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.