VOL. XXXV,] PHILOSOPHICAL TRANSACTIONS. 223 



For the cause of the quantity of the space described, is not barely the quan- 

 tity of the force, but also the continuance of the time wherein the force acts. 

 The force therefore and the time taken together, being necessarily as the space 

 described ; as the velocity and the time taken together, are on all hands ac- 

 knowledged to be; it follows that the velocity and the force are equal, and not 

 the force as the square of the velocity. 



When two unequal bodies, fastened at the ends of the arms of a balance 

 of unequal length, counterpoise each other, and vibrate in equal times; as 

 they must necessarily do, being fastened to the arms of the same balance: 

 which is an observation Mr. Leibnitz lays great stress upon : in that case indeed 

 the forces will be as the spaces described. But not therefore as the squares of 

 the velocities. For in that case, the velocities themselves are as the spaces de- 

 scribed, because the times are equal. 



When a body projected with a double velocity, enters deeper into snow or 

 soft clay, or into a heap of springy or elastic parts, than in proportion to its 

 velocity ; it is not because the force is more than proportional to the velocity ; 

 but because the depth it penetrates into a soft medium, arises partly from the 

 degree of the force or velocity, and partly from the time wherein the force 

 operates before it be spent. 



In the coUision of hard bodies, it is I think agreed on all hands, that it is 

 demonstrated by reason, and confirmed by experience; that when a perfectly 

 hard ball, moved with whatever degree of velocity, strikes full upon another 

 hard ball, equal in size and weight, and without any motion in it ; if the balls 

 be unelastic, they will both go on together the same way, dividing the motion 

 equally between them, with half the velocity the first ball had originally: but 

 if they be perfectly elastic, the moving ball will communicate its whole motion 

 and velocity to the quiescent ball, and itself lie still in the other's place. Were 

 it true now, that the force of the moving ball was as the square of its velocity; 

 these experiments would then show (which is infinitely absurd) that the force 

 or vis inertias in the quiescent ball, the dead force, was always proportional to 

 the square of the velocity (which these gentlemen affect fantastically to call 

 the living force) of the moving ball, whatever its velocity were. Or the force 

 in both might just as reasonably be supposed to be as the cube, or the qua- 

 drato-quadrate, or any other power of the velocity of the moving ball. Which 

 is turning the nature of things into ridicule. Mr. Leibnitz, in some letters 

 which he wrote into England, intimated that he had a prospect of a perpetual 

 motion, founded on the notion of a vital principle, or active power in matter. 

 But from the experiments now mentioned, it is evident that if the force of 

 bodies in motion could be exalted even to the infinit'th power of their velocity; 



