An Essay on the Foundations of Geometry

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 = a sin 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. [47] Elements of Projective Geometry, Second Edition, Oxford, 1893, Chap. IX. [48] Chap. III. Section B. [49] See Nicht-Euklid, I. p. 338 ff. [50] See his Geometrie der Lage, § 8, Harmonische Gebilde. [51] The anharmonic ratio of four numbers, p, q, r, s, is defined as (p - q).(r - s) / (p - r).(q - s). [52] I.e. as transformable into each other by a collineation. See Chap. III. Sec. A, § 110. [53] See Chap. III. Sec. A. [54] It follows from this, that the reduction of metrical to projective properties, even when, as in hyperbolic Geometry, the Absolute is real, is only apparent, and has a merely technical validity. [55] Sir R. Ball does not regard his non-Euclidean content as a possible space (v. op. cit. p. 151). In this important point I disagree with his interpretation, holding such a content to be a space as possible, à priori, as Euclid's, and perhaps actually true within the margin due to errors of observation. [56] See Nicht-Euklid, I. p. 97 ff. and p. 292 ff. [57] Newcomb says (loc. cit. p. 293): "The system here set forth is founded on the following three postulates. "1. I assume that space is triply extended, unbounded, without properties dependent either on position or direction, and possessing such planeness in its smallest parts that both the postulates of the Euclidean Geometry, and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space. "2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2D without losing, in any part of its course, that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it. "3. I assume that if two right lines emanate from the same point, making the indefinitely small angle a with each other, their distance apart at the distance r from the point of intersection will be given by the equation s = 2aD π sin rπ 2D . The right line thus has this property in common with the Euclidean right line that two such lines intersect only in a single point. It may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature, but not being able so to determine it, I assume as a postulate the fundamental property of the Euclidean right line." It is plain that in the absence of the determination spoken of, the possibility of elliptic space is not established. It may be possible, for example, to prove that, in a space where there is a maximum to distance, there must be an infinite number of straight lines joining two points of maximum distance. In this event, elliptic space would become impossible. [58] For an elucidation of this term, see Klein, Nicht-Euklid, I. p. 99 ff. [59] Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom, in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, but which is the representation lying at the bottom of all external experience." [60] The exception to this axiom, in spherical space, presupposes metrical Geometry, and does not destroy the validity of the axiom for projective Geometry. See Chap. III. Sec. B, § 171. [61] Mathematicians of Lie's school have a habit, at first somewhat confusing, of speaking of motions of space instead of motions of bodies, as though space as a whole could move. All that is meant is, of course, the equivalent motion of the coordinate axes, i.e. a change of axes in the usual elementary sense. [62] "Ueber die Grundlagen der Geometrie," Leipziger Berichte, 1890. The problem of these two papers is really metrical, since it is concerned, not with collineations in general, but with motions. The problem, however, is dealt with by the projective method, motions being regarded as collineations which leave the Absolute unchanged. It seemed impossible, therefore, to discuss Lie's work, until some account had been given of the projective method. [63] Lie's premisses, to be accurate, are the following: Let x1 = f (x, y, z, a1, a2...) x2 = φ (x, y, z, a1, a2...) x3 = ψ (x, y, z, a1, a2...) give an infinite family of real transformations of space, as to which we make the following hypotheses: A. The functions f, φ, ψ, are analytical functions of x, y, z, a1, a2.... B. Two points x1y1z1, x2y2z2 possess an invariant, i.e. Ω(x1, y1, z1, x2, y2, z2) = Ω(x1′, y1′, z1′, x2′, y2′, z2′) where x1′..., x2′..., are the transformed coordinates of the two points. C. Free Mobility: i.e., any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞2 positions; when two points are fixed, any other of general position can take up ∞1 positions; when three, no motion is possible—these limitations being results of the equations given by the invariant Ω. [64] On this point, cf. Klein, Höhere Geometrie, Göttingen, 1893, II. pp. 225–244, especially pp. 230–1. [65] Axiom II. of the metrical triad corresponds to Axiom III. of the projective, and vice versâ. [66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: "Die Bearbeiter der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie behauptet."