For some years courses of instruction on the programming of digital computers have been offered at Letchworth College of Technology. The courses of interest to schools and school teachers fall into two categories:

• (a) courses of instruction specifically designed for sixth-formers and

• (b) courses designed for school teachers.

When, as a newcomer to the profession, I started to teach elementary Mechanics, I taught it more or less as I had been taught myself. It was a classroom subject, one which drew on the experience of the pupil certainly, but one in which actual experiment was never attempted. The ideas seemed to me to be fundamentally simple, and to be so interlocked and so few in number that it ought to be possible to communicate them to a class in a very short time. I need hardly say that I was rapidly disillusioned. I began to be faced with “Why?” questions that I could not answer convincingly, and with questions like “What is the difference between momentum and kinetic energy?” and “Is it sine or cosine?” which revealed a complete failure to understand. I found boys dividing their answers by thirty-two, because they were confused by my somewhat haphazard use of gravitational and absolute units. One boy made it plain to me that he believed that my Mechanics had no practical bearing on the world in which he lived. Much later another boy said that Mechanics to the mathematician was substituting in the right formula. I realised that some fundamental re-thinking was called for.

Today, school mathematics syllabuses all over the world are being revised: the teaching world is in a state of ferment. From Illinois to Cracow, from Wellington (N.Z.) to Wellington, Shropshire, there are more than whispers of change. The suggestions for the actual changes vary tremendously: in Cracow, it is for the introduction of lattice structures and axiomatics; in the USA it is logic and truth tables (or was that last year’s model?). In Britain, some introduce groups, while others believe fervently in motion geometry; some ph their faith on vectors, while others tackle the problems of the negative integers. All are less than satisfied with the present syllabuses; but how many are concerned about how it is taught? In SME we are trying to see both sides of the coin and as the work develops it is becoming clear that we need to think carefully about both.

From any given polyhedron another can be derived by reciprocating with respect to some sphere. The most rewarding cases arise when the sphere is either the inscribed or the circumscribed sphere of the first polyhedron. In this way every Archimedean polyhedron gives rise to a new polyhedron, all of whose faces are congruent. Taking the simplest Archimedean solid, the truncated regular tetrahedron, we may derive a solid which has been called* both the “triakis tetrahedron” and the “tristetrahedron”. This solid has twelve isosceles triangles as faces, which meet by threes and sixes in a total of twelve vertices. It enjoys a certain importance in crystallography; for example, the mineral eulytine occurs in crystals which are basically triakis tetrahedra. We shall here consider some of the solids formed by stellating the triakis tetrahedron.

Consider a square array of 25 grid squares, the grid squares being of two types distinguished by the colours black and white. A basic square of four adjacent grid squares can be selected from such a square array in 16 possible ways. If these 16 basic squares are all distinguishably coloured, then the 5x5 square array is said to be pantactic.

Let (α, β) and (X, Y) be two pairs of values satisfying y = xn, where n is a positive integer and where we may at present regard α and β as constant.

We have

where the last bracket contains exactly n terms.

Applying this method we evaluate (see Fig. 1) the projections of the vector (the height at P2 of the given triangle), and also the co-ordinates of the point in which the perpendicular form

the origin meets the plane of the triangle. In this way, as will be shown, we find immediately the co-ordinates of the perpendicular projection of the given point P2(x2, y2, z2) on the straight line passing through P3 and P1, and also the equation of the plane of the triangle.

Suppose that a1, …, an and c1, …, cn are two sets of n positive numbers. If the arithmetic mean of the ci is at least equal to that of the ai, when can we say non-trivially that the geometric mean of the ci is at least equal to that of the ai and that equality will require equality (in some order) of the ai and ci?

A well-known answer of course occurs when each c is equal to the arithmetic mean of the ai. When the two sets each consist of three numbers I can give an answer including this important case. To obtain this result it is sufficient to confine the ci to the interval from the least to the greatest of the ai.

In a Note published in the Mathematics Student, Vol. XXI, (1953), Hansraj Gupta gave a method of determining the centre and axes of a conic without recourse to the usual method of shifting the origin and rotation of the axes of reference. The present author in a Note published in the same journal, Vol. XXIII, (1955), has treated the problem by a still more simple method, and has also derived directly the equations to the asymptotes of a hyperbola. The present paper deals with the determination of the parameter and the location of the parabola by the new method.

A previous note dealt with the summation of the series

and

where n is a fixed positive integer and r = 0, 1, …, n − 1.

The congruence x2 ≡ a (mod p), where p is an odd prime and a is any number not divisible by p, sometimes has a solution, but sometimes it has not. Gauss’s symbol (a/p), also known as Legendre’s symbol, is defined as (a/p) = l if there exists at least one x as a solution to this congruence and (a/p) = − l if no such solution exists.

The “Law of Reciprocity”, the famous theorem by Gauss, then states that, if p and q are odd primes,

where

In a recent paper, under the same title, I proved that inversion of certain functional relations is the characteristic property of the Mobius’ μ-function, by establishing the:

THEOREM I′: f(n), g(n) and μ*(n) (possibly depending on f) are three arithmetical functions with

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities

• (a) matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,

• (b) matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.

By a nearly diagonal matrix is meant a matrix in which the off-diagonal elements are small compared with those on the diagonal. Such matrices occur, for instance, in connection with non-orthogonal experimental designs. Inversion may be performed by methods of general applicability such as the Doolittle or Cholesky procedures (direct methods) or relaxation (an iterative method); many such procedures are described in the literature. Those already familiar with any of these methods may well prefer them to the one described below because of their wider applicability, but anyone not so placed who is often concerned with the inversion of nearly diagonal matrices should find the method described here useful.

When Descartes invented and defined exponents about the year 1637, he did not foresee the use of exponents which are not natural numbers. As a result, the standard treatment found in High School Algebra texts developed. While it is correct mathematically, it is very unsatisfactory psychologically. The following treatment is suggested as an improvement.

It is known that if a solid is turned right-handedly through an angle θ about a line with direction cosines ni, a particle with co-ordinates xi moves to yi, where