_ Methed. 
—— 
many terms as we please. 
~ other 
» 2 2 h , he 
th ; — —— > 
CTA A eanaay 
and, in like manner, 
we ney develope (x-+-/)* into-as 
his expression, and every 
found by this theorem, is merely an identical 
equation, as will appear by reducing all the terms to a 
common denominator, for then it becomes (24)? = 
(apAy?. 
52. Ve might develope each of the functions, log. «, 
a’, sin. x, cos. 2, al ei tet the ard pene! as we 
have developed («+/)"; these, every other 
fanction fat oat Aa arty where « and h 
are indeterminate quantities in ent of each other, 
may be included in one general formula, which we shall 
Let f(z), any function of a variable quantity x, be re- 
u, so that we have f(x)=w, then, when x 
its 
JF (A=u+X(k—2). 
“* . . k : 
Ae bis indepintento et ane 
the u and X(k—x) relatively to x, and divi- 
ding by dx, we in the ft place, k 
u 
o= > en + (k—«) 7 
Xa osx A . 
Considering now dx as a constant quantity, and tak. 
i repeatedly 
fore, we find, as in art. 51, 
dX Hu 
ie Bl 
dz 
et Fryn $e 
(k—«), 
Shou $ (ie) +e ays, 
FLUXIONS. 
Sa+hyaut h+ 
te he 
Sethaut 7h + 
value, and becomes x4, we have seen f(r-+h)=u+ ae + 
413 
Shay Se) 4 Te Cosy Mathod, 
—— 
aX (k—2)3 
qe 2” 
&e. 
and hence again, by substituting r+h for k, but still 
retaining / in the function X, so that 
x — fH—"_f®S@) 
k—x —r 
where & is to be considered as a constant quantity, and 
x alone as variable, we have 
Jla+h=u+ X4, 
d dX, 
ae 
Bui Rs aX hs 
de 2 * de 2’ 
du alee Gu hs 
a2 2" dz 2.3: - 
GX hea 
+ ie 28 
If we suppose the series to proceed ad infinitum, then, 
without paying any regard to the éxpression = 
which enters into each finite developement of the fune- 
tion, we have 
du du h® au h3 
h)= ete: lass mA ESS 
Set da! tas 2 + dx 2:3 ; 
Au 
da 3.307 t 
This formula, remaikable for its elegance and sim~' - 
pny. was first found by Dr Taylor, an eminent Eng- 
ish mathematician, who. published it'in a work. called 
Methodus Incrementorum, about the year 1716, and is - 
called Taylor’s Theorem. Sir Isaac Newton 
ve a similar formula in his Principia, where he treats 
of the theory of comets, but it is icable to a series of 
quantities having finite differences. Newton's formula 
becomes Taylor's theorem, when the differences are in- 
definitely small. This theorem has considerably excited 
the attention of mathematicians ever since the late M. 
to make it the basis of the fluxion- 
al or differential calculus, first in- the Memoirs of the - 
Berlin Academy for 1772, and afterwards in his Theo- 
rie des Fonctions Analytiques. ‘The demonstration of it 
iven here, is taken from a Memoir by M. Ampére, pub- 
fahed in por as th Cahier osha leat Siietens 
ique. Itis simple and very e t. ge 
has also given an el ¥ t pita! bee <n which has 
been improved by M: Poisson, and is now ly com- 
plete as that which we have: here » but more 
diffuse. 
53. We shall now apply. Taylor's theorem to the 
developement of the five elementary functions, which 
have so often come under our consideration. 
Ist. Let,f(z)=u=a", then, (rule (A) art. 26.), 
du 
d x* 
ited =n(n—] ) (n—2) a3, 
=n(n—1)2"—*; 
