[24] Nevertheless, the Geometries of different surfaces of equal curvature are liable to important differences. For example, the cylinder is a surface of zero curvature, but since its lines of curvature in one direction are finite, its Geometry coincides with that of the plane only for lengths smaller than the circumference of its generating circle (see Veronese, op. cit. p. 644). Two geodesics on a cylinder may meet in many points. For surfaces of zero curvature on which this is not possible, the identity with the plane may be allowed to stand. Otherwise, the identity extends only to the properties of figures not exceeding a certain size. [25] For we may consider two different parts of the same surface as corresponding parts of different surfaces; the above proposition then shows that a figure can be reproduced in one part when it has been drawn in another, if the measures of curvature correspond in the two parts. [26] Crelle, Vols, XIX., XX., 1839-40. [27] In this formula, _u_, _v_ may be the lengths of lines, or the angles between lines, drawn on the surface, and having thus no necessary reference to a third dimension. [28] In what follows, I have given rather Klein's exposition of Riemann, than Riemann's own account. The former is much clearer and fuller, and not substantially different in any way. V. Klein, Nicht-Euklid, I. pp. 206 ff. [29] See §§ 69-73. [30] Grundlagen der Geometrie, I. and II., Leipziger Berichte, 1890; v. end of present chapter, § 45. [31] Nicht-Euklid, I. pp. 258-9. [32] Giornale di Matematiche, Vol. VI., 1868. Translated into French by J. Hoüel in the "Annales Scientifiques de l'École Normale Supérieure," Vol. VI. 1869. [33] Crelle's Journal, Vols. XIX. XX., 1839-40. [34] Nicht-Euklid, I. p. 190. [35] This article is more trigonometrical and analytical than the German book, and therefore makes the above interpretation peculiarly evident. [36] Such surfaces are by no means particularly remote. One of them, for example, is formed by the revolution of the common Tractrix x = asin φ, y = a(log tan φ/2 + cos φ). [37] "Teoria fondamentale degli spazii di curvatura costanta," Annali di Matematica, II. Vol. 2, 1868-9. Also translated by J. Hoüel, _loc. cit._ [38] See Klein, Nicht-Euklid, I. p. 47 ff., and the references there given. [39] See quotation below, from his British Association Address. [40] Compare the opening sentence, due to Cayley, of Salmon's Higher Plane Curves. [41] V. Nicht-Euklid, I. Chaps. I. and II. [42] See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a quotation from Klein in Erdmann's Axiome der Geometrie, p. 124 note. [43] Nature, Vol. XLV. p. 407. [44] Nicht-Euklid, I. p. 200. [45] I.e. the equation _AB_ + _BC_ = _AC_, for three points in one straight line. [46] The formula substituted by Klein for Cayley's inverse sine or cosine. The two are equivalent, but Klein's is mathematically much the more convenient.