D'ALEMBERT. 403 



of his composure. The dates these passages bear, of 

 1761 and 1765, long after his admission into the circle 

 of Madame du DefFand, and his participation in the 

 labours and factions of the Encyclopaedists, the Dide- 

 rots, the Holbachs, the Voltaires, show sufficiently that 

 he had exchanged the peace of geometry for the 

 troubled existence of coterie and party. 



We ought, while on this subject, to add the just and 

 judicious remark of Bossut on the circumstance of 

 James Bernouilli having anticipated in some sort 

 D'Alembert's method of treating dynamical problems : 

 " That the latter seemed to prove, by the numerous 

 and important applications which he had made of his 

 Principle, that in all probability he owed the discovery 

 of it solely to himself." (' Hydrodyn,' I., xv.) 



In treating of Hydrodynamics D'Alembert had found 

 the ordinary calculus insufficient, and was under the 

 necessity of making an important addition to its pro- 

 cesses and its powers, already so much extended by 

 the great improvements which Euler had introduced. 

 This was rendered still more necessary when, in 1746, 

 he came to treat of the winds, and in the following 

 year when he handled the very difficult subject of the 

 vibration of cords, hitherto most imperfectly investi- 

 gated by mathematicians.* In all these inquiries the 

 differential equations which resulted from a geometrical 

 examination of the conditions of any problem, proved 



* Taylor ('Methodus Incrementum') had solved the problem of the 

 vibrating cord's movement, but upon three assumptions that it departs 

 very little from the axis or from a straight line, that all its points come to 

 the axis at the same moment, and that it is of a uniform thickness in its 

 whole length. D'Alembert's solution only requires the last and the first 

 supposition, rejecting the second. The first, indeed, is near the truth, 

 and it is absolutely necessary to render the problem soluble at all. The 

 third has been rejected by both Euler and Daniel Bernouilli, in several 

 cases investigated by them. D'Alembert's solution led to an equation of 



(tf y\ /d* y\ 



J = a 9 ( \ in which t is the 



time of the vibration, x and y the co-ordinates of the curve formed by the 

 vibration. 



