A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

[18.] Another effect of the sun’s action, consequent upon this we have now explained, is, that though the moon undisturbed by the sun might move in a circle having the earth for its center; by the sun’s action, if the earth were to be in the very middle or center of the moon’s orbit, yet the moon would be nearer the earth at the new and full, than in the quarters. In this probably will at first appear some difficulty, that the moon should come nearest to the earth, where it is least attracted to it, and be farthest off when most attracted. Which yet will appear evidently to follow from that very cause, by considering what was last shewn, that the orbit of the moon in the conjunction and opposition is rendred less curve; for the less curve the orbit of the moon is, the less will the moon have descended from the place it would move into, without the action of the earth. Now if the moon were to move from any place without farther disturbance from that action, since it would proceed in the line, which would touch its orbit in that place, it would recede continually from the earth; and therefore if the power of the earth upon the moon, be sufficient to retain it at the same distance, this diminution of that power will cause the distance to increase, though in a less degree. But on the other hand in the quarters, the moon, being pressed more towards the earth than by the earth’s single action, will be made to approach it; so that in passing from the conjunction or opposition to the quarters the moon ascends from the earth, and in passing from the quarters to the conjunction and opposition it descends again, becoming nearer in these last mentioned places than in the other.

[19.] All these forementioned inequalities are of different degrees, according as the sun is more or less distant from the earth; greater when the earth is nearest the sun, and less when it is farthest off. For in the quarters, the nearer the moon is to the sun, the greater is the addition to the earth’s action upon it by the power of the sun; and in the conjunction and opposition, the difference between the sun’s action upon the earth and upon the moon is likewise so much the greater.

[20.] This difference in the distance between the earth and the sun produces a farther effect upon the moon’s motion; causing the orbit to dilate when less remote from the sun, and become greater, than when at a farther distance. For it is proved by Sir Isaac Newton, that the action of the sun, by which it diminishes the earth’s power over the moon, in the conjunction or opposition, is about twice as great, as the addition to the earth’s action by the sun in the quarters[188]; so that upon the whole, the power of the earth upon the moon is diminished by the sun, and therefore is most diminished, when the action of the sun is strongest: but as the earth by its approach to the sun has its influence lessened, the moon being less attracted will gradually recede from the earth; and as the earth in its recess from the sun recovers by degrees its former power, the orbit of the moon must again contract. Two consequences follow from hence: the moon will be most remote from the earth, when the earth is nearest the sun; and also will take up a longer time in performing its revolution through the dilated orbit, than through the more contracted.

[21.] These irregularities the sun would produce in the moon, if the moon, without being acted on unequally by the sun, would describe a perfect circle about the earth, and in the plane of the earth’s motion; but though neither of these suppositions obtain in the motion of the moon, yet the forementioned inequalities will take place, only with some difference in respect to the degree of them; but the moon by not moving in this manner is subject to some other inequalities also. For as the moon describes, instead of a circle concentrical to the earth, an ellipsis, with the earth in one focus, that ellipsis will be subjected to various changes. It can neither preserve constantly the same position, nor yet the same figure; and because the plane of this ellipsis is not the same with that of the earth’s orbit, the situation of the plane, wherein the moon moves, will continually change; neither the line in which it intersects the plane of the earth’s orbit, nor the inclination of the planes to each other, will remain for any time the same. All these alterations offer themselves now to be explained.

22. I shall first consider the changes which are made in the plane of the moon’s orbit. The moon not moving in the same plane with the earth, the sun is seldom in the plane of the moon’s orbit, viz. only when the line made by the common intersection of the two planes, if produced, will pass through the sun, as is represented in fig. 97. where S denotes the sun; T the earth; A T B the earth’s orbit described upon the plane of this scheme; C D E F the moon’s orbit, the part C D E being raised above, and the part C F E depressed under the plane of this scheme. Here the line C E, in which the plane of this scheme, that is, the plane of the earth’s orbit and the plane of the moon’s orbit intersect each other, being continued passes through the sun in S. When this happens, the action of the sun is directed in the plane of the moon’s orbit, and cannot draw the moon out of this plane, as will evidently appear to any one that shall consider the present scheme: for suppose the moon in G, and let a straight line be drawn from G to S, the sun draws the moon in the direction of this line from G toward S: but this line lies in the plane of the orbit; and if it be prolonged from S beyond G, the continuation of it will lie on the plane C D E; for the plane itself, if sufficiently extended, will pass through the sun. But in other cases the obliquity of the sun’s action to the plane of the orbit will cause this plane continually to change.

23. Suppose in the first place, the line, in which the two planes intersect each other, to be perpendicular to the line which joins the earth and sun. Let T (in fig. 98, 99, 100, 101.) represent the earth; S the sun; the plane of this scheme the plane of the earth’s motion, in which both the sun and earth are placed. Let A C be perpendicular to S T, which joins the earth and sun; and let the line A C be that, in which the plane of the moon’s orbit intersects the plane of the earth’s motion. To the center T describe in the plane of the earth’s motion the circle A B C D. And in the plane of the moon’s orbit describe the circle A E C F, one half of which A E C will be elevated above the plane of this scheme, the other half A F C as much depressed below it.

24. Now suppose the moon to set forth from the point A (in fig. 98.) in the direction of the plane A E C. Here she will be continually drawn out of this plane by the action of the sun: for this plane A E C, if extended, will not pass through the sun, but above it; so that the sun, by drawing the moon directly toward it self, will force it continually more and more from that plane towards the plane of the earth’s motion, in which it self is; causing it to describe the line A K G H I, which will be convex to the plane A E C, and concave to the plane of the earth’s motion. But here this power of the sun, which is said to draw the moon toward the plane of the earth’s motion, must be understood principally of so much only of the sun’s action upon the moon, as it exceeds the action of the same upon the earth. For suppose the preceding figure to be viewed by the eye, placed in the plane of that scheme, and in the line C T A on the side of A, the plane A B C D will appear as the straight line D T B, (in fig. 102.) and the plane A E C F as another straight line F E; and the curve line A K G H I under the form of the line T K G H I.

Now it is plain, that the earth and moon being both attracted by the sun, if the sun’s action upon both was equally strong, the earth T, and with it the plane A E C F or line F T E in this scheme, would be carried toward the sun with as great a pace as the moon, and therefore the moon not drawn out of it by the sun’s action, excepting only from the small obliquity of the direction of this action upon the moon to that of the sun’s action upon the earth, which arises from the moon’s being out of the plane of the earth’s motion, and is not very considerable; but the action of the sun upon the moon being greater than upon the earth, all the time the moon is nearer to the sun than the earth is, it will be drawn from the plane A E C or the line T E by that excess, and made to describe the curve line A G I or T G I. But it is the custom of astronomers, instead of considering the moon as moving in such a curve line, to refer its motion continually to the plane, which touches the true line wherein it moves, at the point where at any time the moon is. Thus when the moon is in the point A, its motion is considered as being in the plane A E C, in whose direction it then essaies to move; and when in the point K (in fig. 99.) its motion is referred to the plane, which passes through the earth, and touches the line A K G H I in the point K. Thus the moon in passing from A to I will continually change the plane of her motion. In what manner this change proceeds, I shall now particularly explain.