NE WTON 9 



divisible quantities ; not the sums and ratios of 

 determinate parts, but always the limits of sums 

 and ratios : and, that the force of such demonstra- 

 tions always depends on the method laid down in 

 the preceding lemmas. 



14. *'It is objected, that there is no ultimate 



proportion of evanescent quantities ; because the 



proportion, before the quantities bave vanished, is 



not ultimate ; and, when they bave vanished, is 



none. But, by the same argument, it might as 



well. be maintained, that there is no ultimate 



velocity of a body arriving at a certain place, when 



its motion is ended : because the velocity, before 



the body arrives at the place, is not its ultimate 



velocity ; when it has arrived, is none. But the 



answer is easy : for by the ultimate velocity is 



meant that, with which the body is moved, neither 



before it arrives at its last place, when the motion 



ceases, nor after ; but at the very instant when it 



arrives ; that is, that very velocity with which the 



body arrives at its last place, when the motion 



ceases. And, in like manner, by the ultimate ratio 



of evanescent quantities is to be understood the 



ratio of the quantities, not before they vanish, nor 



after, but that with which they vanish. In like 



manner, the first ratio of nascent quantities is that 



with which they begin to be : and the first or last 



sum is that, with which they begin and cease to be, 



or to be augmented or diminished. There is a 



limit, which the veloòity at the end of the motion 



may attain, but cannot exceed. This is the 



