Arithmetic  of  impoffible  Quantities.  323 
become  negative,  and  c~l a—  - £+Vi+£h  Hence  ar  = 
■ — in  like  manner  the  abfcifs  belonging  to  acb,  that 
is  ce  = — - — . Thefe  values  of  the  ordinates  and  abfciffse 
-4 
differ  in  nothing  from  thofe  of  the  fines  and  cofines 
already  found,  except  in  being  free  from  impoffible 
quantities ; for  it  is  evident,  that  the  quantity  a is  related 
in  the  fame  manner  both  to  the  circular,  and  hyperbolic 
fedtors.  If  now  ord.  a and  abf.  b denote  the  ordinate 
and  abfcifs  belonging  to  the  lectors  \b  refpedtively, 
r ca c — a Cb  + C — b ca  + b — C — a~b ca-~b Cb— a 
ord.  a x abl.  b = — - — x — - — = — — — — — — = 
2 2 4 
ord.  a-\~o  ord.  o — b 
6.  The  conclufions  in  both  the  foregoing  cafes  are 
perfectly  coincident,  and  the  methods  by  which  they 
have  been  obtained  are  fimilar;  though  with  this  dif- 
ference between  them,  that  in  the  firft  all  the  fteps  are 
unintelligible,  but  in  the  lafi  fignificant.  If  then,  not- 
withftanding  a difference  which  might  be  expedted  fo 
materially  to  affedt  their  conclufions,  they  have  been 
equally  fuccefsful  in  the  difcovery  of  truth,  it  can  be 
afcribed  only  to  the  analogy  which  takes  place  between 
the  fubjedts  of  inveftigation ; an  analogy  fo  clofe,  that 
every  property  belonging  to  the  one  may,  with  certain 
reflridtions,  be  transferred  to  the  other.  Accordingly, 
T t 2 every 
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