INDUCTIVE EPOCH OF KEPLER. 295 



is, take the cube root of it, and double the proportion so found, that 

 is, square it, he will find the exact proportion of the distances of the 

 Earth and of Saturn from the sun. For the cube root of 1 is 1, and 

 the square of this is 1 ; and the cube root of 30 is greater than 3, and 

 therefore the square of it is greater than 9. And Saturn at his mean 

 distance from the sun is at a little more than 9 times the mean dis- 

 tance of the Earth." 



When we now look back at the time and exertions which the estab- 

 lishment of this law cost Kepler, we are tempted to imagine that he 

 was strangely blind in not seeing it sooner. His object, we might 

 reason, was to discover a law connecting the distances and the periodic 

 times. What law of connection could be more simple and obvious, 

 we might say, than that one of these quantities should vary as some 

 poiver of the other, or as some root ; or as some combination of the 

 two, which in a more general view, may still be called a power ? And 

 if the problem had been viewed in this way, the question must have 

 occurred, to what power of the periodic times are the distances pro- 

 portional ? And the answer must have been, the trial being made, 

 that they are proportional to the square of the cube root. This ex- 

 post-facto obviousness of discoveries is a delusion to which we are 

 liable with regard to many of the most important principles. In the 

 case of Kepler, we may observe, that the process of connecting two 

 classes of quantities by comparing their powers, is obvious only to 

 those who are familiar with general algebraical views ; and that in 

 Kepler's time, algebra had not taken the place of geometry, as the 

 most usual vehicle of mathematical reasoning. It may be added, also, 

 that Kepler always sought Ins, formal laws by means of physical rea- 

 sonings ; and these, though vague or erroneous, determined the nature 

 of the mathematical connection which he assumed. Thus in the 

 Mysterium he had been led by his notions of moving virtue of the 

 sun to this conjecture, among others that, in the planets, the increase 

 of the periods will be double of the difference of the distances ; which 

 supposition he found to give him an approach to the actual proportion 

 of the distances, but one not sufficiently close to satisfy him. 



The greater part of the fifth Book of the Harmonics of the Universe 

 consists in attempts to explain various relations among the distances, 

 times, and eccentricities of the planets, by means of the ratios which 

 belong to certain concords and discords. This portion of the work is 

 so complex and laborious, that probably few modern readers have had 

 courage to go through it. Delambre acknowledged that his patience 



