We are living in an experimental age in the teaching of Mathematics, and new topics and methods by which to introduce them to our pupils exercise the minds of the majority of teachers. Opinions and preferences are as numerous as the alternatives. The following article has been submitted to the Modern Trends Subcommittee and is offered as one way of introducing linear programming and matrices.

Tonight I propose to say something about a man who is now recognised as a pioneer mathematician of his time, but I do not intend to talk about his work as a creative mathematician. I shall leave that to others who are far better qualified to do so. As a young man I studied pure mathematics under Young and G. H. Hardy, but most of my work as a professional mathematician has been in mathematical physics, and I cannot say anything significant about subjects that I ceased to work on some thirty-five years ago.

There has been a great interest in the possibility of teaching ‘modern mathematics’ in schools ever since the Southampton Conference in 1961 when Professor Bryan Thwaites and his team put forward their plan for modernising school mathematics, especially for the period spanning the four years before ‘0’ level. Their reason for the change was that, since during that period the child’s mind is most intellectually alive, a lot of dull, heavy manipulative work would do far more harm than good, whereas they felt that ‘modern mathematics’ would be more intellectually stimulating and enjoyable. Since then, several Examination Boards have produced alternative syllabuses for ‘0’ level, containing set theory, group theory, etc. More and more schools have been doing the two-year course leading up to this examination, and the first papers were set last year.

Statistics is not a strange subject on its own, but should come in naturally in planning and discussing experiments in Physics. Boys must be trained not to expect to get the answer but to get an answer with a certain degree of accuracy. I was lucky to find a group of 15 year olds who didn’t ‘know’ the formula for the period of a simple pendulum. Experiments on this lead to the revision of many of the topics in my statistics programme.

A graph consists of a set of vertices some pairs of which are joined by a single edge. A path is a sequence of distinct vertices (x1, x2, …, xm) such that consecutive vertices xi and xi+1 are joined by an edge (xi, xi+1), for i = 1, 2, .., m — 1. One of the unsolved problems of graph theory is to obtain criteria for determining whether any given graph has a Hamilton path, i.e., a path which passes through each vertex exactly once. (For pertinent references and further definitions, see [1].)

A bag contains m > 0 black balls and n > 0 white balls. A sequence of balls from the bag is discarded in the following manner:

• (i) A ball is chosen at random and discarded.

• (ii) Another ball is chosen at random from the remainder. If its colour is different from the last it is replaced in the bag and the process repeated from the beginning (i.e. (i)). If the second ball is the same colour as the first it is discarded and we proceed from (ii). Thus the balls are sampled and discarded until a change in colour occurs, at which point the last ball is replaced and the process starts afresh.

We consider here three geometrical problems, the solutions of which are not hard, but each involves some rather unobvious point. The reader is invited to attempt them before looking at the solutions. The names of the original proposers are noted in brackets.

A problem arose recently concerning the derivation of Rodrigues’ formula for the generalized Laguerre polynomials from the corresponding formula for the ordinary Laguerre polynomials. The derivation involved a generalized form of Leibniz’s theorem which I think may be of interest.

The method of finding the maxima and minima of a function of two variables f(x, y) in the unrestricted case is straightforward in theory One solves and then substitutes the roots in Hessian function

This note deals with three differential equations referred to in The Mathematical Gazette, Vol. XLVI, No. 356, May 1962.

1. H. E Tester (Mathematical Notes, No 3010) solves the differential equation

from a geometric aspect, stating that it cannot be solved by a standard procedure.

Goodner has studied the equations to conic sections in the elliptic plane by following the methods of Euclidean geometry and using Weierstrassian co-ordinates. In the present note the author has investigated several properties of the elliptic circles from the same analytic standpoint and has shown that some of the well-known properties of the Euclidean circles hold as well for the elliptic circles.

Given a positive continuous increasing function f(x), 0 ≤ x < 1, approximations by a convex function g(x) from below and from above are considered. The approximation is measured by the ratio of the integrals

In elementary courses on differential equations the standard method of obtaining particular integrals is probably that of the variation of parameters or, as it is sometimes rather paradoxically called, the variation of constants. In more sophisticated courses the idea of the Green’s function is introduced. This has many advantages, not the least of which is that the method for finding particular integrals based on such a function is much better motivated. Moreover, it is possible to fit in given boundary conditions, homogeneous or otherwise, as part of the process and this avoids artificially breaking the problem up into two parts, in the second of which a plethora of arbitrary constants is related to given boundary or initial conditions. A disadvantage of Green’s function methods is that, if they are to be properly exploited, it is necessary to introduce the concept of the delta function.