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.

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.

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.

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 approach to plane mechanics by vector methods is often considered to be an unnecessarily sophisticated one and it is true that the preliminary work in vector algebra is lengthy and out of keeping with the elegance achieved. The natural tool, it has been said, is the complex number, and I shall obtain some of the fundamental results by using only complex number theory and vector addition.

Contemporary writers of textbooks on electricity and magnetism and on hydrodynamics make considerable use of the complex variable in two-dimensional problems. It seems strange that two-dimensional dynamics is not taught in this way, even though the application of vectors to three-dimensional dynamics has now become respectable. Mr. Buckley's article preceding this has shown how the basic results may be established. The following examples are further indications of how this powerful technique may be profitably employed in two-dimensional kinematics. The methods are essentially labour-saving.

The International Commission on the Teaching of Mathematics, which is one of the Commissions set up by the International Mathematical Union, has been conducting an enquiry into the teaching of mathematics between the ages of 16 and 21. A full report has already been published on this aspect of teaching so far as Germany is concerned and a questionnaire was sent to all countries participating in the work of the commission. The British National Committee for Mathematics appointed Dr. E. A. Maxwell to represent Britain on the International Commission and a sub-committee consisting of Dr. Maxwell, Miss M. L. Cartwright, A. P. Rollett and G. L. Parsons to prepare the answers to the questionnaire. This sub-committee also received a good deal of help from the Department of Education of Cambridge University in connection with questions relating to the general system of education. The sub-committee confined their replies for the most part to work done in schools. With the replies were sent various relevant pamphlets published by the Ministry of Education, specimen timetables from various types of schools, a large number of examination papers and syllabuses and reports of the Mathematical Association. The questionnaire was framed for dealing with state systems of education similar to that in Germany and covered the whole educational system. We give only those parts of it relating to the technical aspects of the enquiry.