Finally, we discussed the question of absolute magnitude, and found in it no logical obstacle to non-Euclidean spaces. Our conclusion, then, in so far as we are as yet entitled to a conclusion, is that all spaces with a space-constant are _à priori_ justifiable, and that the decision between them must be the work of experience. Spaces without a space-constant, on the other hand, spaces, that is, which are not homogeneous throughout, we found logically unsound and impossible to know, and therefore to be condemned _à priori_. The constructive proof of this thesis will form the argument of the following chapter. FOOTNOTES: [67] The Critical Philosophy of Kant, Vol. I. p. 287. [68] For a discussion of Kant from a less purely mathematical standpoint, see Chap. IV. [69] Cf. Vaihinger's Commentar, II. pp. 202, 265. Also p. 336 ff. [70] E.g. second edition, p. 39: "So werden auch alle geometrischen Grundsätze, z. B. dass in einem Triangel zwei Seiten zusammen grösser sind als die dritte, niemals aus allgemeinen Begriffen von Linie und Triangel, sondern aus der Anschauung, und zwar _à priori_ mit apodiktischer Gewissheit abgeleitet." [71] Cf. Bradley's Logic, Bk. III. Pt. I. Chap. VI.; Bosanquet's Logic, Bk. I. Chap. I. pp. 97-103. [72] Philosophie de la Règle et du Compas, Année Philosophique, II. pp. 1-66. [73] I have stated this doctrine dogmatically, as a proof would require a whole treatise on Logic. I accept the proofs offered by Bradley and Bosanquet, to which the reader is referred. [74] For a further discussion of this point, see Chaps. III. and IV. [75] See Chap. IV. for a discussion of this argument. [76] See Chap. IV. § 185. [77] An Otherness of substance, rather than of attribute, is here intended; an Otherness which may perhaps be called real as opposed to logical diversity. [78] This proposition will be argued at length in Chap. IV. [79] See Psychologie als Wissenschaft, I. Section III. Chap. VII.; II. Section I. Chap. III. and Section II. Chap. III. Compare also Synechologie, Section I. Chaps. II. and III. [80] On the influence of Herbart on Riemann, compare Erdmann, Die Axiome der Geometrie, p. 30. [81] I do not mean that measurement of colours is effected without reference to their relations, since all measurement is essentially comparison. But in colours, it is the elements which are compared, while in space, it is the relations between elements. [82] For a discussion of this point, see Chap. III. Sec. B, § 176.