118 NEWTON S PEINCIPIA. 



increase. These variations are greater in Saturn than in 

 any of the others, considerably greater than the varia 

 tions of Mars, which comes the nearest to them. The 

 variation in the eccentricity of Jupiter s orbit is nearly three 

 times as great as in the Earth s ; that of Saturn between 

 five and six times greater than the Earth s; while the 

 variation in the perihelion longitude of the former is 

 about five-ninths of the Earth s variation; and Saturn s 

 exceeds the Earth s in the ratio of about 25 to 18, and 

 exceeds that of Mars only somewhat more than as 49 

 to 48. 



When the attention of mathematicians and astronomers 

 was first directed closely to examine the disturbances of 

 these planets, it appeared hardly possible to reconcile such 

 vast and numerous irregularities, as were found to exist, 

 with the theory of gravitation, or indeed to reduce them 

 under any fixed rule whatever. The case seemed to be 

 come the more hopeless when so consummate an analyst 

 as Euler, the great improver of the calculus, failed in 

 repeated attempts at investigating the subject, committing 

 several important errors which for a time were not de 

 tected, but which showed, or seemed to show, a wide dis 

 crepancy between the theory and the observations. By 

 one discovery, indeed, to which his researches led him, he 

 may be said to have laid the foundation of the most ex 

 traordinary step which has been made in the knowledge 

 of the planetary system. We allude to his theorem on the 

 periodicity of the eccentricities and aphelia of Jupiter and 

 Saturn. But in most other respects his attempts signally 

 failed. D Alembert made little progress in this inquiry ; 

 but at length Lagrange, and still more Laplace, by apply 

 ing all the resources of the calculus, in its last stage of 

 improvement, and after the method of Variations had been 

 systematised, succeeded in reducing the whole to order, 



