440 HISTORY OF FORMAL ASTRONOMY. 



most certain and exact thing that the proportion 

 which exists between the periodic times of any 

 two planets is precisely the sesquiplicate of the 

 proportion of their mean distances ; that is, of the 

 radii of the orbits. Thus, the period of the earth is 

 one year, that of Saturn thirty years; if any one 

 trisect the proportion, that 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 distance of 

 the earth." 



When we now look back at the time and exer- 

 tions which the establishment of this law cost Kep- 

 ler, 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 dis- 

 tances and the periodic times. What law of con- 

 nexion could be more simple and obvious, we might 

 say, than that one of these quantities should vary 

 as some power 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 proportional ? And 





