56 A Discourse on the Theory of Fluxiens. 
increment of 2" to the increment of x will be na"~1z-En. 
— 37347" oS a 3z3+&c.3z. Dividing by z 
—2 
B22 heecT,: 
If z be diminished by an indefinite sbaivson, the increment 
will approach continually towards nz”"' as its limit. Sup- 
pose z to be less than any seaicabte quantity, it will then 
become equal to o, and all the terms, in which it is found, 
will More Hence the ultimate ratio of the Pigrege is 
x ' 31, ornz" ‘2°: x. If n=4, then oe , the ratio 
of the Tockemnents is 4c? 4604222" 42" t : anid the ulti- 
mate ratio is 42° ? 1, or 42°2° $ a’. Flaxiong Mantis the 
eory by extension and motion, properties which are singu- 
larly adapted to explain the nature of those quantities, to 
which fluxions are applied, and which, when divested of the 
consideration of matter, are with propriety introduced into 
pure mathematics. Thus the two illustrious inventors of 
this science have each taken tenable ground in their mode 
=e ene it, and have placed this branch of the mathe- 
cs on a foundation which cannot be shaken, and which 
ae will never demolish. Each method has its peculiar ex- 
cellencies ; and if either were wanting, the theory would in 
some Hie ge be deficient. 
an illustration of the manner in which fluents are gen- 
dated: Sed an explanation of the nature of that relation, 
which fluxions bear to their fluents, are two distinct things, 
which ought not to be blended together. Whilst the former 
is accomplished in a satisfactory manner, the latter remains, 
in my opinion, unexplaine ut some mathematicians have 
thought differently, and have supposed that the properties of 
the ratio nx" 'zx- } x are sufficient to develop the nature 
of this relation. That this is their view, is manifest from their 
supposing, that ai foundation of fluxions is laid in the tacit 
acknowledgment, that a circle is a polygon of an infinite 
siaipbes of ara (Brewster’ s Encyclopeedia, Art. Fluxions.) 
{t was upon this supposition, that Carnot admitted that an 
error actually arises from the rejection of the quantity, which 
is the difference between the increment and the fluxion; but 
‘iat this error in the course of the operation, is compensated 
by an error of a contrary nature, (Tilloch’s Phil. a8. Vol. 
Sand 9.) By thus applying principles, which are insufficient 
: a n-1 _ —in 
it becomes nz” !-++-n.—>— 2” 2z-4n. “= 
° 
