Page:Mind (New Series) Volume 6.djvu/345

ON THE RELATIONS OF NUMBER AND QUANTITY. 329 tent, but only to objects having a certain property. This property is more complex than that required for negative numbers and fractions, but equally possible. 1 Thus through- out the extensions of number, the truly numerical element is nothing but the positive integers everything else arises from abandoning that complete indifference to the pro- perties of the unit in which pure number consists. So far, however, the properties implied remain perfectly abstract, and our object, being still posited exclusively by thought, is still subject to the laws of thought alone. Throughout this abstract development of number, the unit has become gradually more important and explicit. When it is quite explicit, we get a third kind of number, which, outside our arithmetic books, is alone of impor- tance. This third kind I shall call applied number. In this, there is a reference to a definite unit, and the ap- plied number is always to be regarded as the product of the corresponding pure number with the unit in question. Here we find the justification, in the properties of particular units, for the extension of numbers beyond the positive integers. The unit is some concept, and the number is formed by instances of the concept. Numeration can, of course, be applied to concrete things only by abstraction, for no concrete thing is merely an instance of a concept no man in a set of statistics, for example, is a mere man. But for us here, the important point is that the result, even in its abstraction, can only pretend to accuracy where nature gives a unit of some sort ; where our concept, in other words, finds its extension in a ready-made series. But in most of the applications of number, this is not the case. The matter to which number is to be applied, wherever this consists of sense-data as opposed to conceptions, contains no ready-made divisions, but has to be divided artificially in order to manufacture a unit. Any portion of such a matter is called a quantity. The result of applying number to this quantity I call magnitude, and the operation of apply- ing number I call numerical measurement ; 2 the matter itself is a continuum. But to what extent does number give us information in 1 It is important to observe that irrational numbers do not appear in this development. They arise only in connexion with quantity, and can- not be validly treated by Arithmetic, since their relation to the unit cannot be expressed conceptually. 2 1 shall generally use measure to denote only comparison as to the more or less.