84 THE OCEAN. [Book III. 



square root of the revolving force of the sun's gravi- 

 tation. 



The revolving force of the sun's gravitation is, 

 therefore, as the square of the angular velocity of 

 the motion — that is, the velocity as measured on the 

 surface of the sun which causes the motion.^ 



Now, since, accordmg to the foregoing chapters, 

 the earth is held in equilibrium between opposing 

 forces of gravitation, therefore the retarding force 

 of astral gravitation increases as the square of the 

 velocity of the motion which it opposes (for the 

 revolving force increases in that proportion). 



The action of this retarding force of astral 

 gravitation then makes the squares of the velocities 

 of necessity inversely as the cubes of the relative dis- 

 tances ; because, suppose the revolving force to be 

 increased, then there is a corresponding increase in 

 the opposing force of astral gravitation without any 

 increase hi the direct force of the sun's gravitation, 

 and astral gravitation would then carry the earth 

 back at a tangent to the orbit in which the revolv- 

 ing force tends to carry it, uutil, in consequence of 

 the revolving force decreasing as the cube, and the 

 direct force only as the square of the distance increased, 

 the equilibrium between the force of astral gravitation 

 drawino" from the sun, and that of the direct force of 

 the sun's gravitation, would be restored.^ 



1 See also Proposition XVII., Chapter XXI. 



2 See Proposition XXV., Chapter XXI. 



