The algebra of sets admits a quite elementary presentation in the form of an axiomatic system, which may well be suitable for sixth form school teaching. In this note, however, I am concerned not with axiomatics but with the development of set theory in terms of the membership relation; or to be more precise I am concerned with that part of set theory which deals only with the Boolean operations and the relations of equality and inclusion, but does not seek to generate sets.

There is a widely held view that the jump from school to university mathematics is a big one. Recently James and Fiala have dealt respectively with the interdependence of school and university courses and the teaching of mathematics at university level. It seemed pertinent to obtain the views and opinions of students at the start of their mathematical studies at university.

In a common development of the special theory of relativity one considers two events P and Q, P being the emission of a flash of light and Q the reception of the light. We suppose Q to be infinitesimally close to P and, by using the principle of the invariance of c, the velocity of light, argue that the infinitesimal coordinate differences between P and Q must satisfy

Iand

II

where dx; etc. are the coordinate differences in one inertial frame whilst dx' etc. are those in a second inertial frame (e.g. [1], p. 16).

Finite projective planes are a lively field of research, particularly in America and Germany. SMP Further Mathematics gives them some attention, and recently two American text-books which would serve as suitable introductions to the subject at the beginning undergraduate level have been published [Albert and Sandler, Introduction to Finite Projective Planes, Holt, Binehart and Winston (1968); and Blattner, Projective Plane Geometry, Holden-Day (1968)].

Readers familiar with the nine-point circle property of a triangle (Figure 1) may have wondered whether there is a similar result for a tetrahedron.

In general the answer is “No”: there is no equivalent of the orthocentre for a tetrahedron, since the altitudes from the vertices onto the opposite faces do not intersect. There is, however, a special kind of tetrahedron for which these four lines do have a point H in common.

It is well known that any line through the “mid-point” of a parallelogram bisects the area. What can be discovered about lines which bisect the area of a triangle?

Obviously any median bisects the area. It is also bisected by eachline which is parallel to one side and divides the others in the ratio

Shiu has shown [1] that there are infinitely many triangular numbers that are also square. He remarked that while his theorem did give a list of such numbers, it did not give all of them. It seemed that there might be a reiterative method, working in integers, that gave more of these triangular square numbers, and which included all the known smaller numbers.

In this article, we are going to be looking at certain finite sequences composed of zeros and ones.

Junior School children seem to be able to get the feel of the Klein group by using aptly-prepared paper sheets and wooden blocks. Perhaps, surprisingly, they can also handle such logical implications (when suitably embodied in devices) as:

A→B, ∼B→∼A, B→A, and ∼A→∼B

Here “→” means “implies” and “∼” means “not.”

Given n dimensions, the obvious way of representing n classes is by a hypercube of side 2. Such a hypercube contains 2n unit hypercubes, one for each combination of the n classes. With only one dimension available, it is still possible to represent each of the 2n combinations. Take a line OX with any point P1 marked on it. The interval OP1will represent all combinations excluding the first class A1 and the interval P1X all combinations containing it.

For many practical purposes, Lagrange’s undetermined multipliers can be used for the solution of problems where some function has to be maximised subject to a constraint.

The technique described here arose from a consideration of one such case, where there was a choice of several types of investment, and, within each type, a different scale or level. With a limited total budget, it was necessary to decide how much to spend on each alternative.

The article by Mr. D. C. D. Potter in Gazette LVI, No. 396 (May 1972), pp. 109-10 attracted a number of interesting letters. We cannot print all of these in full, but the main points are included in the following extracts.

First, Dr. K. Szymiczek of the Silesian University, Katowice, Poland (writing from Cambridge) offered some information on the history of the problem. “According to L. E. Dickson, L. Euler was the first to show that there are infinitely many square triangular numbers and S. Roberts proved in 1879 that Euler's formula gives all square triangular numbers ([1], pp. 13, 27).”