What are the units, irreducibles, and primes of the ring ${\mathbb{Z}}\sqrt n $, the set of all numbers $a + b\sqrt n $ where a and b are integers and n is a fixed positive square-free integer? In the ring ${\mathbb{Z}}$, primes and irreducibles are synonymous and its units are ±1. ${\mathbb{Z}}\sqrt n $ is wilder, and our modest goal here is to catalogue all such numbers for ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $, where a and b range from 0 to 10; the result appears in Figure 1. Here are a few teasers that may induce a reader to read on: $3 + \sqrt {10} $ is a unit; 2, 3, 5, and 7 are irreducibles, but not 31; and 7 is the least positive integer that is prime in both ${\mathbb{Z}}$ and ${\mathbb{Z}}\sqrt {{\rm{[10]}}} $.

Both terms in the title of this paper are probably suspect. The mathematician, versed in a subject with hundreds of years of noble tradition, tends to be suspicious of a paper on psychology; the teacher of mathematics, immersed daily in psychological relationships with ordinary children, tends to be suspicious of a paper on mathematical ability. The discussion promises to be unsatisfactory, to the one because psychology lacks mathematical precision, to the other because mathematical ability seems to present no urgent psychological problems. The teacher-mathematician may then be pardoned if he reacts sceptically to our title with the petition, “Psychology, if there be a science of psychology, help me to cope with mathematical disability.” But these suspicions are not well founded. Not only is psychology, as we shall see, becoming increasingly mathematical, but also, if we interpret our terms as mathematicians should, disability is seen to be merely the negative aspect of ability. As such, it will not be excluded from our discussion.

The contents of this paper will, it is hoped, be found to be of value to teachers of physics and radio principles, to their pupils, and to those engineers who prefer a full understanding of phenomena to a superficial knowledge.

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.

The mathematical theory of rocket motion has been given by Rosser, Newton and Gross (1947) for the case of unrotated, and slowly rotated, rockets. A rigorous treatment of the subject has been given also by Rankin (1949) in a recent paper which is applicable to both unrotated and rotated rockets.

One of the main objects of such work is the derivation of formulae which may be used to predict the behaviour of the rocket under the action of various disturbing influences. Thus, an angular deviation of the rocket from its normal trajectory may be caused by the thrust not passing through the centre of gravity, by the action of a crosswind (usually assumed constant in the theory), or by the rocket being launched with an initial yaw or initial angular velocity about an axis at right angles to the rocket axis. Malaligned fins may also cause such a deviation.

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

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.