Arithmetic  of  impojfible  Quantities.  327 
8.  Though  it  might  readily  be  concluded,  that  the 
fame  principle  on  which  the  foregoing  inveftigation  has 
been  found  to  proceed,  extends  itfelf  to  all  thofe  in  which 
imaginary  expreflions  are  put  to  denote  real  quantities, 
it  may  yet  be  proper  to  make  trial  of  its  application  in 
fome  other  inftances. 
Let  ab,  ac,  ad,  ae  (fig.  3.)  be  any  arches  of  a circle 
in  arithmetical  progreffion,  and  let  m be  their  number; 
it  is  required  to  find  the  fum  of  the  fines  bc,  ch,  8cc.  of 
thofe  arches.  Let  the  radius  af  = i,  pc?, -a,  and  the  com- 
mon difference  of  the  arches,  or  bc=a:  the  fum  of  the 
feries  fin.  a + fin.  a + x + fin.  a + 2X  + (ni)  is  to  be 
qCIsJ- — I — ^ — asj — 1 
found.  Now,  becaufe  lin.  a = . ^ and 
Ca+xX  K/—i c—a—xX  \/— I 
fin.  a + x = - 2y/_j 
fin.  a + lin.  a + x + lin.  a + 2 x . . 
2i/— I 
8tc. ; the  feries 
I 1 + cZxV—  I 
I + C' 
v/~  I + c — 2 as/- 
( m ).  But  thefe  feries  are 
both  geometrical  progrelfions,  and  the  fum  of  the  firft  is 
cat/—*  I —cmx*J — 1 1 ~ , r * I-— cr~ 
X — and  of  the  fecond,  - x * 
2i/ — 1 
The  fum  of  the  propofed  feries  therefore 
ca*J — 1 1 — — 1 c — «\/ — 1 x — C — inxtj—i 
2%/ - — I * I- — i,xs/‘  1 — I * I — -C  Xt/  1 
(fisj — ? ca+mx'/Kf~~m  1 — — 1 j 
i -—cxV  1 ——  c~~xs/  1 4~  1 
21/ — 1 
1 
2 /—i 
x 
