrome always occur at some point? Let us do a fe examples to see the sort of thing that happens.

At the end of the nineteenth century David Hilbert made the decisive step in the foundations of geometry (for the historical development see [1]): a road was opened to a theory of geometric structures. In spite of the fact that one spoke (and speaks still today!) strangely enough of ‘implicitly’ defining the geometrical concepts through the axioms, it became clear in the sequel that the axiom system of Hilbert constitutes an ‘explicit’ definition: not of course for ‘point’, ‘line’, etc. but of the concept ‘euclidean geometry’ (or ‘euclidean space’).

Very few teachers ever have the chance of noting the advantage or disadvantage of a change, introduced into their methods of teaching, and of profiting by the same. I desire now to put on record the results of such a change and the method by which they were attained.

The number of systems of teaching dynamics is great, and still the results are deplorable as judged by the knowledge possessed by the average engineer some years after he leaves college. Teachers are either biassed in favour of the Gaussian system or the Perry system, or some other system of their own, and are loath to change because they can train their students to solve the necessary problems, and if a mistake is made by omitting g or some other quantity it is merely a mistake and costs nothing. It is not so with the poor student, however, when he is launched on the world without his teacher to help him.