92 INTRODUCTION 



conceived the effect in too narrow a way, regarding it 

 merely as actual motion (i. e. the momentum acquired by a 

 body) rather than the work done by the force, the kinetic 

 energy it produces (i.e. the vis viva which the body 

 acquires, and which Leibniz calls action motrice). The 

 formula for this action motrice is not mv but mv*. ' In the 

 uniform motions of one and the same body, (i) the action l 

 of traversing two leagues in two hours is double the action 

 of traversing one league in one hour (for the first action 

 contains the second exactly twice) ; (2) the action of tra- 

 versing one league in one hour is double the action of 

 traversing one league in two hours (or, actions which 

 produce one and the same effect are proportional to their 

 velocities) : therefore (3) the action of traversing two 

 leagues in two hours is four times (quadruple) the action 

 of traversing one league in two hours. This demonstration 

 shows that a moving body which receives a double or 

 triple velocity, in order that it may produce a double or 

 triple effect in one and the same time, receives a quadruple 

 or nonuple action. Thus actions are proportional to the 

 squares of the velocities. But most fortunately this 

 happens to agree with my calculation of force, drawn 

 both from experiments and from the pre- supposition that 

 there is no mechanical perpetual motion. For, according 

 to my calculation, forces are proportional to the heights 

 by descending from which heavy bodies might have 

 obtained their velocities, that is to say, as the squares of 

 the velocities. And, as there is always conserved the total 

 force for re-ascending to the same height or for producing 

 some other effect, it follows that there is conserved also 

 the same quantity of motive " force " [action motrice] in the 

 world ; that is to say, to put it definitely, that in any one 

 hour there is as much action motrice in the universe as 

 there is in any other hour. But at every moment 2 the 



1 I. e. the work done or vis viva. For a full explanation of the 

 whole matter, see Stallo, Concepts of Modern Physics, ch. vi, especially 

 pp. 71 sqq. 



1 * A momentary state of a body in motion cannot contain motion, 



