The professional training of the Mathematical Teacher after his degree course is finished is a principle which has come to stay, and it therefore behoves all who are concernedin the practical administration of this principle to do their utmost to make the course of professional training as valuable as possible for the future teacher. So far as I know nothing has as yet been done by the Mathematical Association as a body in helping forwardthis movement, but surely the Mathematical Association will fail in its fundamental aims if it does not take a very active part in helping to decide what are the best courses of instruction to give to the student-teachers while in training. The object of the present paper is not to lay down dogmatic assertions as to what ought to be done, but rather to put forward suggestions, and thus to act as a nucleus round which the discussion can revolve. No attempt is made to deal with the training of anyone except the specialist type of mathematical teacher—the person who in time to come is destined to teach in the Advanced Departments of the Secondary Schools in addition to taking his share in the general mathematical work of the school, and who, if weighed in the balance and not found wanting, will be called upon to act as Head of a Mathematical Department, and thus to guide and mould the mathematical destinies of the institution to which he is attached. In almost every University nowadays there exists a Department of Education, and within that Department of Education a School of Mathematical Pedagogy. It is for us to discuss now what ought to be the type and scope of instruction given in this School of Mathematical Pedagogy.

An arithmetical amusement, said to have been first propounded in 1881, is the expression in the ordinary arithmetic and algebraic notation of the consecutive integers from 1 upwards, as far as practicable, by the use of four “4”s. I have mentioned the problem in my Mathematical Recreations, but my friend Mr. Oscar Eckenstein has now carried the solutions considerably further. I think a bare statement of our procedure may be of interest, and perhaps some readers of the Gazette may amuse themselves by filling in the details of the analysis or applying similar methods to higher numbers.

In view of the popular appeal of Lawn Tennis, and of the fact that a large number of professional mathematicians depend upon this game for much of their recreation, it seems strange that no one has pointed out that a good deal may be learned (as to suitable tactics) by applying a few quite simple principles of dynamics and geometry to the game.