CnAP. XXI.] THE HEAVENS. 235 



squares of those velocities identical with the fractions 

 of the sun's force of gravitation which acts as a motive 

 force along the respective orbits (p. 82) ; and those 

 fractions are inversely as the distance from the sun 

 (Proposition III., Corollary) ; whereas the absolute 

 amount of the revolving force is inversely as the 

 cube of the distance (Proposition IV.), and if not 

 neutralised by the direct force of the sun's gravita- 

 tion, inversely as the square of the distance would 

 give to the planets, moving in comparatively free 

 space, a much greater proportionate velocity with 

 each decrease of distance. 



Therefore, just as any force exerted to move any 

 body along the surface of the earth is resisted by the 

 weight of the body moved, which is the direct force of 

 the gravitation of the earth actmg on it, so also must 

 the planets be held in equilibrium between the sun's 

 direct force of gravitation and opposing force, which 

 jointly resist motion along the orbit, as well as 

 respectively in either direction from the orbit. 



PPvOPOSITION X. 



Theorem. — The direct force of the sun's gravitation 

 tends to retard the sun's axial rotation more than 

 the revolution of the planets. 



