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.