There is an urgent need for the expansion of the R.A.F Many young men who are physically and temperamentally fit for the service are unfortunately lacking in the necessary knowledge of elementary mathematics. For some time the Regional Committees for Adult Education have been giving assistance to such men, but the time has now come when this help must be greatly increased and extended to men of lower initial attainments. Cases differ so much that only very small classes are possible, or individual tuition may be necessary Usually the only time that the men are free is in the evening.

The article by Mrs. Linfoot on the “Teaching of Elementary Inequalities” published in the issue of the Mathematical Gazette for July, 1940, raises a matter of considerable interest and importance.

It is unhappily the case that the elementary treatment of inequalities remains a weak point in English mathematical teaching. While no Higher School Certificate examination paper in Algebra would be considered complete without the inclusion of questions on identities and equations, yet direct questions on inequalities and in equations occur comparatively rarely and, indeed, even when such questions are included in University Entrance Scholarship examinations, they tend to be ignored by the majority of candidates.

The construction, presumably used by Diocles, who flourished in the second century B.C., for the cissoid is as follows.

Let AB, CD be perpendicular diameters of a circle. Let E, F be points on the quadrants BD, BC such that the arcs BE, BF are equal. Draw EG, FH perpendicular to CD. Join CE and let P be the point of intersection of CE and FH The cissoid is the locus of P corresponding to points E, F on the quadrants BD, BC such that the arcs BE, BF are equal.

The usual method of making rugs with cut wool on canvas is to hook a piece of wool into each hole. This process causes the pieces to become very crowded; and, besides being slow in showing results, it is laborious and costly It is usual, moreover, to follow a pattern drawn on the canvas, and the pattern is seldom one to appeal to a mathematician, Victorian in conception as it so often is with its display of flowers and fruits.

In a recent paper (1) formulae were given for the numerical integration of a function in terms of its values at a set of arguments at equal intervals. In this companion paper, formulae for numerical differentiation, using the same data, are collected. Their utility in enabling derivatives of a function given numerically at such a set of arguments to be computed is obvious, the need arises in several approximate methods which are coming more and more into use (2), (3). The formulae avoid the labour of preliminary differencing, and are indeed more convenient than the finite difference formulae when the derivative is required at all the points of subdivision of a limited range.

Professor T. W Chaundy has published two papers on the above subject in Vols. 22 and 25 of the Proceedings of the London Mathematical Society. He discusses the condition necessary that a single infinitude of n-sided polygons may be inscribed in a circle S and circumscribed to another circle S′, basing his discussion on the theory of elliptic functions. A treatment of the topic, within the scope of elementary mathematics, is indicated in this note.