OF THE LAWS OF MOTION. l67 



tional to the space from the beginning of the motion) was not only 

 contradictory to fact, but involved a self-contradiction; and was, 

 therefore, easily disposed of. But this accident did not supersede the 

 necessity of Galileo and his pupils verifying their assertion by refer- 

 ence to experiment, since there were many suppositions which were 

 different from theirs, and still possible, though that of Casrasus was 

 not. 



The mistake of Aristotle and his followers, in maintaining that 

 large bodies fall more quickly than small ones, in exact proportion 

 to their weight, arose from perceiving half of the third law of motion, 

 that the velocity increases with the force which produces it ; and from 

 overlooking the remaining half, that a greater force is required for the 

 same velocity, according as the mass is larger. The ancients never 

 attained to any conception of the force which moves and the body 

 which is moved, as distinct elements to be considered when we en- 

 quire into the subject of motion, and therefore could not even propose 

 to themselves in a clear manner the questions which the third law of 

 motion answered. 



But, when, in more modern times, this distinction was brought into 

 view, the progress of opinion in this case was nearly the same as with 

 regard to the other laws. 



It was allowed at once, and by all, that action and reaction are 

 equal ; but the controversy concerning the sense in which this law is to 

 be interpreted, was one of the longest and fiercest in the history of ma- 

 thematics, and the din of the war has hardly yet died away. The 

 disputes concerning the measure of the force of bodies in motion, 

 or the vis viva, were in fact a dispute which of two measures of action 

 that I have mentioned above should be taken ; the effect in a given 

 time, or the whole effect : in the one case the momentum {MV) in the 

 other the vis viva, {MV'^) was the proper measure. 



20. It may be observed that the word momentum, which one party 

 appropriated to their views, was employed to designate the motive 

 quantity of force, or the action of bodies in motion, before it was 



Y2 



