Chap. XXI.] THE HEAVENS. 231 



the velocities are directly as the ratios which the 

 revolving bears to the direct force. 



Note. — The above is made evident by the table on 

 page 82 ; for the fractions of the sun's direct force, 

 which act as revolving forces, given in the third 

 column of that table, are inversely as the mean of the 

 distances given in the first column approximately ; 

 and the figures in the fourth column, which represent 

 the actual relative velocities of the planets, are the 

 square roots of those fractions of the sun's force. 



PEOPOSITION VI. 



Theorem. — The squares of the relative velocities with 

 which the planets m their orbital motions pass 

 round the sun are to each other directly as the 

 revolving force of the sun's gravitation in their 

 respective orbits. 



For the relative velocities of revolution are m- 

 versely as the times of revolution : 



And the squares of the times are directly as the 

 cubes of the distances (Kepler's law) : 



Therefore the squares of the velocities are in- 

 versely as the cubes of the distances. 



But the revolving force is inversely as the cubes 



