considered
necessary by Ptolemy in order to represent the inequalities in the
motions of the planets around the sun.

The next great advance made in the theory of the planetary motion was
the discovery by Kepler of the celebrated laws which bear his name.
When it was established that each planet moved in an ellipse having the
sun in one focus it became possible to form tables of the motions of
the heavenly bodies much more accurate than had before been known. Such
tables were published by Kepler in 1632, under the name of Rudolphine
Tables, in memory of his patron, the Emperor Rudolph. But the laws of
Kepler took no account of the action of the planets on one another. It
is well known that if each planet moved only under the influence of the
gravitating force of the sun its motion would accord rigorously with
the laws of Kepler, and the problems of theoretical astronomy would be
greatly simplified. When, therefore, the results of Kepler's laws were
compared with ancient and modern observations it was found that they
were not exactly represented by the theory. It was evident that the
elliptic orbits of the planets were subject to change, but it was
entirely beyond the power of investigation, at that time, to assign any
cause for such changes. Notwithstanding the simplicity of the causes
which we now know to produce them, they are in form extremely complex.
Without the knowledge of the theory of gravitation it would be entirely
out of the question to form any tables of the planetary motions which
would at all satisfy our modern astronomers.

When the theory of universal gravitation was propounded by Newton he
showed that a planet subjected only to the gravitation of a central
body, like the sun, would move in exact accordance with Kepler's laws.
But by his theory the planets must attract one another and these
attractions must cause the motions of each to deviate slightly from the
laws in question. Since such deviations were actually observed it was
very natural to conclude that they were due to this cause, but how
shall we prove it? To do this with all the rigor required in a
mathematical investigation it is necessary to calculate the effect of
the mutual action of the planets in changing their orbits. This
calculation must be made with such precision that there shall be no
doubt respecting the results of the theory. Then its results must be
compared with the best observations. If the slightest outstanding
difference i