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 action 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