18 15.] On the Doctrine of Fluxions* 331 



have -77^ = ^^^— . Now it is evident that FG is the limit of ?/', 



y*', &c. ; forj/', y", &c. may approach, by continued bisections, 



nearer the point G than by any assigned difference, how small 



G F 

 soever, but can never pa<;s tlmt point, ^-p is the geometrical limit 



of ^, -^. — . &c. which are successive values of - — ^ , while i is 



diminished by continual bisections j as 7j x" ~ ' is the algebraical 

 limit. 



That 7Z x"~' and p-p, the limits of -^ ~ ^ , are equal to one an- 



■p /-< 

 other, may be proved thus : — If they be not equal, let „ ^ be the 



FG 

 greater, and let D be the difference between -p-w and 7i x"~'; 



then, because ^ — ^-^, or its equal ~ , &c. is always greater than 



F G" 7/* — 11 



-p-pr, - — r— ^ cannot approach nearer to n x"~' than by more than 



D, but ^ . ^ approaches nearer to nx"~'than by any assigned 



FG . 

 difference ; therefore y p is not greater than ?j «"""' ; and in the 



F O 

 same manner it may be proved not to be less; wherefore ■ „ p -is 



equal to « x" ~ '. Q. E. D. 



If .r be put for E F, and j for F G, then ^ — n x"" *. 



4. Observations. — This reasoning seems to me to remove La- 

 grange's objection lately mentioned, and to do so by employing an 



incremental fraction —~~ , &c. equal to an approximating fraction 



If 



■— , &c. of which approximating fraction the denominator is 



always I = v = E F, the numerator continually approaches to 

 F G = j, and the vanishing quantity is the difference between the 

 numerator y\ &c. and F G = _>-. 



This approximating fraction —-^ &c. is always greater than ^ 



1 X 



when the curve, as in fig. 3, is convex to the axis ; if a curve con- 

 cave to the axis be drawn through the point E in the figure, another 



approximating fraction less than 4- will approach ~ from below 



% X 



the tangent, and — is, in the strictest sense of the word, the limit 



which the fractions approximating from above and from below the 

 tangent can never pass. 



