BOOK III.] OF NATURAL PHILOSOPHY. 427 nitely ; PI, PK perpendiculars upon the lines ST, Qq ; Pp a perpendicu lar upon the plane of the ecliptic; A, B the moon's syzygies in the plane of the ecliptic; AZ a perpendicular let fall upon Nil, the line of the nodes ; Q, g the quadratures of the moon in the plane of the ecliptic, and pK a perpendicular on the line Qq lying between the quadratures. The force of the sun to disturb the motion of the moon (by Prop. XXV) is twofold, one proportional to the line LM, the otlier to the line MT, in the scheme of that Proposition ; and the moon by the former force is drawn towards the earth, by the latter towards the sun, in a direction parallel to the right line ST joining the earth and the sun. The former force LM acts in the direction of the plane of the moon's orbit, and therefore makes no change upon the situation thereof, and is upon that account to be neg lected ; the latter force MT, by which the plane of the moon's orbit is dis turbed, is the same with the force 3PK or SIT. And this force (by Prop. XXV) is to the force by which the moon may, in its periodic time, be uni formly revolved in a circle about the earth at rest, as SIT to the radius of the circle multiplied by the number 178,725, or as IT to the radius there of multiplied by 59,575. But in this calculus, and all that follows. I consider all the lines drawn frorri the moon to the sun as parallel to the line which joins the earth and the sun ; because what inclination there is almost as much diminishes all effects in some cases as it augments them in others : and we are now inquiring after the mean motions of the nodes, neglecting such niceties as are of no moment, and would only serve to ren der the calculus more perplexed. Now suppose PM to represent an arc which the moon describes in the least moment of time, and ML a little line, the half of which the moon, by the impulse of the said force SIT, would describe in the same time ; and joining PL, MP, let them be produced to m and /, where they cut the plane of the ecliptic, and upon Tm let fall the perpendicular PH. Now, since the right line ML is parallel to the plane of the ecliptic, and therefore can never meet with the right line ml which lies in that plane, and yet both those right lines lie in one common plane LMPm/, they will be parallel, and upon that account the triangles LMP, ImP will be similar. And seeing MPra lies in the plane of the orbit, in which the moon did move while in the place P, the point m will fall upon the line N//, which passes through the nodes N, n, of that orbit. And because the force by which the half of the little line LM is generated, if the whole had been together, and it once impressed in the point P, would hav^ generated that whole line, and caused the moon to move in the arc whoso chord is LP ; t at is to say, would have transferred the moon from the plane MPwT into the plane LP/T; therefore th* angular motion of the nodes generated by that force will be equal to the angle mTL But n.l is to raP as ML to MP ; and since ML3, because of the time given, is also given, ml will be as the rectan