Had I been called upon to address the Association a few years earlier on a subject of this nature, my task would have been much more difficult. Model atoms were then becoming almost as numerous as the sands of the sea. The very simplicity of the quantitative phenomena shown by the simplest atoms was an obstacle to progress. For example, the simple spectrum of Hydrogen,— the so-called Balmer series,—consisting of a set of frequencies given by the formula

where n and m take all possible integral values, was somewhat too simple if considered by itself instead of in relation to phenomena of a widely different type. And as a consequence, we had at least five completely different forms of suggested structure of a Hydrogen atom, each of which, by suitable mathematical artifices, could be made to yield such a set of frequencies.

Vectors in two dimensions are generally studied in schools; I do not suppose that the study is often pushed into three dimensions. But it is difficult to understand a modern book on electricity without some knowledge of three-dimensional vector notation.

The particular point that I wish to bring before you is that the two-dimensional case is often presented in such a way that its relation to the more general case is obscured.

The Dalton Plan, which has been discussed so much recently, is a change of organisation within school which affects a girl’s whole school life and her general development more than her work in individual subjects. I am obliged in describing it to devote a few minutes to speaking of general school life, and when I turn to the teaching of mathematics there seems so little to say, that could not be said as appropriately of teaching in ordinary circumstances that I do not know if it will justify the consideration of this meeting.

It is now a commonplace of History, that Pure Mathematics had its origin in the practical needs of man, and that at intervals it has renewed its vitality by turning to its practical application. It has, during recent years, become increasingly recognised that the bases of mathematical education are the concrete and the practical; and this Association, by the Reports and Papers published in the Gazette, has emphasised the view that “the utilitarian aspect and application of Mathematics should receive a due share of attention in the mathematical course.” This increasing respect for the concrete and practical has not only led to considerable changes in the teaching of Arithmetic, Geometry and Algebra, but has brought into the Schools closer co-operation between the teachers of Mathematics and Science, and has resulted in a very desirable correlation of the syllabuses in these subjects.