102 LIFE AND WORKS OF KEPLER. 



in presence of sncli a problem require that patience slionld come to the 

 aid of genins. Kei)ler attempted at first the verification of the hypo- 

 theses ])revionsly aihnitted, by seeking- to place all his points on the 

 same circle; but his efforts were futile ; his calculations left errors of 

 sev^en to eight minutes subsisting, and he proved that no better could 

 be done. Eight minutes seem but a small matter; it is about a fourth 

 of the apparent diameter of the sun ; but it is in astronomy especially 

 that it may be said with truth : Jle who despises small thhu/s shall fall hi/ 

 little and little. Kepler knew it, and this little error, which he was 

 unwilling to accept, became considerable by its consequences. ^' The 

 Divine Goodness," he Tsays, " has given us in Tycho an observer so exact 

 that an error of eight minutes is impossible." The hypothesis of a cir- 

 cular orbit was therefore inadmissible; but Kepler does not on that 

 account despair of victory, nor is his confidence at all shaken. He 

 fancies that, like the wanton Galatea (in Virgil,) Mars flies to covert, 

 but in hiding' wishes not to escape unseen : . 



Et fugit ad salices, et se ciqnt ante vhleri. 



This is the first line of his fifty-eighth chapter. 



After numerous attempts and laborious calculations, Kepler at last 

 found that an elliptical orbit satisfies all the observations of Tycho ; 

 then it was that, as he, expresses himself in his preface, he regarded 

 Mars as a prisoner on parole. In a position thenceforth to interrogate 

 the captive at leisure, he continued to press the inquiry more closel}', 

 by marking the places which the new theory indicated for the future, 

 and he had the satisfaction of seeing the planet, punctual to the appoint- 

 ments which he had fixed, respond, so to say, to his summons, as the 

 stars reply to their Creator in the book of Baruch, which La rontaiue 

 so much admired: You have called, me-, behold, I am here! 



This complete and persistent conformity furnished the irresistible 

 evidence of the two celebrated laws which he could at length announce 

 with certainty : Mars describes an ellipsis of which the sun occupies a 

 focus. The areas described by the radius vector are proportional to 

 the time. 



But our statement of the great discovery of Kepler would be incom- 

 plete did we not particularize two remarkable circumstances whicli, 

 coming fortuitously to the aid of his penetration, conducted him with 

 more facility to the goal from which they might otherwise have turned 

 him aside. 



The movement of the earth, the presumed knowledge of which had 

 served as a basis for all his calculations, was theoretically as imi)erfectly 

 known as that of Mars. The circle in which he makes our planet move 

 should be replaced by an ellipsis ; but this ellipsis fortunately differs 

 from a circle in a sufficiently small degree to render the substitution of 

 one for the other a matter of imlifterence for the rate of approximation 

 to be adopted. Had it been otherwise, the method v>"ould have become 

 inexact, and the numbers, by contradicting one another, would have 

 warned and discouraged the accurate and conscientious inquirer. The 

 second circumstance, still more remarkable, perhaps, was the imperfec- 

 tion of the methods of observation and of the instruments of Tycho. 

 Kepler might affirm, it is true, that an error of eight minutes was 

 impossible, and this confidence saved everything ; had he said as much 

 of an error of eight seconds, all would have been lost. The internal 

 organ of judgment, to use au expression of Goethe, would have ceased 

 to be in harmony with the external organ of sight, become too delicate 

 and too piecise. 



