The contributions of France to these revolutions in astronomical science consisted, in 1740, in the determination by experiment of the spheroidal figure of the earth, and in the discovery of the local variations of gravity upon the surface of our planet.  These were two great results; but whenever France is not first in science she has lost her place.  This rank, lost for a moment, was brilliantly regained by the labors of four geometers.  When Newton, giving to his discoveries a generality which the laws of Kepler did not suggest, imagined that the different planets were not only attracted by the sun, but that they also attracted each other, he introduced into the heavens a cause of universal perturbation.  Astronomers then saw at a glance that in no part of the universe would the Keplerian laws suffice for the exact representation of the phenomena of motion; that the simple regular movements with which the imaginations of the ancients were pleased to endow the heavenly bodies must experience numerous, considerable, perpetually changing perturbations.  To discover a few of these perturbations, and to assign their nature and in a few rare cases their numerical value, was the object which Newton proposed to himself in writing his famous book, the ’Principia Mathematica Philosophiae Naturalis’ [Mathematical Principles of Natural Philosophy], Notwithstanding the incomparable sagacity of its author, the ‘Principia’ contained merely a rough outline of planetary perturbations, though not through any lack of ardor or perseverance.  The efforts of the great philosopher were always superhuman, and the questions which he did not solve were simply incapable of solution in his time.

Five geometers—­Clairaut, Euler, D’Alembert, Lagrange, and Laplace—­shared between them the world whose existence Newton had disclosed.  They explored it in all directions, penetrated into regions hitherto inaccessible, and pointed out phenomena hitherto undetected.  Finally—­and it is this which constitutes their imperishable glory—­they brought under the domain of a single principle, a single law, everything that seemed most occult and mysterious in the celestial movements.  Geometry had thus the hardihood to dispose of the future, while the centuries as they unroll scrupulously ratify the decisions of science.

If Newton gave a complete solution of celestial movements where but two bodies attract each other, he did not even attempt the infinitely more difficult problem of three.  The “problem of three bodies” (this is the name by which it has become celebrated)—­the problem of determining the movement of a body subjected to the attractive influence of two others—­was solved for the first time by our countryman, Clairaut.  Though he enumerated the various forces which must result from the mutual action of the planets and satellites of our system, even the great Newton did not venture to investigate the general nature of their effects.  In the midst of the labyrinth formed by increments