144 FROM NEBULA TO NEBULA 



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. Eegarding 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 

 equilibristic lengths of the arcs described in the inverse 

 ratio of the radii, hence the arcs become, 



Rr:Rr (13) 



But the arcs thus related are not plain distances, but 

 spaces, through which the weights (which in the present 

 instance are equal, being, indeed, the same planet) are 

 falling; hence we now find the opposite arcs to be propor- 

 tional, dynamically, thus 



Vl2r : Vl^ (14) 



But, by geometry, the areas of circles are as the squares 

 of their circumferences or of their like arcs. In our ratio 

 we have what we may correctly describe as dynamical 

 arcs, incorporating within them, as they here should, the 

 ideas of geometrical relationship, equilibrium, and accel- 

 erative motion. Squaring the terms, then, we get, 



Rr : Rr q. e. d. (15), meaning, 



