A configuration is a finite collection of points, lines and planes with a number of each on each; any one of the three kinds may be empty.

In this note I classify certain polyhedra which are of interest in that they have in some sense a higher degree of regularity than any other polyhedra except those which are completely regular. The definition of a regular polyhedron is that it shall possess two particular symmetries; one which cyclically permutes the vertices of any face c, and one which cyclically permutes the faces that meet at a vertex C, C being a vertex of c.

There must be many teachers of mathematics today who do not realise the great changes, both in what was taught and in methods of teaching, that were made in the early days of this century and the part that the Mathematical Association took in making them.

The article by C. C. Puckette in a recent issue of the Gazette (Vol. 37, No. 322, pp. 256–260) prompts me to give some results I obtained when investigating the pursuit of a bomber by an attacking fighter.

In this paper, we consider a sequence lr(x) (r = 1, 2, ..) of rational functions of a positive rational variable x, which approach log x as r→∞. Some properties of these functions are analogous to properties of the natural logarithm, and go over into the latter, when the limit r→∞ is taken. Except for pointing out such limiting results, we shall avoid transcendental methods.

Though much work has been done on these most interesting squares, there appear to be a few points, anyhow, that are either new or not published anywhere easily accessible to the majority.

Information Theory was developed to provide electrical engineers with a calculus for comparing telecommunications systems as transmitters of “information”—an ambiguous word used here in a technical sense to be defined. Since 1948 the theory has been rapidly developed and widely applied, but the results of its application to telecommunication engineering have been disappointing in some respects. Although it has been applied indiscriminately to all kinds of communication systems including language, it has nevertheless stimulated new ways of thinking about the storage and transmission of every kind of “information” in the most general sense of that word. It has attracted the attention of workers in fields as diverse as neurology and librarianship, statistical mechanics and psychology, cryptography and the study of social insects. Its terminology and concepts have provided stimulating analogies and have helped to link studies as apparently unrelated as thermodynamics and semantics. It is evidently a branch of applied mathematics of wide general interest.

An ordered pair whose second member (or value) is a number, while its first member (or object) may be anything, will be referred to as a quantity. By a consistent class of quantities—briefly, c.c.q.— we mean a class of quantities that does not contain two quantities with equal objects and unequal values. Reviving Newton’s term, we will refer to c.c.q.’s also as fluents. The class of all objects (of all values) of the quantities belonging to a fluent is called the domain (the range) of the fluent.

Since a link in a plane has three degrees of freedom the simplest closed plane mechanism with constrained motion must have four lower pairs, i.e. one more than the number of degrees of freedom. If the lower pairs are turning pairs, the mechanism is the well-known four-bar linkage. One or two turning pairs may be replaced by sliding pairs without affecting constrained motion, but not less than two turning pairs must remain in a closed link group.

The stereographic projection of the sphere is a perspective projection from a point on the surface onto the opposite diametral plane. Analytical geometry provides a unified method of treating the stereographic projection and problems in spherical trigonometry. The basic result is the equation for the projection of a general circle (eq. 1). This is applied to the polar, equatorial and oblique stereographic projections, and a simple proof of Cayley’s theorem is obtained. Next, the cosine formula for a right-angled spherical triangle and the relation between angles measured along different great circles are derived. Finally, the construction used on astrolabes for determining planetary time is shown to be the stereographic projection of a problem on the sphere, which is solved.

Most mathematicians have encountered the Problem of the Three Prisoners. Kraitchik gives the problem and the solution as follows:

“Three unsuccessful revolutionists managed to escape across the border, where they were interned in the lockup of a small town. The sheriff’s sympathies lay with the prisoners, and he sought a way to let them go free. One day he came into the jail brandishing three white discs and two black ones and said to the prisoners : ‘On the back of each of you I shall pin one of these discs. None of you can see his own disc, though you can see your companions ! If any of you guesses correctly that his own disc is white he will receive his freedom. If not, he will be detained indefinitely.’ With this he pinned a white disc on the back of each and left them in the care of a guard. How can each of the prisoners, A, B, and C, guess whether his disc is white?

Recently a good deal of interest has been taken in the properties of numbers which can be expressed and proved without going beyond the number 1 and the Four Rules, in other words by using only rational numbers. Professor T. S. Broderick has established various theorems about the number and distribution of primes, using this “sub-language”: (Journal of the London Mathematical Society, Vol. 14, pp. 303–310; Proceedings of the Royal Irish Academy, 1940, pp. 17–24).

The purpose of this note is to show at a glance the significance of successive steps in the solutions to some of the quadratic equations that have come down in the cuneiform texts as examples of the mathematical instruction given to Babylonian students c. 1600 B.C. It is due to the translations made by O. Neugebauer in Germany and later, with A. Sachs, in the U.S.A. and to those published by Thureau-Dangin in France that this ancient material is available for general study, and, of course, it is from these sources that the following examples are taken. The texts are rhetorical, they instruct the pupil to perform arithmetical operations with specific numbers derived from the problems; in the translations these numbers retain their sexagesimal form f but here they are in our customary notation although expressed in a somewhat unusual way to facilitate, in particular, ready recognition of the re-entry of a number that has been temporarily left behind; in the text the pupil is reminded that this is the number that “your head held”.

The problem of the “tippe-top” has been discussed in several papers, one by Synge makes the assumption of rolling and discovers instability when the axis is vertical with the peg up, but requires as a necessary condition that the top be not a solid of revolution. Since it appears that as far as is constructionally possible the top is axially symmetric, in which case the rolling motion is always stable, this solution would seem to be unrealistic. In a second group of papers Fokker suggests from observations of peg traces that normal tops roll with no sliding, Braams shows that for the tippe-top sliding will probably take place in the “ rapid precession “, and Hugenholtz deduces general conditions in which such sliding will cause the peg of the top to fall. For the final rise on the peg these authors introduce a “ rolling friction “. Here results substantially in agreement with those of Hugenholtz are found from a simpler result by vector methods. It is further shown that the motion is such that during the final rise sliding must take place until the top is very nearly erect, and that the simple assumption of sliding friction is sufficient to explain the entire motion.