38 DYNAMICAL PEINCIPLE. 



highest rank of geometricians. The theory is deduced with 

 perfect precision, and with as great clearness and simplicity 

 as the subject allows, from a principle which he first laid 

 down and explained, though it be deducible from the equality 

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

 truth, and derived from universal induction, not from abstract 

 reasoning a priori. 



The Principle is this (' Dyn.' part 2, chap. i.). If there 

 are several bodies acting on each other, as by being connected 

 through inflexible rods, or by mutual attraction, or in any 

 other way that may be conceived ; suppose an external force 

 is impressed upon those bodies, they will move not in the 

 direction of that force as they would were they all uncon- 

 nected and free, but in another direction ; then the force 

 acting on the bodies may be decomposed into two, one acting 

 in the direction 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 problems, for example, as 

 to find the locus or velocity of a body sliding or moving 



* 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." (' M6c. An.' vol. i. 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. (See Note in.) 



