If we keep an eye on the many interpretations of the operations of multiplication and addition rather than on the actual operations themselves, we shall need to make several changes of focus.

Let us see what these operations could mean without being concerned with what their interpretations could be.

The expressions “sum of”, “product of” are expressions of functionality.

In the teaching of the Theory of Differentials the really difficult and important problem is not “how shall we teach a given theory?” but “which theory shall we teach?” There are, in fact, a number of rival theories, none of which has met with any general measure of acceptance. In these circumstances, the critical problem of considering the rival merits of these theories is much more urgent than the pedagogical problem of presenting them to students.

The title of this paper is “The Eternal Triangle” ; but titles, like politicians, are sometimes misleading. Only last year, for example, you, Mr. President, skilfully cloaked an address on elliptic arcs under the beguiling title of “The Marquis and the Land Agent : a Tale of the Eighteenth Century”. And so now, Ladies and Gentlemen, any hopes you may have entertained of hearing a new solution of an age-old problem must be abandoned. I cannot even say with George Canning.

It has been pointed out that Mayer's method of solving the differential equation

is quite general and only requires one integration, whereas the other general methods require two, or even thee, integrations. The reason for this is rather obscure in most text-books ; and the proof of the validity of the method rests on an existence theorem which leads to a solution of an apparently different form. I shall prove here that the two forms are identical.

A Logarithm as generally understood and used by the ordinary student is the expression of a number as a power of 10. In order to multiply two numbers together he adds the logarithms of these numbers and takes as the result the anti-logarithm of this sum. Similarly for division he takes the anti-logarithm of the difference of the logarithms of the two numbers.

Perhaps my title will be thought to savour too much of the nursery, but I should like to reassure you. I have no intention of boring you with stories of my own children, although they have been responsible for directing my thoughts from my former pupils in Secondary Schools to the untaught little people whose ideas are so instructive.