396 D'ALEMBERT. 



and explained, though it be deducible from the equality 

 of action and re-action, a physical rather than a mathe- 

 matical truth, and derived from universal induction, 

 not from abstract reasoning a priori. 



The Principle is this, (' Dyn.' pt. 2. ch. i.) If there 

 are several bodies acting on each other, as by being 

 connected through inflexible rods, or by mutual attrac- 

 tion, or in any other way that may be conceived ; sup- 

 pose an external force is impressed upon those bodies, 

 they will move not in the direction of that force as 

 they would were they all unconnected and free, but in 

 another direction ; then the force acting on the bodies 

 may be decomposed into two, one acting in the direc- 

 tion which they actually take, or moving the bodies 

 without at all interfering with their mutual action, the 

 other in such direction as that the forces destroy each 

 other, and are wholly extinguished; being such, that 

 if none other had been impressed upon the system, it 

 would have remained at rest.* This principle reduces 

 all the problems of dynamics to statical problems, and 

 is of great fertility, as well as of admirable service in 

 both assisting our investigations and simplifying them. 

 It is, indeed, deducible from the simplest principles, 

 and especially from the equality of action and re- 

 action ; but though any one might naturally enough 

 have thus hit upon it, how vast a distance lies between 

 the mere principle and its application to such prob- 

 lems, for example, as to find the locus or velocity of a 



^ * Lagrange's statement of the principle is the most concise, but I ques- 

 tion if it is the clearest, of all that have been given. " If there be im- 

 pressed upon several bodies, motions which they are compelled to change 

 by their mutual actions, we may regard these motions as composed of the 

 motions which the bodies will actually have, and of other motions which 

 are destroyed ; from whence it follows, that the bodies, if animated by 

 those motions only, must be in equilibrio." (' Mec. An.' vol. L, p. 239, 

 Ed. 1811.) It is not easy to give a general statement of the principle, 

 and I am by no means wedded to the one given in the text. A learned 

 friend has communicated one which the reader will find in Appendix L, 

 together with a statement, by another excellent geometrician, of the real 

 benefit derived from the Principle. 



