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.

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.

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.

If two segments AB, XX1 of a line have their mid-points coincident, X, X1 are said to be isotomically conjugate with regard to AB, and vice versa. For brevity in this paper A, B and X, X1 will be referred to as “ isotoms ”

If pairs of isotoms XX1, YY1, ZZ1 are taken on the sides BC, CA, AB of a triangle, and if X, Y, Z are collinear, then by Menelaus' theorem X1Y1, Z1 are also collinear, and the two lines are said to be isotomically conjugate with regard to the triangle. For brevity pairs of lines related in this way will be termed “ isotomic lines”. Conversely if two transversals make XX1, YY1 isotoms with regard to two sides of a triangle then ZZ1 will be isotoms on the third side. These simple results, the source of which I have been unable to trace, are the only results on isotomically related figures made use of in the following investigation.

Arithmetic has suffered in the past, and still suffers, from being regarded as merely a tool subject, without interest or content of its own. The three R's have a very definite meaning for many an English business man of to-day, who himself had little or no schooling after the age of 14 : he requires of his would-be employees that they shall be able to read print, to write a legible hand, to add up a column of figures and make certain specialised mental calculations, chiefly concerned with our complicated monetary system.

The University Honours graduate who teaches in our Secondary Schools tacitly admits that these rudiments are necessary, but is in a hurry to turn the reading lesson into literary appreciation, writing into self-expressive composition, and arithmetic into the various branches of mathematics, or the hand-maid of science.