338 A Theory of Fluxions. 



A : B: :C : D, and inversely, B : A: :D : C, and let the ra- 

 tio be r, and -. If, instead of multiplying the third term by 



r 

 the second, and dividing this product by the first term, we 

 multiply the third term by the ratio, the result will be the 



same, that is, Cr=D, and — =C. From the nature of an 



r 



algebraic expression, the ratio here is always given, and in 

 the process, preserved distinct from the other part of the 

 term. Hence, in order to obtain the fourth term, it is ne- 

 cessary only to multiply the third term by the ratio. The 



formula for the ratio in the direct method of fluxions is — ? 



x 

 n representing either a whole number, a fraction, or a mixed 

 number ; and either a positive, or negative quantity. In the 



inverse method of fluxions, the ratio is For ax n X — = 



nx' x 



— X 



nax n ~ l x',a.ndnax n 'rrX — ==ax". 

 nx' 



Scholium. — From the foregoing investigation it is manifest, 

 that, whatever the source may be from which fluxions ema- 

 nated, they are nothing more than certain artificial propor- 

 tional quantities, of a finite magnitude, by the help of which 

 their corresponding fluents may be found. Thus a luminous 

 view of this abstruse branch of the Mathematics is presented, 

 depending on the plain, familiar, and acknowledged princi- 

 ple of the identity of ratios. This principle, as I have been 

 informed, has, in a very concise manner, been touched upon 

 by Dealtry ;* and it appears to me to develope the real na- 

 ture of fluxions. It is important to notice distinctly, that it 

 avoids the seeming error attending the Differential Calculus, 

 arising from the rejection of the infinitely small quantity, 

 which is the difference between the increment and the differ- 

 ential — the Gordian knot, which has long perplexed the most 

 eminent Mathematicians. 



Sec. 5. By describing polygons in the circle, and by con- 

 tinual bisections of the arcs, subtending the sides of the pol- 



* Although Dealtry has alluded to this proportion, between the fluents and 

 fluxions, yet, (in justice to the author of this memoir, it ought to be stated,) he 

 overlooked the object'for which it is now introduced. 



