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.
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