A Discourse on the Theory of Fluaions. 59 
tained, for A=Cr, and c=*. These equations being ap- 
plied to the foregoing proportion, we have aars » and 
A=Cr=ax"X——=naz" x", from whence are derived the 
rules contained in the direct method of fluxions. Again by 
multiplying the fluxion by the reciprocal of ~ we have 
—A snare’ 2 x *.=ax". From this equation the rules 
r nx 
pees quantity of a finite magnitude, and may therefore 
ion stands related, can at any time be brought forth, and their 
places assigne 
JN 
supposed to constitute the third term, and can be detached 
from it; which is done by the rules in the inverse method of 
general formula of the ratio~—> 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- 
end on finite elements. The relation of quantities, resul- 
ting from the principal of proportion, is already known to 
e 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 capa desire of contribu- 
ting something towards the entertainment of those, who take 
a deep interest in mathematical researches, has been m 
tive in entering upon these investigations. And especially, I 
