In the Rules of this Association, there is a declaration that its object shall be

Having regard to such a policy in action, having regard also to the professional duties of what probably is the majority of the members, I must appeal for a measure of your indulgence when my address, in retiring from office as your President, is devoted to some reflections concerning the present range of elementary Applied Mathematics.

Mensuration involves the construction of measures and the methods of using them and the calculation of various quantities from the simplest possible measurements. It is in this last sense that the term is usually employed, and herein problems in heights and distances are solved, the lengths of curves and the areas of plane or other figures and the volumes and surfaces of solids are measured.

Several writers have studied algebras in which multiplication is non-associative, that is, x yz≠xy z. It is necessary in a non-associative algebra to distinguish the possible interpretations of a power xn In a non-commutative non-associative algebra x2 is unique, x3 can mean xx2 or x2x; x4 can mean x xx2, x x2x, x2x2, xx2x or x2x x, x5 has 14 interpretations; x6 has 42; and so on. In a commutative non-associative algebra, the possible interpretations are fewer x3 is unique, x4 can mean xx3 or x2x2, x5 can mean x xx3, x x2x2 or x2x3, x6 has 6 interpretations, and so on. The problem considered here is how many meanings are there for xn (A) in a general non-commutative non-associative algebra ? (B) in a general commutative non-associative algebra ? The answer to (A) is I am not able to find any such simple formula for (B).

As a science becomes mature it usually relies more and more on mathematical ideas. It has followed that in so far as mathematics is not entirely a “pure” subject, its content has been dictated by the more mature, that is, by the physical sciences. Nevertheless, it is slowly being appreciated that the permeation of mathematics through to the non-physical sciences must imply sooner or later a partial reorientation of the existing attitude.

This paper attempts to illustrate the method of representation, which is so powerful a weapon of modern geometry, in terms of elementary Higher Certificate mathematics. The results in their general form are familiar to students of geometry; but the special problem given here has several interesting features of its own.

No originality is claimed for this paper, though I do not know of any other account of the work. It seems, however, worth while to give a brief summary, as it covers a great deal of the ground with which schoolboys should be familiar.