Page:The World and the Individual, First Series (1899).djvu/579

560 you have upon your hands a varying quantity (call it X) which, consistently with these terms, you are able to make, or to assume, as large as you please. In such cases, if some one else is supposed to have predesignated, as the value of X, any definite magnitude that he pleases, say X1 then you are at liberty, under the conditions of the problem, to assume the value of X as larger still, i.e. as greater than any such previously assigned definite value X1. Now, whenever the variable X has this character, in a given problem, then, according to the fashion of speech used in the Calculus, you may define X either simply as infinite, or as capable of being increased to infinity; and in the Calculus you are indeed often enough interested in learning what happens to some quantity whose value depends upon X, when X thus increases without limit, or, as they briefly say, becomes infinite. But in all such cases the term infinite, as used in the modern text-books of the Calculus, is, by definition, simply an abbreviation for the whole conception just defined. The variable X need not even be, at any moment, actually at all large in order to be, in this sense, infinite. It only so varies that, consistently with the conditions of the problem, it can be made larger than a predesignated value, whatever that value may be. And the Calculus is simply often interested in computing the consequences of such a manner of variation on the part of X.

Now, unquestionably a quantity that is called infinite in this sense is not the actually infinite against which Aristotle argued. It is merely the limitlessly increasing variable or the potentially infinite magnitude which he willingly admitted as a valid conception. A parallel definition of the infinitesimal is even more frequently employed in the modern text-books of the Calculus, just because the infinitesimal is mentioned more frequently than the infinite. In this sense, a variable magnitude is infinitesimal merely when it can be made and kept as small as we will, consistently with the conditions of the problem in which it appears. Thus neither the infinite nor the infinitesimal of the modern treatment of the Calculus has any