A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

41. As soon as the nodes, by the action of the sun, are got out of conjunction toward the other quarters, they begin again to recede as before; but the inclination of the orbit in the appulse of the moon to each succeeding node is less than at the preceding, till the nodes come again into the quarters. This will appear as follows. Let A (in fig. 104.) represent one of the moon’s nodes placed between the point of opposition B and the quarter C. Let the plane A D E pass through the earth T, and touch the path of the moon in A. Let the line A F G H be the path of the moon in her passage from A to H, where she crosses again the plane of the earth’s motion. This line will be convex toward the plane A D E, till the moon comes to G, where she is in the quarter; and after this, between G and H, the same line will be concave toward this plane. All the time this line is convex toward the plane A D E, the nodes will recede; and on the contrary proceed, while it is concave to that plane. All this will easily be conceived from what has been before so largely explained. But the moon is longer in passing from A to G, than from G to H; therefore the nodes recede a longer time, than they proceed; consequently upon the whole, when the moon is arrived at H, the nodes will have receded, that is, the point H will fall between B and E. The inclination of the orbit will decrease, till the moon is arrived to the point F, in the middle between A and H. Through the passage between F and G the inclination will increase, but decrease again in the remaining part of the passage from G to H, and consequently at H must be less than at A. The like effects, both in respect to the nodes and inclination of the orbit, will take place in the following passage of the moon on the other side of the plane A B E C, from H, till it comes over that plane again in I.

42. Thus the inclination of the orbit is greatest, when the line drawn between the moon’s nodes will pass through the sun; and least, when this line lies in the quarters, especially if the moon at the same time be in conjunction with the sun, or in the opposition. In the first of these cases the nodes have no motion, in all others, the nodes will each month have receded: and this regressive motion will be greatest, when the nodes are in the quarters; for in that case the nodes have no progressive motion during the whole month, but in all other cases the nodes do at some times proceed forward, viz. whenever the moon is between either quarter, and the node which is less distant from that quarter than a fourth part of a circle.

[43.] It now remains only to explain the irregularities in the moon’s motion, which follow from the elliptical figure of the orbit. By what has been said at the beginning of this chapter it appears, that the power of the earth on the moon acts in the reciprocal duplicate proportion of the distance: therefore the moon, if undisturbed by the sun, would move round the earth in a true ellipsis, and the line drawn from the earth to the moon would pass over equal spaces in equal portions of time. That this description of the spaces is altered by the sun, has been already declared. It has also been shown, that the figure of the orbit is changed each month; that the moon is nearer the earth at the new and full, and more remote in the quarters, than it would be without the sun. Now we must pass by these monthly changes, and consider the effect, which the sun will have in the different situations of the axis of the orbit in respect of that luminary.

44. The action of the sun varies the force, wherewith the moon is drawn toward the earth; in the quarters the force of the earth is directly increased by the sun; at the new and full the same is diminished; and in the intermediate places the influence of the earth is sometimes aided, and sometimes lessened by the sun. In these intermediate places between the quarters and the conjunction or opposition, the sun’s action is so oblique to the action of the earth on the moon, as to produce that alternate acceleration and retardment of the moon’s motion, which I observed above to be stiled the variation. But besides this effect, the power, by which the earth attracts the moon toward itself, will not be at full liberty to act with the same force, as if the sun acted not at all on the moon. And this effect of the sun’s action, whereby it corroborates or weakens the action of the earth, is here only to be considered. And by this influence of the sun it comes to pass, that the power, by which the moon is impelled toward the earth, is not perfectly in the reciprocal duplicate proportion of the distance. Consequently the moon will not describe a perfect ellipsis. One particular, wherein the moon’s orbit will differ from an ellipsis, consists in the places, where the motion of the moon is perpendicular to the line drawn from itself to the earth. In an ellipsis, after the moon should have set out in the direction perpendicular to this line drawn from itself to the earth, and at its greatest distance from the earth, its motion would again become perpendicular to this line drawn between itself and the earth, and the moon be at its nearest distance from the earth, when it should have performed half its period; after performing the other half of its period its motion would again become perpendicular to the forementioned line, and the moon return into the place whence it set out, and have recovered again its greatest distance. But the moon in its real motion, after setting out as before, sometimes makes more than half a revolution, before its motion comes again to be perpendicular to the line drawn from itself to the earth, and the moon is at its nearest distance; and then performs more than another half of an intire revolution before its motion can a second time recover its perpendicular direction to the line drawn from the moon to the earth, and the moon arrive again to its greatest distance from the earth. At other times the moon will descend to its nearest distance, before it has made half a revolution, and recover again its greatest distance, before it has made an intire revolution. The place, where the moon is at its greatest distance from the earth, is called the moon’s apogeon, and the place of the least distance the perigeon. This change of the place, where the moon successively comes to its greatest distance from the earth, is called the motion of the apogeon. In what manner the sun causes the apogeon to move, I shall now endeavour to explain.

45. Our author shews, that if the moon were attracted toward the earth by a composition of two powers, one of which were reciprocally in the duplicate proportion of the distance from the earth, and the other reciprocally in the triplicate proportion of the same distance; then, though the line described by the moon would not be in reality an ellipsis, yet the moon’s motion might be perfectly explained by an ellipsis, whose axis should be made to move round the earth; this motion being in consequence, as astronomers express themselves, that is, the same way as the moon itself moves, if the moon be attracted by the sum of the two powers; but the axis must move in antecedence, or the contrary way, if the moon be acted on by the difference of these powers. What is meant by duplicate proportion has been often explained; namely, that if three magnitudes, as A, B, and C, are so related, that the second B bears the same proportion to the third C, as the first A bears to the second B, then the proportion of the first A to the third C, is the duplicate of the proportion of the first A to the second B. Now if a fourth magnitude, as D, be assumed, to which C shall bear the same proportion as A bears to B, and B to C, then the proportion of A to D is the triplicate of the proportion of A to B.

46. The way of representing the moon’s motion in this case is thus. T denoting the earth (in fig. 105, 106.) suppose the moon in the point A, its apogeon, or greatest distance from the earth, moving in the direction A F perpendicular to A B, and acted upon from the earth by two such forces as have been named. By that power alone, which is reciprocally in the duplicate proportion of the distance, if the moon let out from the point A with a proper degree of velocity, the ellipsis A M B may be described. But if the moon be acted upon by the sum of the forementioned powers, and the velocity of the moon in the point A be augmented in a certain proportion[193]; or if that velocity be diminished in a certain proportion, and the moon be acted upon by the difference of those powers; in both these cases the line A E, which shall be described by the moon, is thus to be determined. Let the point M be that, into which the moon would have arrived in any given space of time, had it moved in the ellipsis A M B. Draw M T, and likewise C T D in such sort, that the angle under A T M shall bear the same proportion to the angle under A T C, as the velocity, with which the ellipsis A M B must have been described, bears to the difference between this velocity, and the velocity, with which the moon must set out from the point A in order to describe the path A E. Let the angle A T C be taken toward the moon (as in fig. 105.) if the moon be attracted by the sum of the powers; but the contrary way (as in fig. 106.) if by their difference. Then let the line A B be moved into the position C D, and the ellipsis A M B into the situation C N D, so that the point M be translated to L: then the point L shall fall upon the path of the moon A E.

47. The angular motion of the line A T, wereby it is removed into the situation C T, represents the motion of the apogeon; by the means of which the motion of the moon might be fully explicated by the ellipsis A M B, if the action of the sun upon it was directed to the center of the earth, and reciprocally in the triplicate proportion of the moon’s distance from it. But that not being so, the apogeon will not move in the regular manner now described. However, it is to be observed here, that in the first of the two preceding cases, where the apogeon moves forward, the whole centripetal power increases faster, with the decrease of distance, than if the intire power were reciprocally in the duplicate proportion of the distance; because one part only is in that proportion, and the other part, which is added to this to make up the whole power, increases faster with the decrease of distance. On the other hand, when the centripetal power is the difference between these two, it increases less with the decrease of the distance, than if it were simply in the reciprocal duplicate proportion of the distance. Therefore if we chuse to explain the moon’s motion by an ellipsis (as is most convenient for astronomical uses to be done, and by reason of the small effect of the sun’s power, the doing so will not be attended with any sensible error;) we may collect in general, that when the power, by which the moon is attracted to the earth, by varying the distance, increases in a greater than in the duplicate proportion of the distance diminished, a motion in consequence must be ascribed to the apogeon; but that when the attraction increases in a less proportion than that named, the apogeon must have given to it a motion in antecedence[194]. It is then observed by Sir Is. Newton, that the first of these cases obtains, when the moon is in the conjunction and opposition; and the latter, when the moon is in the quarters: so that in the first the apogeon moves according to the order of the signs; in the other, the contrary way[195]. But, as was said before, the disturbance given to the action of the earth by the sun in the conjunction and opposition being near twice as great as in the quarters[196], the apogeon will advance with a greater velocity than recede, and in the compass of a whole revolution of the moon will be carried in consequence[197].

48. It is shewn in the next place by our author, that when the line A B coincides with that, which joins the earth and the sun, the progressive motion of the apogeon, when the moon is in the conjunction or opposition, exceeds the regressive in the quadratures more than in any other situation of the line A B[198]. On the contrary, when the line A B makes right angles with that, which joins the earth and sun, the retrograde motion will be more considerable[199], nay is found so great as to exceed the progressive; so that in this case the apogeon in the compass of an intire revolution of the moon is carried in antecedence. Yet from the considerations in the last paragraph the progressive motion exceeds the other; so that in the whole the mean motion of the apogeon is in consequence, according as astronomers find. Moreover, the line A B changes its situation with that, which joins the earth and sun, by such slow degrees, that the inequalities in the motion of the apogeon arising from this last consideration, are much greater than what arises from the other[200].