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.

It is, I think, of happy augury that it should have been fourid possible to arrange a joint meeting of members of the local branches of the Classical and Mathematical Associations, as well as members of the Yorkshire Natural Science Association, to hear a few words about Greek science. No doubt the classical or humane studies can stand by themselves and so (in a way) can the study of science based only on good English : but I venture to think that they are both much stronger when they stand side by side and support one another. The idea that thero is any necessary antagonism or opposition of interests between them is surely unfoimded, and there is room for both in our system of education. It is a mistake to suppose that in our school education there is not time for the one or the other ; the guestion is really one of using the time that thero is to the best atlvantage. No doubt schoolboys in the past have wasted much time in the perfunctory learning of Latin and Greek.

The problem how to express very large numbers is discussed in ‘The Sand-Reckoner’ of Archimedes. Grains of sand being proverbially ‘innumerable’, Archimedes develops a scheme, the equivalent of a 10 n notation, in which the ‘Universe’, a sphere reaching to the Sun and calculated to have a diameter less than 1010 stadia, would contain, if filled with sand, fewer grains than ‘1000 units of the seventh order of numbers’, which is 1051. [A myriad-myriad is 108; this is taken as the base of what we should call exponents, and Archimedes contemplates 108 ‘periods’, each containing 108 ‘orders’ of numbers ; tlle final number in the scheme is 108. 1015.] The problem of expression is bound up with the invention of a suitable notation; Archimedes does not have our ab , with its potential extension to aa a . We return to this question at the end ; the subject is not exhausted.

In Keynes's contribution to the ‘Newton Tercentenary Celebrations’ there is the following passage :

“Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass were concentrated at the centre, and only hit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years.

“Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions.

“His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already.”

Let α, β, γ be three conics, one with foci B, C, one with foci C, A, and one with foci A, B. The specification of a single conic as the locus of the centre of a directed circle which touches two fixed directed circles is not unique but involves both a parameter and a choice of sign. Can we adjust the arbitrary elements in specifying α, β, γ to obtain a simple specification of the focus-sharing set as a whole? Explicitly, if α is derivable from circles η′, ζ′ round B, C, β from circles ζ′′, ξ′′ round C, A, and γ from circles ξ′′′, η′′′ round A, B, what conditions can bc usefully imposed on the concentric pairs ξ′′, ξ′′′; η′′′, η′; ζ′, ζ′′?

A very great amount of scientific effort has been expended in the last few years on Rocket research and development. This work has been carried out in many countries besides Great Britain, notably the U.S.A., Germany and Russia. In the main it has been done by government scientists or by firms and universities under government contract. This is understandable since the chief use of rockets so far has been a military one. There is a very long history of the use of rockets in warfare (they were used at Waterloo), culminating in the production and firing of the German V2. Most of the major rocket weapons produced in the last war have been described in general terms in the Press, but there is, at present, remarkably little mathematical or scientific literature on the subject. Various technical journals,e.g. Flight, have published articles on the various aspects of the rocket problem, and there are one or two books. This lack is due to the fact that most of the papers written during the war years are classified as Secret or Confidential and are not yet available for open publication. Two recent papers are worthy of mention : one by Perring (1) on German Long Range Rocket Development, which gives a considerable amount of data on the V2 and one by Malina and Summerfield (2) on Rocket Escape from the Earth which discusses the possibilities of very large rockets. These may be regarded as the forerunners of very many papers which should eventually find their way into scientific literature.

In these days of examinations and syllabuses it is perhaps not superfluous to remind teachers of the maxim that they ought to teach more than their pupils need to learn. To be an expert on circles it is necessary to study other conics, and to appreciate homographies it is as well to know something about other correspondences.

The ideas of correspondence and characteristics of systems of conics are due to the French mathematician Chasles (c. 1850) and a large collection of illustrations of these subjects may be found in T. Lemoyne's Les Lieux Géométriques (Vuibert, 1923). These illustrations include, though in a summary form, the subject of the present article. Reference may also be made to the last example in C. Taylor's Ancient und Modern Geometry of Conics, and to a note in Salmon's Conic Sections.

If, like David Copperfield, you are on the road to Dover, for the first time and with no one to guide you, and you come to a point where the road forks, then of all your mathematical knowledge, only the method of reductio ad absurdum can help you. For if you choose the wrong turning, and find yourself in Folkestone, you may be quite sure you ought to have taken the other road.