mathematical sense. In whatever way we go to work the expression
comes out in the form of an infinite series of terms, each term
being, on the whole, a little smaller as we increase the number.
So, by increasing the number of these various terms, we can
approach nearer and nearer to a mathematical exactness, but can
never reach it. The mathematician and astronomer have to be
satisfied when they have carried the solution so far that the
neglected quantities are entirely beyond the powers of
observation.

Mathematicians have worked upon the problem in its various phases
for nearly two centuries, and many improvements in detail have,
from time to time, been made, but no general method, applicable to
all cases, has been devised. One plan is to be used in treating
the motion of the moon, another for the interior planets, another
for Jupiter and Saturn, another for the minor planets, and so on.
Under these circumstances it will not surprise you to learn that
our tables of the celestial motions do not, in general, correspond
in accuracy to the present state of practical astronomy. There is
no authority and no office in the world whose duty it is to look
after the preparations of the formulae I have described. The work
of computing them has been almost entirely left to individual
mathematicians whose taste lay in that direction, and who have
sometimes devoted the greater part of their lives to calculations
on a single part of the work. As a striking instance of this, the
last great work on the Motion of the Moon, that of Delaunay, of
Paris, involved some fifteen years of continuous hard labor.

Hansen, of Germany, who died five years ago, devoted almost his
whole life to investigations of this class and to the development