Note 200 — Introductory material

(2) The circle in the definition of straight lines and planes. This difficulty need not long detain us, since we have already, with the material atom, broken through the relativity which caused our circle. Straight lines, in the purely geometrical procedure, are defined only by points, and points only by straight lines. But points, now, are replaced by material atoms: the duality of points and lines, therefore, has disappeared, and the straight line may be defined as the spatial relation between two unextended atoms. These atoms have spatial adjectives, derived from their relations to other atoms; but they have no intrinsic spatial adjectives, such as could belong to them if they had extension or figure. Thus straight lines and planes are the true spatial units, and points result only from the attempt to find, within space, those terms for spatial relations which exist only in a more than spatial matter. Straight lines, planes and volumes are the spatial relations between two, three or four unextended atoms, and points are a merely convenient geometrical fiction, by which possible atoms are replaced. For, since space, as we saw, is a possibility, Geometry deals not with actually realized spatial relations, but with the whole scheme of possible relations.