No history of the Christian Church in the 19th century can ignore the tremendous figure of John Henry Newman (1801-1890), striding across the scene from Calvinism through the Oxford Movement to the Cardinal’s hat. Most biographers of John make some brief reference to his brother, Francis William (1805-1897), poking a little fun at his journey through various branches of nonconformity to an austere agnosticism: his eccentric pilgrimage is contrasted with the contrary progress of his brother by William Robbins in The Newman Brothers (London, 1966), and is very sympathetically studied in one section of Basil Willey’s More Nineteenth Century Studies (London, 1956). But in neither of these books is any reference made to his considerable mathematical ability. G. H. Hardy, in his Divergent Series, asserts (p. 22) that Newman was for a time Professor of Mathematics at University College, London; this seems to be a mistake. Mr. T. L. Wren very kindly looked into this matter, and told me that while Newman was Professor of Latin from 1846 to 1863, there is no evidence that he held the chair of mathematics, though of course it is possible that he had occasionally given lectures on mathematics.

In the extension from the use of natural numbers with zero to integers, i.e. the introduction of directed numbers including negative numbers, much experimental work is now being done with the use of mathematical structure. Some attempts are being made to give this as the initial treatment at about the age of 12, and some teachers are getting their children very interested and consider it a correct method. For other children the formal treatment will come later, when they are more mature and begin to query the rules they are using. This may be when they are starting to work ahead of their O level syllabus, the beginning of a more advanced course with work on groups. It could give work for those doing some mathematics when they are beginning to specialise in some other subjects and want to try to understand more about the mathematics they have learnt.

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”.

It is very desirable when teaching mathematics not merely to state definitions and prove theorems, but also to show the train of thought that would lead us naturally to these definitions and theorems. Students are thus enabled to see mathematics as an alive, continuously developing subject. They are reminded of the interdependence of the various branches of mathematics; the memory of each branch is kept alive in the students’ minds, because it is continually being revisited, and the student finds it easier to remember results when he understands the thinking that produced them. This approach also encourages a student to try to carry the train of thought further and to make his own discoveries.

Notation: Functions which are to be elements of a group will be indicated by small letters, e.g., z′ = f(z), z′ = d(z) etc., the identity-function being denoted by z′ = e(z); z and z′ will be used for the variables. Small letters may also be used for constants (real or complex), with no fear of ambiguity. Groups will be abbreviated: Cn (cyclic group of order n); Dn (dihedral group of order 2n); Sn (symmetric group of order n!); An (alternating group of order .

On the eastern fringe of Bodmin Moor close to the natural rock formation known as the Cheesewring lies a large granite slab with a mathematical figure chiselled on its surface. Close by is another rock on which is carved, “D. Gumb, 1735”. This is all that remains of a rock shelter inhabited by Daniel Gumb, an eighteenth-century recluse. The rock shelter was demolished during quarrying operations and the inscribed stones which now lie a few yards south of the Cheesewring Quarry are about 100 yards south-west of their original site.

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 purpose of this paper is to consider the question: if p is the probability that a player P will win a given rally against a player Q, what is the probability that P will win a given game? The answer, of course, depends upon the method of scoring; and the methods considered here will be those used in squash and, for comparison table tennis.

In table tennis P earns a point each time that he wins a rally. But in squash he must serve the rally that he wins in order to gain a point; and he relinquishes the service when he loses a rally that he serves.

[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.]

We consider the problem of finding necessary and sufficient conditions for a positive integer to be the sum of an arithmetic progression of positive integers with a given common difference, starting with the case when the common difference is unity.

who arrived at the result through discussions in an undergraduate class at the University of Southampton.

Of the various metrics which can be placed on the co-ordinate plane R×R, the Euclidean metric is universally known, and two others are becoming more widely known in schools. With the co-ordinate notation for points x = (x1, x2), these metrics are

Three straight lines are needed to enclose a finite region of a plane, a two-dimensional space. The polygon formed is a triangle. Different types of triangle are described by adjectives; scalene, isosceles, equilateral and acute angled, right angled, obtuse angled. All the triangles have a property in common: they are all convex.

Four lines form a quadrilateral. The convex examples are well known. There are two further types: those which are “re-entrant” having an interior angle greater than 180 degrees, a reflex angle; those in which a pair of opposite sides cross within the quadrilateral, a “crossed” quadrilateral.

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

In the present state of flux of mathematical syllabuses and teaching methods pupils are often exposed to two distinct approaches to the solution of two simultaneous equations in two variables.

Vector geometry is often developed as a subsection of a geometrical structure, much of which is unrelated to the vector properties deduced. The basic results depend on the law which may be stated in the form AB + BC = AC. This requires only that the equality of two vectors is meaningful, and that if AB = XY and BC = YZ, then AC = XZ (if “equals” are added to “equals” the results are equal). We need not assume that A, B, C,… represent points—they could be for example lines in a plane, planes in space or even elements of a group. A vector may be thought of as a “direction” for changing from one element to another: thus AB = CD means that the way of changing from C to D is essentially the same as the way of changing from A to B. We develop a structure which has these properties.

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.

The debate about how best to introduce complex numbers is no doubt perennial. (See for instance [1] for a brief survey of some of the possibilities.) One flaw which many presentations have is that they appear somewhat arbitrary. For example one can define complex numbers as polynomial residues to the modulus y2 +1: this definition ought to provoke the question as to why this particular modulus should be chosen.