ame extent as it
previously diminished, and according to the same laws.

Now, Laplace has shown that the mean motion of the moon around the
earth is connected with the form of the ellipse which the earth
describes around the sun; that a diminution of the eccentricity of the
ellipse inevitably induces an increase in the velocity of our satellite,
and _vice versa_; finally, that this cause suffices to explain the
numerical value of the acceleration which the mean motion of the moon
has experienced from the earliest ages down to the present time.[33]

The origin of the inequalities in the mean motions of Jupiter and Saturn
will be, I hope, as easy to conceive.

Mathematical analysis has not served to represent in finite terms the
values of the derangements which each planet experiences in its movement
from the action of all the other planets. In the present state of
science, this value is exhibited in the form of an indefinite series of
terms diminishing rapidly in magnitude. In calculation, it is usual to
neglect such of those terms as correspond in the order of magnitude to
quantities beneath the errors of observation. But there are cases in
which the order of the term in the series does not decide whether it be
small or great. Certain numerical relations between the primitive
elements of the disturbing and disturbed planets may impart sensible
values to terms which usually admit of being neglected. This case occurs
in the perturbations 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 Saturn, being, in fact, very nearly equal to
twice the mean motion of Jupiter. It happens, in consequence, that
certain 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
acc