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.

Morley's theorem states that the points of intersection of adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle, known as Morley's triangle. See Figure 1.

Concise proofs of the theorem are given in recent papers [1, 2]. A good picture of important previous work can be obtained by looking at [3, p. 1999], and examining references cited there.

If z is a function of y having a differential coefficient at a certain point, and y is a function of x having a differential coefficient at the corresponding point, then z is a function of x having a differential coefficient at that point, and this differential coefficient is given by the formula

This theorem, thought not always stated so carefully, will be found in all texts on the calculus; but, so far as I am aware, no English writer, not even the most exact, offers a valid proof.

Of all mathematical problems, those concerned with probability seem to contain ambiguities more often than any others. In the majority of cases the reason is that too little thought has been given to the wording of the question, but occasionally the question sounds very sensible and it is only when a real attempt is made to solve the problem that difficulties in interpretation appear. I believe that this is mainly due to the loose way that the word ‘probability’ is used.

Solid shapes which pack together to fill space cover a large and varied range. I recently found five such polyhedra which I have not seen described elsewhere. The names which I have used for them here are rather ferocious I am afraid, but that just happens to be the way the names of polyhedra work.