In Classroom Notes No. 178 (Math. Gazette, 1968, pp. 376-380), a problem is set and formally solved on a body falling through a tube in the interior of the Earth. One can condone taking the ideal conditions of a frictionless tube and even overlook the fact that no such tube could withstand the stresses present in the Earth below some 30 kilometres depth. But when, as in this Note, it is further assumed (without even reference to any assumption) that g is proportional to r, the calculations, whatever mathematical interest they might have, cannot be regarded as having any relation whatever to the planet Earth.

If it be granted that the study of trigonometry in the secondary school course should not be limited to the numerical solution of triangles but should include some treatment of form as an introduction to higher mathematical analysis, then the importance of the trigonometrical identity must be admitted. A certain amount of valuable training can be obtained from the proper study of the identity if this study be systematic; but there seems little to be gained by working through the customary set of miscellaneous examples inserted at odd intervals in the trigonometrical text-book.

At some time in the eighties or nineties of the last century the Rev. William Done Bushell, a mathematics master at Harrow School, bought a portrait painting at an auction sale at a house in Harrow. It does not seem to have had any pedigree but bore the inscription “Robt. Recorde M.D. 1556”. It was thus presumed to depict the Tudor physician and mathematician Robert Recorde (1510?–1558) who wrote some of the earliest books in English on mathematics and was the first to use the symbol “=” for equality. No other portrait was (or is) known.

If particles of positive mass lie on a continuous curve whose concavity is always upwards, their centroid clearly lies above the curve and below the chord joining the extreme particles.

This obvious property gives a simple formula which includes several of the standard inequalities of the Algebra text-books.

Many communist countries hold internal olympiads in order to identify and thereafter foster special talents. Since 1959 they have held an annual International Mathematical Olympiad (IMO) for all European communist countries (except Albania) and subsequently Mongolia, Cuba and Vietnam. Non-communist countries were first invited in 1967 and France, Sweden, Italy and Great Britain accepted the invitation. Italy has dropped out and Austria, Netherlands and USA now regularly participate. Belgium, Finland and, last year, Greece have also taken part. Each country sends a maximum of eight competitors and two adults.

(a) The foundation of calculus operations on the circular functions is

The traditional steps by which the inequality

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.