300 HISTORY OF FORMAL ASTRONOMY. 



3. The bisection of the eccentricity, and a stationary point of equa- 

 tions, after the manner of Ptolemy. 



4. The vicarious hypothesis by a free section of the eccentricity 

 made to agree as nearly as possible with the truth. 



5. The physical hypothesis on the supposition of a perfect circle. 



6. The physical hypothesis on the supposition of a perfect ellipse. 

 By the physical hypothesis, he meant the doctrine that the time of 



a planet's describing any part of its orbit is proportional to the distance 

 of the planet from the sun, for -which supposition, as we have said, he 

 conceived that he had assigned physical reasons. 



The two last hypotheses came the nearest to the truth, and differed 

 from it only by about eight minutes, the one in excess and the other 

 in defect. And, after being much perplexed by this remaining error, it 

 at last occurred to him 12 that he might take another ellipsis, exactly in- 

 termediate between the former one and the circle, and that this must 

 give the path and the motion of the planet. Making this assumption, 

 and taking the areas to represent the times, he now saw 13 that both 

 the longitude and the distances of Mars would agree with observation 

 to the requisite degree of accuracy. The rectificatibn of the former 

 hypothesis, when thus stated, may, perhaps, appear obvious. And 

 Kepler informs us that he had nearly been anticipated in this step 

 (c. 55). "David Fabricius, to whom I had communicated my hypoth- 

 esis of cap. 45, was able, by his observations, to show that it erred in 

 making the distances too short at mean longitudes; of which he 

 informed me by letter while I was laboring, 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 con- 

 struction requisite in order to apply it to each point of the orbit, it 

 was far more difficult to see where the error 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, accord- 

 ing 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 



" Be Stella Martis, c. 58. 13 Ibid. p. 235. 



