Some of the group axioms are easy to check. The entries within the table are those outside, so the set is closed under the operation. The first row and column inside the table copy the row and column at the top and side, so 1 is a two-sided identity. A 1 appears in each row and each column of the table, and overall the 1 s appear symmetrically in the table, so each element has a two-sided inverse. The table is a latin square, so that, for given a and b, the equations ax = b and ya = b have unique solutions for x and y.

The article Problems, solved and unsolved in the Gazette for last March (60 (411), 1–11) drew reactions from a number of readers. There is not sufficient space to show in full all the proofs sent in, but I will try to indicate the methods used. For statements of the problems themselves I must ask readers to refer to the earlier article.

In a recent article (this Gazette, 65 (1981), 87-92) Dr Anderson raised the question of the derivation of the well-known formula for the sum of integer squares, in contradistinction to its mere verification by induction. Some time ago, in response to a request from a colleague, I derived each of the well-known formulas

by arguments based upon the decomposition of rectangles. As a complement to Dr Anderson’s article, these may be of interest; especially, in view of the derivation, to those requiring the formulas for elementary integration from first principles.

Individuals who excel in mathematics have always enjoyed a well deserved high reputation. Nevertheless, a few hundred years back, as an honourable occupation with means to social advancement, such an individual would need a patron in order to sustain the creative activities over a long period. Leonhard Euler (1707-1783) had the fortune of being supported successively by Peter the Great (1672-1725), Frederich the Great (1712-1786) and the Great Empress Catherine (1729-1791), enabling him to become the leading mathematician who dominated much of the eighteenth century. In this note celebrating his tercentenary, I shall mention his work in number theory which extended over some fifty years. Although it makes up only a small part of his immense scientific output (it occupies only four volumes out of more than seventy of his complete work) it is mostly through his research in number theory that he will be remembered as a mathematician, and it is clear that arithmetic gave him the most satisfaction and also much frustration. Gazette readers will be familiar with many of his results which are very well explained in H. Davenport's famous text [1], and those who want to know more about the historic background, together with the rest of the subject matter itself, should consult A. Weil's definitive scholarly work [2], on which much of what I write is based. Some of the topics being mentioned here are also set out in Euler's own Introductio in analysin infinitorum (1748), which has now been translated into English [3].

Although high school textbooks from early in the 20th century show that spherical trigonometry was still widely taught then, today very few mathematicians have any familiarity with the subject. The first thing to understand is that all six parts of a spherical triangle are really angles — see Figure 1.

This shows a spherical triangle ABC on a sphere centred at O. The typical side is a = BC is a great circle arc from to that lies in the plane OBC; its length is the angle subtended at O. Similarly, the typical angle between the two sides AB and AC is the angle between the planes OAB and OAC.

When the Mathematical Gazette was founded, statisticians were mainly concerned with the collection and presentation of data. The mathematical side consisted mainly of developments from the work of Gauss and Laplace in the early part of the 19th century, though some surprisingly modern work was done by Edgeworth towards the close of the century. 1889 saw the publication of Galton’s Natural Inheritance, introducing correlation and regression. These two workers influenced Karl Pearson who, in 1900, began the modern era of mathematical statistics with the concept of the significance test usually known by the symbol he used, chi-squared.

Family history research has become a very popular hobby during the last few years due to the ready access to records of births and marriages. Many people have rigorously traced their ancestors back to 1700 and earlier, corresponding to some 12 generations. The principal motivation of most amateur genealogists is to learn about the lives of their forebears and they trust these ‘legal’ records without question and do not usually acknowledge that cuckoldry may have occurred and that their true ancestors may not have been the ones recorded.

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.