Mr.  playfair  on  the 
322 
denote  real  quantities,  the  imaginary  characters  involved 
in  the  different  terms  of  fuch  expreffions  do  then  com- 
penfate  or  deftroy  each  other.  But  befkle  that,  the  man- 
ner in  which  this  compenfation  is  made,  in  expreffions 
ever  fo  little  complicated,  is  extremely  obfcure,  if  it  be 
confidered  that  an  imaginary  character  is  no  more  than 
a mark  of  impoffibility,  fuch  a compenfation  becomes 
altogether  unintelligible:  for  how  can  we  conceive  one 
impoffibility  removing  or  deftroying  another  ? Is  not  this 
to  bring  impoffibility  under  the  predicament  of  quantity, 
and  to  make  it  a fubjeCt  of  arithmetical  computation? 
And  are  we  not  thus  brought  back  to  the  very  difficulty 
to  be  removed?  Their  explanation  cannot  of  confe- 
quence  be  admitted ; but,  on  attempting  another,  it  be- 
hoves us  to  obferve,  that  a more  extenfive  application  of 
this  method,  than  had  been  made  in  their  time,  has  now 
greatly  facilitated  the  inquiry.  We  begin  then  with 
confidering  the  manner  in  which  the  imaginary  expref- 
fions, fuppofed  to  denote  real  quantities,  are  derived ; and 
the  cafes  in.  which  they  prove  ufeful  for  the  purpoles  of 
invefligation. 
4.  Let  a be  an  arch  of  a circle  of  which  the  radius 
is  unity,  and  let  c be  the  number  which  has  unity  for 
its  hyperbolic  logarithm,  then  the  fine  of  the  arch  a , or 
fin. 
