A recent report on the teaching of geometry by a committee of the Mathematical Association (The Teaching of Geometry in Schools) suggests that a coherent geometrical system could be erected on a foundation other than that of Euclid’s axioms and postulates. Indeed, a cogent argument is advanced for the replacement of the principle of superposition and the parallel-postulate by the “principles of congruence and similarity” which are regarded as the expressions of more profound beliefs. The success of the recommendation, to substitute the principle of congruence for the principle of superposition, is undoubted, and the comparative reluctance to adopt the twin “principle of similarity” may be due, in part, to an obscurity in the treatment of parallels with this principle as basis.

Glaisher, I believe, once said that no subject loses more than mathematics by any attempt to dissociate it from its history. It is still customary, however, in schools and most universities to regard the history of mathematics as an “extra” outside the ordinary student’s syllabus—indeed it is usually treated as an exotic luxury. The point I would like to emphasise is that while the primary aim of the tyro should be the mastery of technique, yet in most cases this mastery is liable to become a mere mechanical acquirement unless it is accompanied by some appreciation of the cultural significance of the “Queen of the Sciences”.

A not unnatural consequence of my having complied with a request to act as a Delegate of the Mathematical Association at the International Congress of Mathematicians, which was held at Zürich during the week September 4-12, is that I have been asked to write a short account of the Congress for the Gazette. It is a little difficult to avoid giving merely an epitome of the official programme; and my difficulties are somewhat increased by my having, like many others, deliberately abstained from attending the Congresses at Strassburg and Toronto in 1920 and 1924, for reasons which are now, happily, a matter of past history, and by my having been unavoidably prevented from going to Bologna in 1928; in fact my only previous experience of a Congress was gained at Cambridge in 1912.

1 To prove that a line meeting one given conic in points harmonic to its intersections with another given conic, touches a conic.

2. Lemma I. Let two conies intersect in P, Q, R, S as shown in Fig. 1.

Let UV, FG be a pair of common tangents intersecting in T, and let the tangents at P to the two conies cut UF, VG in the points M, N respectively. Then M, N, T are collinear

For: QR, PS, FU, GV all meet in a point X, and the range MFUX is related to the pencil P(PFUS), which is related to F(PFUS), which is related to PQ′XS, where Q′ is the intersection of SP and FG Similarly, NGVX is related to PQ′XS. Hence MFUX and NGVX are related, and thus MN passes through T

If it was required to divide the area in Fig. 1 into n equal parts—say, for simplicity, three equal parts—how should we proceed?

There is a method, given in books on geometry, of constructing a figure similar to a given figure and n times its area. This method pivots on a property of similar figures as follows: if ABCDE and

FGHKL are similar figures on bases AE and FL respectively, then

A1: A2 = AE2: FL2

For purposes of comparison with what follows, the solution by this method is here recalled.

A difficulty which is frequently met in dealing with examination results is that of bringing to a common standard the marks obtained from two different papers. The most common form of variation occurs when the mark distribution curves are correct in general character but the peaks are at different points of the scale. We describe a machine devised by the author which has been used for the adjustment of marks. Its theory has no statistical justification, but where the distortions are not excessive the results do not differ appreciably from those of the rather lengthy theoretical methods and are usually absorbed in the approximation of a fraction to its nearest unit.

Mr. J. T. Combridge, in opening the discussion, said. My first duty is to give you a brief summary of the Report on Entrance Tests and Initial Degrees, adopted in June 1930 by the Council of the Association of University Teachers. My second duty is to offer such comments on it as may provoke a heated discussion among those who might otherwise be disposed to agree in approving or condemning it. For the benefit of those who have a copy of the Report, I would say that the mysterious reference at the foot of p. 4, “See pp. 4-10”, in reference to the English test, is due to the fact that the Report is an offprint from a certain number of the Universities Review, and the test referred to appears on pp. 4-10 of that number

For a subject whose death has been several times announced, Geometry is at present in rather a thriving state. During this century considerable progress has been made in two kinds of topology, in general differential geometry and in the theory of ovals, and now we have the new development of “Textile Geometry”, largely due to Graf and Sauer, and to Blaschke and his followers. It is the work of the latter writers that is considered in this review