The first thing to be said about what is commonly called “modern” mathematics is that it is not really so modern as one might suppose. A great deal of it was in fact created more than fifty years ago. What gives it an air of novelty is that only in the last generation has it had a serious effect on the undergraduate teaching of mathematics in our universities, and only in the last few years has its impact on school mathematics begun to be felt. Some such time-lag is natural and inevitable ; new mathematics is usually created as a research tool, and so long as it is nothing more there is no compelling reason to seek the simplest way of expounding it or to re-examine more elementary mathematics in its light. It is when one starts to teach a new subject that simplification begins, and when this has been accomplished one begins to appreciate the possible advantages of the new point of view at a more elementary level.