Trigonometry, like most abstract sciences, has both an operational side and a quantitative side. The customary discussion is mainly quantitative, based as it is on the concept of the radian and the definition of sine and cosine as ratios of the sides of a right-angled triangle. The operational side has been touched upon by V. Naylor in his “Note on the Nature of Trigonometry” (1). Starting with the hypothesis that a function λ(x) exists which has the property

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.

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 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.

The main object of this address is not so much to express my own views on the matter, as to stimulate discussion among the many experienced teachers here, which may encourage the inexperienced, or timid, teacher to loose his grim grip on the printed examination syllabus, and refresh himself and his pupils by occasional profitable digressions from the printed path.

By “ anticipating the work of the Sixth Form ” I am assuming that we mean providing the pupil with such mathematical training in the main school that he reaches the Sixth Form with a mind which is eager, alert, and inquisitive, and with a sufficient equipment of mathematical knowledge to enable him to explore some of the mathematical realms which are awaiting his attention.

The question is often asked, “What can a woman do if she has a degree in mathematics and does not wish to teach?” Thirty years ago this was a very real problem, but today no woman mathematician need fear that she will be unable to find suitable employment. Most of the careers mentioned in this article are ones which the writer’s own pupils have taken up (or given up!). It is hoped that this will be the first of a series of articles dealing with careers for boys and girls with a mathematical bent.

Most of us, I suppose, can quote a few phrases read or heard, which have greatly affected our outlook on life in general and our own work in particular. Of about half a dozen such, two came vividly into my mind as I recently attended, at the University of London Institute of Education, a course of lectures in which a closer link between the sixth form and the University courses in mathematics was sought. The first goes back to my youth, and comes from Ruskin—“Know what you have to do—and do it!” Ruskin goes on to say that insufficiency of means is far less frequently a source of failure than an inadequate understanding of the thing to be done. The second is more recent, and came as a gentle rebuke from more than one of the nurses at St. Bartholomew’s Hospital, during my stay there after becoming blind, when I apologetically asked for some small non-routine service—“What are we here for?” Surely Ruskin’s precept and the nurses’ question are very relevant to our work.

This corner is intended for members who are not yet frequent contributors to the Gazette. Regular contributors are invited to write opening sections of articles on elementary topics, and new-comers are invited to continue them in any way that seems to arise naturally.