Note 142 — Introductory material

Metrical Geometry, therefore, though distinct from projective Geometry, is not independent of it, but presupposes it, and arises from its combination with the extraneous idea of quantity. Nevertheless the mathematical form of the axioms, in metrical Geometry, is slightly different from their form in projective Geometry. The homogeneity of space is replaced by its equivalent, the axiom of Free Mobility. The axiom of the straight line is replaced by the axiom of distance: Two points determine a unique quantity, distance, which is unaltered in any motion of the two points as a single figure. This axiom, indeed, will be found to involve the axiom of the straight line—such a quantity could not exist unless the two points determined a unique curve—but its mathematical form is changed. Another important change is the collapse of the principle of duality: quantity can be applied to the straight line, because it is divisible into similar parts, but cannot be applied to the indivisible point. We thus obtain a reason, which was wanting in descriptive Geometry, for preferring points, as spatial elements, to straight lines or planes[146]. Finally, an entirely new idea is introduced with quantity, namely, the idea of Motion. Not that we study motion, or that any of our results have reference to motion, but that they cannot, though in projective Geometry they could, be obtained without at least an ideal motion of our figures through space.