Note 57 — Chapter I · A SHORT HISTORY OF METAGEOMETRY.

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.