25. THUS we have found, how a body may be carried round the figure A B C D E by the action of certain impulses upon it which should all be pointed toward one center. And we likewise see, that when the body is brought back again to the point, whence it first set out; if it there meet with an impulse sufficient to turn it again into the line, wherein it moved at first, its original velocity will be again restored; and by the repetition of the same impulses, the body will be carried again in the same round. Therefore if these impulses, which act on the body at the points B, C, D, E, and A, continue always the same, the body will make round this figure innumerable revolutions. 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.