(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.]
[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