Thus the relativity of space, while it is the essence of the principle of duality, at the same time renders impossible the expression of that principle, or of any other principle of pure Geometry, in a manner which shall be free from contradictions. Nevertheless, if we are to advance at all with our analysis of geometrical reasoning and with our definitions of lines and points, we must, for a while, ignore this contradiction; we must argue as though it did not exist, so as to free our science from any contradictions which are not inevitable.
117. In accordance with this procedure, then, let us define our points as the terms of spatial relations, regarding whatever is not a point as a relation between points. What, on this view, must our points be taken to be? Obviously, if extension is mere relativity, they must be taken to contain no extension; but if they are to supply the terms for spatial relations, e.g. for straight lines, these relations must exhibit them as the terms of the figures they relate. In other words, since what can really be taken, without contradiction, as the term of a spatial relation, is unextended, we must take, as the term to be used in Geometry, where we cannot go outside space, the least spatial thing which Geometry can deal with, the thing which, though in space, contains no space; and this thing we define as the point[127].
Neglecting, then, the fundamental contradiction in this definition, the rest of our definitions follow without difficulty. The straight line is the relation between two points, and the plane is the relation between three. These definitions will be argued and defended at length in section B of this Chapter[128], where we can discuss at the same time the alternative metrical definitions; for our present purpose, it is sufficient to observe that projective Geometry, from the first, regards the straight line as determined by two points, and the plane as determined by three, from which it follows, if we take points as possible terms for spatial relations, that the straight line and the plane may be regarded as relations between two and three points respectively. If we agree on these definitions, we can proceed to discuss the fundamental principle of projective Geometry, and to analyse the axioms implicated in its truth.
118. Projective Geometry, we have seen, does not deal with quantity, and therefore recognizes no difference where the difference is purely quantitative. Now quantitative comparison depends on a recognized identity of quality; the recognition of qualitative identity, therefore, is logically prior to quantity, and presupposed by every judgment of quantity. Hence all figures, whose differences can be exhaustively described by quantity, i.e. by pure measurement, must have an identity of quality, and this must be recognizable without appeal to quantity. It follows that, by defining the word quality in geometrical matters, we shall discover what sets of figures are projectively indiscernible. If our definition is correct, it ought to yield the general projective principle with which we set out.
119. We agreed to regard points as the terms of spatial relations, and we agreed that different points could be distinguished. But we postponed the discussion of the conditions under which this distinction could be effected. This discussion will yield us the definition of quality and the proof of our general projective principle.
Points, to begin with, have been defined as nothing but the terms for spatial relations. They have, therefore, no intrinsic properties; but are distinguished solely by means of their relations. Now the relation between two points, we said, is the straight line on which they lie. This gives that identity of quality for all pairs of points on the same straight line, which is required both by our projective principle and by metrical Geometry. (For only where there is identity of quality can quantity be properly applied.) If only two points are given, they cannot, without the use of quantity, be distinguished from any two other points on the same straight line; for the qualitative relation between any two such points is the same as for the original pair, and only by a difference of relation can points be distinguished from one another.
But conversely, one straight line is nothing but the relation between two of its points, and all points are qualitatively alike. Hence there can be nothing to distinguish one straight line from another except the points through which it passes, and these are distinguished from other points only by the fact that it passes through them. Thus we get the reciprocal transformation: if we are given only one point, any pair of straight lines through that point is qualitatively indistinguishable from any other. This again is, on the one hand, the basis of the second part of our general projective principle, and on the other hand the condition of applying quantity, in the measurement of angles, to the departure of two intersecting straight lines.
120. We can now see the reason for what may have hitherto seemed a somewhat arbitrary fact, namely, the necessity of four collinear points for anharmonic ratio. Recurring to the quadrilateral construction and the consequent introduction of number, we see that anharmonic ratio is an intrinsic projective relation of four collinear points or concurrent straight lines, such that given three terms and the relation, the fourth term can be uniquely determined by projective methods. Now consider first a pair of points. Since all straight lines are projectively equivalent, the relation between one pair of points is precisely equivalent to that between another pair. Given one point only, therefore, no projective relation, to any second point, can be assigned, which shall in any way limit our choice of the second point. Given two points, however, there is such a relation—the third point may be given collinear with the first two. This limits its position to one straight line, but since two points determine nothing but one straight line, the third point cannot be further limited. Thus we see why no intrinsic projective relation can be found between three points, which shall enable us, from two, uniquely to determine the third. With three given collinear points, however, we have more given than a mere straight line, and the quadrilateral construction enables us uniquely to determine any number of fresh collinear points. This shows why anharmonic ratio must be a relation between four points, rather than between three.
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.