One way to make a tetrahedron is to use four equilateral triangles. Let’s see why this works. When we fit two equilateral triangles of the same size together along an edge, the edges match up.

I am not acquainted with any investigation of the problem of dissecting two given rectilineal figures of equal area, by straight lines, so that the parts of either will fit on the other, and the results given here are new to me. The problem is one which suggests a variety of interesting geometrical constructions. It is natural to consider first how, and in how many different ways, a dissection can be made so as to give the smallest possible number of parts ; but types of dissection which do not give the smallest possible number of parts may also be worth attention. I exclude cases of dissections depending on some accidental or unspecified coincidence of dimensions in the given figures.

1. The preceding paper considers the most general theorem on change of variable in a Riemann integral: If g(t) is integrable over [a, b] and f (x) is integrable over G ([a, b]), then (G(t))g(t) dt exists and equals (x)dx.

Finite Galois arithmetics are well-known; finite geometries however, though more interesting to the amateur, have not really acquired professional status and do not appear to any great extent in standard works. The following example arose from a chance remark in Mathematics for T. C. Mits, by L. R. and H. G. Lieber; from this I deduce that a full theory has been worked out, but I have not seen it, and as far as I am concerned what follows is original, and I hope readers of the Gazette may find it new and stimulating.

It is nearly a hundred years since Boole published An Introduction to the Laws of Thought (1). An attempt to reduce the art of logical deduction to algebra was a bold venture, and perhaps not all of Boole’s original hopes have been fulfilled. Nevertheless, Boole algebra has become a well-established branch of mathematics which has given rise to theoretical masterpieces such as Principia Mathematica (2), and has found practical applications in electrical circuitry, switching problems, insurance law and statistics.

A very great amount of scientific effort has been expended in the last few years on Rocket research and development. This work has been carried out in many countries besides Great Britain, notably the U.S.A., Germany and Russia. In the main it has been done by government scientists or by firms and universities under government contract. This is understandable since the chief use of rockets so far has been a military one. There is a very long history of the use of rockets in warfare (they were used at Waterloo), culminating in the production and firing of the German V2. Most of the major rocket weapons produced in the last war have been described in general terms in the Press, but there is, at present, remarkably little mathematical or scientific literature on the subject. Various technical journals,e.g. Flight, have published articles on the various aspects of the rocket problem, and there are one or two books. This lack is due to the fact that most of the papers written during the war years are classified as Secret or Confidential and are not yet available for open publication. Two recent papers are worthy of mention : one by Perring (1) on German Long Range Rocket Development, which gives a considerable amount of data on the V2 and one by Malina and Summerfield (2) on Rocket Escape from the Earth which discusses the possibilities of very large rockets. These may be regarded as the forerunners of very many papers which should eventually find their way into scientific literature.

Expressions like

are known as continued radicals and they occur frequently in the mathematical literature, if only because of their orderly structure and their close links with continued fractions. But it’s worth bringing them out again here because they can make very interesting classroom exercises on sequences, and they lead to unusual ways of expressing π and the golden ratio, etc.

Psychology has been described as the science which tells us what everyone knows, in language that no one can understand. It has a fatal attraction for those whose paucity of ideas is concealed by an exuberance of rocablilary The leading authorities not only ignore each others’ conclusions, but even refuse to use words in the same sense. “When I use a word”, said Humpty Dumpty, “it means just what I choose it to mean-neither more nor less.” The discussion of such a point as the nature of intelligence, sufficiently difficult in itself, becomes hopeless when everyone attaches a different meaning to that term. It is the object of this article to show how, by the application of mathematical methods, Professor C. Spearman has found a way of escape from this jungle of verbal entanglements. As a preliminary we shall require some account of the idea of correlation.

1. It is a commonplace of Biological Science that each individual, in his physical development, reproduces in an abridged form the successive stages through which the race has passed in its remote history. Educational experience suggests that a similar process occurs in mental development, so that, generally speaking, ideas have to be introduced in the order of their historical occurrence, if unnecessary resistance is not to be encountered.

Our Association was founded deliberately to “improve” the teaching of geometry, and its work has extended to cover every branch of the mathematics taught in the pre-undergraduate years. It is no surprise that, among the many distinguished mathematicians who have contributed to this development, a few stand apart clearly for the extent of their influence. Judged by any standards, E. H. Neville was one of the greatest of them all, and this tribute, by one of his earliest pupils at University College, Reading, is written with a sense of deep personal affection and gratitude.

It is more than 91 years since, on January 17th 1871, a group of twenty-six men met in another place and formed the Association for the Improvement of Geometrical Teaching, which twenty-five years later became the Mathematical Association. Most were schoolmasters; some were clergymen; the account of the proceedings in the first annual report adds: “There were also a few gentlemen present who were not members of the Association.” The ladies were allowed to join later.