D'ALEMBERT. 465 



Now nothing can be more certain than that D'Alembert 

 does no such thing as prove this position. He only 

 shews, what never could be doubted, that the deductions 

 from certain assumed facts are necessary and not con- 

 tingent. Assuming the existence of matter, and also its 

 impenetrability, he treats the vis inertise as demonstrated, 

 and also its corollary, the uniformity of motion once 

 begun and not affected by any external causes. But the 

 impenetrability of matter is a contingent truth as well 

 as its existence ; and there is nothing in the definition of 

 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 

 inertise 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 demon- 

 strations of truths deducible and issuing from contingent 

 truths. The propositions of geometry are wholly differ- 

 ent; they result necessarily from the definitions; 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 etirve must have 



2H 



