’ St henctits chaewe of 
400 
’otme- tions of addition, subtraction, multiplication, and divi- 
Peows 
functions we have been 
a 
To the analytically ceeeihie an. 
tended » it will cunvenient. % sdopt, « subtle 
notation ; therefore, like as ve bee denoted the sine 
of an are b the symbol sin. x, and the logarithm of z by 
log. 2, also ) 3() ce we eee any function and 
<FO. quantity x, by the her {), 
‘and bene it sunat bd chieasal, thes the 
i Dee ee et wine 
lamactaiatie, indicating that the ex eesion (=) is 
formed in a determinate manner from quan- 
tity x, and constant quantities. And as, by substitu- 
ting «+A, instead of z, in the expression log. z it be- 
comes log. («+4-/), so, when x+-h mes then J rag 
expression f(.), it becomes r+ 
een cetyl ‘ 
Su wiebe pam E by the increment /, so as 
to er+h; by which the function will change its 
LEAN IO _ 545 H; (M) 
where p is a new dba of x, which is independent 
of A; and H is a function of z and h, which has the 
property of vanishing when h=0: So that p is a limit 
to ED hich the function which expresses the ratio con- 
tinually approaches, as A decreases. 
19. The proposition may be put under another form. 
Por from the formule which express the ratio of the 
increments, we have also these, 
(24h)'=a* paz h4- Hh, 
h 
Log, (2h) =log, «+--+ Hh, 
abe | Aah 4Hh, 
2+h)= sin. 2+h cos. x4 Hh, 
Con tebe | sins} Hh, 
ts pee 
F (e+ =f (2)-+ph-4 Ul. (N.)" 
From these formule we learn, that, if in f(x), any 
function of a variable q » We sul x-4-h in- 
stead of x, h some quantity independent of , then 
SF (+h), the new value of the 
an expression, one term of which pan 
of h by p, some function of x which is quite 
derives i 
Asset aeckr *)+ = MK 
the function p of analytical 
Spvenigntion Gates is also the limit of the ratio 
Se+ 2) p-+H; 
pS ev era ha Beg} find the co-efficient of 
dew power of h in the developement of f(z4.h), 
samme 
The property of a limit, which we have proved to be- 
to every variable function, affords a ion for 
FLUXIONS. 
1. To determine the limit to the ratio of: the inengr Rantane 
ments in any proposed function. 
2. To determine, on the other hand, the 
having given the limit to the ratio of the increments. 
Page & of view, 
is mel Newton, differential 
calculus of Leibnitz. 
Bee ha In explaining the method of Newton, 
writers on the subject in country, 
have scoabate considerations drawn from the theory of 
motion. According to their view of the subject, in or- 
dex. to sephinees Signet it CiNOeE ieee 
mg Faye oh nent nay ere see to 
‘dart at. the exe Inetont and to 
sobre niany Daan D ees 
a I a 
Cc 
A \ x 
One of them, c, is supposed to move uniformly, and 
ao, the distance it has in any time, is taken as a, 
geometrical expression for x: The motion of the other’ 
int, C, is supposed to beso regulated, th that AC, the 
= Eee Se ene fh ea 
ction ‘or exam) 
number which expresses ices Gaus ie 
sitinecd the Lanter which, Ceeer eC 
It is easy to see that there is no function whatever of 
a variable quantity which may not be conceived to be 
in this Sap And it tage that 
, the 
int C that generates the function, f(a) will move fas- 
feat da Sideoaes according to the ae, 
but never uniformly, except in the case when f(x) has a 
constant ratio to z, In some cases it will continually in- 
ot ay fi 
all Dien OO a ee 
te tc, 2 
