During the year 115 new members, including Junior members, have joined the Association. The membership is at present 1784, of whom 9 are honorary members, 1691 are ordinary members and 84 junior members. Thirty libraries are also on the roll. There have been a few resignations, mainly in consequence of war-time difficulties the Council hopes that these members will be able to rejoin later The Council records with regret the deaths of the following members of the Association Mr C. J H. Barr (1902), Mr H. C. Beaven (1906), Mr F C. Boon (1910), Prof. S. L. Loney (1896), Rev J J Milne (1884), Mr A. R. Prestwich (1904), Mr J Leadbeater (1934).

In the discussion held at the annual meeting of the Mathematical Association in January 1936 on “Rider-Work in Geometry”, various interesting proposals were made and examples put forward to show how school geometry might be given a research aspect by means of the “discovery rider” or the “inductive method” in order to develop the full powers of the mind and to stimulate the creative abilities. Teachers who have tried to put into practice the ideas presented on that occasion will have obtained rather varied, if not actually contradictory, results. However many may have been successful there are always quite a number who come to the false conclusion that the average pupil has no creative abilities and that all efforts in that direction will fail, since only the very few mathematical specialists are found by them by benefit by it.

I have long been convinced that the teaching of mathematics suffers from the premature introduction of abstraction without adequate attention to the concrete facts from which the abstractions are derived. Perhaps I cannot better begin my argument than by taking as text a passage from the end of the first chapter of Todhunter’s Differential Calculus :

One of the most agreeable fruits of the renaissance which has followed the achievement of national independence in Ireland has been the decision to make accessible, in a form worthy of their importance, the writings of the greatest Irish mathematician and natural philosopher. The responsibility for the work was undertaken by the Royal Irish Academy, with the cooperation of the University of Dublin and the National University of Ireland, and with assistance from the Royal Society and the editorship was entrusted to Professor A. W Conway, F.R.S., of University College, Dublin, and Professor J L. Synge of Trinity College, Dublin. The first volume, devoted to Hamilton’s writings on Geometrical Optics, was published in 1931, and we have now to congratulate the editors (Professor Synge having meanwhile been succeeded by Professor A. J McConnell) on the appearance of the second volume which contains the papers and manuscripts on Dynamics.

Several mathematicians have studied the question of symmetry in the knight’s tour, one of the earliest being Euler and one of the latest Dr. Kraitchik of Brussels. Their investigations have been in the nature of random experiments, and they have confined themselves chiefly to square boards, thus missing such elegant results as

Fig. 1, which was published anonymously in the German periodical Denken und Raten in 1931.

There is probably no other item of the arithmetic syllabus which is confronted by such a diversity of earnestly supported method as is the case with the multiplication of decimals.

The problems which form the basis of this investigation are :

1. (a) What are the special advantages and disadvantages which bring about the adoption or rejection of these methods?

2. (b) What is the extent of these advantages and disadvantages?

3. (c) And finally a consideration of the methods of overcoming the disadvantages.

In a lecture delivered to the London Branch on 19th October, 1935, and reported in the Gazette, Vol. XX, p. 233, Mr C. V Durell showed a large number of interesting loci suitable for boys and girls up to school certificate standard. It is proposed in this article to pursue the matter further for those of higher certificate standard.

The determination of the errors involved in interpolation or quadrature formulae often involves complicated reasoning. There are many cases, however, where the remainder can be obtained very easily. These cases belong to the class which we call simplex.

In general a formula will be expressible in the form

L(f) =∑rArf(xr) + ∑sBsf'(xs)+ ∑tCtf''(xt) +… ......(1)

to a finite number of terms and it is assumed throughout that f(x) possesses derivatives as far as the remainder may require.

Thales of Miletus (640-546 B.C.) was the first cosmologist. Despite the naïvety of his particular theory, that “everything is water”, he was the first to formulate the hypothesis that, in all the multiplicity and diversity of its aspects, the World is One. Although the Babylonians are believed to have conceived the idea of an “Immutable Necessity” dominating all things, as far as we know they had no clear idea of the unity of the universe.

A closed conic C is completely contained inside a closed conic S. From any point 1 on S we draw a tangent 1, 2 to C meeting S again at 2. From 2 we draw the tangent 2, 3 meeting S again at 3, and so on ; for any n, the tangent to C from the point n on S meets S again at n + 1.

1. The stimulus that provoked this article was the following proof of the fact that the middle points of the diagonals of a complete quadrilateral are collinear.