I begin by expressing my sincere thanks to the Association for the great honour bestowed on me in my election to the office of President. I realised when elected that the honour was indeed great; but it has also happened that, in the course of the year, I have had occasion to look through the list of those who have held the office before me, and to find one’s name in that company is both humbling and—let me be honest—exhilarating. In particular, and this will be relevant for what I have to say later, the records speak not only of men and women famous throughout the scientific world, but also of those whose special genius has led them to the top of the teaching profession, where their memory remains alive to generations of pupils of of whom, doubtless, many are here today. It is a source of real strength to this Association that it honours not only those who create but also those who pass on the fruits of creative labour, sometimes even to those who will themselves become the creators in their turn.

Newton was the greatest mathematician of the seventeenth century. Today, almost three centuries afterwards, we are just beginning to realize the full extent and variety of his achievement. Much of his mathematical work has never been published (though it ranges far through the fields of projective geometry and general point-correspondences to number theory and an exhaustive treatment of interpolation by finite differences) and is now, with few exceptions, to be found only in little known manuscripts in the Cambridge University Library. But Newton is remembered above all, and following his own wish, for his creation of the fluxional calculus and the theory of infinite series, two strands of mathematical technique which he bound inseparably together in his “analytick” method.

Much thought is being given at present to the possibility and desirability of introducing topics of modern mathematics, such as set theory and abstract algebra, into the sixth form curriculum. Very little of this kind of work has as yet been done in schools, chiefly because of its comparatively recent introduction into undergraduate courses, but it seems probable that it will come to play a part in the mathematical training of at least the abler boys in the fairly near future. Recently, during teaching practice, I had the opportunity of giving a course in group theory to four senior boys and I feel that the experience gained may be of interest to others who are contemplating similar kinds of work.

Mechanics is a branch of applied mathematics taught in both school and university. From the mathematical standpoint, the subject is learned in schools primarily by the method of solving examples. Theoretically, this procedure is supposed to help the pupil to grasp the fundamental principles and obtain for him a good examination result. That these seem to be nothing like as successful as they could be is one reason which prompts me to write this article. Others, including the pupil’s lack of appreciation of the subject, will be apparent in the points I shall make. These points have arisen from some years’ experience of lecturing to first year undergraduates and from marking advanced level scripts in applied mathematics. The efforts met with in both instances have led me to believe that there is a case for comparing the knowledge and understanding of the average sixth form pupil with that normally required of a freshman about to read applied mathematics for a general degree. On the one side is the ex-pupil, drilled in the method of solving examples and often lacking the understanding of the background principles involved; on the other is the undergraduate needing to draw upon an understanding and a background wider than he usually possesses. I feel sure that the disparity so defined exists not so much because of a lack of bulk material taught in the schools, but rather because the presentation is misguided and the emphasis misplaced.

In his article “A Problem in Projective Geometry” (Gazette, December 1960) T. G. Room says that he has not been able to find a projective geometric proof of his theorems. May I give the following proof?

The flexagon was defined by Mr R. F. Wheeler (M. G. 1958, p.1) as a strip of equilateral triangles which can be folded in various ways into a hexagon to exhibit different colour combinations. Can this idea be generalised? Suppose a plane figure to consist of m regular n-gons, with a common vertex, which can be folded together in pairs.

In Note 2949 printed in the May 1961 issue of the Gazette, I gave a method for constructing a rounded closed flexless curve, not necessarily a circle, which is self-inverse with respect to two points.

A square matrix is said to be doubly-stochastic if its elements are non-negative and if all row-sums and column-sums are equal to 1. The study of doubly-stochastic matrices was initiated by I. Schur and was subsequently taken up by Hardy, Littlewood, and Polya, who proved the following fundamental proposition [1, Theorem 46].

The following problems are merely different forms of one

1. In how many ways can a convex polygon of n + 1 sides be split into triangles by non-intersecting diagonals? We represent such a splitting by a symbol. Take one side of the polygon as base and letter the other n sides in order by a, b, c,…. If a triangle with sides lettered x, y is cut off, denote the diagonal by (xy). If this diagonal and side z form the sides of another triangle of the dissection, denote the third side of that triangle by ((xy)z). Similarly for two diagonals, as the figure illustrates. Finally the symbol for the unlettered base is the symbol for the dissection.

1. Introduction. While the operational calculus for the solution o initial value problems for linear differential equations with constam coefficients is now commonly introduced in a thoroughly satisfactory manner—either by means of the Laplace transform (e.g. ref. 1) of by the method of J- Mikusinski (ref. 2)—the same cannot be said of the simpler method for the general integration of such equations which is known as the D calculus. In fact, I cannot recall any text which gives a consistent definition of the meaning of expressions of the type [F(D)/G(D)]y, where F and G are polynomials. The reason for this state of affairs seems to be that the scope of the D calculus is rather limited so that the validity of the result can be verified in each case (e.g. ref. 3). In particular, this applies to the decomposition of an operator F-1(D) into partial fractions, which is the central step: in the solution of an equation with nonvanishing right hand side of general form. Nevertheless a rational approach to the entire problem is perhaps not out of place. This is attempted in the present paper.