428 THE MATHEMATICAL PRINCIPLES [BOOK III. gle ML X mP, that is, as the rectangle IT X mP. And if Tml is a right ano-le, the angle mTl will be as 7^—' and therefore as — ^— — that is (beTm Tm ITxPH* cause Tm and mP, TP and PH are proportional), as — FFp~ ' and, there fore, because TP is given, as IT X PH. But if the angle Tml or STN is oblique, the angle mTl will be yet less, in proportion of the sine of the angle STN to the radius, or AZ to AT. And therefore the velocity of the nodes is as IT X PH X AZ, or as the solid content of the sines of the three angles TPI, PTN, and STN. If these are right angles, as happens when the nodes are in the quadra tures, and the moon in the syzygy, the little line ml will be removed to an infinite distance, and the angle mTl will become equal to the angle mPl. But in this case the angle mPl is to the angle PTM, which the moon in the same time by its apparent motion describes about the earth, as 1 to 59,575. For the angle mPl is equal to the angle LPM, that is, to the angle of the moon's deflexion from a rectilinear path; which angle, if the gravity of the moon should have then ceased, the said force of the sun SIT would by itself have generated in that given time : and the angle PTM is equal to the angle of the moon's deflexion from a rectilinear path; which angle, if the force of the sun 31T should have then ceased, the force alone by which the moon is retained in its orbit would have generated in the same time. And. these forces (as we have above shewn) are the one to the other as I to 59,575. Since, therefore, the mean horary motion of the moon (in respect of the fixed stars) is 32' 56" 27'" 12^-iv. the horary motion of the node in this case will be 33" 10"' 331V. 12V. But in other cases the horary motion will be to 33" 10'" 33iv. \2\ as the solid content of the sines of the three angles TPI, PTN, and STN (or of the distances of the moon from the quadrature, of the moon from the node, and of the node from the sun) to the cube of the radius. And as often as the sine of any angle is changed from positive to negative, and from negative to positive, so often must the regressive be changed into a progressive, and the progressive into a regressive motion. Whence it comes to pass that the nodes are pro gressive as often as the moon happens to be placed between either quadra ture, and the node nearest to that quadrature. In other cases they are regressive, and by the excess of the regress above the progress, they are monthly transferred in antecedentia. COR. 1. Hence if from P and M, the extreme points of a least arc PM, on the line Qq joining the quadratures we let fall the perpendiculars PK MA", and produce the same till they cut the line of the nodes Nw in D ana d, the horary motion of the nodes will be as the area MPDd, and the square of the line AZ conjunctly. For let PK, PH, and AZ, be the three said sines, viz., PK the sine of the distance of the moon from the quadra