The death of the greatest English mathematician of our time is no mere national loss, for Hardy was recognised throughout the mathematical world as a master of our science. Of his studies, Hardy himself said, in his Inaugural Lecture at Oxford: “What we do may be small, but it has a certain character of permanence”. Few of us would dare to call Hardy’s contribution to mathematics small, while none of us can have any doubt about the lasting nature of his work. For a full account of the new pathways he opened up, the new territories he explored, reference must be made to the notices being prepared for the Royal Society and the London Mathematical Society. In the Gazette, it is fitting that we should record with special emphasis the debt which teachers of mathematics in this country owe to Hardy for the vast improvements in the teaching of analysis during the past forty years, from vagueness to precision, from obscurity to clarity, along lines mapped out and laid down for us by him.

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show latcr on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.

1. Poncelet’s well-known theorem states that if two conies S and S′ in a plane are such that an n-gon exists whose vertices lie on S′ and whose sides touch S, then an infinite number of such polygons exist. When this is the case, a relation holds between the invariants of the two conies. The explicit determination of this relation is due to Cayley, whose result is quoted by Salmon in a footnote on p. 342 of his Conic Sections (reprint of 1929). With a slight change of notation, Cayley’s result takes the following form.

I. In the May number of the Gazette for 1943, Sir Percy Nunn contributed an excellent article on the celestial cylinder. He called it “an excursion intended to illustrate an application of elementary mathematics to astronomy” My experience has shown that, while many people are interested in the apparent motions of the heavenly bodies, few are prepared to make the necessary effort to obtain that degree of calculation in the subject, which makes it so much more fascinating. One of the main stumbling-blocks against attaining that precision, which Sir John Herschel so strongly emphasises in his Outlines of Astronomy, is the lack of knowledge of spherical trigonometry. The celestial cylinder and cube have been developed, therefore, in the endeavour to convert spherical into plane trigonometry. Mercator rendered signal service in transferring maps from the sphere to the plane. This movement has continued to such an extent that, in the intense concentration on the plane, children often lose sight of the spherical nature of the earth. They appreciate the beauties of the sky, they love to watch the relative positions of the constellations, but plane star maps are much more acceptable to them than spherical. Even with older students, attempts to introduce problems on the astronomical triangle are seriously hampered by a lack of knowledge of spherical trigonometry. To maintain their interest, one is obliged somehow to convert spherical trigonometry into plane trigonometry.

The “curious” property noted by Mr. A. P. Rollett (Note 1261 in the Mathematical Gazette for December 1937), with regard to the centres of circles touching one another and inscribed in the space between two given circles, leads to an interesting general theorem on all such sets of circles.

Given any two circles, and a set of circles touching one another and the two given circles, then, if r1r2, r3 are the radii of three successive circles of the set, of which r2 is known, r1 and r3 are the roots of a quadratic equation of which the coefficients are simple functions of r2 and the radii of the given circles.