32 
Functions, the determination of the term that contains the first 
power of i in the developement of (« + #)", according 
to the powers of i. Now it may be demonstrated by 
the elementary operations of algebra, that whether » be 
positive or negative, whole or fractional, the two first 
terms of the developement of (« +. 4)" are a” 4 na"—*i; 
(See ALcrsna, art. 319; also FLuxrons, art. 7.) ; there- 
fore, the first derivative function of 2” is ma"—, -It is 
now easy to find all the terms of the developement of 
F(a +i) = («472)%. For since from fa= x", we 
have f’x = n x—", from this last we derive 
f’2sn(n—1) 2, 
and hence again f= n(n—1)(n—2) "3, &c. 
So that from the series 
Sle $i) Sfp ifr Eprerg be. 
we get 
(a + i)? 2" 4 nett —— 
which is Newton’s binomial theorem. 
Next let the function be fa = a”, a being eupposed 
constant, and « variable; then, f (« + i)=a*t#. Now 
the common principles of analysis are sufficient to prove 
that the two first terms of the developement of a*+? 
area* + Aa*#; here A is the Napierian log. of a, 
{see ALGEBRA, art. 355; also Fiuxions, art. 14, and 19.) 
Therefore the first derivative function of a” is Aa™s 
that is, fz = Aa”; hence again f” « =A?a", f"2= 
A3 ax &c. These values substituted in the develope- 
ment of f(« +- 7) give 
wre: 
p At A3 
attisa™ 4 Aa"i4 Bae +97g907 3 +&e. 
Hig Gio a tee aby Wo iota tad hee 
tases, that the two first terms of log. (w +7) are 
log. v4 as i, Bbeing put forthe Napierian log. of the ba- 
sis of the system ; (see FLuxrons, art. 18. and] 9.) There- 
fore the first derivative function of log. « isa— r 
x 
because. f’ x =r by the rule for the derivative func- 
and, 
x 4a 1 
tio: f wer, h te — ; a a 
n of a power, we hence find f” x BY =p 
‘ 2 : 
and again f’”" 2 = Bar &* These substitutions being 
made in the general developement of f(x 4. 4), we get 
Me Lp she deride de deoees as te “ 
B: Hbe-e+ po tone + ape & 
It has been shewn, (FLuxions, art. 17, and 19,) that 
the two first terms of the developements of the sine and 
cosine of x + iare 
sin. (« + 7%) = sin. a 4 icos.x - &c. 
cos. (« +4 7)= cos. z— sin. « + &e. 
_ Hence it appears that the first derivative function of 
sin, # 1s cos. @,:and that the first derivative function of 
cos, vis — sin. 2: Since therefore in the case of fc = 
sin, 2, we have f’ «= cos, x, it follows that i w= 
— sin. a, f’” «=m — cos. 2, &c. and since when 
Fe =cos.«, we have Fx = —sin. x, it follows that 
F’2=— cos, a, F’” a=sin, 2, &c, These expres- 
FUNCTIONS. 
sions substituted in the developement of f («+ 7) and Funeti 
F (« + 2) give 
: od 
sin. (x +7) = sin. 2 47 cos. 2—5sin.2— 5, cos. a+ &e. 
2? 
2 
From the brief view we have given of this calculus, 
its intimate analogy with the methiod of fluxions, or dit- 
ferential calculus, must be evident.’ In fact, they all 
rest upon the same analytical principles, and, the ob- 
ject presented to the mind in each is the same; for the 
different orders of derivative functions in Lagrange’s 
calculus are identical with the successive differentials, 
or rather differential coefficients in that of Leibnitz, and 
with the different orders of fluxions in Newton’s theo« 
The peculiarity of each calculus, as delivered ori- 
ginally by the inventor, consists in that relation be- 
tween the original function and its prime function, or 
differential, or fluxion, which the mind selects as a sub- 
ject of contemplation. We have seen that it is a fun~ 
damental proposition in analysis, that if «+47 be sub- 
stituted for x in any function,fa, its new value f(« 4-2) 
has always the form fz + ip + #q + #r + &c. p, q, 
r, &c. being functions of a, which are int dent of ? 
Newton observed, that if « and fx are ted by 
two lines generated by motion, and if i be the velocity 
of the point which generates x, then 7 p, the second term 
of the developement, will be the velocity of the point 
that generates fx; (Fiuxtons, art- 20—22.) hence he 
called ip the fluxion of the function fa. Leibnitz 
again considered, that if « was in by the quan- 
tity i, then f.« was augmented by the increment ¢ p 
#q48r-+4 &c. But supposing ¢ indefinitely smal 
the first term of this series is indefinitely ter than 
the sum of all the following terms; therefore re- 
jecting these, and retaining the term ép alone, he cal« 
led it the differential of the function fx. (FLuxions, 
art. 107—110.) Lagrange, regarding the generation 
of algebraic quantities by motion as incompatible with 
the principles of pure analysis, and also considering the 
doctrine of infinitely small quantities, as too slippery a 
foundation for so sublime an edifice, he rejected both 
views of the subject, and deduced its principles from 
the theory of the developement of functions into series. 
It is in general admitted, that the Theory of Analytic 
Functions has fulfilled the promise of its illustrious au- 
thor, “ to deliver the principles of the differential cal- 
culus dines from the consideration of infinitely 
small or vanishing quantities, also limits and fluxions.” 
We think, however, that he has under-rated the value 
of the theory of limits, as delivered by Maclaurin and 
D’Alembert, when he says that the kind of metaphysique 
that must be employed in it is, if not contrary, at least 
foreign to the spirit of analysis, which ought not to 
have any other metaphysique than that which consists in 
the first principles, and the first fundamental operations 
of algebra. ’ 
The ingenious author, in the discussion of his theory, 
has adopted a new notation. This has been matter of 
regret, (Lacroix Cal. Dif: vol. i. art. 82, 83.) because 
the notation of the differential calculus was quite sufli- 
cient. In the comparison of methods and formule, 
different notations are perplexing, and the number of 
arbitrary characters ly employed in analysis is a 
considerable and increasing evil. This, however, is but 
a small defect, when the luminous views and original 
methods which the work contains are taken into account, 
. ; ‘ 
cos. (w-+2)=cos. — isin. c— Cos, x s3sin. x+ &e. 
