ction of the planets was taken into account. It is, of course,
out of the question to give any technical description or analysis of
the processes which have been invented for solving the problem; but a
brief historical sketch may not be out of place. A complete and
rigorous solution of the problem is out of the question--that is, it is
impossible by any known method to form an algebraic expression for the
co-ordinates of a planet which shall be absolutely exact in a
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 of new
methods of computation. His tables of the moon are those now used for
predicting the places of the moon in all the ephemerides of the world.

The only successful attempt to prepare systematic tables for all the
large planets is that comple