In the second edition of my Contents of the Fifth and Sixth Books of Euclid I have taken the ideas stated in the preceding article as my starting point, and built up the conditions involved in the 5th and 7th definitions in the following way:

1. 1. The ratio of a magnitude A to another magnitude B of the same kind is a real number, rational or irrational, determined in the manner explained in what follows. It is denoted by the symbol A : B.

2. 2. If the magnitudes A and B have a common measure G, so that A = aG, B = bG, where a, b are positive whole numbers, then A : B is defined to be the rational number .

Arithmetic has been until recently the only branch of Mathematics generally taught in the Primary School; and as this subject deals exclusively with numbers in their simplest and most practical form, it must, in the order of teaching, be the first introduced to the child. The premier place must also be awarded to it when considering the relative importance of the various branches composing the unity of Mathematics.

In order to make this brief paper as practical as possible, I shall plunge into the subject with little introduction.

I wish to consider the question from three points of view.

1. (1) What is the ideal course?

2. (2) What is the best practicable course?

3. (3) What effect the best practicable course should have on the teaching of the subject in our secondary schools, and also—since the two are so closely connected—in our elementary schools.