INDUCTIVE EPOCH OF KEPLER. 451 



trieal construction, the problem to which Kepler 

 was thus led, namely, " to divide the area of a semi- 

 circle in a given ratio, by a line drawn from any 

 point of the diameter." This is still termed " Kep- 

 ler'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 

 insolubility of this problem is a mere mathematical 

 difficulty in the deductive process, to which Kepler's 

 inductions gave rise. 



Of Kepler's physical reasonings we shall speak 

 more at length on another occasion. His numerous 

 and fanciful hypotheses had discharged 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 in- 

 duction, since they represented, with sufficient ac- 

 curacy, the best observations. And we 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 



GG2 



