Arithmetic  of  impojjible  Quantities.  3 3'S’ 
14.  The  forms  in  the  Harmonia  Menfurarum  might 
alfo  be  brought  to  confirm  this  theory;  but,  without 
accumulating  inftances  any  farther,  it  may  be  fufficient 
to  remark  two  confequences  that  follow  from  it : 1 . That 
the  only  cafes  in  which  imaginary  expreffions  may  be 
put  to  denote  real  quantities,  are  thofe  in  which  the  mea- 
fures  of  ratios  or  of  angles  are  concerned,  2.  That  the 
property  of  either  of  thofe  meafures,  fo  inveftigated, 
might  have  been  inferred  from  analogy  alone.  Now 
both  thefe  conclufions  are  agreeable  to  experience.  It 
does  not  appear,  that  any  inftance  has  yet  occurred  where 
imaginary  characters  ferve  to  exprefs  real  quantities,  if 
circular  arches  or  hyperbolic  areas  are  not  the  fubjedts  of 
inveftigation ; and  if  the  conclufion  obtained  may  not  be 
transferred  from  the  one  to  the  other,  by  a mere  fubfti- 
tution  of  correfponding  magnitudes ; that  is,  of  fines  for 
ordinates,  cofines  for  abfciffes,  and  circular  arches  for  the 
doubles  of  hyperbolic  fedfors.  The  affinity  between  the 
circle  and  hyperbola  is  not  however  fo  clofe,  but  that  it 
is  fubjedt  to  certain  limitations,  from  confidering  which, 
the  truth  of  what  is  here  afferted  wfill  be  rendered  more 
evident. 
1 . Any  propofition  demonftrated  of  hyperbolic  fee- 
tors  may  be  transferred  to  circular  arches  by  fubftitution 
alone,  without  any  change  in  the  figns,  when  only 
1 abfeiffie 
