A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

[9.] Some of these inequalities would take place, though the moon, if undisturbed by the sun, would have moved in a circle concentrical to the earth, and in the plane of the earth’s motion; others depend on the elliptical figure, and the oblique situation of the moon’s orbit. One of the first kind is, that the moon is caused so to move, as not to describe equal spaces in equal times, but is continually accelerated, as she passes from the quarter to the new or full, and is retarded again by the like degrees in returning from the new and full to the next quarter. Here we consider not so much the absolute, as the apparent motion of the moon in respect to us.

10. The principles of astronomy teach how to distinguish these two motions. Let S (in fig. 95.) represent the sun, A the earth moving in its orbit B C, D E F G the moon’s orbit, the place of the moon H. Suppose the earth to have moved from A to I. Because it has been shewn, that the moon partakes of all the progressive motion of the earth; and likewise that the sun attracts both the earth and moon equally, when they are at the same distance from it, or that the mean action of the sun upon the moon is equal to its action upon the earth: we must therefore consider the earth as carrying about with it the moon’s orbit; so that when the earth is removed from A to I, the moon’s orbit shall likewise be removed from its former situation into that denoted by K L M N. But now the earth being in I, if the moon were found in O, so that O I should be parallel to H A, though the moon would really have moved from H to O, yet it would not have appeared to a spectator upon the earth to have moved at all, because the earth has moved as much it self; so that the moon would still appear in the same place with respect to the fixed stars. But if the moon be observed in P, it will then appear to have moved, its apparent motion being measured by the angle under O I P. And if the angle under P I S be less than the angle under H A S, the moon will have approached nearer to its conjunction with the sun.

11. To come now to the explication of the mentioned inequality in the moon’s motion: let S (in fig. 96.) represent the sun, A the earth, B C D E the moon’s orbit, C the place of the moon, when in the latter quarter. Here it will be nearly at the same distance from the sun, as the earth is. In this case therefore they will both be equally attracted, the earth in the direction A S, and the moon in the direction C S. Whence as the earth in moving round the sun is continually descending toward it, so the moon in this situation must in any equal portion of time descend as much; and therefore the position of the line A C in respect of A S, and the change, which the moon’s motion produces in the angle under C A S, will not be altered by the sun.

12. But now as soon as ever the moon is advanced from the quarter toward the new or conjunction, suppose to G, the action of the sun upon it will have a different effect. Here, were the sun’s action upon the moon to be applied in the direction G H parallel to A S, if its action on the moon were equal to its action on the earth, no change would be wrought by the sun on the apparent motion of the moon round the earth. But the moon receiving a greater impulse in G than the earth receives in A, were the sun to act in the direction G H, yet it would accelerate the description of the space D A G, and cause the angle under G A D to decrease faster, than otherwise it would. The sun’s action will have this effect upon account of the obliquity of its direction to that, in which the earth attracts the moon. For the moon by this means is drawn by two forces oblique to each other, one drawing from G toward A, the other from G toward H, therefore the moon must necessarily be impelled toward D. Again, because the sun does not act in the direction G H parallel to S A, but in the direction G S oblique to it, the sun’s action on the moon will by reason of this obliquity farther contribute to the moon’s acceleration. Suppose the earth in any short space of time would have moved from A to I, if not attracted by the sun; the point I being in the straight line C E, which touches the earth’s orbit in A. Suppose the moon in the same time would have moved in her orbit from G to K, and besides have partook of all the progressive motion of the earth. Then if K L be drawn parallel to A I, and taken equal to it, the moon, if not attracted by the sun, would be found in L. But the earth by the sun’s action is removed from I. Suppose it were moved down to M in the line I M N parallel to S A, and if the moon were attracted but as much, and in the same direction, as the earth is here supposed to be attracted, so as to have descended during the same time in the line L O, parallel also to A S, down as far as P, till L P were equal to I M; the angle under P M N would be equal to that under L I N, that is, the moon will appear advanced no farther forward, than if neither it nor the earth had been subject to the sun’s action. But this is upon the supposition, that the action of the sun upon the moon and earth were equal; whereas the moon being acted upon more than the earth, did the sun’s action draw the moon in the line L O parallel to A S, it would draw it down so far as to make L P greater than I M; whereby the angle under P M N will be rendred less, than that under L I N. But moreover, as the sun draws the earth in a direction oblique to I N, the earth will be found in its orbit somewhat short of the point M; however the moon is attracted by the sun still more out of the line L O, than the earth is out of the line I N; therefore this obliquity of the sun’s action will yet farther diminish the angle under P M N.

13. Thus the moon at the point G receives an impulse from the sun, whereby her motion is accelerated. And the sun producing this effect in every place between the quarter and the conjunction, the moon will move from the quarter with a motion continually more and more accelerated; and therefore by acquiring from time to time additional degrees of velocity in its orbit, the spaces, which are described in equal times by the line drawn from the earth to the moon, will not be every where equal, but those toward the conjunction will be greater, than those toward the quarter. But now in the moon’s passage from the conjunction D to the next quarter the sun’s action will again retard the moon, till at the next quarter in E it be restored to the first velocity, which it had in C.

14. Again as the moon moves from E to the full or opposition to the sun in B, it is again accelerated, the deficiency of the sun’s action upon the moon, from what it has upon the earth, producing here the same effect as before the excess of its action. Consider the moon in Q, moving from E towards B. Here if the moon were attracted by the sun in a direction parallel to A S, yet being acted on less than the earth, as the earth descends toward the sun, the moon will in some measure be left behind. Therefore Q F being drawn parallel to S B, a spectator on the earth would see the moon move, as if attracted from the point Q in the direction Q F with a degree of force equal to that, whereby the sun’s action on the moon falls short of its action on the earth. But the obliquity of the sun’s action has also here an effect. In the time the earth would have moved from A to I without the influence of the sun, let the moon have moved in its orbit from Q to R. Drawing therefore R T parallel to A I, and equal to the same, for the like reason as before, the moon by the motion of its orbit, if not at all attracted by the sun, must be found in T; and therefore, if attracted in a direction parallel to S A, would be in the line T V parallel to A S; suppose in W. But the moon in Q being farther off the sun than the earth, it will be less attracted, that is, T W will be less than I M, and if the line S M be prolonged toward X, the angle under X M W will be less than that under X I T. Thus by the sun’s action the moon’s passage from the quarter to the full would be accelerated, if the sun were to act on the earth and moon in a direction parallel to A S: and the obliquity of the sun’s action will still more increase this acceleration. For the action of the sun on the moon is oblique to the line S A the whole time of the moon’s passage from Q to T, and will carry the moon out of the line T V toward the earth. Here I suppose the time of the moon’s passage from Q to T so short, that it shall not pass beyond the line S A. The earth also will come a little short of the line I N, as was said before. From these causes the angle under X M W will be still farther lessened.

[15.] The moon in passing from the opposition B to the next quarter will be retarded again by the same degrees, as it is accelerated before its appulse to the opposition. Because this action of the sun, which in the moon’s passage from the quarter to the opposition causes it to be extraordinarily accelerated, and diminishes the angle, which measures its distance from the opposition; will make the moon slacken its pace afterwards, and retard the augmentation of the same angle in its passage from the opposition to the following quarter; that is, will prevent that angle from increasing so fast, as otherwise it would. And thus the moon, by the sun’s action upon it, is twice accelerated and twice restored to its first velocity, every circuit it makes round the earth. This inequality of the moon’s motion about the earth is called by astronomers its variation.

16. The next effect of the sun upon the moon is, that it gives the orbit of the moon in the quarters a greater degree of curvature, than it would receive from the action of the earth alone; and on the contrary in the conjunction and opposition the orbit is less inflected.

17. When the moon is in conjunction with the sun in the point D, the sun attracting the moon more forcibly than it does the earth, the moon by that means is impelled less toward the earth, than otherwise it would be, and so the orbit is less incurvated; for the power, by which the moon is impelled toward the earth, being that, by which it is inflected from a rectilinear course, the less that power is, the less it will be inflected. Again, when the moon is in the opposition in B, farther removed from the sun than the earth is; it follows then, though the earth and moon are both continually descending to the sun, that is, are drawn by the sun toward it self out of the place they would otherwise move into, yet the moon descends with less velocity than the earth; insomuch that the moon in any given space of time from its passing the point of opposition will have less approached the earth, than otherwise it would have done, that is, its orbit in respect of the earth will approach nearer to a straight line. In the last place, when the moon is in the quarter in F, and equally distant from the sun as the earth, we observed before, that the earth and moon would descend with equal pace toward the sun, so as to make no change by that descent in the angle under F A S; but the length of the line F A must of necessity be shortned. Therefore the moon in moving from F toward the conjunction with the sun will be impelled more toward the earth by the sun’s action, than it would have been by the earth alone, if neither the earth nor moon had been acted on by the sun; so that by this additional impulse the orbit is rendred more curve, than it would otherwise be. The same effect will also be produced in the other quarter.