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 is
established there is something wrong and the requirements of
astronomical science are not satisfied. The complete solution of
this problem was entirely beyond the power of Newton. When his
methods of research were used he was indeed able to show that the
mutual action of the planets would produce deviations in their
motions of the same general nature with those observed, but he was
not able to calculate these deviations with numerical exactness.
His most successful attempt in this direction was perhaps made in
the case of the moon. He showed that the sun's disturbing force on
this body would produce several inequalities the existence of
which had been established by observation, and he was also able to
give a rough estimate of their amount, but this was as far as his
method could go. A great improvement had to be made, and this was
effected not by English, but by continental mathematicians.

The latter saw, clearly, that it was impossible to effect the
required solution by the geometrical mode of reasoning employed by
Newton. The problem, as it presented itself to their minds, was to