INDUCTIVE EPOCH OF KEPLER. 299 



in proportion as it was supported by the authority of all philosophers, 

 and apparently agreeable to metaphysics." But before he attempts to 

 correct this erroneous part of his hypothesis, he sets about discovering 

 the law according to which the different parts of the orbit are described 

 in the case of the earth, in which case the eccentricity is so small that 

 the effect of the oval form is insensible. The result of this inquiry was 8 

 the Rule, that the time of describing any arc of the orbit is proportional 

 to the area intercepted between the curve and two lines drawn from 

 the sun to the extremities of the arc. It is to be observed that this 

 rule, at first, though it had the recommendation of being selected after 

 the unavoidable abandonment of many, which were suggested by the- 

 notions of those times, was far from being adopted upon any very 

 rigid or cautious grounds. A rule had been proved at the apsides of 

 the orbit, by calculation from observations, and had then been extended 

 by conjecture to other parts of the orbit ; and the rule of the areas 

 was only an approximate and inaccurate mode of representing this 

 rule, employed for the purpose of brevity and convenience, in conse- 

 quence of the difficulty of applying, geometrically, that which Kepler 

 now conceived to be the true rule, and which required him to find the 

 sum of the lines drawn from the sun to every point of the orbit. "When 

 he proceeded to apply this rule to Mars, in whose orbit the oval form 

 is much more marked, additional difficulties came in his way ; and 

 here again the true supposition, that the oval is of that special kind 

 called ellipse, was adopted at first only in order to simplify calculation,'-' 

 and the deviation from exactness in the result was attributed to the 

 inaccuracy of those approximate processes. The supposition of the 

 oval had already been forced upon Purbach in the case of Mercury, 

 and upon Reinhold in the case of the Moon. The centre of the 

 epicycle was made to describe an egg-shaped figure in the former case, 

 and a lenticular figure in the latter. 10 



It may serve to show the kind of labor by which Kepler was led to 

 his result, if we here enumerate, as he does in his forty-seventh Chap- 

 ter," six hypotheses, on which he calculated the longitude of Mars, in 

 order to see Avhich best agreed with observation. 



1. The simple eccentricity. 



2. The bisection of the eccentricity, and the duplication of the 

 superior part of the equation. 



» De Stella Martis, p. 194. » lb. iv. c. 47. 



» L. U. K. Kepler, p. 30. " De Stella Martis, p. 228. 



