A Discourse on the Theory of Fluxions. 59 



tained, for A=Cr, and C = -. These equations being ap- 



A. nx' 



plied to the foregoing proportion, we have T^=r=— 5 and 



A=Cr=ax"X — =wa^"~';c-, from whence are derived the 



X 



rules contained in the direct method of fluxions. Again by 



nx' 

 multiplying the fluxion by the reciprocal of — we have 



C=—=nax''''^x' X —.—ax". From this equation the rules 



r nx 



belonging to the inverse method are derived. According to 

 the present view of the theory, a fluxion is an artificial pro- 

 portional quantity of a finite magnitude, and may thereiore 

 be subjected to examination ; and is so constituted, that, in 

 all its various combinations, it invariably maintains the same 

 relation to its corresponding fluent. Although the fluxional 

 expression, that occurs in a mathematical process, like the 

 straggling boulder in mineralogy, stands alone, and the pro- 

 portion lies concealed ; yet this circumstance does not make 

 it the less real. For the remaining terms, to which the flux- 

 ion stands related, can at any time be brought forth, and their 

 places assigned. But, in practice, this is not necessary. 

 The ratio is contained in the fluxional expression, which is 

 supposed to constitute the third term, and can be detached 

 from it ; which is done by the rules in the inverse method of 

 fluxions. For they are, in reality, nothing more than dividing 

 the third term of four proportionals, or the fluxion, by the 



general formula of the ratio — i by which the fourth term, 



or the fluent, is obtained. A theory of fluxions is here pre- 

 sented to the public, in which the fundamental principles de- 

 pend on finite elements. The relation of quantities, resul- 

 ting from the principal of proportion, is already known to 

 be of very extensive application. If the reasoning, on which 

 the present theory rests, shall be judged to be valid, it will 

 bring into view a chain, by which unknown quantities are 

 connected with those which are known to an almost unlim- 

 ited extent. I have endeavored to give it all the variety of 

 illustration, of which I was capable. A desire of contribu- 

 ting something towards the entertainment of those, who take 

 a deep interest in mathematical researches, has been my mo- 

 tive in entering upon these investigations. And especially, I 



