Note 34 — Introductory material

The view in question has arisen, it would seem, from a natural confusion as to the nature of the coordinates employed. Those who hold the view have not adequately realised, I believe, that their coordinates are not spatial quantities, as in metrical Geometry, but mere conventional signs, by which different points can be distinctly designated. There is no reason, therefore, until we already have metrical Geometry, for regarding one function of the coordinates as a better expression of distance than another, so long as the fundamental addition-equation[45] is preserved. Hence, if our coordinates are regarded as adequate for all Geometry, an indeterminateness arises in the expression of distance, which can only be avoided by a convention. But projective coordinates—so our argument will contend—though perfectly adequate for all projective properties, and entirely free from any metrical presupposition, are inadequate to express metrical properties, just because they have no metrical presupposition. Thus where metrical properties are in question, Beltrami remains justified as against Klein; the reduction of metrical to projective properties is only apparent, though the independence of these last, as against metrical Geometry, is perfectly real.