So much for the objections which may be made to the proposed law of the forces. Let us next enquire into those made to the constitution of the primary elements, deduced from that law.
In the first place, there are many who can by no means be persuaded to admit the existence of points altogether indivisible and inextended, alleging that they cannot possibly form any idea of them. Because all the bodies cognisable by the senses are extended, we are apt to look upon extension as essential to mat ter. But this error may be corrected by reflection ; and the idea of an inextended point may be formed by the help of geometry, and that very idea of con tinued extension, which is so familiar to our senses. 'Thus the common section of two contiguous parts of a plane surface is a line, destitute of breadth, and the meeting of t•o such sections is an indivisible point.
We may here observe that Zeno among the an cients, among the moderns, held that the first elements of matter were simple and inextend ed ; •but they were guilty of an inconsistency in main taining these were contiguous to each other, and thereby comparing a continued extension of indivisible and inextended elements.
There are some who say that if the elements of matter are void of extension, they are in no respect different from .spirits. But the chief difference be tween body and spirit is, that the one is tangible and incapable of thought or volition, the other may think and will, but does not affect the -senses. For sensi bility does not consist in extension, but in impenetra bility, by which the fibres of the body are affected, and the rays of light are reflected.
But if substances capable of cogitation arid volition were endued with the same law of forces, would they not produce to our senses the -same effects as these points ? We answer„that it is not our business to en quire whethr such a conjunction could take place or not.
Such a body would neither be matter nor spirit, s but a thing differing from both ; from the one by its power of cogitation, as from the other by its inertia and impenetrability.
Application of the Theory to Mechanics.
The second department of our snbject is, the ap plication which may be. made of this theory to the explanation of the principal laws of equilibrium, and other parts of elementary mechanics. But, in the
first place, a few observations are to be premised re specting the curve of forces, upon which all the phe nomena depend. These observations relate to the arches of the curve, to the areas intercepted between it and the axis, and to the points in which the curve .cuts the axis.
The arches are either repulsive or attractive, ac cording as they lie on the side of the asymptotic arc EG, or on the opposite side. The arches may touch the axis, or they may bend from it with a con trary flexure, as P efgR, Fig. 1.
The area corresponding to any small portion of the axis may be ever so great, and that which cor responds to a great segment. may be ever so small, according as the curve recedes very far from the -axis, or approaches very near to it. It were easy to demonstrate this, but we shall not occupy the reader's • time with it. The area included between an asymp tote and any ordinate, may be either finite or infinite. The former, when the ordinate increases in a less ratio than the reciprocal simple ratio of the abscisses; the latter, when it increases in that or in a greater ratio, as may be thus proved. Take x the ordinate as A a, Fig. 1, and y the absciss ag, and let yn= 1 or y=x n , then the fluxion of the area yx will be = m ,„ nt X x and its fluent + A, or since x 22 --22: = y, we have — x y + A; A being a constant n-771 quantity. Since the area begins in A, the beginning of the abscisses, if n—ni be a positive number, and therefore the area will be finite, and A= 0 : But the area will be to the rectangle A a to n—ni; which rectangle, since ag may be great or small without limit, is also without limit. -Its value is infinite, if m=n, for then the divisor = 0; much more then if min, hat is, when the ordinate in creases in a greaterthan the reciprocal simple ratio of the abscisses. This observation was•necessary, that we: might have some scale df velocities in the access or recess of one point- from another.' For, aialready observed, -when the spaces are expressed by the ab scisses, and the forces by the' ordifiates, the area de scribed by the ordinate expresses the increment dr decrement of the square of the velocity.