jippUcation of the Fhxional Ratio, Sfc. 299 



principles of fluxions have not yet been so completely disclosed, as 

 to give entire satisfaction. That distinguished Mathematician, Carnotj 

 supposed that something is lost, when a fluxion is introduced into an 

 equation ; and that the error in this defective equation is compensated 

 by an error of equal magnitude, but opposite value, produced when 

 the fluent is taken.* In an essay, contained in Vol. XIV of the Journal 

 of Science, page 297, after mentioning some^ difficulties attending 

 the metaphysics of fluxions, the Author goes on to say ; " For an il- 

 lustration of our remarks, suppse e be an increment of a uniformly 

 varying quantity xi then x-{-e will be the quantity varied by the in- 

 crement. This variation will be uniform in all values of x, but the 

 variation of the variables dependent on x-\-e, or what is denominated 

 the functions of this new value ofx, as (a; + 6)", will not be uniform ; 

 but may be easily investigated by the development of (x+e)". For 

 greater simplicity suppose the function to be (a; + e)^ =a;^+2.re + e^, 

 the increment, or variation of this from its first value, when it had no 

 increment, is 2xe-^ee, which is to the uniform increment of x, 

 as 2xe-\-ee I c, or as 2x+e : 1. Here the ratio of the increment of 

 the function to its base, or root, is ascertained very readily by its 

 algebraic development ; and if this were truly its differential or flux- 

 ion, there would be no ground of questioning the legitimacy of the 

 logic of this science ; but the objection to it rests entirely on casting 

 away the increment e from the expression 2x-\-e of the ratio of the 

 whole increment of the function ; since e must ever constitute a part 

 of it, while it has any finite value. It may be said that the ratio 

 2x ' 1, or what is called the differential coefficient, is independent of e, 

 and has a real value, when e vanishes ; which is true, but it is then at its 

 limit, and the ratio is that of the limit, and not of the increment, con- 

 sequently no new discovery is made by this mode of conception. If 

 the second term of the development be assumed as the true differen- 

 tial, this will be a petitio principii. In short we perceive no logical 

 principles in La Grange's analytical demonstrations, which are not 

 common to the geometrical." 



It is true that, when rightly understood, the principles in the ana- 

 lytical method of La Grange coincide with those in the geometrical 

 method. The error lies, not in the contrariety of the methods, but 

 in taking it for granted, that the ratio between two fluxions is that of 

 the increments ; for instance, that of the increment of a function which 



" Tiiloch's Phil. Mag. Vol. 8, Art. iv. 



