adverting to the researches of some illustrious mathemati cians who have developed the theory of Newton, and by whose investigations physical astronomy has been raised to its present position. Although the law of gravitation, as proposed by Newton, had from the first been admitted by all the most eminent astronomers of Britain, it was for a long time either opposed or neglected on the Continent. In fact, great improvements were required both in analysis and mechanics before it admitted of other applications than had been made by its great author, or could be regarded as anything more than a plausible hypothesis. Newton demonstrated that if two bodies only were pro jected in space, mutually attracting each other with forces proportional directly to their masses and inversely to the squares of their distance, they would each accurately describe an ellipse round the common centre of gravity ; and the spaces described by the straight line joining that centre and the moving body would be proportional to the time of description, according to the second law of Kepler. But when it is attempted to apply Newton s law to the case of the solar system, great difficulties immediately pre sent themselves. Any one planet in the system is not only attracted by the sun, but also, though in a much smaller degree, by all the other planets, in consequence of which it is compelled to deviate from the elliptic path which it would pursue in virtue of the sun s attraction alone. Now, the calculation of the effects of this disturbing force was the problem which geometers had to resolve. In its most general form it greatly transcends the power of analysis ; but there are particular cases of it (and those, too, the cases presented by nature), in which, by reason of certain limitations in the conditions, it is possible to obtain an approximate solution to any required degree of exactness. For example, the Sun, Moon, and Earth form in a manner a system by themselves, which is very slightly affected by the aggregate attractions of the other planets. In the same way the Sun, Jupiter, and Saturn, form another system, in which the motions are very little influenced by the action of any other body. In these two cases, then, the number of bodies to betaken into consideration is only three ; and in this restricted form, the problem, celebrated in the history of analysis under the denomination of " the Problem of Three Bodies," is susceptible of being treated mathematically. With the hope of improving the lunar tables, and of completing the investigations which Newton had commenced in the Principia, three distinguished geometers Clairaut, D Aleinbert, and .Euler about the middle of the last century, undertook, simultaneously, and without the knowledge of each other, the investigation of the problem of three bodies, and commenced that series
of brilliant discoveries which our own times have seen completed.
Clairaut's solution of the problem of three bodies was presented to the Academy of Sciences in 1747, and was applied to the case of the moon. From this solution he deduced with great facility, not only the inequality of the variation, which Newton had obtained by the application of a more complicated though very ingenious method, but also the evection, the annual equation, and many other in equalities which Newton had not succeeded in connecting with his theory. It happened, however, curiously enough, that in the calculation of one effect of the disturbing force, namely, the progression of the moon s apogee, Clairaut was led into an error which produced a result that threatened to overturn the system of gravitation. The error consisted iu the omission of some of the terms of the series express ing the quantity in question, which he wrongly supposed to have only an insensible value ; and by reason of this omission, his first approximation gave only half of the observed progressive motion of the apogee. As this result was confirmed by D Alembert and Euler, who had both fallen into the same error, it seemed to follow, as a necessary consequence, either that the phenomenon depended on some other cause than the disturbing force of the sun. or that the law of gravitation was not exactly proportional to the inverse square of the distance. The triumph which this result gave to the Cartesians was not of long duration. Clairaut soon perceived the cause of his error; and by repeating the process, and carrying the approximations farther, he found the computed to agree exactly with the observed progression, a result which had the effect of dissipating for ever all doubt respecting the law of gravita tion. The researches of Clairaut were followed by a set of lunar tables, much more correct than any which had been previously computed.
The return of the comet of 1682, which Halley had predicted for the end of 1758 or beginning of 1759, afforded an excellent opportunity for putting to the test both the theory of gravitation and the power of the new calculus. Clairaut applied his solution of the problem of three bodies to the perturbations which this comet sustained from Jupiter and Saturn, and, after calculations of enormous labour, announced to the Academy of Sciences, in November 1758, that the comet would return in the beginning of the following year, and pass through its peri helion about the 15th of April. It returned according to the prediction, but passed its perihelion on the 13th of March. The correction of an error of computation reduced the difference to nineteen days ; and if Clairaut had been aware of the existence of the planet Uranus, he might have come still nearer the truth.
Besides these important researches on the system of the universe, Clairaut composed an admirable little treatise on the figure of the earth, in which he gave the differential equations, till then unknown, of the equilibrium of fluids, whether homogeneous or heterogeneous, supposing an attractive force, following any law whatever, to exist among the molecules. He applied these equations to the earth ; demonstrated that the elliptic figure satisfies the conditions of equilibrium ; and assigned the ellipticity of the different strata of which the earth may be supposed to be formed, to gether with the law of gravitation at the exterior surface. He likewise discovered the important theorem which establishes a relation between the oblateness of the terrestrial spheroid and the increase of gravitation towards the poles, on every supposition which can be imagined relative to the interior construction of the earth. By means of this theorem the ellipticity of the spheroid is deduced from observations of the lengths of the seconds pendulum at different points of the earth s surface.
1749 he published his treatise on the precession of the equinoxes, a work remarkable in the history of analysis and mechanics. By means of his newly invented "Calculus of Partial Differences," and the discovery of a fertile prin ciple in dynamics, he determined from theory the rate of the precession, rather more than 50" in a year. He also deter mined the nutation of the earth s axis, which had been discovered by Bradley, and assigned the ratio of the axes of the small ellipse which the true pole of the earth de scribes around its mean place in the same time in which the nodes of the lunar orbit complete a revolution. The solution of this problem led to the determination of the ratio of the attractive forces of the sun and moon, which D Alembert found to be that of seven to three very nearly ; whence he inferred that the mass of the earth is 70 times greater than that of the moon. He proved likewise that
the precession and nutation are the same iu every hypo