104 



LIMITS AND FLUXIONS 



FlG. 



and the tangent AD come to vanish when B reaches 

 A, and their last ratio is unity. Newton " directs 

 Olir imagination, not to these vanishing quantities 

 themselves, but to others which are proportional 

 to them, and always preserve a 

 finite magnitude," such as A<^, 

 the arch Kcb^ Ad. Since at the 

 instant when A and B coincide, 

 ''the angle BAD, or bAd, will 

 vanish ; it is easy to conceive 

 that, . . . the chord Ab must 

 coincide with the tangent Ady 

 . . . consequently, AB, AD 

 must likewise, at the same instant of time, arrive 

 at the same proportion of a perfect equality. " 



124. Proceeding to the last Scholium in Book I, 

 Section I of the Principia, Jurin starts by defining 

 the word liniit. " I apprehend therefore that, by 

 the limit of a variable quantity, is meant some 

 determinate quantity, to which the variable quantity 

 is supposed continually to approach, and to come 

 nearer to it than to have any given difiference, but 

 never to go beyond it. And by the limit of a 

 variable ratio, is meant some determinate ratio, to 

 which the variable ratio is supposed continually to 

 approach, and to come nearer to it than to have any 

 given difference, but never to go beyond it. By 

 arriving at a limit I understand Sir Isaac Newton 

 to mean, that the variable quantity, or ratio, 

 becomes absolutely equal to the determinate quan- 

 tity, or ratio, to which it is supposed to tend." 



