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.

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 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.

During the spring of 1973 we were able to visit the Soviet Union for 14 days to study mathematical education. Our visit was arranged by the British Council under the Anglo/Soviet cultural agreement. One of the most interesting features of the Soviet system which we hoped to study during our short tour was the specialised boarding school. Accounts of some of these have appeared already (for example, Bryan Thwaites’ article in the Gazette for December 1968) and it was intersting to compare our impressions with those which had been recorded by others.

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.

One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:

“For all x, y, z in ℝ:

1. 1. Exactly one of x = y, x < y, y < x holds.

2. 2. If x < y and 0 < z, then xz < yz.

3. 3. If x < y, then x + z < y + z.”

Let G denote a finite non-commutative group. What can we say about the number of ordered pairs (x,y), where x,y ϵ G, such that xy = yx?

In order to prepare ourselves to answer this question, let us examine the Cayley table of S3, the symmetric group on three symbols. We take as generators the reflection (12) and the rotation (123). It is clear that we are examining the Cayley table for symmetry about its main diagonal.

“And though he made many elegant discoveries, he is said to have besought his friends and kinsmen to place on his grave after his death a cylinder enclosing a sphere, with an inscription giving the proportion by which the including solid exceeds the included.” Plutarch, on Archimedes, in The lives of famous men, the Life of Marcellus.