The theorems that follow are mostly well known; the object of this note is to show that they all follow simply from a single argument, whereas they are usually reached from different starting-points.

I Had some difficulty in choosing a title for this paper. I might have called it “A Plea for More Practical Mathematics”, but there would have been a doubt as to whether “more “ qualified “practical “ or “mathematics ”. And I am not very fond of the word “ practical” De Morgan's remark that the self-styled practical man could always be relied on to produce some false logic is too often true. “Practical “ is invariably used as the antithesis of “theoretical “ The boy often regards theoretical chemistry as an unneces¬sary interruption of the practical work ; he does not always regard them as complementary, like bread and butter.

Sir Arthur Eddington died on 22nd November, 1944. The last sixteen years of his life had been occupied chiefly with the discovery and development of certain new principles in physics, which he used in the first place in order to calculate theoretically the “ Constants of Nature ” (such as the ratio of the mass of the proton to the mass of the electron) ; he did in fact succeed in deriving correctly the values of all those constants of Nature which are pure numbers. Later he showed that his ideas could be presented as a coherent system, to which he gave the name Fundamental Theory, and which may be broadly summed up in the statement: the external world is isomorphous with the square of quaternion algebra.

It is not my purpose to expound a new Theory of Differentials in the sense used by Professor G. Temple in his paper to the Mathematical Association in 1936 (G. Temple, “The Rehabilitation of Differentials,” M.G., Vol. XX, p. 120), but merely to embark on what he called the “comparatively trivial business” of giving an interpretation to the differential notation which pre-supposes the definition of derivatives, and to do so from the point of view of School rather than University teaching. A study of elementary textbooks on the Calculus shows two main schools of thought. The first, and older, of these regards differentials as infinitesimals, that is to say as variables which tend simultaneously to zero and whose ultimate ratio is their only interesting property. This presents great difficulties to beginners and savours too much of the early days when the theory of limits had not been developed. It has been largely ousted by the second school, which regards the differentials of independent variables as actual increments and defines the differential of a dependent variable (in terms of them) as what I like to call a “bogus increment”. It seems to me that this second interpretation is not as easy as it pretends to be, and that it suffers from serious drawbacks ; and the purpose of this article is to put forward a third interpretation which so far as I know is original and which is free from the objectionable features of the other two. Of what follows, the earlier part is a criticism of the second interpretation referred to, while the later part is an attempt to show how differentials can be more easily taught, and to demonstrate the adequacy of my interpretation.

Dr. McLachlan and I felt that it would be best to commence the discussion by dealing with the work done in technical colleges and thus lead up from the younger to the older. Mathematics classes in technical colleges can be said to fall into three groups : first, for those in which there are junior and technical schools there are mathematics classes for lads aged from 13 to 16 ; secondly, there are classes for those over 16 which we may group under the heading “ Degree Courses ”, that is to say, Matriculation, Intermediate and Final Degree and Special Honours classes for science or engineering; thirdly, there is what is probably the largest group, taking the colleges throughout the country, namely, the National Certificate classes leading up to the Higher National Certificate.

1. Certain steps in the solution of linear differential equations involving the operatoṙ nearly always cause the student some confusion. It may therefore sometimes be helpful to have an alternative way of proceeding. None of the results of this note is original, but some of the familiar difficulties are presented in a new light, which may prove useful, especially in a revision course.

Algebraic magic squares of order 4, constructed with the numbers 1, 2,… 16, are of three main types.

From the 4 × 4 frame, nine 2 × 2 quadrants may be cut, and the sum of the numbers in these quadrants furnishes the criteria for distinguishing between the various types of magic squares.

While a full discussion of nomograms requires the use of determinants and other ideas beyond the School Certificate stage, many practical needs can be met using two simple types. These can be understood and applied with only a knowledge of similar triangles; and, besides being of interest in themselves, produce as a by-product a working knowledge of logarithmic and other scales.

Before considering the method of construction, use of one or two types should be made by the pupil. This will help him to appreciate their simplicity. Their practical value, too, becomes apparent by using types (e.g. the focal length of a lens) that tie up with his work in the laboratory. Construction on squared paper can be done quickly and with fair accuracy. One or two carefully done examples with subdivisions inserted should be attempted also.

This article contains a discussion of the problem considered by Routh and other writers under the general heading “Rocking Stones”, but is restricted to cases which can be treated as problems in two dimensions. Thus Routh writes: “A perfectly rough heavy body rests in equilibrium on a fixed surface; it is required to determine whether the equilibrium is stable or unstable. We shall suppose the body to be displaced in a plane of symmetry so that the problem may be considered to be one in two dimensions.”