A joint paper by Tranter and the present author concerning the heat flow in an infinite medium heated by a sphere has necessitated a proof that no complex zeros of the equation

Recently, in connection with some work on Diophantine approximations, I encountered the following problem on triangles.

Let T be a triangle with vertices A, B, C which are, respectively, inner points of the sides a, b, c of a second triangle t. Is it always possible to move T into a new position where its vertices are inner points of t?

I give here an affirmative answer to the problem and prove, moreover, that it suffices to apply to T an arbitrarily small rotation about a suitably chosen point of the plane. I am indebted to my Manchester colleagues for a number of simplifications of this solution, arrived at when discussing the problem with them.

The purpose of this article is to describe some apparatus which is easy to construct from materials that are reasonably cheap, and which has been found useful as a visual aid in many branches of elementary mathematics. It is thought that this apparatus will satisfy the need to which attention is drawn in § 4.8 of the Trigonometry Report, although it was being used before the report was published. After the apparatus has been described, some of its applications will be briefly indicated and a more detailed account of one application will follow.

The elementary theorem that, if the triangles ΩAB, ΩPQ are directly similar, so also are the triangles ΩAP, ΩBQ, can be elevated into a general principle:

Given a point Ω and any number of points P1, P2,.. lying on a curve Γ, let Q1, Q2, … be points such that the triangles ΩP1Q1, ΩP2Q2, .. are all directly similar. Then Q1, Q2, … lie on a curve similar to Γ, obtainable by rotating Γ through the angle PrΩQr and applying the scale-factor ΩQr/ΩPr.

The earlier sections of this account are concerned with some standard properties of pure recurring decimals. These include the cyclic property, illustrated for example by the decimals representing the six proper fractions with denominator 7, each of which has a period using a cyclic permutation of the digits 142857 It was a question concerning this property, raised by a student, which led to the writer's interest in the properties considered in the later sections. References are given in Dickson's History of the Theory of Numbers, Vol. I, Ch. 6.

The purpose of this article is to apply a method, previously discussed in some detail, to the solving of standard elementary problems in the Calculus of Variations and to show how rapidly the solutions can be derived by introducing (p, r) and certain other coordinates. In each case a Lagrange undetermined multiplier is introduced in the course of the solution, so that the problem if not already an isoperimetrical problem is converted to one.

The study of mechanics and the solution of problems in mechanics are an intrinsic part of the subject of mathematics as taught in British schools and universities; the subject is associated with such names as those of Routh, Lamb, Loney, Love. It is perhaps a peculiarly British tradition, well worth preserving as a discipline of the first order, comparable with Latin and Euclidean geometry; many examination candidates, even perhaps theoretical physicists of distinction, can write learnedly on atomic physics, and yet find the greatest difficulty in solving dynamical problems requiring a firm grasp of principle together with facility in applying it. The present logical unsatisfactoriness of quantum mechanics in certain aspects may be due to want of interest in the foundations of mechanics amongst its exponents.

The question about the teaching of Mathematics considered most frequently and with most detail is How? This is the main preoccupation of the Mathematical Association Reports on separate subjects : it is of special interest to note that in the Report on the Teaching of Calculus oft repetition of the phrase “ the teacher will know ” implies that the topic is as rich as human nature. Yet for writers in the Mathematical Gazette, and for most scholarship winners, the question is superfluous, though all appreciate a good teacher. (For these too the answer to Why? is simple—the urge of joy or duty.) On the other hand, it would scarcely be denied that for more than two-thirds of the pupils in our schools the question How? is unanswerable under present conditions : for them we are attempting the impossible.

1. In a seismological context, it was recently noticed that the use of (p, r) coordinates in a variation problem where polar coordinates had previously been used presented some unusual features. The method followed is one of some generality, and the purpose of the present note is to illustrate it in a suitable elementary problem: “A heavy uniform inelastic string has its ends fastened to two assigned points on the surface of a smooth right circular cone fixed with its axis vertical and vertex O uppermost, and rests in equilibrium on the surface of the cone. Prove that if this surface is developed into a plane, the curve on which the string lay is given by the form p (a + br) = 1.”