A surface terminates a solid; a line terminates a surface; a point terminates a line; but I assert, that if the _ideas_ of a point, line, or surface, were not indivisible, 'tis impossible we should ever conceive these terminations. For let these ideas be supposed infinitely divisible, and then let the fancy endeavour to fix itself on the idea of the last surface, line, or point, it immediately finds this idea to break into parts; and upon its seizing the last of these parts it loses its hold by a new division, and so on _in infinitum_, without any possibility of its arriving at a concluding idea. The number of fractions bring it no nearer the last division than the first idea it formed. Every particle eludes the grasp by a new fraction, like quicksilver, when we endeavour to seize it. But as in fact there must be something which terminates the idea of every finite quantity, and as this terminating idea cannot itself consist of parts or inferior ideas, otherwise it would be the last of its parts, which finished the idea, and so on; this is a clear proof, that the ideas of surfaces, lines, and points, admit not of any division; those of surfaces in depth, of lines in breadth and depth, and of points in any dimension. The _schoolmen_ were so sensible of the force of this argument, that some of them maintained that nature has mixed among those particles of matter, which are divisible _in infinitum_, a number of mathematical points in order to give a termination to bodies; and others eluded the force of this reasoning by a heap of unintelligible cavils and distinctions. Both these adversaries equally yield the victory. A man who hides himself confesses as evidently the superiority of his enemy, as another, who fairly delivers his arms. Thus it appears, that the definitions of mathematics destroy the pretended demonstrations; and that if we have the idea of indivisible points, lines, and surfaces, conformable to the definition, their existence is certainly possible; but if we have no such idea, 'tis impossible we can ever conceive the termination of any figure, without which conception there can be no geometrical demonstration. But I go farther, and maintain, that none of these demonstrations can have sufficient weight to establish such a principle as this of infinite divisibility; and that because with regard to such minute objects, they are not properly demonstrations, being built on ideas which are not exact, and maxims which are not precisely true. When geometry decides any thing concerning the proportions of quantity, we ought not to look for the utmost _precision_ and exactness. None of its proofs extend so far: it takes the dimensions and proportions of figures justly; but roughly, and with some liberty. Its errors are never considerable, nor would it err at all, did it not aspire to such an absolute perfection.