Recent Notes in the Mathematical Gazette have suggested directly or by implication the surprise felt by some students of mathematics when Σx3 turns out to be (Σx)2, Perhaps the results appear less surprising if they are seen, not in isolation, but as two particular cases of summation formulae for odd powers of the integers expressed in powers of {x(x + 1)}.

The fact that the sums of odd powers of the integers are expressible in powers of {x(x + 1)} can be used as a demonstration of the distinctive pattern of the difference tables of these sums, which facilitates their speedy calculation.

Nowadays, when so much computing is done with the use of the binary scale, it is arguable that some knowledge of binary arithmetic should be part of the mathematical equipment of the normal grammar-school pupil. The basic electric circuits for binary addition are comparatively simple, and a satisfactory machine for demonstrating the principles can be constructed with standard multiple switches and a flash-lamp battery and bulbs. In the U.S.A. the progressive firm of Berkeley Enterprises, Inc., of Newtonville Mass. has popularized “Geniac” kits from which various simple computing devices can be manufactured from interchangeable components. I am indebted to the Geniac Manual for some of the basic circuits I have used, though I have modified them considerably

In two articles in the Mathematical Gazette, Pedal Triangles and Pedal Circles, July 1919, and Some Properties Relative to a Tetrastigm, Oct. 1920, J. H. Lawlor gave proofs of a number of properties of the pedal circles of the four vertices of a plane quadrangle ABCD with respect to the triangles formed by the remaining three vertices. The articles concluded with the statement of ten results, the first nine of which were fairly easily deducible from those already proved, but No. (10) was given as a conjecture. This was that the common chord of the pedal circles of A and D bisects BH1 and CH2, where H1and H2 are the orthocentres of the triangles BAD and CAD, and hence that BH2, CH1 meet on the common chord of the pedal circles of A and D. It can be seen that the second result would imply the former, for if h is the rectangular hyperbola through ABCD, which is unique unless the points are orthocentric, and if O is the centre of h, then H1 and H2 also lie on h, and it is known that the pedal circles pass through O. The meet of BH2, CH1 is a diagonal point of the quadrangle BH1H2C inscribed in h, and its join to the centre of h is the diameter conjugate to the parallel chords BH1CH2 and therefore bisects them.

It forms an interesting and topical example in particle dynamics to investigate the effect of a small frictional perturbation on an elliptic orbit. In this paper the frictional force is assumed to be proportional to the square of the velocity and the eccentricity of the orbit is assumed to be small. Formulae are deduced for the rates of change of the major axis and eccentricity of the ellipse in terms of the rate of change of periodic time, numerical values being compared with the rather sparse information on Sputnik I.

The purpose of this article is twofold. In the first place it aims to give a general picture of the content and method of school mathematics teaching in a country in whose scientific and technical education the Western world is becoming increasingly interested. This general picture is here based entirely on publications of the Ministry of Education of the RSFSR (Russian Soviet Federal Socialist Republic) and on text- and other books in general use, mostly from the Russian Republic but a few from the Ukraine. It is not based on personal contact with the system or with teachers who have worked within it, but it is hoped to extend this article at a later date on the basis of first-hand experience. Aware of the diversity and flexibility of our own system of mathematics teaching, we would not expect a foreigner to understand it fully from publications alone. The Russian System, however, is less extensive and more unified, and we may therefore hope to obtain a useful understanding of it from books alone. We must remember, though, that it is in a process of change, and also that, as in England, there will be considerable variability from one school to another depending on the quality of the teachers. However oversimplified this introductory article may be, it should at least serve to give an orientation towards the subject.

The process of producing the facial planes of a polyhedron to form stellated polyhedra is well known. When applied to the icosahedron it leads to a series which is limited to eight solids if we make the condition that every face of each must be covered in forming the next of the series. If this condition is dropped a large number of combinations of the parts of these eight solids is possible. The total number of solids, limited by symmetry considerations, is 59, and they are all illustrated in The 59 Icosahedra by Coxeter, Du Val, Flather, and Petrie. The corresponding basic series of solids starting with the Rhombic Triacontahedron numbers 13, and the total number of combinations is very great.

There are four methods which are normally taught in schools for solving quadratic equations:

(i) By factorisation.

(ii) By the use of graphs.

(iii) By completing the square.

(iv) By the use of the formula.

In outlining the following course planned for girls of sixteen to eighteen years of age, I should make it clear that it occupied only a subordinate pla in school organisation. The orthodox General Certificate of education ordina level course is taken by all girls in the school for the first five ye (with the exception of perhaps two or three in the fifth year) a we have always had a strong group for Advanced level in the Six Indeed in the four girls’ schools which I have known intimately, the Mathematics course has always been highly popular and more successful (if judg by examination results) than most other school subjects. I do not believe on such facts as I have—that there is a sex difference in mathematical abili The lower total number of girls taking the General Certificate of Educati examination is, I fancy, explained on other grounds.

It is just on twenty years since I was first permitted to air in public some of my views on school algebra. That talk, in 1937, was officially part of a British Association discussion, but since the other speakers were yourself, Mr. President, Professor Broadbent and Mr. G. L. Parsons, with Professor Neville in the chair, it was quite a Mathematical Association family party. That was also the occasion of my becoming well acquainted with Alan Robson, who during the intervening years did so much, by his splendid text-books and by the influence of his pupils, to bring new ideas into school mathematics, and particularly into school algebra. It was a great pleasure to meet and correspond with him after so long an interval, during the years in which the Report that we are to discuss to-day was in preparation. His sudden death at the very time when the Report was first presented to the Teaching Committee of the Association was a most sad and unexpected blow. His thoughts had been occupied a great deal with it after his retirement, and it owes to him not only the chapters for which he was mainly responsible, but innumerable improvements at all the points that his sharp critical eye had lighted upon.