26. The proof, which we have here made use of, holds the same in any number of straight lines, whereof the figure A B D should be composed; and therefore by the method of reasoning referred to above[91] we are to conclude, that what has here been said upon this rectilinear figure, will remain true, if this figure were changed into one of a continued curvature, and instead of distinct impulses acting by intervals at the angles of this figure, we had a continual centripetal force. We have therefore shewn, that a body may be carried round in any curve figure A B C ( fig. 82.) which shall every where be concave towards any one point as D, by the continual action of a centripetal power directed to that point, and when it is returned to the point, from whence it set out, it shall recover again the velocity, with which it departed from that point. It is not indeed always necessary, that it should return again into its first course; for the curve line may have some such figure as the line A B C D B E in fig. 83. In this curve line, if the body set out from B in the direction B F, and moved through the line B C D, till it returned to B; here the body would not enter again into the line B C D, because the two parts B D and B C of the curve line make an angle at the point B: so that the centripetal power, which at the point B could turn the body from the line B F into the curve, will not be able to turn the body into the line B C from the direction, in which it returns to the point B; a forceable impulse must be given the body in the point B to produce that effect.
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.
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.
32. Let E F be drawn parallel to C D. Then, from what has been said upon the second law of motion[92], it is evident, that since the body was moving in the line A B, when it received the impulse in the direction D C; it will have moved after that impulse through the line D E in the same time, as it would have taken up in moving through D F, provided it had received no disturbance in D. But the time of the body’s moving from D to E is supposed to be equal to the time of its moving through A D; therefore the time, which the body would have imployed in moving through D F, had it not been disturbed in D, is equal to the time, wherein it moved through A D: consequently D F is equal in length to A D; for if the body had gone on to move through the line A B without interruption, it would have moved through all parts thereof with the same velocity, and have passed over equal parts of that line in equal portions of time. Now C F being drawn, since A D and D F are equal, the triangular space C D F is equal to the triangular space C A D. Farther, the line E F being parallel to C D, it is proved by Euclid, that the triangle C E D is equal to the triangle C F D[93]: therefore the triangle C E D is equal to the triangle C A D.
33. After the same manner, if the body receive at E another impulse directed toward the point C, and be turned by that impulse into the line E G; if it move afterwards from E to G in the same space of time, as was taken up by its motion from D to E, or from A to D; then C G being drawn, the triangle C E G is equal to C D E. A third impulse at G directed as the two former to C, whereby the body shall be turned into the line G H, will have also the like effect with the rest. If the body move over G H in the same time, as it took up in moving over E G, the triangle C G H will be equal to the triangle C E G. Lastly, if the body at H be turned by a fresh impulse directed toward C into the line H I, and at I by another impulse directed also to C be turned into the line I K; and if the body move over each of the lines H I, and I K in the same time, as it imployed in moving over each of the preceding lines A D, D E, E G, and G H: then each of the triangles C H I, and C I K will be equal to each of the preceding. Likewise as the time, in which the body moves over A D E, is equal to the time of its moving over E G H, and to the time of its moving over H I K; the space C A D E will be equal to the space C E G H, and to the space C H I K. In the same manner as the time, in which the body moved over A D E G is equal to the time of its moving over G H I K, so the space C A D E G will be equal to the space C G H I K.
34. From this principle Sir Isaac Newton demonstrates the proposition mentioned above, by that method of arguing introduced by him into geometry, whereof we have before taken notice[94], by making according to the principles of that method a transition from this incurvated figure composed of straight lines, to a figure of continued curvature; and by shewing, that since equal spaces are described in equal times in this present figure composed of straight lines, the same relation between the spaces described and the times of their description will also have place in a figure of one continued curvature. He also deduces from this proposition the reverse of it; and proves, that whenever equal spaces are continually described; the body is acted upon by a centripetal force directed to the center, at which the spaces terminate.