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.

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.

The reports of the London University on the General School Examinations throw an interesting light on the knowledge of candidates of the subject of approximations. The 1928-30 reports make it quite clear that the subject is badly taught and that glaring errors occur very frequently The 1929 report states that this fault is general and is not confined to particular schools. When it is considered that the University examines annually about 25,000 pupils from schools all over the country there is justification for the suggestion in the title of this paper.

At the beginning of the seventeenth century π had already been calculated to 35 places of decimals by the Archimedean method of finding in succession the perimeters of inscribed and circumscribed polygons, each having twice as many sides as its predecessor.

But at this time several mathematicians, notably Snell, were seeking shorter methods of approximation, and finally Huygens in thirteen propositions proved or disproved the assumptions they made, and in three more propositions, making use of the known position of the centre of gravity of a parabolic segment and a theorem of his own on the centre of gravity of a segment of a circle, showed how to find a closer upper limit for π from any polygon than had yet been obtained, and by a seventeenth theorem “on centre of gravity”, of which he does not give the proof or even the enunciation, he found also a closer lower limit.

Among many English writers of text-books on arithmetic and algebra there appears to exist a state of confusion regarding the use of the sign of multiplication. Here are three instances. Godfrey and Price’s Arithmetic (1915) contains the following statement:

“The sign × means multiplied by; thus 3 ft. × 7 = 21 ft., and (7 × 3) ft. = 21 ft.; but 7 × 3 ft. is unmeaning ” (p. 13). Yet, on page 147, we find the expressions

in which multipliers precede the corresponding multiplicands.

1. PSQ is a focal chord of an ellipse and the normals at P and Q intersect at U.

Theorem I. The locus of the foot of the perpendicular from U to PSQ is a similar coaxal conic.

Let the tangents at P and Q meet at T: then T lies on the directrix and TS is perpendicular to PSQ. Draw UH perpendicular to PSQ and AR parallel to PSQ to meet the given conic at R and CVT at W Since TU is the diameter of the circle TQUP, and UH, TS are perpendicular to PQ, V, the mid-point of PQ is also the mid-point of SH But W is the mid-point of AR and hence C, H, R are collinear. Hence CH/CR = CS/CA = e, the eccentricity

Mr. Hickling’s article on “Mathematics in the Schools” is a challenge to all those of us who are following the modern tendencies in the teaching of Mathematics. Many good and sufficient reasons have been advanced why Trigonometry, Mechanics and even Calculus should be made available to the average boy before he has passed School Certificate. Experiments have been made, suitable text-books have been published, and the idea has been recognised by examining boards by the introduction of “Additional Mathematics” as an optional subject. Mr Hickling asks us to go back on all this, to damp down the “unfortunate demand for Higher Mathematics”, and to restrict the work to the Arithmetic, Algebra and Geometry of thirty years ago. The burden of proof therefore lies with him, and we are entitled to ask whether he is facing the facts.

[When the activities of the International Commission on the Teaching of Mathematics were suspended by the War, an enquiry into the training of school teachers was in progress, and when the Central Committee of the Commission decided that whole-hearted cooperation was again possible, advantage was naturally taken of the fact that there was interrupted work to be resumed. Some of the participating Sub-Commissions had already presented reports; to the others a revised version of the questionnaire on which reports were to be based was issued. The reports were considered by Prof. Gino Loria, the Italian mathematician and historian, whom we are proud to number among the honorary members of our own Association, and he presented at the Zürich Congress a comprehensive survey of the whole subject.

Communication with the United Kingdom was made through Prof. E. H. Neville—who has since been offered a seat on the Central Committee—and the following Report was compiled for the Commission by the Board of Education, for whose cooperation, as willing as it was indispensable, the Commission is most thankful.]