Uranus requires no
less than eighty-four years to accomplish his mighty revolution around
the sun. The planet has completed one entire revolution since its
discovery, and up to the present time (1900) has accomplished more than
one-third of another. For the careful study of the nature of the orbit,
it was desirable to have as many measurements as possible, and extending
over the widest possible interval. This was in a great measure secured
by the identification of the early observations of Uranus. An
approximate knowledge of the orbit was quite capable of giving the
places of the planet with sufficient accuracy to identify it when met
with in the catalogues. But when by their aid the actual observations
have been discovered, they tell us precisely the place of Uranus; and
hence, instead of our knowledge of the planet being limited to but
little more than one revolution, we have at the present time information
with regard to it extending over considerably more than two revolutions.

From the observations of the planet the ellipse in which it moves can be
ascertained. We can compute this ellipse from the observations made
during the time since the discovery. We can also compute the ellipse
from the early observations made before the discovery. If Kepler's laws
were rigorously verified, then, of course, the ellipse performed in the
present revolution must differ in no respect from the ellipse performed
in the preceding, or indeed in any other revolution. We can test this
point in an interesting manner by comparing the ellipse derived from the
ancient observations with that deduced from the modern ones. These
ellipses closely resemble each other; they are nearly the same; but it
is most important to observe that they are not _exactly_ the same, even
when allowance has been made for every known source of disturbance in
accordance with the principles explained in the next chapter. The law of
Kepler seems thus not absolutely true in the case of Uranus. Here is,
indeed, a matter demanding our most earnest and careful attention. Have
we not repeatedly laid down the universality of the laws of Kepler in
controlling the planetary motions? How then can we reconcile this law
with the irregularities proved beyond a doubt to exist in the motions of
Uranus?

Let us look a little more closely into the matter. We know that the laws
of Kepler are a consequence of the laws of gravitation. We know that the
planet moves in an elliptic path