Note 150 — Introductory material

(1) How do we measure the equality of solids? These could only be brought into actual congruence if we had a fourth dimension to operate in[157], and from what I have said before of the absolute necessity of this test, it might seem as though we should be left here in utter ignorance. Euclid is silent on the subject, and in all works on Geometry it is assumed as self-evident that two cubes of equal side are equal. This assumption suggests that we are not so badly off as we should have been without congruence, as a test of equality in one or two dimensions; for now we can at least be sure that two cubes have all their sides and all their faces equal. Two such cubes differ, then, in no sensible spatial quality save position, for volume, in this case at any rate, is not a sensible quality. They are, therefore, as far as such qualities are concerned, indiscernible. If their places were interchanged, we might know the change by their colour, or by some other non-geometrical property; but so far as any property of which Geometry can take cognisance is concerned, everything would seem as before. To suppose a difference of volume, then, would be to ascribe an effect to mere position, which we saw to be inadmissible while discussing Free Mobility. Except as regards position, they are geometrically indiscernible, and we may call to our aid the Identity of Indiscernibles to establish their agreement in the one remaining geometrical property of volume. This may seem rather a strange principle to use in Mathematics, and for Geometry their equality is, perhaps, best regarded as a definition; but if we demand a philosophical ground for this definition, it is, I believe, only to be found in the Identity of Indiscernibles. We can, without error, make our definition of three-dimensional equality rest on two-dimensional congruence. For since direct comparison as to volume is impossible, we are at liberty to define two volumes as equal, when all their various lines, surfaces, angles and solid angles are congruent, since there remains, in such a case, no measurable difference between the figures composing the two volumes. Of course, as soon as we have established this one case of equality of volumes, the rest of the theory follows; as appears from the ordinary method of integrating volumes, by dividing them into small cubes.