8. THUS is proved, that the sun acts upon the secondary planets, as much as upon the primary at the same distance: but it was found in the last chapter, that the action of the sun upon bodies is reciprocally in the duplicate proportion of the distance; therefore the secondary planets being sometimes nearer to the sun than the primary, and sometimes more remote, they are not alway acted upon in the same degree with their primary, but when nearer to the sun, are attracted more, and when farther distant, are attracted less. Hence arise various inequalities in the motion of the secondary planets[187]. 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.