The Council has done me a great honour in making me your president, but has given me a difficult assignment in following our last president Professor Coulson, who achieved outstanding success in providing interest and stimulus for all of us. I played with the idea of the title “On not liking Mathematics” and there are indeed those who do not, but I felt that most of us here probably cannot remember a time when we did not like numbers and, moreover, my teaching life has been spent with those who have chosen Mathematics as their study at the University and this must mean that they like the subject at least better than others. A possible title would have been “Let ‘em be moderate as well as intelligent” which I found in a letter written in London by Sir Isaac Newton (1) (who represented the University of Cambridge in Parliament) on 30 April, 1689, to the Vice-Chancellor of Cambridge University concerning a Bill to settle ye Charters & Privileges of ye Universities: “And if at that distance you are from hence you cannot communicate your advice so easily as might be desired, you may perhaps do well to send up one or two intelligent persons (with such instructions as you shall see fit) for us to consult with here in drawing ye Bill. But if you send up anybody let ‘em be moderate as well as intelligent and let ‘em be sent as soon as may be”.

This Association came into existence nearly one hundred years ago with the prime object of improving the teaching of geometry and although its field of interest has widened considerably the teaching of geometry remains a very important issue. Of course, the question “Whither Geometry?” is rather meaningless unless one specifies whether one is talking about geometry as taught to primary, junior or secondary school levels, to Colleges of Education and the Polytechnics, to undergraduates and postgraduates at University, or geometry as understood by professional research mathematicians. Moreover the motivation for studying geometry may well be quite different at these different levels. I shall try to deal with the question as it affects all these various levels of mathematics.

[The following paper is a slightly modified version of one that was presented to the Universities and Schools Committee of the Mathematical Association. It was intended as an attempt to outline the qualities of mind that are valuable in the studyof mathematics, and that one would ideally like to be possessed by potential entrants to university mathematics courses. In practice, of course, the extent to which these qualities are developed will vary a good deal from one student to another, but it is believed that they all deserve attention.

The Committee would be glad to receive comments on this paper, either criticisms or amplifications, and would also be interested in receiving illustrative material for any of the points made in the paper. It is hoped ultimately to use some version of the paper as a preamble to an outline scheme of mathematical teaching for sixth forms.

Comments or other material should be sent to the Secretary of the Committee, Mr. S. L. Parsonson, 57 High Street, Harrow on the Hill, HA1 3HT.]

An important contribution to the history of mathematics has appeared in the Australian Journal of Science, October 1967, Vol. 30 No. 4.

The aims, methods and content of secondary school mathematics are undergoing quite searching examination at the present time to the dismay of some teachers and the satisfaction of others. One of the products of this examination has been a growing dissatisfaction with the content of the older textbooks and a consequent increase in the number of books containing so-called “modern” mathematics. Typically these textbooks attempt to provide a terminal mathematical education, to the extent that they say that they cater for pupils who are not necessarily going on to do more advanced maths. It is appropriate at this time to ask just how much the consumers of the educational product are in agreement with the producers of that product over the qualities that the product should have in order to satisfy future requirements. Accordingly, this investigation compared the opinions of employers and secondary teachers over the mathematical content of a terminal course for 15-16 year olds.

It has been my pleasure lately to be involved in the organizing and teaching of elementary computer techniques in a college of further education. Whilst the courses were not intended to give tuition in the electronic side of computing it was felt, nevertheless, that some simple explanation of logic circuits and their adaptability to binary adders would not only be an advantage to the students but, if supported by practical classes, would create some considerable interest.

The Fifteen puzzle is well known. In its original form it consists of a shallow square box that can just hold sixteen small squares. Squares, numbered 1 to 15, are placed in any order in the box, leaving the sixteenth position empty (e.g. Fig. 1). The problem is to slide the squares, without lifting them out, until the numbers are in their natural order (Fig. 2).

A selection process used by some compilers of big-prize competitions is to present a triangle (usually equilateral) divided into (say) nine congruent sub triangles by the lines joining corresponding trisection points of the sides, and to ask how many triangles there are in the figure. It is amusing to obtain the general result when each side of the original triangle A is divided into n equal parts and pairs of dividing points are joined by parallels to the sides in all possible ways.

One is sometimes asked for an elementary introduction to the principles of information theory. I suggest that this need may be met by showing its application to the well-known problem of twelve balls. The problem is as follows:

Some time ago I asked what is the average rectilinear distance D between the ends of a string of length L when thrown down at random on a table (The Mathematical Gazette, LII (1968) 165). I have to thank a number of correspondents (J. Clemow, R. 0. Davies, A. R. Everett, M. Fine, J. R. Hunt, G. S. Light and E. Sike) for letters about the problem. Some offered solutions based on simple assumptions regarding randomness. The simplest (Everett) is that all values of D in the range (0, L) are equally probable, and so the answer is \L. I do not find this assumption acceptable, nor the other assumptions put forward.

1. L. J. Mordell has shown very simply (Math. Gaz. Lii (1968), p. 135) that the minimum value of the definite integral

Mr. A. M. Cohen in his article (The Mathematical Gazette, LIII, Feb. 1969, p. 41) makes some suggestions for school work in numerical methods and number theory. These prompt me to describe some work actually done in which, for reasons which will appear, the emphasis is rather different from his.

Let xi , i = 1, 2, ... , n, be the roots of the polynomial equation

we first determine the value of the generalized Vandermonde determinant

In Note 2590 (Gazette 1956) I gave a very simple method for constructing a ring of 2n 0’s and 1’s in such a way that each set of consecutive n digits formed a unique number in the scale of 2. For instance with n = 4, the cycle 0111101100101000 provides in turn the numbers 0111 = 7, 1111 = 15, 1110 = 14, 1101 = 13, 1011 = 11, 0110 = 6, 1100 = 12, 1001 = 9, 0010 = 2, 0101 = 6, 1010 = 10, 0100 = 4, 1000 = 8, 0000 = 0, 0001 = 1, 0011 = 3.

In a previous article on the Spirograph, published in The Mathematical Gazette of December, 1966, I suggested that it could be used at sixth form level to illustrate some advanced work on cartesian coordinates. Following this up, I have found a number of links with calculus, algebra and trigonometry.