58 LIMITS AND FLUXIONS 



and much more so to comprehend the moments, or 

 those increments of the flowing quantities in staiu 

 nascenti, in their very first origin or beginning to 

 exist, before they become finite particles. And it 

 seems stili more difficult to conceive the abstracted 

 velocities of such nascent imperfect entities. But 

 the velocities of the velocities — the second, third, 

 fourth, and fifth velocities, etc. — exceed, if 1 mistake 

 not, ali human understanding " {Analyst, § 4). . . . 



75. '*In the calculus differentialis . . . our 

 modem analysts are not content to consider only 

 the differences of finite quantities : they also 

 consider the differences of those differences, and 

 the differences of the differences of the first differ- 

 ences : and so on ad infiniturn. That is, they 

 consider quantities infinitely less than the least 

 discernible quantity ; and others infinitely less than 

 those infinitely small ones ; and stili others infinitely 

 less than the preceding infinitesimals, and so on 

 without end or limit " (§ 6). 



'j^. "■ I proceed to consider the principles of this 

 new analysis. . . . Suppose the product or rectangle 

 AB increased by continuai motion : and that the 

 momentaneous increments of the sides A and B are 

 a and b. When the sides A and B are deficient, or 

 lesser by one-half of their moments, the rectangle 

 was À-4^xB-i ^, i.e. h.^-\ a^-l bK^-\ ab. 

 And as soon as the sides A and B are increased by 

 the other two halves of their moments, the rectangle 

 becomes A + -| ^ x B + J ^ or AB + | aV> + \ bA + | ab. 

 From the latter rectangle subduct the former, and 



