ABSTRACT
In descriptive Geometry, we start, as before, with points, and as before, any two points determine a class of points. But this class now consists only of the points between the two given points. What is to be understood by between is not explained by any writer on this subject except Vailati, in terms of a transitive asymmetrical relation of two points; and Vailati’s explanation is condemned by Peano,* on the ground that between is a relation of three points, not of two only. This ground, as we know from Part IV, is inadequate and even irrelevant. But on the subject of relations, even the best mathematicians go astray, for want, I think, of familiarity with the Logic of Relations. In the present case, as in that of projective Geometry, we may start either with a relation of two points, or with a relation between a pair and a class of points: either method is equally legitimate, and leads to the same results, but the former is far simpler. Let us examine first the method of Pasch and Peano, then that of Vailati.
