=124.= Now when we consider what is involved in such absolute qualitative equivalence, we find at once, as its most obvious prerequisite, the perfect homogeneity of space. For it is assumed that a figure can be completely defined by its internal relations, and that the external relations, which constitute its position, though they suffice to distinguish it from other figures, in no way affect its internal properties, which are regarded as qualitatively identical with those of figures with quite different external relations. If this were not the case, anything analogous to projective transformation would be impossible. For such transformation always alters the position, _i.e._ the external relations, of a figure, and could not, therefore, if figures were dependent on their relations to other figures or to empty space, be studied without reference to other figures, or to the absolute position of the original figure. We require for our principle, in short, what may be called the mutual passivity and reciprocal independence of two parts or figures of space. This passivity and this independence involve the homogeneity of space, or its equivalent, the relativity of position. For if the internal properties of a figure are the same, whatever its external relations may be, it follows that all parts of space are qualitatively similar, since a change of external relation is a change in the part of space occupied. It follows, also, that all position is relative and extrinsic, _i.e._, that the position of a point, or the part of space occupied by a figure, is not, and has no effect upon, any intrinsic property of the point or figure, but is exclusively a relation to other points or figures in space, and remains without effect except where such relations are considered. =125.= The homogeneity of space and the relativity of position, therefore, are presupposed in the qualitative spatial comparison with which projective Geometry deals. The latter, as we saw, is also the basis of the principle of duality. But these properties, as I shall now endeavour to prove, belong of necessity to any form of externality, and are thus _à priori_ properties of all possible spaces. To prove this, however, we must first define the notion of a form of externality in general. Let us observe, to begin with, that the distinction between Euclidean and non-Euclidean Geometries, so important in metrical investigations, disappears in projective Geometry proper. This suggests that projective Geometry, though originally invented as the science of Euclidean space, and subsequently of non-Euclidean spaces also, deals really with a wider conception, a conception which includes both, and neglects the attributes in which they differ. This conception I shall speak of as a form of externality.