Page:Mind (New Series) Volume 8.djvu/418

 404 CRITICAL NOTICES : conclusive. A fuller explanation of these points might, perhaps, make Mr. Russell's work more useful for philosophy. As it is it seems to be too mathematical for philosophers, even if it is not also too philosophical for mathematicians. The first point, on which I think a fuller explanation would be desirable, may be raised in connexion with the argument on page 85. Mr. Russell here asserts that ' There is no qualitatively similar unit in the three kinds of space, by which quantitative comparison can be effected '. 'A debt of 300 ' he goes on ' may be represented as the asset of - 300, and the height of the Eiffel Tower is + 300 metres ; but it does not follow that the two are quantitatively comparable.' This analogy seems to me at least unfortunate. For though - 300 is not quantitatively com- parable with + 300 metres, yet, in order that - 300 may have any meaning at all (and Mr. Russell has admitted that one measure of curvature may be regarded as negative), there must be some conceivable positive quantity, 300 with which - 300 is quantitatively comparable, and what this is, in the case of a negative measure of curvature, Mr. Russell does not explain. His argument would seem rather to exclude the possibility of any such ; and thus we should be left with a negative quantity which is a negative quantity of nothing. Moreover Mr. Russell repeatedly asserts that our actual space ' may have a very small space-constant ' (p. 115), and in order that any meaning may be given to this assertion, it would seem that one space-constant must be quantitatively comparable with another. Unless one denotes a smaller, and the other a larger quantity of some common quality, none could be ' very small '. That two cannot be suffered ' to coexist in the same world ' (p. 85) is surely utterly irrelevant, since we are quite well able to compare non-existent with existent magnitudes, provided only we know of what they are magnitudes. My next point may perhaps involve nothing but a verbal error. It concerns the statement made on page 159, with reference to the geometry of non-congruent surfaces, that ' The fundamental formula, that for the length of an infinitesimal arc, is only obtained on the assumption that such an arc may be treated as a straight line, and that Euclidean Plane Geometry may be applied in the immediate neighbourhood of any point '. But on page 19, where Mr. Russell is explaining how we can find a ' sense of the measure of curvature in which it can be extended to space,' it appears that the formula given involves a reference to ' small arcs '. Now if by ' small ' is here meant the same as by ' infini- tesimal ' in the former passage, it would seem either (1) that in any space which is defined by means of this formula ' Euclidean Plane Geometry may be applied in the immediate neighbourhood of any point,' or else (2) that on page 159 'the Plane Geometry of any congruent space ' should be substituted for Euclidean Plane Geometry. The former alternative would seem to contradict Mr.