| The length is inseparable from the breadth both in nature and in our minds; but this excludes not a partial consideration, and a distinction of reason, after the manner above explained. |
| In refuting this answer I shall not insist on the argument, which I have already sufficiently explained, that if it be impossible for the mind to arrive at a minimum in its ideas, its capacity must be infinite, in order to comprehend the infinite number of parts, of which its idea of any extension would be composed. |
| I shall here endeavour to find some new absurdities in this reasoning. |
| 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, it is 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 school 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, it is 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 anything 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. |
| I first ask mathematicians, what they mean when they say one line or surface is EQUAL to, or GREATER or LESS than another? Let any of them give an answer, to whatever sect he belongs, and whether he maintains the composition of extension by indivisible points, or by quantities divisible in infinitum. |
| This question will embarrass both of them. |
| There are few or no mathematicians, who defend the hypothesis of indivisible points; and yet these have the readiest and justest answer to the present question. |
| They need only reply, that lines or surfaces are equal, when the numbers of points in each are equal; and that as the proportion of the numbers varies, the proportion of the lines and surfaces is also varyed. |
| But though this answer be just, as well as obvious; yet I may affirm, that this standard of equality is entirely useless, and that it never is from such a comparison we determine objects to be equal or unequal with respect to each other. |