=140.= 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. Section B. THE AXIOMS OF METRICAL GEOMETRY. =141.= We have now reviewed the axioms of projective Geometry, and have seen that they are _à priori_ deductions from the fact that we can experience externality, _i.e._ a coexistent multiplicity of different but interrelated things. But projective Geometry, in spite of its claims, is not the whole science of space, as is sufficiently proved by the fact that it cannot discriminate between Euclidean and non-Euclidean spaces[145]. For this purpose, spatial measurement is required: metrical Geometry, with its quantitative tests, can alone effect the discrimination. For all application of Geometry to physics, also, measurement is required; the law of gravitation, for example, requires the determination of actual distances. For many purposes, in short, projective Geometry is wholly insufficient: thus it is unable to distinguish between different kinds of conics, though their distinction is of fundamental importance in many departments of knowledge.