The axioms which have been assumed in the above analysis, and which, it would seem, suffice to found projective Geometry, may be roughly stated as follows:
I. We can distinguish different parts of space, but all parts are qualitatively similar, and are distinguished only by the immediate fact that they lie outside one another.
II. Space is continuous and infinitely divisible; the result of infinite division, the zero of extension, is called a point[129].
III. Any two points determine a unique figure, called a straight line, any three in general determine a unique figure, the plane. Any four determine a corresponding figure of three dimensions, and for aught that appears to the contrary, the same may be true of any number of points. But this process comes to an end, sooner or later, with some number of points which determine the whole of space. For if this were not the case, no number of relations of a point to a collection of given points could ever determine its relation to fresh points, and Geometry would become impossible[130].
This statement of the axioms is not intended to have any exclusive precision: other statements equally valid could easily be made. For all these axioms, as we shall see hereafter, are philosophically interdependent, and may, therefore, be enunciated in many ways. The above statement, however, includes, if I am not mistaken, everything essential to projective Geometry, and everything required to prove the principle of projective transformation. Before discussing the apriority of these axioms, let us once more briefly recapitulate the ends which they are intended to attain.
123. From the exclusively mathematical standpoint, as we have seen, projective Geometry discusses only what figures can be obtained from each other by projective transformations, i.e. by the operations of projection and section. These operations, in all their forms, presuppose the point, straight line, and plane[131], whose necessity for projective Geometry, from the purely mathematical point of view, is thus self-evident from the start. But philosophically, projective Geometry has, as we saw, a wider aim. This wider aim, which gives, to the investigation of projectively equivalent figures, its chief importance, consists in the determination of qualitative spatial similarity, in the determination, that is, of all the figures which, when any one figure is given, can be distinguished from the given figure, so long as quantity is excluded, only by the mere fact that they are external to it.
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.