47. THE angular motion of the line A T, wereby it is removed into the situation C T, represents the motion of the apogeon; by the means of which the motion of the moon might be fully explicated by the ellipsis A M B, if the action of the sun upon it was directed to the center of the earth, and reciprocally in the triplicate proportion of the moon’s distance from it. But that not being so, the apogeon will not move in the regular manner now described. However, it is to be observed here, that in the first of the two preceding cases, where the apogeon moves forward, the whole centripetal power increases faster, with the decrease of distance, than if the intire power were reciprocally in the duplicate proportion of the distance; because one part only is in that proportion, and the other part, which is added to this to make up the whole power, increases faster with the decrease of distance. On the other hand, when the centripetal power is the difference between these two, it increases less with the decrease of the distance, than if it were simply in the reciprocal duplicate proportion of the distance. Therefore if we chuse to explain the moon’s motion by an ellipsis (as is most convenient for astronomical uses to be done, and by reason of the small effect of the sun’s power, the doing so will not be attended with any sensible error;) we may collect in general, that when the power, by which the moon is attracted to the earth, by varying the distance, increases in a greater than in the duplicate proportion of the distance diminished, a motion in consequence must be ascribed to the apogeon; but that when the attraction increases in a less proportion than that named, the apogeon must have given to it a motion in antecedence[194]. It is then observed by Sir IS. NEWTON, that the first of these cases obtains, when the moon is in the conjunction and opposition; and the latter, when the moon is in the quarters: so that in the first the apogeon moves according to the order of the signs; in the other, the contrary way[195]. But, as was said before, the disturbance given to the action of the earth by the sun in the conjunction and opposition being near twice as great as in the quarters[196], the apogeon will advance with a greater velocity than recede, and in the compass of a whole revolution of the moon will be carried in consequence[197].