448 



matter or of motion to make it impossible that a motion 

 once begun should cease at a time proportioned for 

 example to its quickness, or should be accelerated by 

 the very nature of the original impulse and so of the 

 equality of action and reaction. No doubt, if the vis 

 inertias be granted, and the equality of action and re- 

 action, the composition of forces may be demonstrated, 

 and so may the proposition of equal areas in equal 

 times, and the principle of equilibrium first discovered 

 by D'Alembert. But these are only mathematical de- 

 monstrations of truths deducible and issuing from con- 

 tingent truths. The propositions of geometry are 

 wholly different ; they result necessarily from the de- 

 finitions ; they are indeed involved in those definitions. 

 Thus, if a circle is defined as the curve described by 

 the extremity of a given straight line revolving round 

 a fixed point, in this definition there is really contained 

 the proposition that its length is proportional to the 

 describing line's length, and its surface to the square 

 of that line. We affirm in these two propositions only 

 that if there be a curve line such as to have all the 

 lines equal, which are drawn to it from a given point, 

 that curve must have a certain measure of its length 

 and surface. When we affirm that a body moves in 

 the diagonal if solicited by two impulses along the two 

 sides of a parallelogram, we assume, not merely that 

 there is a body and that there is motion, but that the 

 body has certain qualities and that motion has certain 

 laws, and these are facts which exist, not mere supposi- 

 tions which we make. D'Alembert has only the merit, 

 and a great one it is, of having, first in his i Dyna- 

 mique,' and afterwards in his * Elemens,' reduced the 

 whole laws of motion and equilibrium to the fewest 

 and simplest possible fundamental principles, and there- 

 fore generalized those principles. 



All D'Alembert's writings have now passed under 

 our review : it remains to form a more general esti- 

 mate of his merits in the two capacities with a detailed 



