In many books on elementary mechanics, it is assumed that velocity is a vector quantity on the grounds that it has magnitude and direction. It is, however, very far from obvious that these are sufficient reasons and a rather more thorough investigation of the nature of velocity may provide not only a clearer idea of what velocity is but may also clear up some of the difficulties associated with “relative” velocity.

The profile of a familiar type of key is shown in Fig. la. The shape of the key profile is determined by the depths of successive cuts. Each cut is of wedge shape. The centres of the cuts lie at N regular well-defined positions. In the key shown there are 6 cuts (N = 6). The depths d of these cuts, which distinguish one key from another, must have one of a set D of well-defined values. If we regard a cut of zero depth as defined by d = 1, the key shown may have values d = 1, 2, …, 6, 7 (D = 7).

This evening, on the occasion of our Annual General Meeting, I propose that we look at the following problem, which I shall call “the AGM problem.”

The object of this article is to show how the problems of transformation geometry, and in particular the determination of the composition of two or more successive transformations, can be solved far more easily by Argand diagram methods leading to algebraic equations involving complex numbers, rather than by purely geometric considerations. It must be conceded, of course, that the methods advocated here cannot be extended to three dimensions.

The growth of language

Someone once said, in answer to an invitation to draw up a chair, “It depends what you mean by ‘chair’; do you mean the thing, the word, or the concept?” There seems to be some relevance in this to the idea of a vector as a physical object, as against its “name” as a triad of numbers, or as an algebraic entity. In illustration, one could point to a table, and give the word for it in several languages. It is clear that communication is only possible at any reasonable level by having first established a language, and then using it as a vehicle for thought. Some would say we cannot think without language; whether this be true or not, the articulate speaker seems brimful of ideas, and the tongue-tied individual dull.

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

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.

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, …)