The Special Theory of Relativity leads to the following conclusions :

1. (i) The three dimensions of space and one of time constitute an isotropic fourfold, in which there is no unique time-direction, just as there is no unique space direction.

2. (ii) Specifying the velocity of a body is equivalent to specifying its “ time-direction”, that is, the direction of its path through space-time (its “ world-line ”).

3. (iii) If the units of distance and time are taken to correspond with each other, so that the velocity of light is unity, then the lines in our diagram (L'OL, M'OM, and lines parallel to these) which represent the paths of light pulses (Robb's “ optical lines ”) are at right angles for light going in opposite directions, and in this case the time direction (T'OT) of a particle and its corresponding space-direction (X'OX) are equally inclined to the optical line (L'OL) between them, so that setting a particle in motion involves “ rotating ” its time-direction and its space-direction through equal angles (T1OT2, X1OX2 towards the optical line which goes in the direction of motion of the particle.

If we keep an eye on the many interpretations of the operations of multiplication and addition rather than on the actual operations themselves, we shall need to make several changes of focus.

Let us see what these operations could mean without being concerned with what their interpretations could be.

The expressions “sum of”, “product of” are expressions of functionality.

The process of randomisation has in recent years come to play such a central part in experimental design that it is of some interest to find that it affords a means of resolving one of the oldest paradoxes which arose in discussions of gaming.

Readers of Todhunter’s The Mathematical Theory of Probability will recall his account (sections 187-190, pp. 106-110) of the correspondence between Montmort and Nicolas Bernoulli on the rule by which the players might guide themselves most advantageously in the game called “le Her”.

The ideal aimed at in Saracenic designs is that each “ pattern space ” should have either radial or bilateral symmetry. It is surprising that this ideal is so nearly achieved in a pattern in which the very awkward constituent of seven-rayed stars is combined with octagons and regular square-shaped outlines.

There are certain topics connected with the teaching of mathematics which seem to present themselves periodically for review. The aims or objectives of mathematical teaching is one of them ; the foundations of school mathematics is another. I have selected for review to-day one of the fundamental concepts of mathematics, that of functionality.

May I begin my discussion with a brief historical survey ? The idea that the function concept should have a prominent, even a dominant, place in the mathematical programme of the schools may be said to have originated with Klein. Others before him, in France, England and America, as well as in Germany, had advocated the inclusion of “ variables ” and “ functions ” in the school programme, but Klein was the first to press the view that “ functional thinking ” should be made the binding or unifying principle of school mathematics. Klein was equally distinguished as a mathematician and as a leader in the pedagogy of mathematics in Germany, but some years elapsed before his views became generally accepted, even in his own country.

In the teaching of the Theory of Differentials the really difficult and important problem is not “how shall we teach a given theory?” but “which theory shall we teach?” There are, in fact, a number of rival theories, none of which has met with any general measure of acceptance. In these circumstances, the critical problem of considering the rival merits of these theories is much more urgent than the pedagogical problem of presenting them to students.

1. Nowadays the beginning of conics is a blend of pure and coordinate geometry, but the starting point is generally the focus-directrix definition. From this the standard equations and properties are deduced, but a good deal of time is taken before some of the more powerful methods can be applied.

The object here is to indicate compact proofs of some well-known results, and to obtain, at an early stage, such results as Newton’s Theorem and the cross-ratio property from the focus-directrix definition.

Recent notes in the Gazette on the discrimination of the roots of the cubic and quartic suggest that the following method, if not new, is not so widely known as it deserves to be. The only tool required in addition to a facility in reading graphs is a knowledge of differentiation and so the method lies within the scope of the work done by evening technical students and others whose grasp of ordinary algebra is so weak as to prevent their following up the normal methods of discussing the roots of equations. The treatment is also sufficiently general to be of real interest to a type of student whose body of mathematical knowledge is always, from the pressure of other subjects, threatening to disintegrate into a collection of scraps of special methods each applicable to one type of problem. The method yields very easily the discriminants for the quadratic and cubic and it also offers a nomographic device for the solution of numerical equations. In practice, however, even with large scale drawings, students are not able to obtain results as accurate as they can readily get by other graphical methods.

It seems to me that, no matter what doubts and perplexities may have assailed me in the selection of the subject and the contents of this Address, my primary duty on this occasion is obvious and inevitable ; it is to express my appreciation of the impartiality of the Council in so arranging this afternoon’s proceedings that my Address is subsequent to the discussion on Professor Neville’s motion, so that supporters of the motion were unable to use the Address as an argument in favour of limiting each President to one year of office.