450 HISTORY OF FORMAL ASTRONOMY. 



making the distances too short at mean longitudes ; 

 of which he informed me by letter while I was 

 labouring, by repeated efforts, to discover the true 

 hypothesis. So nearly did he get the start of me 

 in detecting the truth." But this was less easy 

 than it might seem. When Kepler's first hypothesis 

 was enveloped in the complex construction re- 

 quisite in order to apply it to each point of the 

 orbit, it was far more difficult to see where the 

 errour lay, and Kepler hit upon it only by noticing 

 the coincidences of certain numbers, which, as he 

 says, raised him as if from sleep, and gave him 

 a new light. We may observe, also, that he was 

 perplexed to reconcile this new view, according to 

 which the planet described an exact ellipse, with 

 his former opinion, which represented the motion 

 by means of libration in an epicycle. "This," he 

 says, "was my greatest trouble, that, though I 

 considered and reflected till I was almost mad, I 

 could not find why the planet to which, with so 

 much probability, and with such an exact accord- 

 ance of the distances, the libration in the diameter 

 of the epicycle was attributed, should, according 

 to the indication of the equations, go in an ellip- 

 tical 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 geome- 



