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 baflled 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 QR, P/ Q! RI 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 
