Page:Mind (New Series) Volume 8.djvu/356

342 BOBEET LATTA: By this view that the infinitely little is the basis of the finite the older doctrine of limits is transcended. According to this negative doctrine of limits, an infinitely little differ- ence between two figures (say) is negligible. But if an infinitely little difference is negligible, it must be for some reason. Infinite littleness is a matter of degree. An in- finitely small quantity is a quantity less than any that can be assigned. But such a conception has no meaning unless we are speaking of an infinitely small thing or unity of dif- ferences, at the very least an infinitely small element in a numerical series which is not a bare addition or subtraction of homogeneous units but has some characteristic law of increment or decrement. It is the law or principle of the series, the nature or character of the whole, which enables us to say that the infinitely little difference may be neglected. Thus, adopting a phrase from Grandi, Leibniz writes to him in 1713 : " Infinite parva concipimus non ut nihila simpliciter et absolute, sed ut nihila respectiva (ut ipse bene notas), id est ut evanescentia quidem in nihilum, retinentia tamen characterem ejus quod evanescit ". l Accordingly, when it can be shown that two things ultimately " run into " one another or are continuous with one another, that is to say that the ultimate difference between them is infinitely little, it is presupposed that they are differences of a unity or that their difference is one of degree and not of kind. Thus the negative doctrine of limits im- plicitly presupposes a system within which its various objects are related, while the positive method, of which the fullest expression is to be found in the Calculus, explicitly recognises this system and regards the various objects or elements as necessarily determined by it. The method of limits was a true method so far as it went ; but it was inadequate because it did not think out its presuppositions. The advance that was made by Leibniz and his contemporaries consisted in investigating these presuppositions by inquiries (direct and indirect) into the true meaning of mathematical infinity. We are now in a position to consider the agreement and the difference between the scientific standpoint of Spinoza and that of Leibniz. The mathematics of Spinoza are the mathe- matics of Descartes. Spinoza is at the negative point of view implied in the method of limits, while Leibniz is at the positive point of view implied by the method of infinitesimals. In mathematics the method of limits is logically dependent upon the method of infinitesimals ; it assumes, without 1 Gerhardt, Leibniz's Math. Schriften, iv., 218. So also the conception of "infinities of infinity" is a favourite one with Leibniz, who frequently argues against the possibility of an absolute quantitative infinite.