The subject of the figure of the earth, which Clairavt had treated with such distinguished ability, was success fully resumed by D'Alembert. Hitherto it had been sup posed, that the fluid strata were all on a level, or that the direction of gravity was perpendicular to each of the strata. D'Alembert, however, has determined the of different strata of the spheroid, on the supposition that gravity does not act perpendicularly to Inny of these beds, but the superior one. His solution is also applica ble to an infinity of other figures besides the elliptical figure, and is founded on the supposition that the meri dians are not similar, and that there is a difference of density in the points of the same stratum. The genius of D'Alembert, however, shone with greater splendour in his admirable solution of the problem of the precession of the equinoxes. The imperfect state of geometry aniT dynamics prevented Newton from succeeding in this im portant investigation. He obtained, indeed, from theory, the true quantity of the precession of the equinoxes ; but his assumptions were not legitimate, and it was reserved for D'Alembert to treat this subject without the intro duction of hypotheses. By the application of his new principle of dynamics, he has determined, in the most rigorous manner, all the forces which affect the paral lelism of the earth's axis, and which give it a retrograde motion round the poles, and a libration towards the plane of the ecliptic ; and the results of these researches ac cord completely with the observations of Bradley. He has demonstrated, that the precession and nutation remain the same, whatever be the density of the interior strata ; and he has obtained a correct determination of the dimen sions of the small ellipse described by the pole of the ecliptic,—a result which the observations of Bradley had not accurately fixed. D'Alembert afterwards resumed this subject, and by integrating more rigorously the dif ferential equations, by correcting the numerical coeffi cients, and by supposing the meridians to be dissimilar, he obtained some slight corrections on his former con clusions. In the year 1768, D'Alembert published two ingenious memoirs, sheaving, from the principle which he had established in his solution of the precession of the equinoxes, that the elongation of the diameter of the moon which is turned towards the earth, produced a rotation equal to its monthly revolution, and occasioned the singular coincidence which was observed by Cassini in the motion of the nodes of her orbit and her equator. La Grange, however, had the merit of solving this pro blem before D'Alembert. The solution of the problem of three bodies, and that of the precession of the equi noxes ; the discovery of the new calculus of partial dif ferences, and of a fertile principle in dynamics,—are the brilliant additions which the sciences have received from the genius of D'Alembert.
We have already seen that Euler shared with Clai raut and D'Alembert the high honour of having solved the problem of the three bodies. From his own solution of it, he constructed a set of lunar tables, which, at the suggestion of Turgot, were rewarded by the Board of Longitude in France. The application of this impor tant problem to the perturbations produced by the mu tual action of the primary plaidip., was reserved for the genius of this distinguished nrrthematician. The de rangements in the motions of Jupiter and Saturn formed the subject of the two prizes which were offered by the academy of sciences for the years 1748 and 1751, and both of them were carried off by Euler alone. This in
vestigation presented peculiar difficulties, which did not exist in determining the lunar inequalities. The frac tion, which in that case expressed the ratio of the dis tance of the sun to that of the moon, was so small as to produce a rapid convergency in the series ; but in the case of the primary planets, the mutual distance of Jupiter and Saturn is often nearly equal to the distance of Jupiter from the sun, and as the fraction which ex presses the ratio of the distances, is almost equal to unity, the series which represents the action of the one planet upon the other will converge with extreme slowness. The method with which Euler overcame this alarming difficulty, was a new proof of the fertility of his genius, and has been followed in all subsequent researches of the same kind. He concluded, from these investigations, that the inequalities of Jupiter and Saturn, which turned out very considerable, were all periodical, and increa sed, diminished, and vanished alter stated intervals of twenty or thirty years. By a comparison with actual observations, Mr Mayer of Gottingen obtained a com plete confirmation of Euler's theoretical results. It was naturally imagined by astronomers, that the radius of the earth was disturbed in a similar way by the motion of Jupiter, Venus, and the moon ; and the academy of sciences offered their prize in 1756 for the best theory of these derangements. Euler again carried off the prize, and established a general theory of the action of two planets upon one another. It appeared from these inves tigations, that the change in the obliquity of the ecliptic, was owing to the action of the other planets upon the earth ; and Euler demonstrated, that this change, amoun ting to 50" in a century, wa., periodical, and alternately increased and diminished in certain periods of unequal length. The problem of the precession of the equi noxes, which was so happily solved by D'Alembert, re ceived another solution from Euler nearly about the same time ; but the priority certainly belonged to the French philosopher. Euler was the first who suggested that the oblate spheroidal form of Jupiter might produce a sensible change in the law of his attraction ; and it was afterwards found by Walmsley, that from this cause there resulted a perceptible motion in the nodes and the apsides of his satellites. The subject of the nodes was also treated by Euler, but he only shared the prize of the year 1740 with Maclaurin and D. Bernoulli, and we are sorry to add with Father Cavelleri, who explained this phenomenon by means of the vortices of Descartes. Such are the valuable labours for which astronomy is indebted to this illustrious mathematician. In the his tory of geometry, of analysis, and of all the physical sciences, we shall have occasion to admire the unwearied industry, and the inventive genius of Euler.
The nineteenth century was distinguished by a num ber of other astronomers, of whose labours we have not room to give a detailed account, and must therefore refer to the biographical sketches which will be given in the course of our work. The names of Maclaurin, D. Bernoulli, Simpson, Wahnsley, Boscovich, Wargentin, De Lisle, Le Gentil, Mathew Stewart, Albert Euler, Horrebow, Le Monnier, Sejour, Bailly, Frisi, Lambert, Condorcet, Mason, Zanotti, and La Londe, will be asso ciated with the history of astronomy, and will live in the remembrance of posterity, while the human mind con tinues to derive satisfaction from the contemplation of nature.