obtained by Means of imaginary Quantities. 113 
either o, 2 7r s/ — t, ^ttv/ — 1 , 6tt s/ — 1 ... . .or (2 m n s/ — 1], 
the equation 1 = 1 -j- x 4" 7—^ &c. becomes identical ; and if, in 
the same equation, for x be substituted either^ s/ — r,.girs/ — 1, 
5 tt s/ — 1, . . ... or (2 m + 1) *r v/ — 1 , the equation — 1 = 1 
&c. becomes identical ; for, substitute 2 m it \/ — 1 
for x in series for e x , then it becomes 
. , / x (2m<n-) z (2 mi) 5 V — i . (zrmr)* 
1 HH (2W7TV l) — ~ 7 ~ - ,.,. 3 — + — 3 — 
(2mir) z , (znhr)* 0 , / / 
or 1 — - — — 4- — &c. + v — 1 2 m 
X . 2 ' I . 2 . 3 . 4 1 \ 
(2 m it ) 1 
1.2.3 
&c. 
or cos. 2 m 7 t4- v 7 ■ — 1 sin. am tt ; but sin. 2 m tt = o cos. 2 m tt- 
In like manner it will appear, that — 1 = — 1, if 
(2 m -f 1)7 r/ — 1 be substituted for x. 
zm.'ir'J — 1 
Since 1 — e 
= *, and since. 
1 = (__ iy==el 2m+1 ) z ” V ~ l ; but, in 
the developement of e^~ m + ^ 2 " ^ ~ the rule of — x — = + 
is observed .-.it must be observed bn the other side of the equation; 
hence, if (1) 2 = (— 1) 1 , the only strict conclusion that can be 
drawn is this, that and f 1 m + 1 1 2 v v ~ 1 * developed, 
produce the same series. It is a false consequence that, since 
(i) 2 = ( — 1) 2 the logarithm of (.!■)*= logarithm ( — 1) 2 , the 
logarithms are the indices 4m n \/ — 1 and ( 2 m -f . 1 ) 2 n \/ — 1 . 
* 1 is equal 
are each = 1 5 
sti-V' — 1 4 vV — 1 lirf-i , 
e or e &c. or e , and e expanded 
no consequence can be drawn from what is true for quantities elevated 
xV~i 
to real powers, since t can only have the meaning assigned it, that of being 
an abridged symbol. 
MDCCCI. 
2 
