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].

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.

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.

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:

[This is part of a lecture delivered at the Annual Conference of the Association on 31st March 1967. It contains an elementary teaching approach to matrices.]

Let us look at this diagram and consider what it might depict. It might be a map of airline routes between airports in three countries; with two airports, x 1 and x 2, in the first country, three airports y 1, y 2 and y 3, in the second country, and two, z 1 and z 2, in the third. The numbers beside the lines in the diagram might indicate the number of alternative flights (perhaps planes departing at different times) along the routes.

As an introduction to elementary programming for digital computers, it has been found advantageous to make a start with a desk machine. A very simple code was devised for this purpose. Interested sixth-formers tackled problems prominent in their ordinary mathematical work by this method and were soon able to make programmes and get results themselves. They were later able to visit a digital computer installation and to try their hands at using an established “autocode” (Sirius). They found that their previous experience with the desk machine was most valuable in initiating, very quickly, the kind of thinking required for the more sophisticated procedures. The total experience, although very brief, appears to have been well worth while, as a means of showing them something of the potentialities of modern computers in relation to problems within their own competence.

For convenience we shall repeat here the definition by Hawkins, Hill, Reeve and Tyrrell (1, hereafter referred to as HHRT) of a k-map:

To continue the debate (Mansfield and Bruckheimer [1], Broomhead [2]) :

1. Scalar Quantities are things like length, area and mass. Vector Quantities are things like force, velocity and acceleration (in 2 or 3D). But let us not, for the present anyway, call vector quantities “vectors.”