Arithmetic  of  impojjible  Quantities.  3.21 
Accordingly  geometry,  which  has  its  modes  of  reafon- 
ing  that  correfpond  to  every  other  part  of  the  algebraic 
calculus,  has  nothing  fimilar  to  the  method  we  are  now 
confidering ; for  the  arithmetic  of  mere  characters  can 
have  no  place  in  a fcience  which  is  immediately  conveiv 
fant  with  ideas. 
But  though  geometry  rejects  this  method  of  inveftigar 
tion,  it  admits,  on  many  occafions,  the  conclufions  de- 
rived from  it,  and  has  confirmed  them  by  the  moft  ri- 
gorous demonftration.  Here  then  is  a paradox  which 
remains  to  be  explained.  If. the  operations  of  this  imagir 
nary  arithmetic  are  unintelligible,  why  are.  they  not  alfo 
ufelefs?  Is  inveftigation  an  art  fo  mechanical,  that  it  may 
be  conducted  by  certain  manual  operations  \ or  is  truth 
fo  eafily  difcovered,  that  intelligence  is  not  neceflary  to 
give  fuccefs  to  our  refearches  ? Thefe  are  difficulties 
which  it  is  of  fome  importance  to  refolve,  and  on  which 
much  attention  has  not  hitherto  been  bellowed.  Two 
celebrated  mathematicians,  Bernoulli  and  maclaurin, 
have  indeed  touched  on  this  fubjedl;  but  being  more  in- 
tent on  applying  their  calculus,  than  on  explaining  the 
grounds  of  it,  they  have  only  fuggefted  a folution  of  the 
difficulty,  and  one  too  by  no  means  fatisfa£tory.  They 
alledge M,,  that  when  imaginary  expreffions  are  put  to . 
(a)  Op.  j.  BERN.  tom.  I.  N°  70.  maclaur.  Flux.  art.  699—763. 
denote. 
