268 DISCOURSE ON THE STUDY 



analogy to all the planets, was found to be verified in 

 the case of each. This with the other remarkable 

 laws which are usually cited in physical astronomy 

 by the name of Kepler's laws, constitute undoubt- 

 edly the most important and beautiful system of 

 geometrical relations which have ever been disco- 

 vered by a mere inductive process, independent of 

 any consideration of a theoretical kind. They com- 

 prise within them a compendium of the motions of 

 all the planets, and enable us to assign their places 

 in their orbits at any instant of time past or to come 

 (disregarding their mutual perturbations), provided 

 certain purely geometrical problems can be numeri- 

 cally resolved. 



(297.) It was not, however, till long after Kepler's 

 time that the real importance of these laws could be 

 felt. Regarded in themselves, they offered, it is true, 

 a fine example of regular and harmonious disposition 

 in the greatest of all the works of creation, and a 

 striking contrast to the cumbersome mechanism of 

 the cycles and epicycles which preceded them; but 

 there their utility seemed to terminate, and, indeed, 

 Kepler was reproached, and not without a semblance 

 of reason, with having rendered the actual cal- 

 culation of the places of the planets more difficult 

 than before, the resources of geometry being then 

 inadequate to resolve the problems to which the 

 strict application of his laws gave rise. 



(298.) The first result of the invention of the 

 telescope and its application to astronomical pur- 

 poses, by Galileo, was the discovery of Jupiter's 

 disc and satellites, of a system offering a beautiful 

 miniature of that greater one of which it forms a 



