One has only to glance down the list of past presidents, which includes Sylvester, Whitehead, Nunn, Hardy and Eddington, to be acutely conscious of the honour of occupying this chair. It would be pleasant to be able to sing, with the King of Barataria, that “the duties are delightful and the privileges great.” The chief duty appears to be the delivery of a presidential address—hardly a pleasure even for the deliverer—and privileges there are none. I notice, however, that I am the fiftieth president of this Association, and that if the A.I.G.T. is included I represent the 57th variety. I am only the third former member of Her Majesty's Inspectorate (which is much older than this Association) to hold office, the others being Mr. W. C. Fletcher and Miss L. D. Adams, in 1939 and 1960 respectively.

Any two distinct circles a and b (in the real inversive plane) have one or two mid-circles which may be described as the locus of the point of contact of a variable pair of tangent circles that touch a and b [8, p. 31; 4, p. 74]. We easily see by inversion that two non-intersecting or tangent circles have a unique mid-circle, but two intersecting circles have two orthogonal mid-circles, bisecting the angles of intersection. The circles a and b are interchanged by inversion in their mid-circle (or in either of their mid-circles). This subject has been surprisingly neglected, possibly because of the terrifying name “circles of antisimilitude” [1, p. 28].

The National Council of Teachers of Mathematics in the USA took the occasion of the 1966 International Congress of Mathematicians as the opportunity for arranging with the Russian authorities a tour of Soviet mathematical centres by a small party of selected American mathematicians with special experience and interests in various aspects of education. The tour covered Kiev, Moscow and Leningrad and it was hoped that, in each city, a full range of mathematical education would be seen, from the department of mathematics in the University to the famous special boarding schools for pupils of outstanding mathematical ability and finally to the elementary level.

I was honoured to be invited to join the party as the only non-American and am very grateful to the Schools Council for enabling me to go by offering to meet the costs of travel and subsistence.

The way in which people spend their money is a matter of considerable interest. It is a topic which can be studied from many points of view and with varying degrees of sophistication, but it is well worth considering with the aid of a few graphs or Engel curves. Ernst Engels published his first account of Belgian working class family budgets in 1857 and his name is still associated with this study, though many of his conclusions concerning expenditure patterns have been falsified by subsequent changes in economic and social conditions.

These curves are used to illustrate the way in which expenditure on particular goods and services changes as household income increases. Data for their construction is obtained from household budget inquiries and is usually given in the form of average expenditure per household on a wide range of commodities for a limited number of income groups.

There is grave difficulty in showing elementary school students how mathematics is applied in practical situations (at least in Canadian schools). Since current emphasis is on pure mathematics, applied mathematics (and applying mathematics)—procedures of model building, approximation, analytical manipulation and interpretation—may remain obscure or unknown to students leaving school despite essential simplicity of principles.

Thus it is important to keep a look-out for problems which have an obvious practical interest or economic moment, but which can be tackled with the most elementary mathematics. Emphasis on the strategies of model building in the schools seems all the more sensible (and feasible) now that automatic numeration is permitting easy command of even complicated analyses.

The Mathematical Association’s News-letter No. 11, February 1967, contained this paragraph:

Magic squares might reasonably be supposed to be amongst the most useless of mathematical diversions. Yet, like other apparently fruitless amusements such as Eulerian squares and Kirkman’s school-girls problem, they have found applications in the field of the design of experiments. In an investigation of the effects of two variables upon a phenomenon, where the observations are obtained one after the other, magic squares make it possible to take account of linear trend in the results. For example (the writer is only competent to offer illustrations in the field of psychology : others in bis own field will doubtless occur to the reader) Miller and Selfridge investigated how much subjects could remember of series of words which varied both in their length and the degree to which they resembled everyday English. Again Fitts and Annett, Golby and Kay investigated how fast a subject could alternately tap with a metal stylus two separate metal plates, and how his speed was affected by the width of the plates and the distance between them. In both cases the subjects performed the task a number of times under different combinations of levels of the variables (length and resemblance to English, and width and distance respectively), so that any analysis of the data is liable to be vitiated by the fact that each subject will probably improve with practice throughout the experiment, unless special account is taken of this trend.

The name ANTELOPE stands for “A Nor (Transistorised Element) Logic Operations Patchboard Experiment.” It was originally designed by Mr. R. L. Lindsay, a mathematics master at Reading School, in the early part of 1965, to demonstrate how a complicated computer can be built up from a number of simple circuits. Mr. Lindsay left Reading School before it was fully operational for logic experiments, although it had been used for binary arithmetic.

After a series of lessons on Symbolic Logic by Mr. J. Periton as part of the second year sixth form English course, two sixth formers developed a method of representing logical arguments by electronic circuits.

I feel that many of the descriptions of linear programming are either too elementary or else expressed in terms of n variables with m constraints using matrices and innumerable Σ's. This is an attempt to arrive at a description somewhere between the two and in it I take n = 7, m = 3.

Suppose that the function to be minimized is given by

and that in addition to xq ≥ 0 for all q, the equations of constraint are

Solving equations (2) for any three of the variables, say x1, x2, x3, gives equations like

Suppose that solving for the variables x1, x2, x3 gives non-negative values of β1, β2, β3; if it does not, three other variables must be chosen until positive numbers are obtained on the right-hand side.

One approach used by text books to introduce sets is to define a set to be a collection of objects and then to explain that there are two types of sets, finite sets and infinite sets. It will be seen below that unless it is stressed that the two types of sets are fundamentally different, an unfortunate misconception of mathematics could be produced among the pupils.

In all the ferment of ideas for syllabus reform that have come to the notice of the main body of mathematics teachers in this country over the last ten years or so, there is the least agreement on those concerned with the teaching of geometry. Largely through the efforts of the Mathematical Association in its earlier years, the cry “Euclid must go” is an empty phrase in Britain today; for forty years or more we have managed with a pseudo-Euclidean compromise, but now many teachers, dissatisfied with this, are experimenting with fresh approaches. The purpose of this article is to state some of the objectives of geometry teaching, and to examine some of the possible treatments in the light of these objectives. It is confined to a consideration of the CSE/GCE O Level course, both for those whose formal mathematical education ends there and for those who will go on to further study.

There is a natural argument about teaching topology in schools which goes something like this. Here is a modern subject that is flourishing at the research level, it is truly geometrical in flavour (more so than geometry itself we gather) and its elementary notions are more primitive in the learning of the young child (according to Piaget)f than the ideas of triangle, rectangle or circle. Moreover, we understand that, as well as having an intrinsic interest and beauty in itself, this subject has elegant applications in other branches of mathematics, such as the calculus, which we think of as part of the traditional syllabus in schools. Surely, then, there is the strongest possible case for bringing this topic to the attention of the children while their geometrical intuition is still uninhibited by perfectly straight lines and perfectly round circles and to give them the advantage of the more intuitive proofs of the theorems in calculus which topology can furnish.

As the name suggests, the Multinomial Theorem is an extension of the Binomial theorem, and it was when I first met the latter that I began to consider the trinomial and the possibility of a corresponding Pascal's triangle.

My mathematics master suggested that I construct the triangle myself. It became apparent that such a triangle may not be drawn, as the result is a pyramid or a tetrahedron where the apexes of the tetrahedron are occupied by the coefficients of xn, yn, zn in the expansion of (x + y + z)n.

This may be shown by forming expansion coefficient triangles for a trinomial of power 0, 1, 2, 3.

In a technological university most undergraduates study mathematics. For engineers this is post A-level G.C.E. and in many cases is taken to a considerable depth. However, some departments admit students with only O-level G.C.E. mathematics and these people are taught the pure mathematics corresponding to that for A-level in the first session. Within the University of Aston this system is used on the Environmental Health Course. This is an undergraduate, four year, sandwich course for the training of public health officers. The students are sponsored by Local Authorities, who provide their practical training. Most of the students enter the University with at least two A-levels, which are usually a selection from Chemistry, Physics, Biology, Botany, Zoology.

The length of a piece of string is of course given by a real number. While this is a complete answer to the question posed, it is amusing to consider the converse problem of constructing a piece of string whose length (in terms of an appropriate unit) is a given real number and, further, to consider which real numbers can be constructed in this way.

In the last paragraph of an article entitled “Octahedra Inscribed in a Cube” by T. Bakos, Math. Gazette, Vol. XLIII pp. 17-20, (1959), the following statement is made: “About a cube (±b, ±b, ±b) one can circumscribe 3 regular octahedra (the edges of the octahedra through the vertices of the cube); their vertices are:

§1. Our first theory concerns a set of elements, denoted by letters a, b, c, … and words derived from them. The simplest word, apart from a single letter, has the form ab; other words are formed by adjoining a letter on the right to a previous word. Thus ab, abc, abcd are words. We shall say e.g. d is “hooked on” to abc.

The only statements about words in the theory are of the form X = Y, where X, Y are words, and the sign = has the usual properties. The statements are general in the sense that if any word be substituted for a letter in X or Y, and X', Y' be the new words so got, then X' = Y'. Furthermore if X = Y, then Xa = Ya. Note that if e be hooked on to abcd, we get (abcd)e, which is written abcde.