Page:Mind (New Series) Volume 8.djvu/411

 V. CRITICAL NOTICES. An Essay on the Foundations of Geometry. By BERTRAND A. W. EUSSELL, M.A., Fellow of Trinity College, Cambridge. Cambridge : At the University Press, 1897. Pp. xvi., 201. THIS book is in the main an attempt to determine the logical relations of the most elementary constituents of space. For this purpose Mr. Eussell makes an extensive use of the modern mathematical developments of Geometry ; and it is, therefore^ very difficult to criticise his results without a considerable know- ledge of Mathematics. This, I must confess, I do not possess ; and, therefore, any statements I may make, involving mathematical technicalities, must be understood to be liable to correction. It seems impossible, however, to avoid such statements, since the arguments by which Mr. Eussell endeavours to establish that certain axioms are a priori and others merely empirical always involve some reference to purely mathematical notions. And the chief value of the book undoubtedly lies in these discussions, since the purely philosophical questions as to how in general the a priori is to be distinguished from the empirical, and as to the relation of space to other objects of knowledge, are not treated at any great length ; while the treatment of them, such as it is, seems liable to grave objections. The book is divided into an introduction and four chapters. The introduction (pp. 1-6) defines the problem with particular reference to Kant, attempting to distinguish more clearly than he had done between the a priori and the subjective, and announcing that the former notion alone will be used in the following discussion. Chapter i. (pp. 7-53) is entitled ' A Short History of Meta- geometry '. In this Mr. Eussell distinguishes three periods. In the first of these, which is very briefly handled, it was proved by Lobatchewsky and Bolyai that a system of geometry, in which the axiom of parallels was denied, could be consistently worked out. In the second period the chief names are Eiemann and Helmholtz, whose object was largely philosophical. This period is marked by the use of analytical methods, particularly by the analytical conception of ' measure of curvature,' invented by Gauss, but first extended by Eiemann and Helmholtz to a mani- fold of more than two dimensions, and by the discovery of a second system of Metageometry. The conception of measure of curvature as developed in this period is all-important for Meta-