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B. A. w. RUSSELL, Essay on the Foundations of Geometry. 403 other things. As it is, we certainly have discovered it : we do distinguish different positions, as Mr. Kussell says, in ' intuition ' ; and to assign a cause for the existence of this intuition does not impugn but rather presupposes its validity. The assertion of absolute position in this sense obviously does not involve ' an action of mere space, per se, on things,' which Mr. Russell seems to regard as equivalent to it (p. 151). And thus the necessary connexion which he asserts between the axiom of free mobility and the relativity of position disappears. The assertion of this connexion seems, indeed, to be partly due to the way in which this first of Mr. Russell's a priori axioms is stated a form which is indicated by the name. For Mr. Russell conceives this axiom, as distinguished from the corresponding projective axiom of the homogeneity of space, to introduce ' an entirely new idea,' ' namely, the idea of Motion ' ; for, he says, our results cannot be obtained ' without at least an ideal motion of our figures through space ' (p. 149). It seems desirable to protest against this conception of the movement of spatial figures. It is implied in the view accepted by Mr. Russell, that super- position is the criterion of equality between spatial figures. But, in fact, ' the ideal motion ' by which superposition is affected implies that we already know the figures in question to be equal. For to move a spatial figure can mean nothing more than to construct a figure equal to it in another part of space ; and this can not be done, unless we know immediately when two figures in different parts of space are equal. This, therefore, ' that there are equal figures in all parts of space,' would seem to be the true form of the so-called ' Axiom of Free Mobility,' which is presupposed alike in the motion and in the measurement of things which occupy space. To conceive spatial figures as them- selves moving, we must conceive them as things, whose identity in motion means no more than that they successively occupy equal spaces. For the equality of these spaces themselves there is then no possible criterion. Its necessity is proved by the fact that it is presupposed alike in the assumed equality or inequality of any things whatsoever that can occupy space. The only other point in Mr. Russell's book on which I desire to touch is the possibility of non-Euclidean spaces. Mr. Russell agrees ' with Helmholtz in thinking the distinction between Euclidean and non-Euclidean spaces empirical ' (p. 73). His arguments in favour of this contention seem to involve a consider- able knowledge of mathematics, and yet his result is philosophical. Philosophers therefore would seem to be placed in an awkward dilemma : either they must accept the result merely on the author's authority a proceeding not generally to be recommended in philosophy ; or else they must be liable to be told that their criticisms are vitiated by mere mistakes in mathematics. In view of this dilemma, I shall only venture to mention some points which make Mr. Russell's arguments appear to me to be in-