A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

2. These powers or forces are by Sir Isaac Newton called centripetal; and their first effect is to cause the body, on which they act, to quit the straight course, wherein it would proceed if undisturbed, and to describe an incurvated line, which shall always be bent towards the center of the force. It is not necessary, that such a power should cause the body to approach that center. The body may continue to recede from the center of the power, notwithstanding its being drawn by the power; but this property must always belong to its motion, that the line, in which it moves, will continually be concave towards the center, to which the power is directed. Suppose A (in fig. 72.) to be the center of a force. Let a body in B be moving in the direction of the straight line B C, in which line it would continue to move, if undisturbed; but being attracted by the centripetal force towards A, the body must necessarily depart from this line B C, and being drawn into the curve line B D, must pass between the lines A B and B C. It is evident therefore, that the body in B being gradually turned off from the straight line B C, it will at first be convex toward the line B C, and consequently concave towards the point A: for these centripetal powers are supposed to be in strength proportional to the power of gravity, and, like that, not to be able after the manner of an impulse to turn the body sensibly out of its course into a different one in an instant, but to take up some space of time in producing a visible effect. That the curve will always continue to have its concavity towards A may thus appear. In the line B C near to B take any point as E, from which the line E F G may be so drawn, as to touch the curve line B D in some point as F. Now when the body is come to F, if the centripetal power were immediately to be suspended, the body would no longer continue to move in a curve line, but being left to it self would forthwith reassume a straight course; and that straight course would be in the line F G: for that line is in the direction of the body’s motion at the point F. But the centripetal force continuing its energy, the body will be gradually drawn from this line F G so as to keep in the line F D, and make that line near the point F to be convex toward F G, and concave toward A. After the same manner the body may be followed on in its course through the line B D, and every part of that line be shewn to be concave toward the point A.

3. This then is the constant character belonging to those motions, which are carried on by centripetal forces; that the line, wherein the body moves, is throughout concave towards the center of the force. In respect to the successive distances of the body from the center there is no general rule to be laid down; for the distance of the body from the center may either increase, or decrease, or even keep always the same. The point A (in fig. 73.) being the center of a centripetal force, let a body at B set out in the direction of the straight line B C perpendicular to the line A B drawn from A to B. It will be easily conceived, that there is no other point in the line B C so near to A, as the point B; that A B is the shortest of all the lines, which can be drawn from A to any part of the line B C; all other lines, as A D, or A E, drawn from A to the line B C being longer than A B. Hence it follows, that the body setting out from B, if it moved in the line B C, it would recede more and more from the point A. Now as the operation of a centripetal force is to draw a body towards the center of the force: if such a force act upon a resting body, it must necessarily put that body so into motion, as to cause it to move towards the center of the force: if the body were of it self moving towards that center, the centripetal force would accelerate that motion, and cause it to move faster down: but if the body were in such a motion, as being left to itself it would recede from this center, it is not necessary, that the action of a centripetal power upon it should immediately compel the body to approach the center, from which it would otherwise have receded; the centripetal power is not without effect, if it cause the body to recede more slowly from that center, than otherwise it would have done. Thus in the case before us, the smallest centripetal power, if it act on the body, will force it out of the line B C, and cause it to pass in a bent line between B C and the point A, as has been before explained. When the body, for instance, has advanced to the line A D, the effect of the centripetal force discovers it self by having removed the body out of the line B C, and brought it to cross the line A D somewhere between A and D: suppose at F. Now A D being longer than A B, A F may also be longer than A B. The centripetal power may indeed be so strong, that A F shall be shorter than A B; or it may be so evenly balanced with the progressive motion of the body, that A F and A B shall be just equal: and in this last case, when the centripetal force is of that strength, as constantly to draw the body as much toward the center, as the progressive motion would carry it off, the body will describe a circle about the center A, this center of the force being also the center of the circle.

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.

7. But we may observe farther, that if the centripetal power, while the body increases its distance from the center, retain sufficient strength to make the lines drawn from the center to the body to become at length less oblique to the curve; then if this diminution of the obliquity continue, till at last the line drawn from the center to the body shall cease to be obliquely inclined to the curve, and shall become perpendicular thereto; from this instant the body shall no longer recede from the center, but in its following motion it shall again descend, and shall describe a curve line in all respects like to that, which it has described already; provided the centripetal power, every where at the same distance from the center, acts with the same strength. So we observed in the preceding chapter, that, when the motion of a projectile became parallel to the horizon, the projectile no longer ascended, but forthwith directed its course downwards, descending in a line altogether like that, wherein it had before ascended[83].

8. This return of the body may be proved by the following proposition: that if the body in any place, suppose at I, were to be stopt, and be thrown directly backward with the velocity, wherewith it was moving forward in that point I; then the body, by the action of the centripetal force upon it, would move back again over the path I H B, in which it had before advanced forward, and would arrive again at the point B in the same space of time, as was taken up in its passage from B to I; the velocity of the body at its return to the point B being the same, as that wherewith it first set out from that point. To give a full demonstration of this proposition, would require that use of mathematics, which I here purpose to avoid; but, I believe, it will appear in great measure evident from the following considerations.

9. Suppose (in fig. 75.) that a body were carried after the following manner through the bent figure A B C D E F, composed of the straight lines A B, B C, C D, D E, E F. First let it be moving in the line A B, from A towards B, with any uniform velocity. At B let the body receive an impulse directed toward some point, as G, taken within the concavity of the figure. Now whereas this body, when once moving in the straight line A B, will continue to move on in this line, so long as it shall be left to it self; but being disturbed at the point B in its motion by the impulse, which there acts upon it, it will be turned out of this line A B into some other straight line, wherein it will afterwards continue to move, as long as it shall be left to itself. Therefore let this impulse have strength sufficient to turn the body into the line B C. Then let the body move on undisturbed from B to C, but at C let it receive another impulse pointed toward the same point G, and of sufficient strength to turn the body into the line C D. At D let a third impulse, directed like the rest to the point G, turn the body into the line D E. And at E let another impulse, directed likewise to the point G, turn the body into the line E F. Now, I say, if the body while moving in the line E F be stopt, and turned back again in this line with the same velocity, as that wherewith it was moving forward in this line; then by the repetition of the former impulse at E the body will be turned into the line E D, and move in it from E to D with the same velocity as before it moved with from D to E; by the repetition of the impulse at D, when the body shall have returned to that point, it will be turned into the line D C; and by the repetition of the other impulses at C and B the body will be brought back again into the line B A, with the velocity, wherewith it first moved in that line.

10. This I prove as follows. Let D E and F E be continued beyond E. In D E thus continued take at pleasure the length E H, and let H I be so drawn, as to be equidistant from the line G E. Then, by what has been written upon the second law of motion[84], it follows, that after the impulse on the body in E it will move through E I in the same time, as it would have imployed in moving from E to H, with the velocity which it had in the line D E. In F E prolonged take E K equal to E I, and draw K L equidistant from G E. Then, because the body is thrown back in the line F E with the same velocity as that wherewith it went forward in that line; if, when the body was returned to E, it were permitted to go straight on, it would pass through E K in the same time, as it took up in passing through E I, when it went forward in the line E F. But, if at the body’s return to the point E, such an impulse directed toward the point D were to be given it, whereby it should be turned into the line D E; I say, that the impulse necessary to produce this effect must be equal to that, which turned the body out of the line D E into E F; and that the velocity, with which the body will return into the line E D, is the same, as that wherewith it before moved through this line from D to E. Because E K is equal to E I, and K L and H I, being each equidistant from G E, are by consequence equidistant from each other; it follows, that the two triangular figures I E H and K E L are altogether like and equal to each other. If I were writing to mathematicians, I might refer them to some proportions in the elements of Euclid for the proof of this[85] but as I do not here address my self to such, so I think this assertion will be evident enough without a proof in form; at least I must desire my readers to receive it as a proposition true in geometry. But these two triangular figures being altogether like each other and equal; as E K is equal to E I, so E L is equal to E H, and K L equal to H I. Now the body after its return to E being turned out of the line F E into E D by an impulse acting upon it in E, after the manner above expressed; the body will receive such a velocity by this impulse, as will carry it through E L in the same time, as it would have imployed in passing through E K, if it had gone on in that line undisturbed. And it has already been observed, that the time, in which the body would pass over E K with the velocity wherewith it returns, is equal to the time it took up in going forward from E to I; that is, equal to the time, in which it would have gone through E H with the velocity, wherewith it moved from D to E. Therefore the time, in which the body will pass through E L after its return into the line E D, is the same, as would have been taken up by the body in passing through E H with the velocity, wherewith the body first moved in the line D E. Since therefore E L and E H are equal, the body returns into the line D E with the velocity, which it had before in that line. Again I say, the second impulse in E is equal to the first. By what has been said on the second law of motion concerning the effect of oblique impulses[86], it will be understood, that the impulse in E, whereby the body was turned out of the line D E into the line E F, is of such strength, that if the body had been at rest, when this impulse had acted upon it, this impulse would have communicated so much motion to the body, as would have carried it through a length equal to H I, in the time wherein the body would have passed from E to H, or in the time wherein it passed from E to I. In the same manner, on the return of the body, the impulse in E, whereby the body is turned out of the line F E into E D, is of such strength, that if it had acted on the body at rest, it would have caused the body to move through a length equal to K L, in the same time, as the body would imploy in passing through E K with the velocity, wherewith it returns in the line F E. Therefore the second impulse, had it acted on the body at rest, would have caused it to move through a length equal to K L in the same space of time, as would be taken up by the body in passing through a length equal to H I, were the first impulse to act on the body when at rest. That is, the effects of the first and second impulse on the body when at rest would be the same; for K L and H I are equal: consequently the second impulse is equal to the first.