But what is this possibility? A thing is possible, according to Bradley (Logic, p. 187), when it would follow from a certain number of conditions, some of which are known to be realized. Now the conditions to which a form of externality must conform, in order to be affirmed, are: first, of course, that it should be experienced, or legitimately inferred from something experienced; but secondly, that it should conform to certain logical conditions, detailed in Chapter III., which may be summed up in the relativity of position. Now what Metageometry has done, in any case, is to suggest the proof that the second of these conditions is fulfilled by non-Euclidean spaces. Euclid is affirmed, therefore, on the ground of immediate experience alone, and his truth, as unmediated by logical necessity, is merely assertorical, or, if we prefer it, empirical. This is the most important sense, it seems to me, in which non-Euclidean spaces are possible. They are, in short, a step in a philosophical argument, rather than in the investigation of fact: they throw light on the nature of the grounds for Euclid, rather than on the actual conformation of space[102]. This import of Metageometry is denied by Lotze, on the ground that non-Euclidean logic is faulty, a ground which he endeavours, by much detail and through many pages, to make good--with what success, we will now proceed to examine. =90.= Lotze's attack on Metageometry--although it remains, so far as I know, the best hostile criticism extant, and although its arguments have become part of the regular stock-in-trade of Euclidean philosophers--contains, if I am not mistaken, several misunderstandings due to insufficient mathematical knowledge of the subject. As these misunderstandings have been widely spread among philosophers, and cannot be easily removed except by a critic who has gone into non-Euclidean Geometry with some care, it seems desirable to discuss Lotze's strictures point by point. [Illustration]