Geometry of Four Dimensions

Abstract

PERHAPS the main interest of this work is that it treats the subject in a way that is comparatively new, and one that is likely enough to be generally adopted. Until quite recently, most works on hypergeometry might be roughly divided into three classes: popular or semi-popular outlines, which, however stimulating or suggestive, have little or no scientific value; frankly analytical disquisitions, such as those of Riemann, etc.; and works which, although couched in geometrical language, give the impression of being, so to speak, translations of previous analytical demonstrations. It must be admitted, of course, that some authors (such as Segre) have obtained new and valuable results for surfaces in three dimensions by considering them as sections pi hyper-surfaces, and have pursued other four-dimensional researches in a way which has much more the aspect of being purely geometrical. But since it is a psychological fact that so far we have no true intuition of four-dimensional space, the inference seems to be that these authors have become so familiar with the analytical arguments underlying their theorems that they pass without an effort to the corresponding geometrical form of statement; much in the same way as dualisation of a projective theorem becomes almost mechanical after sufficient practice.

Geometry of Four Dimensions.

By Prof. H. P. Manning. Pp. ix + 348. (New York: The Macmillan Co.; London: Macmillan and Co., Ltd., 1914.) Price 8s. 6d. net.