=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.