Furthermore, he discovered the long inequality of Jupiter and Saturn,
whose period is 929 years. For an investigation of this also the
Academy of Sciences offered their prize. This led Euler to write a
valuable essay disclosing a new method of computing perturbations,
called the instantaneous ellipse with variable elements. The method
was much developed by Lagrange.

But again it was Laplace who solved the problem of the inequalities of
Jupiter and Saturn by the theory of gravitation, reducing the errors
of the tables from 20' down to 12", thus abolishing the use of
empirical corrections to the planetary tables, and providing another
glorious triumph for the law of gravitation. As Laplace justly said:
"These inequalities appeared formerly to be inexplicable by the law of
gravitation--they now form one of its most striking proofs."

Let us take one more discovery of Halley, furnishing directly a new
triumph for the theory. He noticed that Newton ascribed parabolic
orbits to the comets which he studied, so that they come from
infinity, sweep round the sun, and go off to infinity for ever, after
having been visible a few weeks or months. He collected all the
reliable observations of comets he could find, to the number of
twenty-four, and computed their parabolic orbits by the rules laid
down by Newton. His object was to find out if any of them really
travelled in elongated ellipses, practically undistinguishable, in the
visible part of their paths, from parabolae, in which case they would
be seen more than once. He found two old comets whose orbits, in shape
and position, resembled the orbit of a comet observed by himself in
1682. Apian observed one in 1531; Kepler the other in 1607. The
intervals between these appearances is seventy-five or seventy-six
years. He then examined and found old records of similar appearance in