Probably many members of the Mathematical Association are being called upon to help with the instruction of an Air Training Corps, or of attested men who have been accepted as pilots or observers and then sent home on “deferred service”. It may be of interest to give the experiences of a teacher who came into contact with this work at an early stage and so encountered some of the inherent difficulties in their most awkward form. As Klein remarked, mathematical teaching is a function of two variables, the subject and the pupil. The difficulties with the subject are not serious, for there are few ideas in air navigation which are not already known to those who teach elementary mathematics and geography. However, these ideas are clothed in unfamiliar dress, and it is necessary to learn a few technical terms, which will be given at the end of this article.

Broadly speaking, the development of mathematical thought has proceeded along two main channels. On the one hand mathematicians have been studying properties of objects which in some obvious and intuitive sense form what is called a continuum, while others were concerned with objects which one conceived as being discrete and well separated from each other. An example of the former type of mathematics is the differential calculus; and the theory of the distribution of the prime numbers 2, 3, 5, 7, 11, … belongs to the mathematics of the discrete. The mysteries of the epsilontic—so dear to us all: “If epsilon is sufficiently small, then delta will be as small as we please”—have no direct counterpart in the theory of the natural numbers 1, 2, 3, …. The underlying idea of the continuum seems to be that of motion, while the discrete and separable appears to be more of a static nature.

I Give below a topological proof of the following theorem, based on a well-known conundrum:

In a letter to the Gazette (XXV, February, 1941, p. 36) Mr. Siddons discussed my article in the Gazette for October 1940; he gave a method of long multiplication of money and suggested that I might carry out further tests using his method. I was at once attracted to Mr. Siddons’ method, and carried out some tests, though I was unable to make them as extensive or as exhaustive as I desired. The present article contains a report on these tests. No very definite conclusions can therefore be drawn. This record of the work done may, however, serve as a useful pointer for those who may be sufficiently interested to carry the investigations further. I should have been pleased to have done so myself but the circumstances of my professional life have altered somewhat since carrying out the investigations recorded in recent numbers of the Mathematical Gazette.

A few sessions ago the writer was presented with the problem of providing for one hour per week a course outside the normal Intermediate syllabus in Mathematics for a group of First Year engineering students. Knowing how the engineer thinks with pencil in hand, and uses the drawing-board for a variety of design calculations, the solution adopted was that of putting as much elementary mathematics as possible on the drawing-board. This article outlines the course, along with some additional related matter, and among the reasons which seem to justify publication are

(a) the author—and some at least of his students!—have enjoyed the work described;

(b) much of the subject-matter is not easy to find in English books;

(c) a number of illuminating sidelights are thrown upon topics of the “normal” mathematics course;

(d) it is believed that portions at least could be introduced with advantage into both pre- and post-certificate courses.