s 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 find algebraic
expressions for the positions of the planets at any time. The latitude,
longitude, and radius-vector of each planet are constantly varying, but
they each have a determined value at each moment of time. They may
therefore be regarded as functions of the time, and the problem was to
express these functions by algebraic formulae. These algebraic
expressions would contain, besides the time, the elements of the
planetary orbits to be derived from observation. The time which we may
suppose to be represented algebraically by the symbol t, would remain
as an unknown quantity to the end. What the mathematician sought to do
was to present the astronomer with a series of algebraic expressions
containing t as an indeterminate quantity, and so, by simply
substituting for t any year and fraction of a year whatever--1600,
1700, 1800, for example, the result would give the latitude, longitude,
or radius-vector of a planet.

The problem as thus presented was one of the most difficult we can
perceive of, but the difficulty was only an incentive to attacking it
with all the greater energy. So long as the motion was supposed purely
elliptical, so long as the action of the planets was neglected, the
problem was a simple one, requiring for its solution only the analytic
geometry of the ellipse. The real difficulties commenced when the
mutual a