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330 B. KUSSELL : such a case '? For the ideal application of number, we require concepts embodied in a series of really diverse in- stances. Everything relevant must be supposed known about the instances before numeration begins, for this gives no fresh information as to any of them. The only connexion between them implied in numeration is the purely formal connexion involved in their being thought of simultaneously, and there is therefore only a purely formal reaction of the numerical whole on the instances which constitute it. But though no information is derived, by counting, as to the instances, complete information is derived as to the whole, since the numerical whole is nothing but the mere aggregate of the instances. In order, however, that information as to the whole may result from numeration, the instances must be definite and distinct. The ideal, for number, would be Leibnitzian independent monads. As soon as our unit ceases to be definite, and ceases to be known independently of all numerical considerations, counting ceases to give definite information. To be told that a town consists of 10,000 souls is real information, for a soul is a natural unit, concerning which most of us profess considerable knowledge ; but to be told that a foot consists of twelve inches is no in- formation, for a man might study space for ever without discovering the inch as a natural unit. Number, in short, demands a unit not itself numerical, a unit to whose nature number is wholly external. We require a subject-matter with real divisions. Where there are several ways of making these divisions, as e.g. into pounds and shillings, the number to be applied begins to be arbitrary, and to give only incomplete information as to the whole. Where the divisions are wholly fictitious, as with continua, the number is wholly arbitrary, and the information it gives is nil. Number is now only available for measurement, i.e., for comparison of one quantity with another. The unit of measurement, the inch or the foot, is already a quantity. Comparison with another quantity, therefore, which is all that number can effect, involves, if taken as exhausting the nature of quantitj 7 , an obvious vicious circle. Quantity must be investigated so it would seem by analysis of the unit itself ; it must be an intrinsic property of our arbitrary unit, and not a particular or extreme case of number. From this conclusion, if quantity is to have any meaning, there would seem to be no escape. But if we adopt this view, new obstacles, no less formid- able than those which led us to declare quantity independent of number, bar our further progress. Quantity, we have