234 



LIMITS AND FLUXIONS 



X-\-X *^ —X*^ X + X ** —X' 



x+x' —X 



=x-\-x' 



-1 



which, when x vanishes, is manìfestly equal to 



7/^ ^f - 1 



— ;r« , the limit of the said ratio." 

 n 



The explanation of the method of drawing tan- 

 gents is too long for quotation, and we shall limit 

 ourselves to the following outline of it, as given 

 by Landen : 



*' I consider the curve as already described, with- 

 out any regard to its generation, and find the value 

 of a certain line (terminated by the curve and 

 its tangent), in algebraic terms involving {s) the 

 subtangent with other quantities ; which algebraic 

 expression I observe, from an obvious property of 

 the line it is found to denote, must have a certain 

 property with respect to being positive or negative 

 in certain cases. I therefore assume that expres- 

 sion equal to another which is known to have that 

 very property ; and from thence, by means of the 

 theorem mentioned in page 5, readily find the re- 

 quired value o{ s" (p. io). 



204. Landen's Discourse was attacked by. an 

 anonymous writer in the Monthfy Review for June 

 1759, who claims that the Residuai Analysis " is no 



