﻿206 
  REMARKABLE 
  ARRANGEMENT 
  OF 
  NUMBERS. 
  

  

  Fourthly. 
  Every 
  four 
  adjacent 
  numbers 
  in 
  any 
  two 
  consecutive 
  

   rings, 
  with 
  half 
  the 
  auxiliary 
  number, 
  give 
  the 
  same 
  amount, 
  180. 
  

  

  As 
  to 
  the 
  four 
  remaining 
  sets 
  of 
  circles 
  and 
  the 
  rings 
  which 
  they 
  

   form, 
  their 
  centres 
  are 
  at 
  the 
  four 
  points 
  in 
  which 
  the 
  principal 
  diam- 
  

   eter, 
  and 
  a 
  conjugate 
  perpendicular 
  to 
  it, 
  intersect 
  the 
  least 
  and 
  interior 
  

   circumference. 
  If 
  our 
  attention, 
  for 
  the 
  instant, 
  be 
  confined 
  to 
  any 
  

   one 
  of 
  these 
  centres, 
  and 
  to 
  the 
  corresponding 
  set 
  of 
  circles, 
  the 
  bound- 
  

   ing 
  circumferences 
  of 
  the 
  exterior 
  and 
  interior 
  rings 
  will 
  be 
  seen 
  to 
  

   touch 
  the 
  greatest 
  and 
  least 
  of 
  the 
  nine 
  principal 
  circumferences, 
  at 
  

   points 
  in 
  the 
  principal 
  diameter 
  or 
  its 
  conjugate. 
  According 
  to 
  this 
  

   construction 
  there 
  are 
  Jive 
  rings 
  between 
  the 
  bounding 
  circumferences 
  

   of 
  each 
  of 
  the 
  four 
  sets 
  of 
  circles 
  under 
  consideration 
  ; 
  and 
  all 
  the 
  

   twenty 
  rings 
  thus 
  constituted 
  possess 
  the 
  same 
  property 
  with 
  the 
  eight 
  

   rings 
  first 
  mentioned 
  ; 
  or 
  in 
  more 
  specific 
  terms, 
  the 
  eight 
  numbers 
  in 
  

   each 
  of 
  the 
  twenty 
  secondary 
  rings, 
  with 
  the 
  auxiliary 
  number 
  at 
  the 
  

   principal 
  centre, 
  form 
  the 
  sum 
  360. 
  

  

  These 
  are 
  the 
  different 
  properties 
  comprised 
  in 
  the 
  Magic 
  Circle, 
  

   left 
  by 
  its 
  original 
  and 
  sagacious 
  author. 
  They 
  certainly 
  must 
  be 
  

   regarded 
  as 
  not 
  a 
  little 
  curious, 
  and 
  would 
  seem 
  to 
  require 
  a 
  consider- 
  

   able 
  familiarity 
  with 
  the 
  powers 
  of 
  numbers. 
  As 
  to 
  the 
  mode 
  of 
  

   investigation 
  by 
  which 
  they 
  were 
  first 
  discovered, 
  we 
  have 
  seen 
  no 
  

   account 
  sufficient 
  to 
  enable 
  us 
  to 
  pronounce 
  with 
  any 
  degree 
  of 
  con- 
  

   fidence. 
  We 
  should 
  not, 
  however, 
  be 
  inclined 
  to 
  think 
  that 
  they 
  

   resulted 
  either 
  from 
  conjecture 
  or 
  trial, 
  although 
  they 
  are 
  by 
  no 
  means 
  

   confined 
  to 
  the 
  particular 
  distribution 
  of 
  numbers 
  published. 
  We 
  

   should 
  rather 
  be 
  disposed 
  to 
  join 
  in 
  the 
  opinion 
  that 
  they 
  were 
  sug- 
  

   gested 
  by 
  remarks 
  made 
  on 
  other 
  arrangements 
  previously 
  formed. 
  

   But 
  still 
  we 
  are 
  forced 
  to 
  believe 
  that 
  they 
  must 
  have 
  been 
  deduced 
  

   from 
  views 
  which 
  were 
  incapable 
  of 
  embracing 
  in 
  its 
  full 
  extent 
  the 
  

   general 
  problem, 
  whence 
  originated 
  the 
  present 
  observations. 
  The 
  

   reasons 
  which 
  justify 
  this 
  conclusion 
  will 
  immediately 
  appear 
  on 
  a 
  

   glance 
  at 
  the 
  drawing 
  which 
  accompanies 
  this 
  paper, 
  and 
  which 
  may 
  

   be 
  regarded 
  as 
  a 
  generalization 
  of 
  Dr 
  Franklin's 
  Magic 
  Circle. 
  The 
  

   additions 
  made 
  are 
  Volutes, 
  commencing 
  at 
  the 
  extremities 
  of 
  the 
  

   diameters 
  between 
  the 
  numbered 
  radii; 
  and 
  on 
  which 
  account 
  the 
  

   drawing 
  may 
  not 
  inappropriately 
  be 
  termed 
  a 
  Magic 
  Cyctovolute. 
  

  

  