THE LAW OF EQUILIBRIUM 143 



The periodic times of planetary bodies are to each 

 other as their respective orbital radii (distances) into 

 the square roots of those radii. 



Notice that this gives the ratio of the simple periodic 

 times, not the ratio of their squares, hence is logically 

 better ; but if you wish to identify the ratio with Kepler's 

 law, all you need do is to square the separate terms, thus : 



r*r:R*Ror (5) 



r*:R s q.e.d. (6) 



Again, since under the third law given above, veloci- 

 ties are proportional to the square roots of the spaces 

 (here circumferences, or orbits), they are necessarily pro- 

 portional in like manner with respect to the radii, or as 



VF.-Vlf (7) 



But we are not to forget that these velocities are di- 

 rectly proportional to the number of units of weight that 

 run the course, or, in other words, they are inversely pro- 

 portional to the radii, or lever arms, whence : 



R V r : r \/ R (8) meaning, 



The velocities of planets are proportional to their 

 respective distances into the square roots of their OP- 

 POSITES. 



With this formula (8) before us, it is easy to derive 

 the law of gravitation, gravitation being a form of 

 ENERGY. Energy being proportional to the square of the 

 velocities, we have, then, 



R 2 r :r*R (9) 



Now, there is this further rule regarding energy of 

 motion, namely, that it varies directly as the load that has 

 been successfully carried; and, as we have seen, the 

 heavier load on a balanced lever is at the end of the 

 shorter arm. We therefore multiply the first term by R 

 and the second by r, obtaining : 



R*r:r*R,or, (10) 



R*:r* q.e.d. (11) 



