Mr.  playfair  on  the 
3«o 
which  isimpoffible.  The  fame  holds  wherever  expreffions 
of  this  kind  occur.  Thus,  when  it  is  afferted  that  unity 
has  the  three  cube  roots  I,  — no 
more  is  meant  than  that  when  the  general  equation 
X^-ax^  + bx-r-o  is,  by  a change  in  the  data,  reduced  to 
the  particular  ftate  x 3-  i =o,  xis  then  equal  to  unity  only, 
and  admits  not  of  any  other  value,  as  it  does  in  more 
general  forms  of  the  equation.  The  natural  office  of 
imaginary  expreffions  is,  therefore,  to  point  out  when 
the  conditions,  from  which  a general  formula  is  derived, 
become  inconliftent  with  each  other;  and  they  cor- 
refpond  in  the  algebraic  calculus  to  that  part  of  the  geo- 
metrical analylis,  which  is  nfually  ftyled  the  determina- 
tion of  problems. 
3.  This,  however,  is  not  the  only  ufe  to  which  imagi- 
nary expreffions  have  been  applied.  When  combined 
according  to  certain  rules,  they  have  been  put  to  denote 
real  quantities,  and  though  they  are  in  fa<5t  no  more  than 
marks  of  impoffibility,  they  have  been  made  the  fubjecfls 
of  arithmetical  operations;  their  ratios,  their  products, 
and  their  fums,  have  been  computed,  and,  what  may 
feem  ftrange/juft  conclulions  have  in  that  way  been  de- 
duced. Neverthelefs,  the  name  of  reafoning  cannot  be 
given  to  a procefs  into  which  no  idea  is  introduced. 
Accordingly 
