In this lecture we shall see how natural philosophy—or what we traditionally call applied mathematics—is continually changing. I hope to show that the subject is far livelier and more adventurous than we sometimes suppose, and that each advance opens ever wider vistas of further progress.

We are concerned with the relation between mind and experience of the physical world. Our minds are such as to be most strongly persuaded that there is a significant relationship by their being able to predict experience that no one has ever had before.

The reader will be familiar with the golden rectangle, i.e. that rectangle which leaves another rectangle of the same shape when the largest possible square is cut from it (Fig. 1). In this article the effect of successively cutting squares from rectangles of other shapes will be investigated.

This article is based on students’ remarks about the introduction of matrices and linear transformations; remarks which provide interesting support to those recorded by M. L. Cornelius [1] in a recent Gazette. However, the purpose of the article is not merely to record students’ opinions, but to consider what implications such responses have for us as teachers.

Among the topics emphasized at the Second International Congress on Mathematical Education held at Exeter in 1972 were the increased interest in approximation due to computers and the pedagogical value of geometric figures. This article brings together these two themes.

The following notes arose from a talk given by Professor Halberstam to the South West Wales branch of the Association. In the marriage bureau problem we imagine that n boys and n girls are to be paired off in marriage, that we are able to establish preference lists for each boy of all the girls and vice versa. These preference lists may be displayed in a table such as that given below, which shows the rankings for four boys α, β, γ and δ, and for four girls A, B, C, and D. The boys’ rankings are the first members of the number pairs, in rows, and the girls’ are the second members, in columns.

The increasing use of simple computing languages in schools gives an opportunity for rationalisation of our existing mathematical notation as follows:

It is time we abandoned “÷”, which is used in primary schools and causes great difficulties later on when children have to cope with the fact that or, worse still, If we used “|” from the start, what could be easier than (Nor is there a case for restricting ÷ to ‘sharing’ situations, since + and − are used for both natural numbers and rationals.)

The intersection of two circular cylinders of equal radius is not only of mathematical interest but also has application in both engineering and architecture. The joining of pipes of circular cross-section at a variety of given angles is an obvious example. The Romans and Normans, in using the barrel vault to span their buildings, were familiar with the geometry of intersecting cylinders where two such vaults crossed one another to form a cross vault. Larger numbers of equal intersecting cylinders arise in the following way.

In Note 2566 (Gazette XXXIX, 305) Professor Arscott quoted some results on divisibility, generalised to cover the cases of odd numbers of the form N = 10M + z, where z = 1, 3, 7 or 9. He stated that the proofs are “based upon simple congruences . . .” and the proof for divisor 53 is given.

This article develops the four cases from one central proof, and then indicates how these may be applied as tests for all odd divisors. If the need is to resolve a given number into its prime factors, the appropriate powers of 2 and 5 may easily be removed; if, however, all that is required is to establish whether (or not) a given odd number is a factor, this preliminary operation is unnecessary.

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.

A typical sixth form book which is introducing vectors may approach the subject in the following manner. Of necessity, this particular introduction is rather condensed, but the reader will almost certainly recognise certain key phrases and be familiar with the general spirit of what is intended.

Any simple transformation that leaves the origin unchanged corresponds to multiplication by a 2 × 2 matrix which can generally be written down at sight. Transformations such as translations, under which the origin moves, cannot be represented in this simple manner. However, by moving into three dimensions this deficiency can be made good. Furthermore, the exercise gives useful experience of thinking in three dimensions and a useful addition to the repertoire of matrices.

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.