130 FKOM NEBULA TO NEBULA 



THE LAW OF GRAVITATION 



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*r:r 2 R (9) 



Now, there is this further rule regarding energy of 

 motion, namely, that it varies directly as the load, 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 K and the second by r, obtaining, 



R* r : r 3 R, or, (10) 



R 2 :r 2 q.e.d. (11) 



Which is the law of gravitation itself, namely, that the 

 energy of planets' motions varies inversely as the squares 

 of the radii (distances). This method really gives the 

 centrifugal force, but as the centripetal is, by the law of 

 reaction, its equal, the expression given is true for both. 



THE LAW OF AREAS 



Kepler's second law declares that the radius vector 

 sweeps over equal areas in equal times. This can be 

 proven, on the principle of the falling lever, in this way : 



Suppose a planet to describe a certain arc at peri- 

 helion in the space of one hour and, later on, an arc at any 

 other part of its orbit, say aphelion, in a like space of 

 time. Regarding the arcs thus described as arcs of 

 circles, and the differing distances as radii, r and R, the 

 arcs, geometrically, will be in the ratio of 



r:R (12) 



We are not dealing with plain circles, however, but with 

 the rotations of a lever. According to this principle, 

 what is gained in power is lost in velocity, and vice versa. 

 Moving the planet out to aphelion, therefore, modifies the 



