INDUCTIVE EPOCH OF KEPLER. 301 



accordance of the distances, libration in the diameter of the epicycle was 

 attributed, should, according to the indication of the equations, go in 

 an elliptical path. What an absurdity on my part ! as if libration in 

 the diameter might not be a way to the ellipse !" 



Another scruple respecting this theory arose from the impossibility 

 of solving, by any geometrical construction, the problem to which 

 Kepler was thus led, namely, " To divide the area of a semicircle in a 

 given ratio, by a line drawn from any point of the diameter." This is 

 still termed "Kepler's Problem," and is, in fact, incapable of exact 

 geometrical solution. As, however, the calculation can be performed, 

 and, indeed, was performed by Kepler himself, with a sufficient degree 

 of accuracy to show that the elliptical hypothesis is true, the insolu- 

 bility of this problem is a mere mathematical difficulty in the deduc- 

 tive process, to which Kepler's induction gave rise. 



Of Kepler's physical reasonings we shall speak more at length on 

 another occasion. His numerous and fanciful hypotheses had dis- 

 charged their office, when they had suggested to him his many lines 

 of laborious calculation, and encouraged him under the exertions and 

 disappointments to which these led. The result of this work was the 

 formal laws of the motion of Mars, established by a clear induction, 

 since they represented, with sufficient accuracy, the best observations. 

 And wc may allow that Kepler was entitled to the praise which he 

 claims in the motto on his first leaf. Ramus had said that if any one 

 would construct an astronomy without hypothesis, he would be ready 

 to resign to him his professorship in the University of Paris. Kepler 

 quotes this passage, and adds, " it is well, Ramus, that you have run 

 from this pledge, by quitting life and your professorship ; 14 if you held 

 it still, I should, with justice, claim it." This was not saying too 

 much, since he had entirely overturned the hypothesis of eccentrics 

 and epicycles, and had obtained a theory which was a mere represen- 

 tation of the motions and distances as they were observed. 



14 Ramus perished in the Massacre of St. Bartholomew. 



