We cannot know, it is true, what _psychological_ theory of space-perception would apply to such beings: they might have a sense different from any of ours, and they might have no sense in any way resembling ours, but yet their Geometry would have points of resemblance to ours, as that of the blind coincides with that of the seeing. If space has any objective counterpart whatever, in short, and if any inference is possible, as Lotze holds it to be, from space to its counterpart, then a converse argument is also possible, though it may give some only of the qualities of Euclidean space, since some only of these qualities may be found to have a necessary analogue in the counterpart. =87.= Admitting, then, in Lotze's sense, the subjectivity of space, the above possibility does not seem so empty as he imagines. He discusses it briefly, however, in order to pass on to what he regards as the real meaning of Metageometry. In this he is guilty of a mathematical mistake, which causes much irrelevant reasoning. For he believes that Metageometry constructs its spaces out of straight lines and angles in all respects similar to Euclid's, whence he derives an easy victory in proving that these elements can lead only to the one space. In this he has been misled by the phraseology of non-Euclideans, as well as by Euclid's separation of definitions and axioms. For the fact is, of course, that straight lines are only fully defined when we add to the formal definition the axioms of the straight line and of parallels. Within Euclidean space, Euclid's definition suffices to distinguish the straight line from all other curves; the two axioms referred to are then absorbed into the definition of space. But apart from the restriction to Euclidean space, the definition has to be supplemented by the two axioms, in order to define completely the Euclidean straight line. Thus Lotze has misconceived the bearing of non-Euclidean constructions, and has simply missed the point in arguing as he does. The possibility contemplated by a non-Euclidean, if it fell under any of Lotze's cases, would fall under the second case discussed above. =88.= But the bearing of Metageometry is really, I think, different from anything imagined by Lotze; and as few writers seem clear on this point, I will enter somewhat fully into what I conceive to be its purpose. In the first place, there are some writers--notably Clifford--who, being naïve realists as regards space, hold that our evidence is wholly insufficient, as yet, to decide as to its nature in the infinite or in the infinitesimal (cf. Essays, Vol. I. p. 320): these writers are not concerned with any possibility of beings different from ourselves, but simply with the everyday space we know, which they investigate in the spirit of a chemist discussing whether hydrogen is a metal, or an astronomer discussing the nebular hypothesis.