1814.] Biographical Account of M. Lagrange. 3 



ii'ould wish to see in a work of pure analysis. The author admitted 

 this himself. He allowed the necessity of a demonstration inde- 

 pendent of geometry. " He appeared to doubt the resources of 

 analysis, and terminated his work by saying, If my principle be not 

 sufficiently demonstrated, yet as it is conformable to truth, I have no 

 doubt that by means of a rigid metaphysical explanation it may be 

 put in the clearest light, and I leave that task to the metaphysicians. 



" This appeal, to which the metaphysicians paid no attention, was 

 listened to by Lagrange, and excited his emulatio?i." In a short time 

 the young man found the solution of which Euler had despaired. He 

 found it by analysis. " And in giving an account of the xvay in 

 which he had been led to that discovery, he said expressly, and as it 

 were in answer to Eider's doubt, that he regarded it not as a meta- 

 physical principle, but as a necessary result of the laws of mecha- 

 nics, as a simple corollary from a more general law which he 

 afterwards made the foundation of his Mechanique Analytique." — 

 (See that work, p. 189 of the first edition.)* 



Let us now give some exact details. 



Euler, in his famous work on isoperimetres (Methodus Inve- 

 niendi, &c.) expressed a wish that a purely analytical solution of 

 the general question could be found. He expresses himself as follows 

 in the 56th page of that work : " Desideratur itaque methodus a 

 resolutione geometrica et lineari lihera, qua pateat in tali investiga- 

 tione maximi minimive, loco P dp scribi debere — pd P." This 

 was the appeal to which Lagrange listened, and which he answered 

 by the discovery of the method of variations. He has said nothing 

 respecting the ideas which led to the discovery. It was not pub- 

 lished till 1762, though he had communicated it to Euler by letter 

 as early as 1755. (See the Melanges de Turin, t. iv. p. 163.) 



On the other hand, at the end of his work on isoperimetres, 

 Euler had introduced two appendixes ; one on elastic curves, the 

 other on the motion of projectiles. In this last he demonstrated 

 that " in the trajectories described by central forces the integral of 

 the velocity multiplied by the element of the curve is always either 

 a maximum or a minimum." But he only perceived this property 

 in the motion of isolated bodies, and made vain attempts to extend 

 it to the motion of those which act upon each other in any manner 

 whatever. All he could do was to satisfy himself, by a metaphysical 

 argument, that it ought to apply in these cases also ; and he termi- 

 nated his dissertation in the following manner: " Cujus ratiocinii 

 vis, etiamsi nondum satis pcrsp'uiatur, tamen quia cum veritate 

 eongruit, non dubito quin, ope principiorum sanioris mctaphysicae, 

 ad majorem evidentiam evehi queat ; quod negotium aliis, qui 

 metaphysieam profitentur, relinquo." 



Lagrange, being in possession of the method of variations, did 



* In transcribing tUeie paragraph* llie passngfi Hint will not Apply to the 

 Mirtliuil of vurialiout bave b«tn narked With inverted commut. 



A 2 



