[7] Klein's first account of elliptic Geometry, as a result of Cayley's projective theory of distance, appeared in two articles entitled "Ueber die sogenannte Nicht-Euklidische Geometrie, I, II," Math. Annalen 4, 6 (1871-2). It was afterwards independently discovered by Newcomb, in an article entitled "Elementary Theorems relating to the geometry of a space of three dimensions, and of uniform positive curvature in the fourth dimension," Crelle's Journal für die reine und angewandte Mathematik, Vol. 83 (1877). For an account of the mathematical controversies concerning elliptic Geometry, see Klein's "Vorlesungen über Nicht-Euklidische Geometrie," Göttingen 1893, I. p. 284 ff. A bibliography of the relevant literature up to the year 1878 was given by Halsted in the American Journal of Mathematics, Vols. 1, 2. [8] Veronese (op. cit. p. 638) denies the priority of Gauss in the invention of a non-Euclidean system, though he admits him to have been the first to regard the axiom of parallels as indemonstrable. His grounds for the former assertion seem scarcely adequate: on the evidence against it, see Klein, Nicht-Euklid, I. pp. 171-174. [9] V. Briefwechsel mit Schumacher, Bd. II. p. 268. [10] f. Helmholtz, Wiss. Abh. II. p. 611. [11] Crelle's Journal, 1837. [12] Theorie der Parallellinien, Berlin, 1840. Republished, Berlin, 1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892. [13] Frischauf, Absolute Geometrie, nach Johann Bolyai, Leipzig, 1872. Halsted, The Science Absolute of Space, translated from the Latin, 4th edition, Austin, Texas, U.S.A. 1896. [14] Both Lobatchewsky and Bolyai, as Veronese remarks, start rather from the point-pair than from distance. See Frischauf, Absolute Geometrie, Anhang. [15] Compare Stallo, Concepts of Modern Physics, p. 248. [16] Gesammelte Werke, pp. 255-268. [17] On the history of this word, see Stallo, Concepts of Modern Physics, p. 258. It was used by Kant, and adapted by Herbart to almost the same meaning as it bears in Riemann. Herbart, however, also uses the word _Reihenform_ to express a similar idea. See Psychologie als Wissenschaft, I. § 100 and II. § 139, where Riemann's analogy with colours is also suggested. [18] Compare Erdmann's "Grössenbegriff vom Raum." [19] Compare Veronese, op. cit. p. 642: "Riemann ist in seiner Definition des Begriffs Grösse dunkel." See also Veronese's whole following criticism. [20] Vorträge und Reden, Vol. II. p. 18. [21] Cf. Klein, Nicht-Euklid, I. p. 160. [22] Since we are considering the curvature at a point, we are only concerned with the first infinitesimal elements of the geodesics that start from such a point. [23] Disquisitiones generales circa superficies curvas, Werke, Bd. IV. SS. 219-258, 1827.