particularly as given by French Writers. 35 



Let a body be acted on by a force 2 F. This is equiva- 

 lent to two forces, F and F. The first would produce a ve- 

 locity V, and therefore the second may be considered as acting 

 upon the body already moving with the velocity V. But 

 hence by the second laiv of motion it would add a velocity V, 

 and hence the whole velocity would be 2 V ; and similarly 

 we might prove that a force m F would produce a velocity 

 m V, which is easily seen to be an expression of the third law 

 of motion. 



The preceding are the principal objections which I wish at 

 present to propose. Perhaps some may be at first disposed 

 to think the question rather one of words and definitions than 

 of principles. It will, however, appear on close consideration 

 to be something more. For two different principles obtained 

 from observation cannot be made one by any mere alteration 

 of phraseology. To those who feel an interest in the strict- 

 ness of scientific logic the question will not appear unimpor- 

 tant. It is in fact the question, whether, after carrying the 

 process of induction and generalization as far as is requisite, 

 in order to obtain the three laws of Newton, we can carry it 

 one step farther, and include one of these laws in the other. It, 

 is the question, whether the third law of motion be capable of 

 being considered as a particular case of the second. The pre- 

 ceding considerations seem to disprove such a dependence. 



The manner in which the French writers have tried to esta- 

 bhsh such a connection has been by including both in the 

 principle that Jbrce is as velocity. 



But that this apparent simplification is made at the expence 

 of accuracy of language seems certain. To express the two 

 cases clearly, we might say, 1st, that when force acts on a 

 body already in motion, the velocity compounded with the for- 

 mer velocity is proportional to the force ; and, 2d, that when 

 different forces act on the same body, the velocity generated 

 is proportional to the force. And these two propositions are 

 different, and should be established separately, as appears by 

 considering that each might be true though the other were 

 false. Thus the proof given by Laplace, that velocity is pro- 

 portional to force, depends upon this fact, that a body struck 

 upon the surface of the earth moves in the direction in which 



