The whole of ordinary algebra, dealing with real or complex numbers a, b, c, is based on three fundamental laws which are usually called:

1. I. The Associative Law,

2. II. The Distributive Law,

3. III. The Commutative Law.

Elsewhere I have shown how the study of compound interest leads to the conception of a “growth curve” (the exponential curve) from whose properties we can derive, first, the theory of logarithms, then the conception of e as the limit of (1 + 1/n)n, and finally the differential coefficients of ax and logax. The method of differentiating ax and of introducing e to be described here follows entirely different lines, and may be useful to students who have approached logarithms in the ordinary way through the theory of indices and have begun the study of the calculus before reading what is commonly called “higher algebra.”

The following elementary way of obtaining this expansion is interesting in connection with my paper on the Gamma-function in the May issue of the Mathematical Gazette.

My address last year was devoted to the consideration of a few rules or statements made in teaching, with, as it seemed to me, insufficient explanation, in the hope that I might suggest some line of action, in a few particular cases, which would be helpful to members of our Association engaged in elementary teaching.

“We are all pacifists now”; or if not, it is difficult to see what excuse we have for living in the modern world. I feel therefore that a word of apology is needed for a paper on a military subject. But I hope you will acquit me of any intention to revive the hatred and bloodthirsty panic that poisoned the years 1914 to 1918, because the activity that I have to tell you about is not concerned with the destruction of life but rather with the saving of it.

A Well-Known treatise on Physics written in 1892 begins with the words:

Such was the accepted doctrine in the latter part of the nineteenth century. I want to show how it originated and to what extent we have now departed from it.

If ab is the numerical value of the product of two factors a and b, and if h is added to a (for the purpose of bringing the product up to an assigned value P, or for any other reason), then bh must be added to ab. If, now, k be added to b (for the same or any purpose), the product must be increased by k (a + h) where (a + h) is the new value of the factor previously a. These facts are so elementary that they merit quoting only on account of the curious neglect of their systematic alternate use by the arithmetician.