THE LAW OF EQUILIBKIUM 129 



At this point, let us not forget that we are not dealing 

 with a simple bar, but with a lever, whose arms, in order 

 that they may balance, must be weighted inversely as 

 their length; that is to say, a weight R must be attached 

 to the end of arm r, and a weight r to the end of arm R. 

 Pound for pound, then, the time required to complete the 

 running of the smaller coil as compared with that re- 

 quired for the larger is inversely in the proportion of 

 R : r. Amending our last equation, then, a second time, 

 we get for the ratio of the "periodic times" of our two 

 weights (or planets) 



A/~O ~^ (3) whence, (parts of circles being 



7T / V fC TT JL\J , , , , , 



. to each other as their 



. 



like parts) 



r V^* R V^ (4) meaning, 



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, which 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 2 r :R 2 R or (5) 



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



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

 ties are proportional to the square roots of the heights 

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

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



V^: y~R (7) 



But do not forget that these velocities are inversely pro- 

 portional to the size of the weights carried, whence 



Ry7~:r~ V~]R~ (8) meaning, 



The velocities of planets are proportional to their 

 respective distances into the square roots of their OP- 

 POSITES. 



