332 B. RUSSELL : We may say, with M. Couturat, 1 that quantity is a wholly independent category, of which, just because it is a category, no account can be given in terms of other categories. Or we may say that quantity is an immediate datum in sense, which, like colours and sounds, can be indicated but not described. I shall discuss these two views successively. Taking the view that quantity is an independent category, we shall say : Quantity is distinct from and independent of quality ; its changes do not produce changes of quality, and it must be taken as immediately apprehended, without reduction to other terms. When we are given a foot-rule, we apprehend its quantity (length) along with it. All we can say of this quantity, beyond mere apprehension, con- sists, it is true, of comparison with other quantities. But this no longer constitutes an antinomy, since the quantity, ex hypothesi, is an irreducible property of our foot-rule. Let us seek, on this basis, for a characterisation of quantity. This will have to proceed entirely from the nature of quantitative comparison, since we have admitted that, apart from comparison, there is nothing to be said about quantity except that it is quantitative. We have, at the outset, a fundamental division of quanti- ties into two kinds, extensive and intensive, according as a change of quantity is, or is not, a quantity of the same kind as the quantity changed. A change of length is itself a length, but a change of temperature or illumination is not itself hot or bright. Of these two kinds, extensive quantity is the more important, and I shall consider it first. A quantity is extensive when a change in it is a quantity of the same kind. From this definition we can, I think, construct all the ordinary properties of extensive quantity. In the first place, we have addition (and subtraction). The original quantity, together with its change, is equal to the final quantity. When two quantities together are thus equal to a third, each of the two is said to be a part of the third. Thus an extensive quantity is always susceptible of division into extensive quantities, which are therefore in turn so divisible. This gives infinite divisibility. Again, since quantities may be increased as well as diminished, any extensive quantity may be regarded as part of a larger extensive quantity, and so on ad infinitum. This gives infinite quantity, as the negation of a limit to the growth of quantity. Again, by definition, all quantities of a kind are qualitatively alike ; therefore our collection of quantities 1 De Vlnfini Mathematique, Paris, Alcan, 1896, passim.
