330 LAPLACE. 



or great. Certain numerical relations between the prim- 

 itive elements of the disturbing and disturbed planets 

 may impart sensible values to terms which usually admit 

 of being neglected. This case occurs in the perturba- 

 tions of Saturn produced by Jupiter, and in those of 

 Jupiter produced by Saturn. There exists between the 

 mean motions of these two great planets a simple relation 

 of commensurability, five times the mean motion of Sa- 

 turn, being, in fact, very nearly equal to twice the mean 

 motion of Jupiter. It happens, in consequence, that cer- 

 tain terms, which would otherwise be very small, acquire 

 from this circumstance considerable values. Hence arise 

 in the movements of these two planets, inequalities of 

 long duration which require more than 900 years for 

 their complete development, and which represent with 

 marvellous accuracy all the irregularities disclosed by 

 observation. 



Is it not astonishing to find in the commensurability of 

 the mean motions of two planets, a cause of perturbation 

 of so influential a nature ; to discover that the definitive 

 solution of an immense difficulty which baffled the 

 genius of Euler, and which even led persons to doubt 

 whether the theory of gravitation was capable of account- 

 ing for all the phenomena of the heavens should depend 

 upon the fortuitous circumstance of five times the mean 

 motion of Saturn being equal to twice the mean motion 

 of Jupiter? The beauty of the conception and the ulti- 

 mate result are here equally worthy of admiration.* 



* The origin of this famous inequality may be best understood by 

 reference to the mode in which the disturbing forces operate. Let 

 p Q R, p' Q' Tcil repsesent the orbits of Jupiter and Saturn, and let us 

 suppose, for the sake of illustration, that they are both situate in the 

 same plane. Let the planets be in conjunction at p, p', and let them 

 both be revolving around the sun s, in the direction represented by 



