238 LIMITS AND FLUXIONS 



(not used by him in the development of his funda- 

 mental formulas). He says (p. ii) : '* It may not 

 perhaps be improper to add, that, if to the ex- 

 pressions delivered above for the excess of the 

 magnitude, which has to B a ratio, having to the 

 ratio of A + N to B, the ratio of R to O, above the 

 magnitude, which has to B a ratio, having to the 

 Ratio of A to B the same ratio of R to O ; and for 

 the excess of the magnitude, which has to B a ratio, 

 having to the ratio of A to B the ratio of R to Q, 

 above the magnitude, which has to B a ratio, having 

 to the ratio of A — N to B the ratio of R to O, be 

 prefixed the magnitude, which has to B a ratio, 

 having to the ratio of A to B the ratio of R to Q, 

 we get a geometrical Binomial, of which, when it is 

 supposed to become numerical, the famous Binomial 

 Theorem of Sir Isaac Newton is only a particular 

 case. " 



Reuiarks 



206. The classic treatment of fluxions in Great 

 Britain, during the eighteenth century, rests prim- 

 arily on geometrical and mechanical conceptions. 

 Attempts to found the calculus upon more purely 

 arithmetical and algebraical processes are described 

 in this chapter. Ali these attempts are either a com- 

 plete failure or so complicated as to be prohibitive. 

 Easily the ablest among these authors was John 

 Landen. De Morgan says of his Analysis^: " It 

 is the limit of D'Alembert supposed to be attaìned, 



^ Penny CycloJ>iedia, Art. " Differential Calculus." 



