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. 31. HE reasons thus. Suppose a body set out from the point A (in fig. 85.) to move in the straight line A B; and after it had moved for some time in that line, it were to receive an impulse directed to some point as C. Let it receive that impulse at D; and thereby be turned into the line D E; and let the body after this impulse take the same length of time in passing from D to E, as it imployed in the passing from A to D. Then the straight lines C A, C D, and C E being drawn, Sir ~ISAAC NEWTON~ proves, that the and triangular spaces C A D and C D E are equal. This he does in the following manner.