Contemporary with La Caille were John James Cas sini, Bouguer, Condamine, and Maupertuis. Cassini pub lished a treatise on astronomy, with astronomical tables, founded on the observations of his father. Bouguer wrote h valuable work on the figure of the earth. In conjunction with Condamine, he went to South America, to measure a degree of the meridian in Peru ; and they were the first who ascertained, by actual experiment, the attraction of mountains, by determining the deflec tion of the plumbline produced by the action of Chim boraco. (See ATTRACTION of Mountains). Maupertuis was at the head of the French astronomers who under took a journey to Lapland to measure a degree of the meridian.
The great errors in astronomical tables, and particu larly in those of the moon, turned the attention of ma thematicians to the subject of physical astronomy. The valuable improvements which the higher analysis had received since the time of Newton, enabled his succes sors to complete the great work of which he had laid the foundation. In the year 1747, Clairaut, D'Alembert, and Euler, three of the first mathematicians of the age, un dertook, without any knowledge of each other's inten tions, to solve the famous problem of three bodies, or to determine the curves described by three bodies project ed from three points given in position, and with veloci ties given in quantity and direction, the force with which they gravitate being directly as their quantities of mat ter, and inversely as the squares of their distance. The object of this problem, which is susceptible only of 'CI approximate solution, was to investigate all the lunar inequalities which could arise from the theory of gravity. Newton had explained only five of the principal equa tions of the moon's orbit, and it was obvious that there were many other irregularities, which observation alone could never have detected. The solution of the problem obtained by the three philosophers, seemed at first to overturn the whole theory of gravitation. In conse quence of all of them having neglected some small quan tities in the approximation of the series which repre sented the motion of the apogee, it appeared that gravity did not diminish in the inverse ratio of the square of the distance. Clairaut had the honour of being the first who detected this error. He therefore rectified his solu tion, and finding that the result was in perfect accor dance with the theory of gravity, he computed a set of lunar tables, which were vastly superior, in point of ac curacy, to those which had hitherto appeared.
The celebrated Tobias Mayer of Gottingen, who had directed his whole attention to the improvement of the solar and lunar tables, computed a new set, founded on the solution of Euler, and carefully corrected by a num ber of his own accurate observations. These tables
gave the longitude of the moon within thirty seconds. A copy ol them was sent to the lords of the Admiralty in 1755 ; and they were found to be extremely correct, by a comparison with the observations of Bradley. Until the day ol his death, however, Mayer continued to ren der them more perfect ; and he left behind him a com plete set of solar and lunar tables, for which his widow received the sum or L3000, a part of the reward that was offered for the discovery of the longitude. In addition to these labours, Mayer constructed an accurate cata logue of 992 fixed stars ; he observed the proper mo tions of several of these bodies ; and he delineated an excellent map of the moon's disc, from a number of ac curate observations on the longitude and latitude of the lunar spots.
The labours of Clairaut were not confined to the solu tion or the problem of three bodies. At a very early age, he accompanied Maupertuis in his journey to mea sure a degree in Lapland, and his attention was na turally turned to the figure of the earth. In the ex cellent work which he has published on this subject, he determined the figure which the earth would assume if it were composed of homogeneous materials; he found that the diminution of gravity, on different parts ol its surface, was always proportional to the square of the cosine of the latitude, if the spheroid were composed of concentric strata, whose density and ellipticity followed any law whatever from the centre to the circumference; and he ascertained the figure which the earth would, assume if it consisted entirely of a number of concentric fluid strata of different densities. Clairaut had the merit of applying to the motion of comets his solution of the problem of three bodies. He surmounted the difficulty arising from the great eccentricity of their orbits ; and he computed the action ol Jupiter and Saturn on the motion of the comet which, according to the prediction of Hal ley, was expected to return about the beginning of 1759. These calculations led him to expect, that the comet would pass its perihelion on the 13th April 1759 ; but it had reached that point of its orbit on the 13th March. By revising his calculations, M. Clairaut reduced this error of thirty days to twenty-two days ; and, in a paper on comets, which shared the prize of the Academy of St Petersburg with Albert Euler, he afterwards reduced it to nineteen days.