22^ I'HILUSUPIIICAL TRANSACTIONS. [aNNO J 728, 



the same space, as double the force would do in one part of time. The space 

 described therefore by a body in motion, is not as the force; but as the force 

 and the time taken together. A body, with any the least assignable force, 

 will move through infinite space, if it meets with no resistance, in an infinite 

 time. And in spaces where there is a uniform resistance to motion, the space 

 described before the motion ceases, must needs be as the force and as the time 

 together : because a double force will carry a body twice as far in the same 

 time, and will also cause the motion to be twice as long time in destroying by 

 a uniform resistance. The space described therefore before the motion ceases, 

 is in this case demonstrably as the square of the force. A body thrown up- 

 wards with double force, will be carried four times as high, before its motion 

 be stopped by the uniform resistance of gravity ; because the double force will 

 carry it twice as high in the same time, and moreover require twice the time 

 for the uniform resistance to destroy the motion. The case is the same in ac- 

 celerated motion ; in bodies accelerated by a succession of elastic impressions, 

 or falling with a motion accelerated by the uniform power of gravity, or by any 

 other uniform power whatever. The space described must needs be as the 

 force, and as the time wherein the force operates. 



What I have thus demonstrated concerning any force, considered as the 

 cause producing an eiFect; and concerning the time, during which the force 

 operates ; is on all hands acknowledged to be true concerning velocity. And 

 therefore velocity and force, in this case, are one and the same thing. So that 

 to affirm force to be as the square of the velocity, is to affirm that the force is 

 equal to the square of itself. 



Now from hence appears very clearly the ground of the error these gentle- 

 men have fallen into, and of their misapplication of the experiments they build 

 upon. 



The effect of a force impressed on a moveable body, is the motion of that 

 body from one place to another. Now forasmuch as the effect cannot but be 

 proportional to its cause, hence Mr. Leibnitz, whom the other gentlemen have 

 followed, contends that the space described by a body in falling, is proportional 

 to the force by which it is impelled during its fall ; and that the force acquired 

 by a body in falling, is proportional to the space it has described in its fall. 

 Which space being agreed to be as the square of the velocity, as being pro- 

 portional to the velocity and to the time taken together, hence they infer that 

 the force likewise is as the square of the velocity. 



But from what has been said, it is plain, that the space described in these 

 and all other the like cases, is not as the force only, but as the force and as 

 the time wherein the force acts; that is to say, as the square of the force. 



