114 NEWTON S PRINCIPIA. 



use of the calculus, as it were, by a new triumph, in 

 another part of the Mecanique Celeste.* For the equation 



t V , 

 of the mean anular motion is shown to be n = -, 





t being the time, a the transverse axis, and ju, the sum of 

 the masses of the two bodies, in this case the moon and 

 the earth. Therefore n, the mean motion, must neces 

 sarily be accelerated as a, the axis, is diminished. 



And here in passing, we also observe how Kepler s law 

 of the sesquiplicate ratio may be anew proved, but only if 

 we make p, = S (the sun), and neglect the mass of the 

 planet. For take two planets whose mean motions are n 

 and n* round a third body, and their mean motions be 

 ing as and \^~&amp;gt; an( ^ because (2 TT being 360), 

 az a? 



Q n 



n t = 2 TT, therefore t = , and ?= -^- 9 or t = 



3 2 TT v i 



w l_ ag , and ^JLf ; consequently 2 : ^ 2 :: a 3 : &amp;gt;3 



being Kepler s law, which is thus demonstrated. But 



it is only demonstrated and is only true if V ^ is the same 

 to both planets, that is, if ju. = S in each case. Now, this 

 may be assumed in the case of those bodies revolving round 

 the sun, or of the satellites of Jupiter and Saturn revolving 

 round those primary planets, because of the great dispro 

 portion between the central body and the others, (the largest 

 of them, Jupiter, being less than a thousandth part of the 

 sun.) But the law would not hold true if p were taken, 

 which in strictness it ought to be, as S + P, the sum of 

 the masses of the central and the revolving body ; for then 

 p. would differ in each instance, and the sesquiplicate pro- 



* Liv. ii. ch. 3. 



