=121.= We can now prove, I think, that two figures, which are projectively related, are qualitatively similar. Let us begin with a collection of points on a straight line. So long as these are considered without reference to other points or figures, they are all qualitatively similar. They can be distinguished by immediate intuition, but when we endeavour, without quantity, to distinguish them conceptually, we find the task impossible, since the only qualitative relation of any two of them, the straight line, is the same for any other two. But now let us choose, at hap-hazard, some point outside the straight line. The points of our line now acquire new adjectives, namely their relations to the new point, _i.e._ the straight lines joining them to this new point. But these straight lines, reciprocally, alone define our external point, and all straight lines are qualitatively similar. If we take some other external point, therefore, and join it to the same points of our original straight line, we obtain a figure in which, so long as quantity is excluded, there is no conceptual difference from the former figure. Immediate intuition can distinguish the two figures, but qualitative discrimination cannot do so. Thus we obtain a projective transformation of four lines into four other lines, as giving a figure qualitatively indistinguishable from the original figure. A similar argument applies to the other projective transformations. Thus the only reason, within projective Geometry, for not regarding projective figures as actually identical, is the intuitive perception of difference of position. This is fundamental, and must be accepted as a _datum_. It is presupposed in the distinction of various points, and forms the very life of Geometry. It is, in fact, the essence of the notion of a form of externality, which notion forms the subject-matter of projective Geometry. =122.= We may now sum up the results of our analysis of projective Geometry, and state the axioms on which its reasoning is based. We shall then have to prove that these axioms are necessary to any form of externality, with which we shall pass, from mere analysis, to a transcendental argument. 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].