Given points P1 P2, …, Pn on a circle centre 0, they have a centroid G. We shall show that they have also a generalised orthocentre H, a generalised nine point circle with centre N, and that OGNH lie on a line forming a harmonic range.

David Hilbert (1862–1943) was one of the most outstanding mathematicians of the modern era. At the International Congress of Mathematicians in Paris in 1900 he presented twenty-three major research problems which he felt would be important for the development of mathematics in the twentieth century. These problems all seemed very hard, but in bringing them to the attention of other mathematicians Hilbert felt the need to stress that this should not be used as an excuse to put off trying to solve them.

The dramatic increase in the speed of numerical calculations made possible by electronic wizardry has opened up new vistas in school mathematics in the last few years. One application is the solution of equations by iterative procedures. Interest so far has been only on the speed of convergence of the iteration and not on how a given iteration will succeed or fail—an equally interesting area. At school level this is a relatively uncharted area, and one where a sixth-former with a calculator could produce results of interest.

In Sir Thomas Heath’s entrancing Manual of Greek Mathematics it is stated on p. 52 that the Babylonians discovered “a most perfect or musical proportion” between two numbers, which Pythagoras introduced into Greece. This is the ratio a ½(a + b):: 2ab/(a + b) b. The two middle terms are the arithmetic and harmonic means of the extreme terms. Not knowing of this when considering the fundamental principle of the logarithm, it seemed to me that the use of these two means should assist in evaluating the natural logarithm of a ratio in which the numerator and denominator are the first and last terms. In this way I found the surprisingly exact but approximate expression for the numerical value of such a logarithm which appeared in Nature of 14th March, 1931.

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

The following, written at the reguest of the Editor of the Gazette, gives an account of the mathematical part of the remarks made by the lecturer at the Annual Meeting of the Association last January. There are added some words dealing with geometry on a sphere, in regard to which some questions were raised at the meeting.

In a previous article I have outlined my conception of the rôle of history in the exposition of mathematical technique. In this article I attempt to provide my bare thesis with a respectable clothing of practicability The treatment of Number and Continuity which follows is a short and very sketchy historical supplement to the technical treatment of the same subjects in, say, the earlier chapters of Hardy’s Pure Mathematics.

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.

One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:

“For all x, y, z in ℝ:

1. 1. Exactly one of x = y, x < y, y < x holds.

2. 2. If x < y and 0 < z, then xz < yz.

3. 3. If x < y, then x + z < y + z.”