230  Mr.  playfair  on  the 
of  continual  proportionals,  can  be  exhibited  geometri- 
cally. For,  from  the  points  a,  b,  c,  and  d,  let  am,  bn, 
co,  dp,  be  drawn  at  right  angles  to  the  affymptote  fp  ; 
let  gb  produced  meet  fp  in  Q_,  and  let  br  be  perpendicu- 
lar to  the  conjugate  axis  fr.  Then,  becaufe  the  triangles 
frs,  fma,  are  equiangular,  af  : fm  ::  fs  : fr;  hence  fr  = 
x fs  = — x fn  - nb.  For  the  fame  reafon  ch  = 
FA  FA 
——  x fo— oc,  and  dk=  — xfp-pd.  Therefore,  bg  + ch  + dk 
fa  7 FA 
=™xFN  + fo  + fp-—  xbn+co  + dp;  now,  fn, fo,  fp,  are 
FA  FA 
continual  proportionals,  and  fo  alfo  are  bn,  fo,  fp,  be- 
caufe the  fedtors  fbc,  fcd,  are  equal.  But  in  the  circle 
no  fuch  refolution  of  the  propofed  feries  of  lines  can 
take  place,  that  feries  being  fubjedt  to  alternate  increafe 
and  diminution ; on  which  account  it  is,  that  imaginary 
characters  enter  into  the  exponential  value  ox  the  line. 
Thofe  characters  are  therefore  fo  far  from  compenfating 
each  other  in  the  prefent  cafe,  as  they  ought  to  do,  on  the 
fuppolitionof  Bernoulli  and  maclaurin, that  they  ma- 
nifellly  ferve  as  marks  of  impoffibility.  There  remains, 
of  confequence,  the  affinity  between  circular  arches  and 
hyperbolic  areas,  or  between  the  meafures  of  angles  and 
of  ratios,  as  the  only  principle  on  which  the  imaginary 
inveltigation  can  proceed.  It  need  fcarcely  be  obferved, 
that 
