Ever since the war, the methods, techniques and results of operational research have increasingly been affecting the organisation and functioning of modern industry and commerce. Diverse in character and embracing general and unsophisticated approaches as well as specialised mathematical ones, operational research is initially concerned with the analysis of situations and the identification of the factors affecting them. Once such analysis has taken place, certain standard optimisation procedures are used, or new ones devised, so as to assign a set of values to the parameters involved that will yield maximum efficiency, minimum cost, greatest speed, etc. One of the simpler and better known examples of the effects of such operational analysis was the institution of the penny post in 1840. Two factors were considered to affect the cost of a postal system: the collection and distribution of mail on the one hand and its actual transportation on the other. The fact that the cost of the latter was only a fraction of the former enabled penny postage to be established. This process of isolating relevant factors and making them interact in certain ways to yield a model of a real situation is one which today’s teachers of mathematics cannot afford to ignore. It is contended here that, if this approach is properly introduced into the classroom and applied to simplified industrial or local problems, it can help

The use of matrices in connection with geometric transformations is now well established in the literature devoted to the teaching of “modern mathematics” in schools. A possible way of motivating an interest in projective transformations is to be found in the problem of constructing an accurate perspective drawing of an object as seen from a selected viewing point. One of the first artists to master the technique was Albrecht Dürer, and the woodcut by him (Figure 1) shows clearly the basic idea, that such a drawing is a projective transformation from “3D” space onto a plane. These transformations can be represented by 3 = 3 matrices. This note gives an elementary derivation of such a matrix representation, and shows how it may be used to construct perspective drawings.

In a paper with the same title, Carlitz [1] proves two inequalities about a triangle ABC and an internal point O:

Throughout this article, all numerals are assumed to be in base ten notation, unless otherwise stated.

Under many of the new mathematics programmes currently in use in secondary schools, the topic “Numbers to different bases” is introduced. Indeed, provided such an elementary textbook series has chapters on Sets, Counting to different bases, Symmetry and/or Transformation Geometry, and, possibly, Vectors and Matrices, it is then allowed to write “Modern Mathematics” on its title page almost regardless of the methods employed in the book. However, my topic does not concern the vagaries of modern mathematics textbooks, their content and their methodology, but rather the question of why ten is used as a number base. Undoubtedly many of you already feel that you know, and that the reason is concerned with numbers of fingers and toes. However, one might try to decide more rationally whether ten is a sufficient base to represent the wide range of numbers with which we are normally concerned.

Engineers photographing oscilloscope traces often want to know how much film is left in a cassette after some of the original film has been exposed and cut off. The obvious way to find out is to open the cassette in a dark room and measure the length left inside. Alternatively of course the length can be found by weighing or if the amounts of film originally present and used are known. Another way, unfortunately only of academic interest, is suggested below.

The problem is to join two given straight pipes by a curved pipe made up of two circular arcs of equal radii. The case when the two given pipes are coplanar has been discussed by E. Kosko in the Gazette of May 1964, pp. 192-196. Purely geometrical constructions for this case have been provided by S. N. Collings in the Gazette of October 1965, p. 302, and by me in the Gazette of February 1966, pp. 37-39. (Unfortunately Mr. Collings’s construction does not always work. On his diagram one can put in the two straight pipes in four different ways. For two of these ways the construction gives valid solutions, but for the other two ways the joining curve has a discontinuity of sense.)

The following note develops ideas in fields outside the normal school syllabus and if its contents are introduced at school, will, it is hoped, serve as a stimulus to encourage the brighter student to carry out further investigations in these fields.

It is the purpose of this note to remark that the mean deviation is quite as respectable and therefore useful a test of dispersion of data as the x2 or other widely used tests—at least in simple statistical situations. That it is not used appears to result from some misunderstanding of its efficiency and perhaps also of its bias as an estimator of parameters of a frequency function.

1. Introduction

In [1] the following three questions have been asked, and answered: For which values of f is there a convex polyhedron whose faces consist of

• (i) f triangles? (Ans f = 4, 6, 8, 10, 12, 14, …)

• (ii) f quadrilaterals? (Ans f = 6, 8, 9, 10, 11, …)

• (iii) f pentagons? (Ans f = 12, 16, 18, 20, 22, …)

Correspondence on the ‘Antelope’ article has indicated that there is an error in the “Mr. Moneybags” argument on pages 234-235. The error lies in the translation of the argument from its English form to the corresponding Symbolic Logic form.

“About, about in reel and rout,
The death fires danced at night:
The water, like a witch’s oils,
Burnt green and blue and white.”
Coleridge

(1) If a small light source, S, is observed through a parallel sided glass sheet, G, such as a window, the only result is that S appears to be less brilliant. But if G is misted over by moisture, the apparent brilliancy of S is further decreased, and diffuse haloes are seen concentric with S. If S moves at a steady speed parallel with G, the haloes appear to move with it. But if S is a discharge lamp, so that it flickers steadily, a new phenomenon is noticed as it moves: these are evenly spaced parallel light and dark bands at right angles to the direction of motion of S. When S is a polychromatic source, such as a mercury-discharge lamp, both the haloes and the bands are coloured.

The best separation of substances by adsorption chromatography requires a de-sorbing solvent whose concentration-volume graph is concave. Various methods have been devised to achieve this, sometimes ignoring the density difference between the dilute and concentrated solvents. A particular method of mathematical interest is that of Baker, Ellis and Hawley [1] where standard components can be used and density differences are reduced by using a floating siphon.

As far as the author is aware, there does not seem to be currently available in the literature a proof of the little-publicised pivotal condensation method of evaluating determinants. The purpose of this note is to supply a simple proof of this.

A weight is suspended by a rigid rod from frictionless bearings which allow it unimpeded travel of 360° in either direction. The pendulum swings in vacuo, and is kept in motion by electro-magnetic impulses at its lowest point, monitored (as regards direction of impulse) by two photo-electric cells, on either side of bottom-centre.

We propose to construct some triangles somewhat like the Pascal triangle, but more general. The only essential knowledge is that of the binomial theorem, though for some of the results we assume a knowledge of number theory, up to Fermat’s theorem. At the end, we show how matrix algebra could be used to advantage.