A Theory of Fluxions. 343 



which " is neither before those quantities cease to move, nor 

 after ; but at the very instant in which they arrive at their 

 last place, and the motion ceases. 1 ' (See Newton's Princip. 

 Lem. XL) 



Corresponding with these ideas of the illustrious inventor 

 of fluxions, variable or flowing quantities are considered, 

 first, as in the act of arising into existence ; quantities in this 

 incipient state are called nascent quantities : secondly, as in 

 the act of disappearing or ceasing to be ; in this vanishing 

 state they are called evanescent quantities. Mathematicians 

 have taken different views of nascent and evanescent quan- 

 tities. Whilst some have considered them as quantities at 

 which we can arrive, and which have a real existence ; oth- 

 ers have held them to be non-entities, and have complained 

 of Sir Isaac Newton, for having first supposed certain quan- 

 tities to exist, having certain properties and relations, and 

 these properties and relations still to remain, after the quan- 

 tities themselves have vanished. Whilst some have held, 

 that they were real quantities, but small beyond the reach of 

 imagination ; others, with Cavalerius, have supposed them 

 to be entirely divested of magnitude, and have called them 

 indivisibles. In this diversity of opinions, it will be well to 

 attend to their nature, and to the station which they hold in 

 the system of mathematical quantities. Here it must be 

 considered, that we can never arrive at quantities, which are 

 less than any assignable ones, as is evidently the case with 

 nascent and evanescent quantities. They are therefore in- 

 finitely small. And of an infinite quantity, we can form only 

 a negative idea. To obtain that quantity, which is less than 

 any assignable one, is contradictory ; to overtake that, which 

 recedes as fast as it is pursued, is impossible. Therefore 

 whenever quantities are said to be infinitely great, or infinite- 

 ly small, they must be considered as moveable, ever retreat- 

 ing, unassignable quantities ; for a positive infinitive quanti- 

 ty is beyond the grasp of any created being. But, notwith- 

 standing we can form no positive idea of infinite quantities "; 

 yet, in some cases, we can have a clear conception of cer- 

 tain relations, which belong to them. For instance, let two 

 cylinders, whose bases are to each other as 2 : 1, be extend- 

 ed to any assignable equal lengths, the proportion will al- 

 ways be as 2:1. Now suppose this length to be greater 

 than any assignable one, that is, let no bounds be set to their 

 length, then, since it cannot be affirmed that the length of 



