48. IT is shewn in the next place by our author, that when the line A B coincides with that, which joins the earth and the sun, the progressive motion of the apogeon, when the moon is in the conjunction or opposition, exceeds the regressive in the quadratures more than in any other situation of the line A B[198]. On the contrary, when the line A B makes right angles with that, which joins the earth and sun, the retrograde motion will be more considerable[199], nay is found so great as to exceed the progressive; so that in this case the apogeon in the compass of an intire revolution of the moon is carried in antecedence. Yet from the considerations in the last paragraph the progressive motion exceeds the other; so that in the whole the mean motion of the apogeon is in consequence, according as astronomers find. Moreover, the line A B changes its situation with that, which joins the earth and sun, by such slow degrees, that the inequalities in the motion of the apogeon arising from this last consideration, are much greater than what arises from the other[200].
49. FARTHER, this unsteady motion in the apogeon is attended with another inequality in the motion of the moon, that it cannot be explained at all times by the same ellipsis. The ellipsis in general is called by astronomers an eccentric orbit. The point, in which the two axis’s cross, is called the center of the figure; because all lines drawn through this point within the ellipsis, from side to side, are divided in the middle by this point. But the center, about which the heavenly bodies revolve, lying out of this center of the figure in one focus, these orbits are said to be eccentric; and where the distance of the focus from this center bears the greatest proportion to the whole axis, that orbit is called the most eccentric: and in such an orbit the distance from the focus to the remoter extremity of the axis bears the greatest proportion to the distance of the nearer extremity. Now whenever the apogeon of the moon moves in consequence, the moon’s motion must be referred to an orbit more eccentric, than what the moon would describe, if the whole power, by which the moon was acted on in its passing from the apogeon, changed according to the reciprocal duplicate proportion of the distance from the earth, and by that means the moon did describe an immoveable ellipsis; and when the apogeon moves in antecedence, the moon’s motion must be referred to an orbit less eccentric. In the first of the two figures last referred to, the true place of the moon L falls without the orbit A M B, to which its motion is referred: whence the orbit A L E, truly described by the moon, is less incurvated in the point A, than is the orbit A M B; therefore the orbit A M B is more oblong, and differs farther from a circle, than the ellipsis would, whose curvature in A were equal to that of the line A L B, that is, the proportion of the distance of the earth T from the center of the ellipsis to its axis will be greater in the ellipsis A M B, than in the other; but that other is the ellipsis, which the moon would describe, if the power acting upon it in the point A were altered in the reciprocal duplicate proportion of the distance. In the second figure, when the apogeon recedes, the place of the moon L falls within the orbit A M B, and therefore that orbit is less eccentric, than the immoveable orbit which the moon should describe. The truth of this is evident; for, when the apogeon moves forward, the power, by which the moon is influenced in its descent from the apogeon, increases faster with the decrease of distance, than in the duplicate proportion of the distance; and consequently the moon being drawn more forcibly toward the earth, it will descend nearer to it. On the other hand, when the apogeon recedes, the power acting on the moon increases with the decrease of distance in less than the duplicate proportion of the distance; and therefore the moon is less impelled toward the earth, and will not descend so low.