THE LAW or EQUILIBRIUM 131 



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 

 heights, 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 



\r~W : V~Kr (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, 



that the areas of the respective segments at perihelion 

 and at aphelion covered in equal times are equal. Of 

 course the rule holds good whatever parts of the orbit be 

 compared. 



Kepler's first law, namely, that planets travel in el- 

 lipses, is true only in the most superficial of senses. It 

 is not true geometrically, dynamically, philosophically, or 

 actually. Planets travel, instead, in open elliptical 

 spirals of an exceedingly complicated design. Astron- 

 omers, indeed, acknowledge that the orbits are not re- 

 entering, inasmuch as the translation of the sun is well 

 established; but thej- assume that, even if the sun were 

 stationary, the planets would, nevertheless, continue re- 

 volving, and that these orbits would then veritably be 

 closed curves. They picture the sun as possessing a 

 unique rectilinear motion not primevally shared by his 

 planets, and that the reason they accompany him is be- 

 cause he is dragging them along as a horse does a vehicle, 

 whether or no. Thus have the Newtonians, in a way be- 

 come habitual with them, underestimated the most signifi- 

 cant and potential facts of the cosmos, facts, too, gained 

 by their own herculean labor and expense. 



