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

1. The title of my paper is, perhaps, a little misleading. The word Elementary is intended to apply to the word Mathematics, not to the word School, and I will keep strictly to Elementary Mathematics.

The treatment of Geometry in feigned independence of numerical measurement was theoretically abandoned as soon as the supremacy of Euclid collapsed under the onslaught of Professor Perry in the year 1901, and gradually, largely through the efforts of a few outstanding educationists, the teaching of School Geometry in this country has undergone a very remarkable change.

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.

1. In a recent paper Mammana considered the equation

z″ + pz′ + qz = 0,.....(1)

where x is the independent variable and p and q are functions of x, real, finite and continuous in the interval (a, b), and investigated the possibilities of its decomposition into the form

(D + α)(D + β)z = 0,.....(2)

where D ≡ d/dx and α and β are functions of x.

[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.]