The formation of a Joint Mathematical Council for the United Kingdom has recently been announced, and readers of the Gazette may be interested to know the reasons for setting up such a body, and what it hopes to achieve.

The problem is to join two given straight pipes by a curved pipe made up of two circular arcs of equal radii. The case when the two given pipes are coplanar has been discussed by E. Kosko in the Gazette of May 1964, pp. 192-196. Purely geometrical constructions for this case have been provided by S. N. Collings in the Gazette of October 1965, p. 302, and by me in the Gazette of February 1966, pp. 37-39. (Unfortunately Mr. Collings’s construction does not always work. On his diagram one can put in the two straight pipes in four different ways. For two of these ways the construction gives valid solutions, but for the other two ways the joining curve has a discontinuity of sense.)

We propose to construct some triangles somewhat like the Pascal triangle, but more general. The only essential knowledge is that of the binomial theorem, though for some of the results we assume a knowledge of number theory, up to Fermat’s theorem. At the end, we show how matrix algebra could be used to advantage.

1. Introduction

In [1] the following three questions have been asked, and answered: For which values of f is there a convex polyhedron whose faces consist of

• (i) f triangles? (Ans f = 4, 6, 8, 10, 12, 14, …)

• (ii) f quadrilaterals? (Ans f = 6, 8, 9, 10, 11, …)

• (iii) f pentagons? (Ans f = 12, 16, 18, 20, 22, …)

In a paper with the same title, Carlitz [1] proves two inequalities about a triangle ABC and an internal point O:

Geometry in a Mathematical Syllabus seems to me to occupy much the same position that Greek does in the Classical. The two have similar merits and similar defects, and I am doubtful whether either deserves its place except in specialist classes.

The following note develops ideas in fields outside the normal school syllabus and if its contents are introduced at school, will, it is hoped, serve as a stimulus to encourage the brighter student to carry out further investigations in these fields.

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not alwaysdefine, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.

Oxbridge Tripos examination questions originating from the turn of the century have had a marked effect on the teaching of mechanics both at School and at University. These questions, reproduced in the standard textbooks of the day, were the stable diet of the undergraduate population over so many years and set the pattern for a style of question in which the student applies mechanical principles by rote without ever questioning the validity or relevance of those principles. An unfortunate consequence of this state of affairs is that mechanics has been largely overlooked as a vehicle for the teaching of mathematical modelling. There have been moves recently to make questions more “relevant” and it is not unknown to see illustrations of “real” boats, shotputters etc. However, the student is still expected to answer the question “mechanically”, if the reader will excuse the pun, without any discussion of the modelling process. How refreshing it is, even with first year undergraduates, to look at the problem of a tin can being knocked off a wall by a bullet and, instead of asking the students to apply two conservation laws in order to determine the speed with which the can hits the ground, to ask under what assumptions the two conservation laws would appropriately model the physical situation? It is quite staggering how the lecture theatre erupts with ideas, suggestions and interest.