298 Application of the Fluxional Ratio, &fc. 



Art. XI. — On the application of the Fluxional Ratio to particular 

 cases ; and the coincidence of the several orders of Fluxions, with 

 the binomial theorem; by Ellzur Wright, Esq. 



The object of this essay is to develope more fully the nature of 

 the fluxional ratio ; and to apply it to particular cases. To do this 

 it will be necessary to bring into view certain particulars, familiar 

 probably to every one, who has paid any attention to the subject. 

 When the method, used in algebra, of determining unknown quan- 

 tities is extended, by considering the unknown quantity as capable of 

 increase or decrease, that is, of taking all possible values, from to 

 a;", it is called fluxions. If this principle be applied to quantities, in 

 which the rate of increase is uniform in all its parts, no advantage is 

 gained. But if it be applied to those, in which the rate is variable 

 according to some given law, it affects quantities, which require an 

 artful management, and involves problems, that are sometimes of 

 very difficult solution. It may therefore be understood, that fluxions 

 have respect to those quantities, which increase or decrease by de- 

 grees that are less than any assignable one ; that is, in which the 

 alteration of magnitude is produced by one continued increment or 

 decrement, and in which the rate of increase is variable. The 

 laws, by which the rate of increase is regulated, result from the va- 

 rious dimensions of the variable, and are intimately connected with 

 the development of the binomial series. In the year 1772 La Grange 

 in the Berlin Memoirs proposed to show, that the theory of the de- 

 velopment of functions into series, contains the true principles of the 

 Differential Calculus (Fluxions) independently of the consideration 

 of infinitely small quantities ; and he demonstrated by this theory 

 the theorem of Taylor, which he regarded as the fundamental princi- 

 ple of the Calculus, and which had been demonstrated only by the 

 help of the Calculus itself, or else by the consideration of infinitely little 

 quantities. La Grange, in my opinion, has hit upon the true method 

 of explaining the theory of fluxions. He doubtless gave a just 

 view of the elements of a fluxion, when he said, that, in the devel- 

 opment of a binomial, the fluxion is expressed by the second term of 

 the series.* But notwithstanding all that La Grange has done, the 



Ryan's Dif. and Int. Calculus, Art. 19. 



