Francis Bacon held every man a debtor to his profession, and there are few of us who would not discharge this obligation if we could. I feel particularly indebted because I have been fortunate, over a long period, in having been able to attend Association and Branch meetings, where I have met many of the best informed, most active and aesthetically inspiring teachers of mathematics of all ages and at all levels. When, in addition, the Association honoured me by making me its President, I knew that I was, in more senses than one, permanently overdrawn. I know that this distinction would never have come to me had it not been for the accidents of time and place that made me Honorary Treasurer for several years. I am an assistant master, and, since 1954, a House Master, so that I could describe myself, in a light-hearted way, as a full-time research worker in the educational field. My researches are not entirely mathematical, but close contact with boys of 13 to 18 has been an interesting and instructive experience.

Mathematical confidence and appreciation cannot be fostered unless each technique of the subject is established in an understandable fashion. It can hardly be claimed that this condition is satisfied in connection with division of fractions, for the rationale of “invert the divisor and multiply” is hardly ever grasped by children or, for that matter, by adults, except as a very loose induction from a few non-typical examples. It is interesting, therefore, to note that a straightforward procedure does, in fact, exist for this operation, adoption of which would serve to make the operation appear much more reasonable than it usually does at present.

This note is due to the fact that the identity of two analytic functions, with the same values for every z, is not obvious.

As a subject of study statistics differs from other subjects in being non-existent by itself. A “statistic” is a measure—e.g. mean, variance, standard deviation, median—of a sample, and therefore of a sample of “something”, and that original “something” pertains to some other “subject”—e.g. biology, agriculture, medicine, industry, economics. Or, if we investigate or research in the theory on which our techniques are based, we are at once plunged into mathematics. This somewhat nebulous quality of the subject, together with the peculiarly subtle mixture of down-to-earth algebra and logic on the one hand and subjective intuition and philosophical approach on the other, all wrapped together in a sort of cotton-wool of probability, is often for the beginner a real difficulty and thorn-in-the-flesh, though largely contributing later, it is hoped, to the fascination of the subject. I have found in over ten years of teaching statistics at Sheffield University that the student can be much helped over this initial period by simple experiments in which he collects his own data, in the course of which he builds up an idea for himself of the underlying probability situations and in some of which he learns something of the relevance of design and the practical difficulties associated with the carrying out of such simple-sounding instructions as “randomise.” When it comes to analysing his data he experiences a certain “vested interest” in the results just because they are his own observations.

The purpose of this article is to offer a new approach to the solution of the equation

In a recent competition (see M.R. 23 (1962), A1482) it was required to show that the set {2, 3, 4} has the property, here called Pv that every number x in (0, 1] (i.e. 0 < x ≤ 1) can be expressed in the form , where m1 is an integer and belongs to the set for all i.

Professor Forder’s elegantly simple method in Note 3068 (Math. Gazette 47 (1963), 237-8) can be extended to prove a more general result. Let k be any positive integer, ξ any irrational number, ar the (r + l)-th quotient in the simple continued fraction for ξ and pr/qr the (r+ l)-th convergent, so that

The object of this note is to give a simple geometrical proof of the following well-known

Readers may be interested in an elementary exposition of the convergence properties of the sequence of iterates obtained by applying the Newton-Raphson process to an arbitrary polynomial. The coefficients of the polynomial may be real or complex. By restricting attention to polynomials we will give some precise results using the least possible analysis, while proofs for more general functions would necessarily require more high-powered methods. The arguments we will use require only an elementary knowledge of limiting processes.

It has been noticed that heavy wear takes place on the wheel flanges of electric locomotives in countries where mountain railways exist and where curves are numerous and severe. Locomotives have to return to shops after a comparatively small mileage to have their wheels attended to unless steps are taken to reduce flangewear.

Desargues’ theorem is concerned with two triangles in perspective for some point O. Let us denote the vertices of the first triangle by , it is then natural to denote the sides by where is the side containing . Now bringing in the second triangle we get sides . Moreover, one pair of corresponding sides is ; it is therefore natural to christen their intersection . The other two meets are ; and, by Desargues' theorem, these three meets are collinear; we therefore christen the axis of perspective .