ELISHA MITCHELL SCIENTIFIC SOCIETY. 7 



Definition of the Limit of a J ^ariable. When a variable 

 magnitude takes successively, values which approach more 

 and more that of a constant magnitude, so that the differ- 

 ence with this last can become and remain less than any 

 designated fixed magnitude of the same species, however 

 small, whether the variable is always above or always below 

 or sometimes above and sometimes below the constant, we 

 say that the first approaches indefinitely the second and that 

 the constant magnitude is the limit of the variable mag- 

 nitude. 



More briefly, this is often stated thus: The limit of a 

 variable is the constant, which it indefinitely approaches 

 but never reaches. 



Definition of an Infinitesimal. An infinitely small quan- 

 tity or an infinitesimal, is a finite quantity whose limit is 

 zero. Hence the infinitesimal approaches zero indefinitely, 

 but can never attain it, since zero is its "limit." As an 

 illustration, take two straight lines incommensurable to 

 each other. Mark the ends of the first line A\ B\ the 

 ends of the second A, B. Now as we can always find a 

 unit of measure that will go into A^B^ an integral num- 

 ber of times, apply such a unit to AB from A to C, as 

 many times as possible, leaving a remainder over CB less 

 than one of the parts. Then the ratio, 



AC 



A^B^ 



is less than the ratio of the two lines, but approaches it 

 indefinitely as the unit of measure decreases indefinitely, 

 since CB being always less than the unit, tends towards 

 zero but can never reach zero; hence CB is an infinitesimal 

 and AC approaches AB indefinitely without ever being 

 able to reach it. By the definition therefore, the limit of 

 CB is zero and the limit, of AC is AB, hence the limit 

 of the ratio above, 



