Note 176 — Introductory material

In connection with the straight line, it will be convenient to discuss the conditions of a metrical coordinate system. The projective coordinate system, as we have seen, aims only at a convenient nomenclature for different points, and can be set up without introducing the notion of spatial quantity. But a metrical coordinate system does much more than this. It defines every point quantitatively, by its quantitative spatial relations to a certain coordinate figure. Only when the system of coordinates is thus metrical, i.e., when every coordinate represents some spatial magnitude, which is itself a relation of the point defined to some other point or figure—can operations with coordinates lead to a metrical result. When, as in projective Geometry, the coordinates are not spatial magnitudes, no amount of transformation can give a metrical result. I wish to prove, here, that a metrical coordinate system necessarily involves the straight line, and cannot, without a logical fallacy, be set up on any other basis. The projective system of coordinates, as we saw, is entirely based on the straight line; but the metrical system is more important, since its quantities embody actual information as to spatial magnitudes, which, in projective Geometry, is not the case.