The usual method of making rugs with cut wool on canvas is to hook a piece of wool into each hole. This process causes the pieces to become very crowded; and, besides being slow in showing results, it is laborious and costly It is usual, moreover, to follow a pattern drawn on the canvas, and the pattern is seldom one to appeal to a mathematician, Victorian in conception as it so often is with its display of flowers and fruits.

In Routh’s Elementary Rigid Dynamics (Sixth edition, 1897, § 300) a section is given to the application of the principle of linear momentum to determine the gradual changes produced by alterations of mass, with examples; but I cannot recall that any modern textbook explicitly points out that the formulae

M dV/dt = F .....(1)

and d(MV)/dt = F, .....(2)

which serve to determine the motion of a body of constant mass, are not necessarily true when the mass is changing.

I Give below a topological proof of the following theorem, based on a well-known conundrum:

Preliminary note. Beginners find two ways of regarding an ellipse easy to understand. The first presents it as the curve traced out by a point P (Fig. 1) when r + r′, the sum of P’s distances from two fixed points F and F′, is a constant, 2a. From this definition come the equivalences AA′ = 2a, CA = CA′ = a, and CF = CF′ = ae, where e is a fraction less than unity C is the “centre” of the ellipse, F and F′ its “foci”, CA = a the “semi-axis major”, CB = b the “semi-axis minor”, and e the “eccentricity”

In a recent paper (1) formulae were given for the numerical integration of a function in terms of its values at a set of arguments at equal intervals. In this companion paper, formulae for numerical differentiation, using the same data, are collected. Their utility in enabling derivatives of a function given numerically at such a set of arguments to be computed is obvious, the need arises in several approximate methods which are coming more and more into use (2), (3). The formulae avoid the labour of preliminary differencing, and are indeed more convenient than the finite difference formulae when the derivative is required at all the points of subdivision of a limited range.

In Analytical Geometry there is a most useful equation which is very rarely used and seems to be very unfamiliar to candidates in school examinations. It is the equation of the chord of a conic whose mid-point is (x1, y1). Speaking from a long experience as an examiner, I am always astonished at the unfamiliarity of candidates with this particular equation. If one sets a question asking the candidates to obtain this equation and follows this with a simple rider designed to show its usefulness, even if the candidates know how to find the equation, they rarely solve the rider by means of it.

In a letter to the Gazette (XXV, February, 1941, p. 36) Mr. Siddons discussed my article in the Gazette for October 1940; he gave a method of long multiplication of money and suggested that I might carry out further tests using his method. I was at once attracted to Mr. Siddons’ method, and carried out some tests, though I was unable to make them as extensive or as exhaustive as I desired. The present article contains a report on these tests. No very definite conclusions can therefore be drawn. This record of the work done may, however, serve as a useful pointer for those who may be sufficiently interested to carry the investigations further. I should have been pleased to have done so myself but the circumstances of my professional life have altered somewhat since carrying out the investigations recorded in recent numbers of the Mathematical Gazette.

It was quite clear that the company were much more interested in what the Baron had to say about the marriage customs of the people of Octonaria than in his remarks about their system of numbering. So I bided my time until he left the hospitable parlour of The Dolphin and, attaching myself to him, with little difficulty persuaded him to accompany me to my lodgings. Here, comfortably ensconced in my own armchair, he proceeded to fill the great octagonal pipe which had excited such curiosity at The Dolphin, while I set about collecting the ingredients for a bowl of punch.

The following method of introducing the ideas of rotation and strain into the study of Elasticity depends upon the identity

m · n grad u=n · (m grad u + m × curl u).....(1)

and upon the fact that if the straight line PQ undergoes a small rotation ω about an axis through P, the displacement of Q is

Consider first of all a rigid body which undergoes a small displacement. Let the point P of the body move to P′. We shall represent PP′ by u and regard u as typical of a vector field extending throughout the space originally occupied by the body. Let Q be a point near to P and represent by m.

Professor T. W Chaundy has published two papers on the above subject in Vols. 22 and 25 of the Proceedings of the London Mathematical Society. He discusses the condition necessary that a single infinitude of n-sided polygons may be inscribed in a circle S and circumscribed to another circle S′, basing his discussion on the theory of elliptic functions. A treatment of the topic, within the scope of elementary mathematics, is indicated in this note.

Problems about the American city road-system with its streets running east and west and its avenues running north and south have appeared from time to time in examination papers and are interesting because they give relations between binomial coefficients which are harder to get by straightforward algebra. After illustrating some of the methods by which difficult-looking results can be obtained simply, I shall obtain a simple-looking result with difficulty and ask for an easier way of getting it.

A few sessions ago the writer was presented with the problem of providing for one hour per week a course outside the normal Intermediate syllabus in Mathematics for a group of First Year engineering students. Knowing how the engineer thinks with pencil in hand, and uses the drawing-board for a variety of design calculations, the solution adopted was that of putting as much elementary mathematics as possible on the drawing-board. This article outlines the course, along with some additional related matter, and among the reasons which seem to justify publication are

(a) the author—and some at least of his students!—have enjoyed the work described;

(b) much of the subject-matter is not easy to find in English books;

(c) a number of illuminating sidelights are thrown upon topics of the “normal” mathematics course;

(d) it is believed that portions at least could be introduced with advantage into both pre- and post-certificate courses.

The points A, B, C on the circle x2 + y2=R2 are

(R cos θi, R sin θi) (i = 1, 2, 3).

We may without loss of generality assume 0 ≤ θ1 < θ2 < θ3 < 2π.

Definition: ABC is a triangle; A′B′C′ is its medial triangle; L, M, N are the feet of the perpendiculars from A, B, C respectively on a straight line, σ1 σ2, σ3 are three circles having their centres at A′, B′, C′ respectively such that σ1 passes through M and N, σ2 through N and L, and σ3 through L and M Then it is readily seen that the radical axes of the three circles taken in pairs meet in a point W—the radical centre of the circles. Again, since L is a common point of σ2 and σ3, their radical axis is the line through L at right angles to their line of centres B′C′, and hence to BC. Hence the lines through L, M, N at right angles to BC, CA, AB respectively meet at W, which is said to be the orthopole of the line LMN The common radical circle of σ1, σ2, σ3 is called the Orthopolar circle of the line LMN. Let us consider a few applications of the properties of this circle and its centre.