Note 177 — Introductory material

The above three axioms, we have seen, are à priori necessary to metrical Geometry. No others can be necessary, since metrical systems, logically as unassailable as Euclid's, and dealing with spaces equally homogeneous and equally relational, have been constructed by the metageometers, without the help of any other axioms. The remaining axioms of Euclidean Geometry—the axiom of parallels, the axiom that the number of dimensions is three, and Euclid's form of the axiom of the straight line (two straight lines cannot enclose a space)—are not essential to the possibility of metrical Geometry, i.e., are not deducible from the fact that a science of spatial magnitudes is possible. They are rather to be regarded as empirical laws, obtained, like the empirical laws of other sciences, by actual investigation of the given subject-matter—in this instance, experienced space.