We may give the argument a more concrete, and perhaps a more convincing shape, by considering the matter arranged in our form. If two things are mutually external, they must since they belong to the same world, have some relation of externality; there is, therefore, a relation of externality between two things. But since our form is homogeneous, the same relation of externality may subsist in other parts of the form, i.e. while the two things considered alter their relations of externality to other things. The relation of externality between two things is, therefore, independent of other things. Hence, when we return to the abstract language of the form, two positions have a relation determined by those two positions alone, and independent of other positions.
Precisely the same argument applies to the relations of three positions, and in each case the relation must appear in the form as not a mere inference from the positions it relates. For relations, as we have seen, actually constitute a form of externality, and are not mere inferences from terms, which are nowhere to be found in the form[144].
To sum up: Since position is relative, two positions must have some relation to each other; and since our form of externality is homogeneous, this relation can be kept unchanged while the two positions change their relations to other positions. Hence their relation is intrinsic, and independent of other positions. Since the form is a mere complex of relations, the relation in question must, if the form is sensuous or intuitive, be itself sensuous or intuitive, and not a mere inference. In this case, a unique relation must be a unique figure—in spatial terms, the straight line joining the two points.
139. With this, our deduction of projective Geometry from the à priori conceptual properties of a form of externality is completed. That such a form, when regarded as an independent thing, is self-contradictory, has been abundantly evident throughout the discussion. But the science of the form has been founded on the opposite way of regarding it: we have held it throughout to be a mere complex of relations, and have deduced its properties exclusively from this view of it. The many difficulties, in applying such an à priori deduction to intuitive space, and in explaining, as logical necessities, properties which appear as sensuous or intuitional data, must be postponed to Chapter IV. For the present, I wish to point out that projective Geometry is wholly à priori; that it deals with an object whose properties are logically deduced from its definition, not empirically discovered from data; that its definition, again, is founded on the possibility of experiencing diversity in relation, or multiplicity in unity; and that our whole science, therefore, is logically implied in, and deducible from, the possibility of such experience.
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.
Metrical Geometry is, then, a necessary part of the science of space, and a part not included in descriptive Geometry. Its à priori element, nevertheless, so far as this is spatial and not arithmetical, is the same as the postulate of projective Geometry, namely, the homogeneity of space, or its equivalent, the relativity of position. We can see, in fact, that the à priori element in both is likely to be the same. For the à priori in metrical Geometry will be whatever is presupposed in the possibility of spatial measurement, i.e. of quantitative spatial comparison. But such comparison presupposes simply a known identity of quality, the determination of which is precisely the problem of projective Geometry. Hence the conditions for the possibility of measurement, in so far as they are not arithmetical, will be precisely the same as those for projective Geometry.