Page:Mind (New Series) Volume 8.djvu/412

 398 CRITICAL NOTICES : geometry, since it makes it possible to define the quality which dis- tinguishes non-Euclidean spaces from the Euclidean as an intrinsic property of the space in question. The third period finally is marked by the invention of Projective Geometry, a department of mathe- matics not hitherto discussed by philosophers, but which Mr. Eussell considers to be of the utmost importance for the philosophy of space. He accordingly treats this period at greater length, endeavouring to show that Projective Geometry deals exclusively with the qualitative properties of space, and involves no reference to quantity. In this period the chief names are Cayley and Klein, the latter of whom invented a third system of Metageometry. Chapter ii. (pp. 54-116) contains a critical discussion of ' some previous philosophical theories of Geometry '. In this chapter Mr. Eussell contends that Kant's arguments suffice to prove the apriority of some form of externality, but are insufficient to determine which of the geometrical axioms are necessary to this ; and he points out that Lotze's attempted refutation of Meta- geometry is entirely based on mathematical mistakes. But by far the greater part of the chapter is occupied by a refutation of the views of Biemann, Helmholtz and Benno Erdmann, who wished to maintain that all the axioms peculiar to geometry, that is, all those that involve other conceptions than those of quantity, are empirical. Chapter iii. (pp. 117-177) is divided into two sections, in the first of which Mr. Russell attempts to formulate the axioms of Projective Geometry, and to deduce their necessity from the conception of a form of externality. He finds three axioms whicl are thus sufficient and a priori. In the second section he does the same for Metrical Geometry, finding the three corresponding a priori axioms, but emphasising the fact that for Metrics Geometry a fourth axiom is needed, which, however, he main- tains to be of an empirical nature. The a priori axioms, whethe in their projective or their metrical form, are involved in Euclideai and non-Euclidean Geometries alike. The fourth purely metric axiom serves to distinguish the one from the other. Chapter iv. (pp. 178-201) contains a short survey of ' Philc sophical Consequences '. The subjects discussed in it are twc The first of these is the general relation of a ' form of externality to 'experience,' in connexion with which Mr. Eussell attempts show that time alone is not sufficient for experience. The seconc subject of discussion is the contradictions in space, of which Mr. Eussell enumerates three, deciding that each of them may overcome if we interpret space as consisting merely of relations between unextended atoms that are in no sense in space. Mr. Eussell's general philosophical position does not seem verj much to affect the main value of his book. He does not indee discuss it at any great length, referring us constantly for a fuller treatment of important points to the Logics of Bradley anc Bosanquet. Only one philosophical distinction is absolutely