SEQUEL TO THE GENERALIZATION. 365 



ciple in 1*743, in a great degree superseded their interest. The 

 Transactions of the Academies of Paris and Berlin, as -well as St. 

 Petersburg, are filled, up to this time, with various questions of this 

 kind. They require, for the most part, the determination of the mo- 

 tions of several bodies, with or without weight, which pull or push 

 each other by means of threads, or levers, to which they are fastened, 

 or along which they can slide ; and which, having a certain impulse 

 given them at first, are then left to themselves, or are compelled to 

 move in given lines and surfaces. The postulate of Huyghens, respect- 

 ing the motion of the centre of gravity, was generally one of the 

 principles of the solution ; but other principles were always needed in 

 addition to this ; and it required the exercise of ingenuity and skill to 

 detect the most suitable in each case. Such problems were, for some 

 time, a sort of trial of strength among mathematicians : the principle 

 or D'Alembert put an end to this kind of challenges, by supplying a 

 direct and general method of resolving, or at least of throwing into 

 equations, any imaginable problem. The mechanical difficulties were 

 in this way reduced to difficulties of pure mathematics. 



A. D 'Alemberi 's Principle. — D'Alembert's Principle is only the ex- 

 pression, in the most general form, of the principle upon which John 

 Bernoulli, Hermann, and others, had solved the problem of the centre 

 of oscillation. It was thus stated, " The motion impressed on each par- 

 ticle of any system by the forces which act upon it, may be resolved into 

 two, the effective motion, and the motion gained or lost : the effective 

 motions will be the real motions of the parts, and the motions gained 

 and lost will be such as would keep the system at rest." The distinc- 

 tion of statics, the doctrine of equilibrium, and dynamics, the doctrine 

 of motion, was, as we have seen, fundamental ; and the difference of 

 difficulty and complexity in the two subjects was well understood, and 

 generally recognized by mathematicians. D'Alembert's principle re- 

 duces every dynamical question to a statical one ; and hence, by means 

 of the conditions which connect the possible motions of the system, 

 we can determine what the actual motions must be. The difficulty of 

 determining the laws of equilibrium, in the application of this prin- 

 ciple in complex cases is, however, often as great as if we apply more 

 simple and direct considerations. 



5. Motion in Resisting Media. Ballistics. — We shall notice more 

 particularly the history of some of the problems of mechanics. Though 

 John Bernoulli always spoke with admiration of Newton's Principia. 

 and of its author, he appears to have been well disposed to point out 



