Note 140 — Introductory material

In metrical Geometry, on the contrary, we shall find a very different result. Although the geometrical conditions which render spatial measurement possible, will be found identical, except for slight differences in the form of statement, with the à priori axioms discussed above, yet the actual measurement—which deals with actually given space, not the mere intellectual construction we have been just discussing—gives results which can only be known empirically and approximately, and can be deduced by no necessity of thought. The Euclidean and non-Euclidean spaces give the various results which are à priori possible; the axioms peculiar to Euclid—which are properly not axioms, but empirical results of measurement—determine, within the errors of observation, which of these à priori possibilities is realized in our actual space. Thus measurement deals throughout with an empirically given matter, not with a creature of the intellect, and its à priori elements are only the conditions presupposed in the possibility of measurement. What these conditions are, we shall see in the second section of this chapter.