﻿CONSTITUTING 
  A 
  MAGIC 
  CYCLOVOLUTE. 
  207 
  

  

  To 
  trace 
  one 
  of 
  these 
  curves, 
  commence 
  at 
  the 
  extremity 
  A 
  of 
  the 
  

   principal 
  diameter 
  AA 
  7 
  , 
  and 
  continue 
  along 
  the 
  circle, 
  of 
  which 
  the 
  

   centre 
  is 
  a, 
  nearly 
  to 
  the 
  extent 
  of 
  a 
  semicircle 
  ; 
  then 
  incline 
  towards 
  

   the 
  least 
  interior 
  circle, 
  ad 
  bb 
  1 
  , 
  and 
  terminate 
  in 
  its 
  circumference. 
  

   In 
  like 
  manner 
  another 
  volute 
  may 
  be 
  traced 
  in 
  the 
  opposite 
  direction, 
  

   and 
  thus 
  will 
  appear 
  two 
  of 
  the 
  volutes 
  originating 
  in 
  the 
  point 
  A. 
  

   Six 
  similar 
  volutes 
  may 
  be 
  traced 
  from 
  the 
  extreme 
  points 
  A', 
  B, 
  B' 
  ; 
  

   and 
  all 
  the 
  eight 
  viewed 
  in 
  pairs 
  may 
  be 
  easily 
  recognized 
  by 
  the 
  four 
  

   different 
  colours 
  in 
  which 
  they 
  are 
  delineated. 
  Besides 
  these 
  volutes, 
  

   we 
  may 
  trace 
  eight 
  analogous 
  curves, 
  from 
  the 
  extremities 
  of 
  the 
  

   diameters 
  intermediate 
  between 
  the 
  conjugates 
  AA', 
  BB'. 
  In 
  the 
  

   drawing 
  they 
  may 
  be 
  traced 
  by 
  passing 
  along 
  circular 
  segments, 
  de- 
  

   creasing 
  and 
  changing 
  their 
  colours, 
  whilst 
  verging 
  towards 
  the 
  interior 
  

   circumference 
  aa! 
  bb'. 
  There 
  will 
  thus 
  appear 
  sixteen 
  similar 
  volutes, 
  

   in 
  addition 
  to 
  the 
  circles 
  first 
  described 
  ; 
  and 
  all 
  these 
  have 
  precisely 
  

   the 
  same 
  property 
  relatively 
  to 
  the 
  number 
  360, 
  which 
  forms 
  the 
  com- 
  

   mon 
  result 
  of 
  the 
  auxiliary 
  12, 
  and 
  every 
  eight 
  numbers 
  within 
  any 
  

   two 
  consecutive 
  boundaries. 
  

  

  These, 
  we 
  believe, 
  are 
  all 
  the 
  properties 
  of 
  which 
  the 
  arrangement 
  

   of 
  numbers 
  constituting 
  the 
  cyclovolute 
  appears 
  susceptible 
  ; 
  and 
  we 
  

   intended 
  to 
  subjoin 
  here 
  the 
  investigation 
  which 
  led 
  to 
  them, 
  and 
  to 
  

   the 
  different 
  changes 
  that 
  may 
  be 
  made 
  in 
  disposing 
  the 
  numbers 
  in 
  

   the 
  drawing. 
  We 
  have, 
  however, 
  concluded 
  to 
  omit 
  this 
  investigation 
  

   for4;he 
  moment, 
  and 
  make 
  it 
  the 
  subject 
  of 
  a 
  supplementary 
  note 
  to 
  

   be 
  read 
  at 
  a 
  future 
  meeting. 
  

  

  Regarding 
  the 
  objects 
  of 
  the 
  Society, 
  this 
  paper 
  is 
  presented 
  without 
  

   any 
  desire 
  for 
  its 
  publication, 
  and 
  chiefly 
  in 
  compliance 
  with 
  the 
  wishes 
  

   of 
  a 
  friend. 
  But 
  as 
  the 
  Magic 
  Circle 
  originated, 
  and 
  has, 
  I 
  presume, 
  

   been 
  completed 
  in 
  Philadelphia; 
  and 
  as 
  it 
  has 
  been 
  considered 
  in 
  

   Europe 
  as 
  the 
  most 
  ingenious 
  arrangement 
  of 
  numbers 
  ever 
  imagined, 
  

   the 
  Society 
  may 
  not 
  be 
  disinclined 
  to 
  insert 
  some 
  notice 
  of 
  the 
  subject 
  

   in 
  their 
  records. 
  

  

  