ATTEMPTS AT ARITHMETISATION 233 



tion, which is founded on those borrowed prin- 

 ciples, may be done as well, by another method 

 founded entirely on the anciently-received principles 

 of algebra. . . . It is by means of the foUowing 

 theorem [p. 5], viz. 





in nt 



Xn —Vn ^JI^i X Xj X 



= ;l'« X 



x — v 



v\'^ v\^^ v\^^ 



(where m and n are any integers) that we are 

 enabled to perform ali the principal operations in 

 our said Analysis." 



His Residuai Analysis is a method involving 

 vanishing fractions and therefore not free from con- 

 troversial questions. That the fluxion of x^ is ^x'^ 

 is explained according to the Residuai Analysis by 

 the consideration that {j^ — x^) — {}> —x)=x'^^xy-\-y^, 

 which is equal to 3;^^ when j;/ = ;r. We proceed to 

 give an application in Landen's own words : 



203. (Page 5) " Fluxionists, in determining the 



limit of the ratio of the increments of x and x"" , 

 commonly have recourse to the binomial theorem 

 (which is mudi more diffìcult to investigate than the 

 limit they are seeking) : But how easily may that 

 limit be found, without the help of that theorem, by 

 the equation exhibited in page 5 ! Thus, the incre- 

 ment of x being denoted by x\ the increment of 



x"* is x-{-x'» —x'\ and the ratio of those incre- 

 ments is 



