NEWTON S PRINCIPIA. 185 



The process that is now used to determine the motion 

 is founded on the following reasoning. Let us suppose 

 the particle moving in any curved line ; let s be the arc 

 described measured from any point at the time t. The 

 time is supposed to be measured from any epoch anterior 

 to the commencement of the motion. Then in the small 

 time S t, the particle will, according to the notation of the 

 differential calculus, describe a small arc s, hence the 

 mean velocity of the particle during this interval will be 



g . Now let 8 t diminish without limit, the mean velocity 



will become the actual velocity (y) at the instant t, and 

 hence 



ds 



V = Tf 



Similarly, the velocity being v at the time t, that at the 

 time t + $t will be v + 8 v ; hence the acceleration is such 

 that in time t a velocity S v has been added to the motion ; 

 hence the mean acceleration in that interval, measured by 

 the velocity that would have been added in a unit of time 



v 



if it had remained constant during that time, will be = 



Now let 5 t diminish without limit, and the mean accelera 

 tion becomes the actual acceleration (f) at the instant, and 



d t 



But an accelerating force is measured by the quantity of 

 velocity it would add to the body in a unit of time, if it 

 remained constant during that interval, so that we have 

 merely to equate the accelerating force as given by the 

 question, to the acceleration as given by the preceding 



