[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.]
[83] The works of Helmholtz on geometrical philosophy comprise, in addition to the articles quoted in Chap. I., the following articles: "Ursprung und Sinn der geometrischen Axiome, gegen Land," Wiss. Abh. Vol. II. p. 640, 1878. (Also Mind, Vol. III.: an answer to Land in Mind, Vol. II.) "Ursprung und Bedeutung der geometrischen Axiome," 1870, Vorträge und Reden, Vol. II. p. 1. (Also Mind, Vol. I.) Two Appendices to "Die Thatsachen in der Wahrnehmung," entitled: II. "Der Raum kann transcendental sein, ohne dass es die Axiome sind"; and III. "Die Anwendbarkeit der Axiome auf die physische Welt," 1878, Vorträge und Reden, Vol. II. p. 256 ff.
The two Appendices last mentioned are popularizings and expansions of the article in Mind, Vol. III. The most widely read, though also, to my mind, the least valuable, of all Helmholtz's writings on Geometry, is the article in Mind, Vol. I. This contains the famous and much misunderstood analogies of Flatland and Sphereland, which will be discussed, and as far as possible defended, in answering Lotze's attack on Metageometry—an attack based, apparently, almost entirely on this one popular article. The present discussion, therefore, may be confined almost entirely to Mind, Vol. III., and the philosophical portions of the two papers quoted in Chap. I., i.e. to the articles in Wiss. Abh. Vol. II. pp. 610–660. His other works are popular, and important only because of the large public to which they appeal.
[84] In the answer to Land, Mind, Vol. III. and Wiss. Abh. II. p. 640.
[85] See also Die Thatsachen in der Wahrnehmung, Zusatz II., Der Raum kann transcendental sein, ohne dass es die Axiome sind. Vorträge und Reden, Vol. II.