About this time, Clairaut laid the foundation of his ex cellent work Sur les courbes a double eourbure, which he finished at the age of 13, with an assiduity which threw him into a violent fever. He continued his investigations on this subject, and in 1730 they were published, with a certificate from the Academy, which expressly stated, that the author had accomplished, at the age of 16, what would have done honour to the most celebrated geome ters.
The academicians were now desirous to rank Clairaut among their number ; but, by the rules of the Academy, this could not be accomplished till he had reached his 20th year. Application was, however, made to the king, and a special dispensation was granted in favour of the young candidate, who, on the 14th July 1731, was admitted adjunct mechanician at the early age of 18.
The joy with which this event naturally inspired M. Clairaut and his family, was painfully alloyed by the death of the younger brother, whom we have already mention ed, and who was carried off in two days by the small-pox.
After Clairaut had recovered from this dreadful shock, which for a while threatened his own life, he chearfully accepted an offer from M. 1\Iaupertuis, to accompany him to Basle on a visit to John Bernoulli. On his re turn from this excursion, with which he was peculiarly delighted, he found his associates in the Academy occu pied with the question of the figure of the earth, and he resolved to devote his attention to this important subject. With this view, he and M. Maupertuis retired to Mont Valerien, where they formed the project of a voyage to Lapland, in the execution of which our author was of the most essential service. The Marchioness of Chatelet having taken a fancy for the study of geometry, went fre quently on horseback to visit Clairaut at Mont Valerien ; and it was for this lady that he composed his Elemens dc Geometric, which appeared in 1741.
After his return from measuring a degree of the me ridian in Lapland, the king granted him a pension of a thousand livres, which in less than a year was followed by another that had been vacated by M. Chevalier.
In the year 1758, on the death of Bouguer, his pension of 3000 livres was divided between Clairaut and Lc Monnier; and in consequence of this encouragement, he laid before the Academy an excellent Memoir on Naval Tactics.
It would lead us much beyond the limits of a biogra phical article, were we to follow Clairaut through the numerous memoirs which he communicated to the Aca demy of Sciences. The progress of his researches, and the various subjects which attracted his attention, will he more readily seen from the list of his Memoirs which is given at the end of the present article. We shall there fore only notice at present such of his works as have been most conducive to the advancement of science.
In his Theorie de la Figure de la Terre suivant lee principes de l'Hydrostatulue, which appeared in 1743, he has treated this interesting subject with unusual ability. He has determined, upon the Newtonian theory, what would be the figure of the earth if it were wholly fluid ; and either uniformly dense, or consisting of concentric strata of variable density : Ile has examined what would take place if a part of it were rendered solid ; and by an application of the same principles, he has given the first philosophical theory of capillary attraction.* In the year 1747, Clairaut shared the honour, along with Euler and d'Alembert, of having solved the problem of three bodies. In the three solutions of this great problem which had been separately obtained, without any communication, by these illustrious mathematicians, it turned out, to the astonishment of them all, that the calculus gave only half the observed motion of the moon's apogee. Clairaut proposed to make a slight alteration upon the law of gravity, in order to produce a corres• pondence between the theory and the observations ; but this idea was warmly attacked by Buffon, and Clairaut at last discovered that he and his learned associates had neglected some small quantities in the series, and that the result thus corrected was perfectly consonant to the theory of gravitation. Ile gained the prize for this sub ject which was given in 1751 by the Academy of St Petersburgh, and in 1754 he published his Lunar Tables, which were founded upon his own solution of the pro blem of three bodies, and which greatly exceeded in ac curacy those that had been formerly in use. A new and correct edition of this work was published in 1765.