SEQUEL TO THE GENERALIZATION. 97 



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 pro- 

 blems were, for some time, a sort of trial of strength 

 among mathematicians : the principle of 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 pro- 

 blem. The mechanical difficulties were in this way 

 reduced to difficulties of pure mathematics. 



4. D' Alembert' s Principle. D'Alembert's Prin- 

 ciple is only the expression, in the most general 

 form, of the principle upon which John Bernoulli, 

 Hermann, and others, had solved the problem of 

 the center of oscillation. It was thus stated, " The 

 motion impressed on each particle 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 distinction of statics, the doctrine of equili- 

 brium, 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 ma- 

 thematicians. D'Alembert's principle reduces 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 

 VOL. ir. H 



