An Introduction to Analytical Geometry

Abstract

THIS is a continuation of an earlier volume published in 1940. The most original part of Vol. 2 is Chapter 19, on general geometry. This, with its references to G6, G6 and G7, is at first sight difficult to understand, but becomes clear and very interesting if we refer back to Chapter 8 of Vol. 1. The leading idea, expanded from a suggestion of the late Prof. G. H. Hardy, is that what is usually called analytical geometry of two dimensions is not one subject, but several. The simplest kind, called G2 corresponds closely to Euclidean geometry G1. G3 is similar to G2, but regards a ‘point' as merely a term for an ordered pair of real numbers x and y. In G4 these numbers are no longer real, but complex. In G5 we keep the pair real, and supplement them by a third real number z, introduced to make the equations homogeneous. Points at which z = 0 are called points of infinity. G6 has triplets of complex numbers, thus combining the advantages of G4 and G5. In both G5 and G6 the points of infinity and the line on which they lie are exceptional. In G7 all points and lines are on an equal footing. Theorems which are true in one kind of geometry may be untrue or even absurd in another. This discussion clears up the paradoxes which used to puzzle us concerning lines at right angles to themselves and separate points the distance apart of which is zero. However, the schoolboy, for whom the author professes to write, may not find this part of the book easy. An Introduction to Analytical Geometry By A. Robson. Vol. 2. Pp. viii + 215. (Cambridge: At the University Press, 1947.) 10s. 6d.