430 RISE AND PROGRESS OF 



the length of a simple pendulum which will perform its vibrations 

 in the same time. Huygens was the first to give a complete 

 solution of this important problem. James Bernoulli afterwards 

 treated the subject with great elegance ; and derived its solution 

 from the general and self-evident principle that the motions 

 lost by the different parts of the body, in virtue of their mutual 

 connexion, must be in equilibrio. This important and funda- 

 mental principle was afterwards adopted by D'Alembert, as the 

 basis of his treatise on dynamics, and it is now connected with 

 his name. The science of motion was thus reduced to that of 

 equilibrium, and was made to assume the perfect form in which 

 we find it at present. Its application to particular problems is 

 now made by regular and consistent methods, and the success 

 of such applications is limited only by the powers of the 

 analytical instrument which is employed in their development. 

 In the elegance and generality of its methods, the Mecanique 

 Analytique of Lagrange now leaves little to desire ; and we cannot 

 hesitate to rank this great work among the most perfect creations 

 of human genius. 



But though the general methods of dynamical science may be 

 fixed, and its doctrines complete, yet, when we come to apply these 

 to particular problems, we find abundant room for ingenuity and 

 analytical skill. The cases are comparatively few, in which we can 

 proceed to the direct solution of a dynamical problem by a complete 

 integration of the equations on which it depends. The resources of 

 analysis are, in most cases, unequal to such an attempt, and we are 

 generally obliged to content ourselves with approximate determina- 

 tions. This is especially the case in celestial mechanics, when we 

 introduce the consideration of even a third body into the system 

 whose motions we seek to determine ; and in this department of 

 dynamical science, accordingly, there was abundant scope for the 

 highest exercise of the reasoning powers. We have already spoken 

 of the rise of this science in the mind of Newton, and of the vast 

 steps which he made towards its completion : it now remains to 

 say a few words of the additions which it has since received. 



Clairaut was the first writer who made any important addition 

 to physical astronomy, as it had been left by Newton. Observa- 

 tions had shown that the apogee of the moon's orbit was not fixed 

 in space, but moved according to the order of the signs with a 

 motion of about 3 yearly. Newton proved that such a progres- 



