The talents of D'Alembert were in some measure known by some corrections on the 4nalyse Denzontree of Father Rcinau, which had occurred to him when stu dying that work; and in 174 i , he was thought worthy, at the age of 24, to be admitted a member of the Acade my of Sciences.
In the year 1743, appeared the Traite de Dyna»zique of D'Alembert, founded on a new and fertile principle of mechanics. He has shown, that in whatever way the bodies of one system act upon those of another, the ac tion of the moving forces may, at every instant, be de composed into two parts, the one producing the motion of the body in the second instant, and the other destroy ing that which it had in the first. This simple principle had been in some measure employed by James Bee nouilli, in his solution of the problem of the centre of oscillation ; but D'Alembert made the discovery his own, by simplifying arid generalizing it, and by applying it to the resolution of many beautiful problems, and par ticularly that of the procession of the equinoxes.
The mathematical sciences were enriched with an other discovery by this illustrious geometer. The equa tions to which he was conducted in the application of his new principle to the theory of fluids, often resisted all the resources of the common analysis. A new calculus, therefore, seemed necessary her their resolution, and D'Alembert had the high honour of making the dis covery. In his Rejlexion sur trs causes generates des Vents which was crowned by the AcatIciny of Berlin in 1746, he gave the first details of the integral calculus with partial differences; and in 1717, he applied it to the solution of Dr Taylor's problem of librating chords. Euler seems to have given some hints concerning this calculus in a paper published in the Pete•sburgh T•ans actions for 1734 ; but D'Alembert has evidently the honour of the discovery, though the nature and algo rithm of the calculus was afterwards completely ex plained by Euler, in his paper, entitled, "investigatio Ea»ctionunz, Sze." 1762.
Ills treatise on Dynamics was followed, in 1752, by his ESN(ti3 d'une nouvelle Thcorie du mouvenzent ilex Fluides a work of singular merit, by his P:tenzeii8 Theorique et Pratique,suivant les Principes de Ranzeau, 1752 ; and by his Recherches sir le Calcul Int(gral, in the Memoirs of the Berlin Academy for the same year.
Hitherto the talents of D'Alembert had been applied chiefly to geometry and physics ; and though both these' branches of knowledge were deeply indebted to his genius, yet the reputation which he enjoyed had scarcely extended beyond the small circle of friends, with whom he was daily associated. The only persons in office with whom he was acquainted, were the two ministers, Messrs D'Argenson, men who were distin guished by their abilities, as well as by the most amia ble qualities of the heart. His fame, however, began to extend, and, as if it had been confined by some unnatu ral restraint, it now spread with astonishing rapidity. The dedication of his treatise on Winds, to the king of Prussia, had procured him the notice of that patron of merit ; and he was now rewarded with a pension from the French king, a favour which he owed more to the friendship of the count D'Argenson, than to the wisdom and generosity of his sovereign.
While D'Alembert was treading in the thorny paths of geometry, lie had few rivals, and therefore few en e mies. But when the Encyclopaedia, which he had under taken in conjunction with Diderot, gave him frequent opportunities of displaying Ids fine taste for literature and the arts, a crowd of inferior authors, who felt their reputation obscured by the splendour of his name, con spired to disturb his repose, and to bring him down to their own level, by the formidable weapons of calumny and detraction. The discourse, which lie prefixed to this immortal work, and the other articles, with which he enriched it, exhibited a depth and fertility of under standing, an elegance and correctness of taste, a bril liancy and luxuriance of imagination, a richness and splendour of eloquence, a clearness and energy of style, which are seldom united in the same character, and which have raised him to a most distinguished place among those illustrious philosophers, who have been the brightest ornaments of their species.