UNEXTENDED POINTS.
166. There are two strong arguments against the existence of unextended points: the first is, that we must suppose them infinite in number, for otherwise it does not seem possible to arrive at the simple, starting from the extended: the second is, that even supposing them infinite in number they are incapable of producing extension. These arguments are so powerful as to excuse all the aberrations of the contrary opinion, which, however strange they may seem, are not more strange than the simple forming extension, and the smallest portion of matter containing an infinite number of parts.
167. It does not seem possible to arrive at unextended points unless by an infinite division. The unextended is zero in the order of extension, and in order to arrive at zero by a decreasing geometrical progression it must be continuedad infinitum. Mathematical calculation presents a sensible image of this. When two parts are united they must have a side where they touch, and another wherethey are not in contact. If we separate the interior side from the exterior we have two new sides, one which touches and another which does not. Continuing the division the same thing happens again; we must, therefore, pass through an infinite series in order to arrive at the unextended, which is equivalent to saying that we shall never arrive there. To continue the divisionad infinitumwe must suppose infinite parts, and consequently the existence of an actual infinite number. From the moment that we suppose this infinite number to exist it seems to become finite, since we already see a limit to the division, and also other numbers greater than it. Let us suppose that this infinite number of parts is found in a cubic inch; there are numbers which are greater than this which we suppose infinite; a cubic foot, for example, will contain 1,728 times the infinite number of parts contained in the cubic inch.
Thus the opinion of unextended points seeking to avoid infinite division, runs into it; just as its adversaries trying to escape from unextended points are forced to acknowledge their existence. The imagination loses itself and the understanding is confused.
168. The other objection is not less unanswerable. Suppose we have arrived at unextended points, how shall we reconstitute extension? The unextended has no dimensions; therefore, no matter how many unextended points we may take, we can never form extension with them. Let us imagine two points to be united, as neither of them alone occupies any place, neither will they both together. We cannot say that they penetrate each other; for penetration cannot exist without extension. We must admit that these parts being zero in the order of extension, their sum can never give extension, no matter how many of them we may add together.
169. It is certain that a sum of zeros can give only zerofor the result, but mathematicians admit that there are certain expressions equal to zero, which multiplied by an infinite quantity will give a finite quantity for the product. 0 + 0 + 0 + 0 + N × 0 = 0; but if we take 0/M = 0, and multiply it by the expression M/0 = 0, we shall have (0/M) × (M/0) = (0 × M)/(M × 0) = 0/0 which is equal to any finite quantity, which we may express by A. This is shown by the principles of elementary algebra only; if we pass to the transcendental we have dz/dx = o/o = B; B expressing the differential coefficient which may be equal to a finite value. Can these mathematical doctrines serve to explain the generation of the extended from unextended points? I think not.
It is evident that, multiplication being only addition shortened, if an infinite addition of zeros can give only zero; multiplication can give no other result, although the other factor be infinite. Why then do mathematical results say the contrary? This contradiction is not true, but only apparent. In the multiplication of the infinitesimal by the infinite we may obtain a finite quantity for product, because the infinitesimal is not regarded as a true zero, but as a quantity less than all imaginable quantities, but still it is something. If this condition were wanting, all the operations would be absurd, because they would turn upon a pure nothing. Shall we therefore say that the equation, dz/dx = o/o, is only approximate? No; for it expresses the relation of the limit of the decrement, which is equal to B only when the differentials are equal to zero. But as geometricians only consider the limit in itself, they passover all the intervals of the decrement, and place themselves at once at the point of true exactness. Why then operate on these quantities? Because the operations are a sort of algebraic language, and mark the course that has been followed in the calculations, and recall the connection of the limit with the quantity to which it refers.
170. Unity which is not number produces number; why then cannot points without extension produce extension? There is a great disparity between the two cases. The unextended, as such, involves only the negative idea of extension; but in unity, although number is denied, this negation does not constitute its nature. No one ever defined unity to be the negation of number, yet we always define the unextended to be that which has no extension. Unity is any being taken in general, without considering its divisibility; number is a collection of unities; therefore the idea of number involves the idea of unity, of anundividedbeing, number being nothing more than the repetition of this unity. It belongs to the essence of all number that it can be resolved into unity; it contains unity in a determinate manner. But the extended can not be resolved into the unextended, unless by proceedingad infinitum, or else by some process of decomposition which we know nothing of.