58 DISSERTATION SECOND. [part ii. 



annihilated another, and these impulses, when properly 

 collected into one sum, must evidently be equal to the 

 force which they have extinguished. It happens, however, 

 that there are two ways of computing the amount of these 

 retarding forces, which lead to different results, both of 

 them just, and neither of them to be assumed to the exclu- 

 sion of the other. 



Suppose the body, the force of which is to be measur- 

 ed, to be projected perpendicularly upward with any velo- 

 city, then, if we would compute the quantity of the force 

 of gravity which is employed in reducing it to rest, we may 

 either inquire into the retardation which that force produ- 

 ces during a given time, or while the body is moving 

 over a given space. In other words, we may either in- 

 quire how long the motion will continue, or how far it will 

 carry the body before it be entirely exhausted. If the 

 length of the time that the uniform resistance must act 

 before it reduce the body to rest be taken for the effect, 

 and consequently for the measure of the force of the body, 

 that force must be proportional to the velocity, for to this 

 the time is confessedly proportional. If, on the other 

 hand, the length of the line which the moving body de- 

 scribes, while subjected to this uniform resistance, be taken 

 for the effect and the measure of the force, the force must 

 be as the square of the velocity, because to that quantity the 

 line in question is known to be proportional. Here, there- 

 fore, are two results, or two values of the same thing, the 

 force of a moving body, which are quite different from one 

 another ; an inconsistency which evidently arises from this, 

 that the thing denoted by the term force, is too vague and 

 inderinite to be capable of measurement, unless some farther 

 condition be annexed. This condition is no other than a 

 specification of the work to be performed, or of the effect to 

 be produced by the action of the moving body. Thus, when 



