130 THE MATHEMATICAL PRINCIPLES [BOOK 1IJ If, therefore, the nodes went on with that velocity uniformly continued which they acquire in the moon's syzygies, they would describe a space double of that which they describe in fact ; and, therefore, the mean motion, by which, if Uniformly continued, they would describe the same space with that which they do in fact describe by an unequal motion, is but one-half of that motion which they are possessed of in the moon's syzygies. Where fore since their greatest horary motion, if the nodes are in the quadratures, is 33" 10'" 33iv. 12V. their mean horary motion in this case will be 16" 35'" 16iv. 36V. And seeing the horary motion of the nodes is every where as AZ2 and the area PDdM conjunctly, and. therefore, in the moon's syzygies, the horary motion of the nodes is as AZ2 and the area PDdM conjunctly, that is (because the area PDdNL described in the syzygies is given), as AZ2, therefore the mean motion also will be as AZ2 ; and, there fore, when the nodes are without the quadratures, this motion will be to 16" 35'" I6'v. 36V. as AZ2 to AT2. Q.E.D. PROPOSITION XXXI. PROBLEM XII. To find the horary motion of the nodes of the moon in an elliptic orbit Let Qjpmaq represent an ellipsis described with the greater axis Qy, am the lesser axis ab : QA^B a circle circumscribed ; T the earth in the com mon centre of both ; S the sun ; p the moon moving in this ellipsis ; and