The Annual Meeting of the British Association in 1938 will be held at Cambridge, August 17-24. A programme notice appears on the opposite cover.

When it is said of the Axioms of Projective Geometry that, for the “points”, “lines” and “planes” of which the Axioms treat, we may take what elements we please (provided only that the Axioms be satisfied), it is, I believe, tacitly assumed that the elements so to be taken are classes of algebraic symbols. That in point of fact the “elements” most commonly used, at least in the introductory stages of learning geometry, are not “points”, “lines” and “planes”, but dots, dashes and blackboards, is rarely acknowledged. Only too often the so-called “figure” of a theorem is apologised for and treated as a mere concession to human weakness. Whereas in fact the “dot”, “dash” and “blackboard”, as elements satisfying the Axioms, provide, in many cases, a most excellent symbolism for the expression of geometrical theorems.

1,1. This contribution to a fundamentally important subject has a multiple origin. Its immediate occasion is the address by Professor Temple, on “The Theory of Complex Numbers”, published in the Gazette of July 1937; and its theme is another alternative to the two types of “theory” there considered (“descriptive” and “constructive”)—which may be designated descriptive-in-terms-of-discovery.

The principles underlying my proposals with regard to this matter were published in an article in the Mathematical Gazette (May 1922). They may be briefly restated as follows.

The first principle is that school geometry should be organised not as a purely logical system but rather on the lines of the natural sciences. This means that it should be based explicitly upon the pupils’ spatial experiences, which it should seek to extend, to make precise, to analyse and to co-ordinate. Under this description one includes the important field in which geometrical observations and conclusions have practical utility.

Recently, in the course of conversation with a friend, I was asked in what ways was it possible to liven up the mathematical teaching of a school. I made a few vague suggestions at the time, but it occurred to me later that I might consider the question, and then apply the results of my deliberations not only to my own work, but to put them down on paper in case others might believe them worth while.

Operational methods in pure mathematics, e.g. the work of Boole, Graves and Murphy, existed many years before Oliver Heaviside introduced new ideas into mathematical physics. His first paper using such ideas was published in the Proceedings of the Royal Society in 1893. Remarks therein indicate his unfamiliarity with some of the literature of the subject, owing to lack of reference facilities, for example, Liouville’s work on fractional differentiation.

The point of the question, “What is the envelope of the principal axes of conics touching any four lines?” posed by Prof. Forder on p. 83 of the current volume of the Gazette has puzzled me not a little; because (i) though my ignorance of the literature of coniCs is profound, yet that literature is so extensive that it seems unlikely that no solution of the problem has ever been published; and (ii) it seems even more unlikely that Prof. Forder is not in possession of a solution. However that may be, here is a solution which might have been submitted by a well-prepared candidate, had the question been set in a Cambridge Entrance Scholarship Examination, whether in December 1937 or in December 1906—or even in April 1877.

1. The reviewer of Berg’s Operational Calculus as applied to Engineering and Physics (2nd ed. 1936) in the Gazette (Oct. 1937, p. 309) remarks that “Heaviside’s methods deserve more attention in this country than they appear to receive”, and a reference is made to articles contributed by Bromwich, Jeffreys and myself to Vol. XIV in 1928.