It cannot be admitted, however, that Locke has given a correct account of the origin of the notion of the quantitative infinite, or--to speak philosophically, and not after the use of terms convenient in mathematics--the indefinite. According to him, its origin is the mere empirical repeating of a sensuous datum of time and space. According to Leibniz, this repetition, however long continued, can give no idea beyond itself; it can never generate the idea that the process of repetition may be continued without a limit. Here, as elsewhere, he objects that experience cannot guarantee notions beyond the limits of experience. Locke's process of repetition could tell us that a number _had_ been extended up to a given point; not that it could be extended without limit. The source of this latter idea must be found, therefore, where we find the origin of all extra-empirical notions,--in reason. "Its origin is the same as that of universal and necessary truths." It is not the empirical process of multiplying, but the fact that the _same reason_ for multiplying always exists, that originates and guarantees the idea. "Take a straight line and prolong it in such a way that it is double the first. It is evident that the second, being perfectly _similar_ to the first, can be itself doubled; and we have a third, which in turn is _similar_ to the preceding. The _same reason_ always being present, it is not possible that the process should ever be brought to a stop. Thus the line can be prolonged 'to infinity.' Therefore the idea of 'infinity' comes from the consideration of the identity of relation or of reason." The considerations which we have grouped together in this chapter serve to show the fundamental philosophical difference between Locke and Leibniz. Although, taken in detail, they are self-explanatory, a few words may be permitted upon their unity and ultimate bearing. It is characteristic of Locke that he uses the same principle of explanation with reference to the conceptions of substance, identity and diversity, and infinity, and that this principle is that of spatial and temporal relation. Infinity is conceived as quantitative, as the successive addition of times and spaces; identity and diversity are oneness and difference of existence as determined by space and time; substance is the underlying static substratum of qualities, and, as such, is considered after the analogy of things existing in space and through time. It must not be forgotten that Locke believed as thoroughly as Leibniz in the substantial existence of the world, of the human soul, and of God; in the objective continuity of the world, and the personal identity of man, and in the true infinity of God. Whatever negative or sceptical inferences may have afterwards been drawn from Locke's premises were neither drawn nor dreamed of by him. His purpose was in essence one with that of Leibniz.