) 
Arithmetic  of  impojjible  Quantities. 
323 
ct»j— I 1 , - -f  i — I 
fin.  <2= — — ; and  cof.  a = . Thefe 
exponential  and  imaginary  values  of  the  line  and  cofine 
are  already  well  known  to  geometers ; and  the  inveftiga- 
tion  of  them,  according  to  the  received  arithmetic  of 
impoflible  quantities,  may  be  as  follows. 
Let  fin.  a=z,  then  a- 
s/*-; 
To  bring  this  fluxion 
under  fuch  a form  that  its  fluent  may  be  found  by  loga- 
rithms, both  numerator  and  denominator  are  to  be  mul- 
tiplied by  V - 1 ; then  a —\/—  i-x  —= — , and  (by  form. 
%/  Z — I 
6. harm. Men.) -ix  1 . Hence  — — 01 
1 x ~~i  - log , and  becaufe  1 is  the  log,  of  c. 
V— '-AA” 1 . 
C ~ 
V- 
; wherefore,  if  both  parts  of  the  fractional 
index  of  c be  multiplied  by  \/— 1,  c~~a A/~ 1 • 
Again,  if  the  arch  a be  conlidered  as  negative,  its  fine  be- 
comes alfo  negative,  and  therefore  -a-V  - 1 x log. 
i or>  -as/ - 1 = - log.^Lt^  ~~lf  and  as/- 1 = 
log.  — ; whence  alfo,  cai/~l  — • If  from 
this  equation  the  former  be  taken  away,  there  remains 
= ca/~~I  - £—V— whence  dividing  by  iV - 1 we 
i n cas/~~ 1 — -r-W—1 
have  z = fin.  <3:  = - 
Vol.  LXVIIL 
zV- 
. By  adding  together  the 
T t equations 
