4. IF the body, instead of setting out in the line B C perpendicular to A B, had set out in another line B G more inclined towards the line A B, moving in the curve line B H; then as the body, if it were to continue its motion in the line B G, would for some time approach the center A; the centripetal force would cause it to make greater advances toward that center. But if the body were to set out in the line B I reclined the other way from the perpendicular B C, and were to be drawn by the centripetal force into the curve line B K; the body, notwithstanding any centripetal force, would for some time recede from the center; since some part at least of the curve line B K lies between the line B I and the perpendicular B C.
5. THUS far we have explained such effects, as attend every centripetal force. But as these forces may be very different in regard to the different degrees of strength, wherewith they act upon bodies in different places; I shall now proceed to make mention in general of some of the differences attending these centripetal motions.
6. TO reassume the consideration of the last mentioned case. Suppose a centripetal power directed toward the point A (in fig. 74.) to act on a body in B, which is moving in the direction of the straight line B C, the line B C reclining off from A B. If from A the straight lines A D, A E, A F are drawn at pleasure to the line C B; the line C B being prolonged beyond B to G, it appears that A D is inclined to the line G C more obliquely, than A B is inclined to it, A E is inclined more obliquely than A D, and A F more than A E. To speak more correctly, the angle under A D G is less than that under A B G, the angle under A E G less than that under A D G, and the angle under A F G less than that under A E G. Now suppose the body to move in the curve line B H I K. Then it is here likewise evident, that the line B H I K being concave towards A, and convex towards the line B C, it is more and more turned off from the line B C; so that in the point H the line A H will be less obliquely inclined to the curve line B H I K, than the same line A H D is inclined to B C at the point D; at the point I the inclination of the line A I to the curve line will be more different from the inclination of the same line A I E to the line B C, at the point E; and in the points K and F the difference of inclination will be still greater; and in both the inclination at the curve will be less oblique, than at the straight line B C. But the straight line A B is less obliquely inclined to B G, than A D is inclined towards D G: therefore although the line A H be less obliquely inclined towards the curve H B, than the same line A H D is inclined towards D G; yet it is possible, that the inclination at H may be more oblique, than the inclination at B. The inclination at H may indeed be less oblique than the other, or they may be both the same. This depends upon the degree of strength, wherewith the centripetal force exerts it self, during the passage of the body from B to H. After the same manner the inclinations at I and K depend entirely on the degree of strength, wherewith the centripetal force acts on the body in its passage from H to K: if the centripetal force be weak enough, the lines A H and A I drawn from the center A to the body at H and at I shall be more obliquely inclined to the curve, than the line A B is inclined towards B G. The centripetal force may be of that strength as to render all these inclinations equal, or if stronger, the inclinations at I and K will be less oblique than at B. Sir ~ISAAC NEWTON~ has particularly shewn, that if the centripetal power decreases after a certain manner with the increase of distance, a body may describe such a curve line, that all the lines drawn from the center to the body shall be equally inclined to that curve line.[82] But I do not here enter into any particulars, my present intention being only to shew, that it is possible for a body to be acted upon by a force continually drawing it down towards a center, and yet that the body shall continue to recede from that center; for here as long as the lines A H, A I, &c drawn from the center A to the body do not become less oblique to the curve, in which the body moves; so long shall those lines perpetually increase, and consequently the body shall more and more recede from the center.