Page:Mind (New Series) Volume 8.djvu/419

B. A. W. RUSSELL, Essay on the Foundations of Geometry. 405 Eussell's assertion of the qualitative dissimilarity of Euclidean and non-Euclidean spaces. Again I find a difficulty in what seems to be implied on page 111, in the argument against M. Delboeuf, that in non-Euclidean spaces the space-constant may have a definite ratio to other magnitudes in the same space. If this is the case, it would seem it must be possible to measure the magnitude of the space- constant against other magnitudes in the same space ; and measurement, as Mr. Russell often tells us, presupposes the straight line. But the straight line in any space means not that which is in common between Euclidean and non-Euclidean spaces, but the straight line of that space ; and hence we should be meas- uring our space-constant by means of that which presupposes it. Or else, if by the straight line presupposed in measurement is meant the common quality of Euclidean and non-Euclidean straight lines it would seem that we could only use it in measurement if the things measured are nothing but a certain quantity of it ; but in that case the space-constant of one space would differ only quantitatively from that of another, and there seems no room for the qualitative difference between spaces which Mr. Eussell asserts to be the only one. The conclusion which suggests itself to rne from these con- siderations, is that all spatial magnitudes can only be regarded as magnitudes of some one quality. If we are to allow that the straight line has two qualities, one that is in common between Euclidean and non-Euclidean spaces, and another that is found in any given non-Euclidean space, then it seems that measure- ment must be impossible in such a space, since we have two measures of magnitude, which are consequently not comparable inter se. Measurement seems to presuppose only one measure of magnitude, and hence the truth of Euclid would be involved in the possibility of measurement, and so would be a priori in the same sense as the axiom of ' Free Mobility ' : since ' in Metageometry we have, while in Euclid we have not, a standard of comparison involved in the nature of our space as a whole ' (p. 111). That the axiom of parallels is logically independent of the other axioms would seem to make nothing against this argument, since, if Mr. Eussell's three a priori axioms differ from one another at all, they likewise must be logically independent of one another, in the sense that some part, at least, of one may be denied, without compelling us, on pain of contradiction, to deny the others also. G. E. MOORE.