[173] In Frischauf's "Absolute Geometrie nach Johann Bolyai," Anhang, there is a series of definitions, starting from the sphere, as the locus of congruent point-pairs when one point of the pair is fixed, and hence obtaining the circle and the straight line. From the above it follows, that the sphere so defined already involves a curve between the points of the point-pair, by which various point-pairs can be known as congruent; and it will appear, as we proceed, that this curve must be a straight line. Frischauf's definition by means of the sphere involves, therefore, a vicious circle, since the sphere presupposes the straight line, as the test of congruent point-pairs. [174] Nor in any argument which, like those of projective Geometry, avoids the notion of magnitude or distance altogether. It follows that the propositions of projective Geometry apply, without reserve, to spherical space, since the exception to the axiom of the straight line arises only on metrical ground. [175] Psychology, Vol. II. pp. 149-150. [176] This step in the argument has been put very briefly, since it is a mere repetition of the corresponding argument in Section A, and is inserted here only for the sake of logical completeness. See § 137 ff. [177] Cf. Hannequin, Essai critique sur l'hypothèse des atomes, Paris, 1895, passim. CHAPTER IV. PHILOSOPHICAL CONSEQUENCES. =180.= In the present chapter, we have to discuss two questions which, though scarcely geometrical, are of fundamental importance to the theory of Geometry propounded above. The first of these questions is this: What relation can a purely logical and deductive proof, like that from the nature of a form of externality, bear to an experienced subject-matter such as space? You have merely framed a general conception, I may be told, containing space as a particular species, and you have then shown, what should have been obvious from the beginning, that this general conception contained some of the attributes of space. But what ground does this give for regarding these attributes as _à priori_? The conception Mammal has some of the attributes of a horse; but are these attributes therefore _à priori_ adjectives of the horse? The answer to this obvious objection is so difficult, and involves so much general philosophy, that I have kept it for a final chapter, in order not to interrupt the argument on specially geometrical topics.