414 
Direa, Therefore, by the formula, f(s), that is, (244/)'= 
— ze ht + &e. 
6. x, then supposing 5 to be at 
and putting B for |. (4), we have, 
bye (8) a (ya pets 
u i 
F 
pressions in the general 
+h), orlog. (z+h)= 
1h he is i 
bog. 2 + ef — st gaat *} 
By transposing the first term ] EN idle 
log. (z4+-h)— log. x= log. it's log. (1+ —), 
this formula becomes 
log. (1+ =5 {2-5 +4 - &e.} 
From this » lati the calcula 
ere ead shewn in ies 
ALGEBRA. 
$d. Next, let f(x) =u=a’, let A=1(a); then, by 
prin) oes 
du a’, Ae Gu z, 
—s As, fe aNe ’ ate 
du _- = 
a At » &e. 
Therefore, substituting co fie gene Beate A find 
at zh 
S (t+h) ora"  =aae = 
Ath Ath 
1.2.3" 1.2.3.4 
+8} 
Let both spre methane bedivided by a’, and 
then changing h into z, we have 
a a1 +Arp AF 4 ae 
It we suppose 21, then, 
o=i+Ay Vy A 
and if we make x= =i a 
+ A* F ond 
2.3.4 
+ &e. 
» 
oe 1 1 
aeSt+otastisa : + +f 
I 
Thus the quantit: a “is a constant number, which jis 
the value of a, when A=1. By taking the sum of a 
L 
sufficient number of terms of the series, we find a*—= 
2.71828 1828459045. 
Let this number be denoted by ¢, and then a! =*% 
anda=e*. AsA isthe Napierean logarithm of ,e 
must be the Jasis or radical number in N; : 
tem. We have found its value by a different poikpa 
ime ting ing Since a=e*, therefore a™— 
qauntity of which the of Ax; hence, an exponen- 
basis is any number a, may 
FLUXIONS. 
be transformed into another, determinate 
robe for by whch such hat er “ 
this scopy Sie See ree important element 
ih, Spon ow tat (== si, by rule @) 
8, Vo» Fethew 
—— = SiN. 2, 
dei axe 
in this case, the general fila oh. 
tf Ee sin. (ex++-h)= ‘ 
sin 2 — hy c08. 2 
sin. xh eos.0— A 
it —_— 
$l sine + coe 2 hes 
5th, Let f(x) =u=cos. , then, rule (D) art 26, 
au es u_ Bu sin, % 
‘ 
~ 
and hence f (*4-h)=cos. Sa ier ty 
“cos, a — sin. s— = 008, 2 r+ sin. z 
3 2.3 wail 
ie he re eh eh. io 
se 9) WH ‘hee pan) geese 2B 
ae; ay . 
eae 
Pao aka neck ahs 
hs hs 
a xh! BR &¢ 
Qh 54 xehb 2S be Co + ee 
edi ai ents of the functions sin, GH: and 
a C4 =P sin, 24+Q cos. 2, 
Cos. («-++h)=P cos. r—Q sin. 2; 
But, by the Arrrumeric of Sines, 
Sin. ab =cos. A sin, e+sin. h cos. x, 
— 
Cos. (#++-h)= hice = Phe HAA i 
From these equations we bxas! aaa 
(P—cos. /) sin. x+( me a 
(P—cos. h) cos. eam #=0; 
and hence, ae 
P—cos. i=0, Q—sin. h=0; ; 
therefore, cos. h=P 5 sin, h=Q, tai ping 2 ine 
stead of h, 
maps Th + +s S7a373.4,516" 
2 Sen he? + 
F845. 640 ot 
; x 
pe Ts ag o.a6 
&e. 
54. Resuming Taylor re 4 i ‘“ 
3 u 
Seth) sup Sth pF 5 gt aves &e. 
ce ge hen #= 0, then f (2), oF, be 
w = u, De- 
comes Ua tha mpee the. rs nig 
ah old ', 
spectively. The theorem then’ vig a=0, 
s@)aU +UA4UrS i oil in 
