330 On the Doctrine of Flvxions. (May, 



fluxions bear to their fluents ; nor of exponential and logarithmic 

 quantities ; nor of the arithmetic of sines ; but 1 refer tor informa- 

 tion to some of the authors afterwards mentioned. 



3. With regard to the demonstration, 1 think that the view 

 already exhibited leaves no doubt in the mind of the learner ; but a 

 rigorous demonstration should be given, in order to enable him to 

 reply to every objection. 



Newton's second lemma of the second book seems to afford a 

 demonstration that, while it is brief and comprehensive, is con- 

 vincing, if the reduction to absurdity by Robins, vol, il. ; or by 

 Hales, in the Logarithmic Writers, vol. v. p. 133 and J 34; be 

 subjoined to Newton's case first ; if the process from particular to 

 general be admitted in his case third ; and if the momentum be 

 admitted instead of the fluxion. 



In a department of science so important and extensive as fluxions, 

 the demonstrations of various authors should, I think, be studied by 

 the learner ; such as those of Simpson, Maclaurin (though ex- 

 tremely tedious), Euler, I'Huilier, Bossut, Vince, Dealtry, La- 

 croix, and Lagrange. 



Any function of a variable quantity may be represented by the 

 ordinate of a curve of which x is the abscissa. Let y = a," be a 

 function of .r, and let x become a: + i ; then y = a," will become 



n . n ~ \ jr"-' 



r = 



TTi" = 



x" + n .r I + 



1 .2 



f- + &c. If we 



subtract the first equation from the second, and divide both sides by 



we shall have - — ^ = 



n X 



+ 



n . n — 1 x" 



+ &c. Now 



t 1-2 



it is evident that i, -And consequently y^ — y, which depends on ?", 



may be so diminished that n x"~^ may differ less from rax""' + 



~ — :: — ^ i 4- &c. than by any assigned quantity how small 



soever ; and when in ^ — ^-^, i and y^ — y vanish, -rj- seems equal 

 But this conclusion, says Lagrange, *' presents no 



to 71 x" 



idea." 



FG 



Prop, .p-rr is equal to n x"~", fig. 3. Construct the figure in 



E 1- 

 which T G 

 AI = ^: I 



KH = y'; 



is the tangent. Let 

 K = i ; E I = y ; 

 bisect E F in O, and 

 through O draw P N parallel to E I 

 or K H ; and through P draw EPM; 

 let P O = 'z/; E O = 'i; F M 

 — y' : then by similar triangles 

 P O : OE :: MF :F E, or'y : 



'i :: y' : i; therefore -~ = ~'. 



«nd if E O be bisected, we shall 



AT 



