Newton had failed to surmount.

Five geometers, Clairaut, Euler, D'Alembert, Lagrange, and Laplace,
shared between them the world of which Newton had disclosed the
existence. They explored it in all directions, penetrated into regions
which had been supposed inaccessible, pointed out there a multitude of
phenomena which observation had not yet detected; finally, and it is
this which constitutes their imperishable glory, they reduced under the
domain of a single principle, a single law, every thing that was most
refined and mysterious in the celestial movements. Geometry had thus the
boldness to dispose of the future; the evolutions of ages are
scrupulously ratifying the decisions of science.

We shall not occupy our attention with the magnificent labours of Euler,
we shall, on the contrary, present the reader with a rapid analysis of
the discoveries of his four rivals, our countrymen.[25]

If a celestial body, the moon, for example, gravitated solely towards
the centre of the earth, it would describe a mathematical ellipse; it
would strictly obey the laws of Kepler, or, which is the same thing, the
principles of mechanics expounded by Newton in the first sections of his
immortal work.

Let us now consider the action of a second force. Let us take into
account the attraction which the sun exercises upon the moon, in other
words, instead of two bodies, let us suppose three to operate on each
other, the Keplerian ellipse will now furnish merely a rough indication
of the motion of our satellite. In some parts the attraction of the sun
will tend to enlarge the orbit, and will in reality do so; in other
parts the effect will be the reverse of this. In a word, by the
introduction of a third attractive body, the greatest complication will
succeed to a simple regular movement upon which the mind reposed with
complacency.

If Newton gave a complete solution of the question of the celestial
movements in the case wherein two bodies attract each other, he did not
even attempt an analytical investigation of the infinitely more
difficult problem of three bodies. The problem of three bodies (this is
the name by which it has become celebrated), the problem for determining
the movement of a body subjected to the attractive influence of two
other bodies, was solved for the first time, by our countryman
Clairaut.[26] From this solution we may date the important improvements
of the lunar tables effected in the last century.

The most