[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 x{1} = f(x, y, z, a{1}, a{2}...) x{2} = φ(x, y, z, a{1}, a{2}...) x{3} = ψ(x, y, z, a{1}, a{2}...) 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, a{1}, a{2}.... B. Two points x{1}y{1}z{1}, x{2}y{2}z{2} possess an invariant, _i.e._ Ω(x{1}, y{1}, z{1}, x{2}, y{2}, z{2}) = Ω(x{1′}, y{1′}, z{1′}, x{2′}, y{2′}, z{2′}) where x{1′}..., x{2′}..., 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." CHAPTER II. CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.