The mid-points of the sides of any quadrilateral form a parallelogram. Repeated applications of this inscription process will lead to two families of similar and similarly situated figures. It is proved below that the general n-sided polygon lying in space of h dimensions gives rise to similar sequences in the limit. If the initial polygon is not re-entrant convergence is rapid, as will be seen from Figures 1 and 5. The parallelogram is replaced by the affine transformation of a regular polygon lying in a plane.

The reciprocal nomogram described in Note 2108 (February, 1950) for the solution of 1/R=1/R1 + 1/R2 is a particular case of a more general property.

This note gives a general 2nd order solution, with the corresponding 3rd order correction, by means of which a good numerical answer can be obtained from an approximate estimate in one operation, and results of high accuracy by one or two iterations. The method is applied to integral roots of numbers, and to common types of equations.

For the extraction of square roots and cube roots of high accuracy, the general method is developed in the form of series designed to simplify the computation, and one example is given in full with an accuracy of over 30 significant figures.

For the last 75 years the largest known prime has been the Mersenne number 2127 – 1, proved prime by Lucas. In two recent notes (Eureka, October 1951 and Nature, 168 (1951), 838), Miller and Wheeler announced their identification of several larger primes, of which the greatest is

In the Journal of the London Mathematical Society (July, 1951, Vol. 26, pp. 221-8), proofs are given of a theorem of McCoy's and of various results that follow from it. McCoy's theorem establishes the necessary and sufficient condition that a number of n × n matrices should be reducible to triangular form by the same transformation T-1( )T, where the matrix T is unitary, (We define a triangular matrix to be one in which all the elements below the diagonal are zero.)

Beginning a new study involves considerations both of strategy and of tactics. The strategic objective is to impart to the study a sense of purpose, to arouse interest in it, to encourage the pupil to accept the challenge which is implicit in the new situation which confronts him. The tactical question involves the choice of method.

So far as strategy is concerned, the older student beginning algebra has no difficulty in recognising its practical importance in science and technology, where the power of its language to exhibit and clarify relational patterns is manifested. The appeal by way of vocational and cultural interests has been made successfully in various popular books for adults and “second-tryers”; these interests simplify the teaching problem, and the teacher can proceed at an early stage to give training in analysis and expression.

There is no doubt that Professor D. E. Littlewood’s stimulating and instructive article on the foundations of Euclidean Geometry will be widely read by teachers of geometry both in the Schools and in the Universities. It is almost equally certain that many of these readers, unfamiliar in the main with the language of transformation groups, will be in danger of being misled by an ambiguous usage of the term motion.

If it proves to be the case that no reader has in fact been misled then this note will harm no one but the critic himself, and I venture to hope that Professor Littlewood will accept this as evidence of my good intentions.