remarkable that he introduced an epicycle which was not 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