Mr.  flayfair  on  the 
34  2 
to  the  circle  and  hyperbola,  and  that  it  was  into  the  in- 
Veffigation  of  thofe  theorems,,  that  the  imaginary  expref- 
fions  were  introduced. 
The  concluuons  therefore  from  the  whole  are  thefe: 
that  imaginary  expreffions  are  never  of  ufe  in  inveftiga- 
tion  but  when  the  fubjecf  is  a property  common  to  the  mea- 
fures  both  of  ratios  and  of  angles ; that  they  never  lead 
to  any  confequence  which  might  not  be  drawn  from  the 
affinity  between  thofe  meafures;  and  that  they  are  in- 
deed no  more  than  a particular  method  of  tracing  that 
affinity.  The  deductions  into  which  they  enter  are  thus 
reduced  to  an  argument  from  analogy,  but  the  force  of 
them  is  not  diminifhed  on  that  account.  The  laws  to 
which  this  analogy  is  fubjeCt;  the  cafes  in  which  it  is 
perfect,  in  which  it  fuffers  certain  alterations,  and  in 
which  it  is  wholly  interrupted,  are  capable,  as  may  be 
concluded  from  the  fpecimens  above,  of  being  precifely 
afcertaihed.  Supported  on  fb  fure  a foundation,  the  arith- 
metic of  impoffible  quantities  will  always  remain  an  ufe- 
ful  inftrument  in  the  difcovery  of  truth,  and  may  be  of 
fervice  when  a more  rigid  analyiis  can  hardly  be  applied. 
For  this  reafon,  many  refearches  concerning  it,  which  in 
themfelves  might  be  deemed  abfurd,  are  neverthelefs  not 
deffitute  of  utility.  M.  Bernoulli  has  found,  for  exam- 
ple, that  if  r be  the  radius  of  a circle,  the  circumference 
i 
