27. IF at the point B, whence the body sets out, the curve line return into it self (as in fig. 82;) then the body, upon its arrival again at B, may return into its former course, and thus make an endless circuit about the center of the centripetal power.
28. WHAT has here been said, I hope, will in some measure enable my readers to form a just idea of the nature of these centripetal motions.
29. I HAVE not attempted to shew, how to find particularly, what kind of centripetal force is necessary to carry a body in any curve line proposed. This is to be deduced from the degree of curvature, which the figure has in each point of it, and requires a long and complex mathematical reasoning. However I shall speak a little to the first proportion, which Sir ~ISAAC NEWTON~ lays down for this purpose. By this proposition, when a body is found moving in a curve line, it may be known, whether the body be kept in its course by a power always pointed toward the same center; and if it be so, where that center is placed. The proposition is this: that if a line be drawn from some fixed point to the body, and remaining by one extream united to that point, it be carried round along with the body; then, if the power, whereby the body is kept in its course, be always pointed to this fixed point as a center, this line will move over equal spaces in equal portions of time. Suppose a body were moving through the curve line A B C D (in fig. 84.) and passed over the arches A B, B C, C D in equal portions of time; then if a point, as E, can be found, from whence the line E A being drawn to the body in A, and accompanying the body in its motion, it shall make the spaces E A B, E B C, and E C D equal, over which it passes, while the body describes the arches A B, B C, and C D: and if this hold the same in all other arches, both great and small, of the curve line A B C D, that these spaces are always equal, where the times are equal; then is the body kept in this line by a power always pointed to E as a center.
30. THE principle, upon which Sir ~ISAAC NEWTON~ has demonstrated this, requires but small skill in geometry to comprehend. I shall therefore take the liberty to close the present chapter with an explication of it; because such an example will give the clearest notion of our author’s method of applying mathematical reasoning to these philosophical subjects.