The aim of this note is simply to encourage dialogue between mathematics and physics teachers. In most school physics and chemistry courses, any mention of quantum theory has been limited to a brief qualitative discussion, or has been restricted to the simple Bohr theory. The new Nuffield Advanced Physics material [1] goes much further than this, with the opportunity to apply some quite simple mathematics, supported by a computer analysis. We shall here give a gross simplification of the end of a long story in the physics course.

Many teachers in junior schools will know the interest which children show in drawing patterns of multiples on a 10 × 10 lattice. If those for the primes 2, 3, 5, 7 and 11 are drawn on separate transparent sheets and are then superimposed, the picture produced by the overhead projector will have some of the appearances of a sieve. It will then be worthwhile to refer to an article by R. A. Fisher in Gazette XIV, 564–6 (No. 204, December 1929) which carries out the process in a slightly different way to reveal all the odd primes less than 1600, by drawing the factor lines of the primes up to p = 37; this same set will, of course, isolate all the primes less than 1681.

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved:

In the (n + l)th row of Pascal’s triangle, there are

Because of ‘modern mathematics’ most people are aware of different number scales. Ordinarily there is the decimal scale, while computers use the binary system and sometimes, I believe, the octal system based on 8s. Other scales may appear to be freakish, or mere excuses for arithmetical manipulations; yet at times they can be useful.

The paper begins with a description of a diagrammatical model which I used when lecturing on elementary calculus to a class of engineers. The second part suggests how the use of the model might be extended to provide a framework on which one could build up the theory of differentiation of functions of one or more real variables. For the purposes of this article it is assumed that the reader is familiar with this theory—the paper, therefore, merely gives the outline of a possible approach. The keynote of the lecture course to the engineers was ‘plausibility’ rather than ‘rigour’ and my use of ‘function boxes’ was sparked off by the fact that the class were using flow diagrams and ‘black boxes’ in a number of other courses—most noticeably in that given by the person who lectured immediately before me and who, in his progress up the technological scale to the electronic computer, had clearly not encountered that mundane educational aid, the blackboard duster. The grand finale of the one-term course was to be an introduction to partial differentiation and, in particular, the chain rule—a piece of mathematics which students often find difficult and which I hoped to make easier by means of an approach using a simple type of ‘flow’ diagram.

Newton was fascinated by the fact “which is very remarkable” that the two “principal cases of attractions” (i.e. the inverse square and direct distance laws) are distinguished from others in two independent ways:

1. (i) the attraction of a spherically symmetric body at a point outside “preserves the same law of increase or decrease” as that of a point mass (call this the spherical property);

2. (ii) a particle can move in an elliptical orbit, though with different centres of force in the two cases (call this the elliptical property).

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.