﻿240 
  J. 
  M. 
  Blake 
  — 
  Plotting 
  Crystal 
  Zones 
  on 
  the 
  Sphere. 
  

  

  made 
  to 
  the 
  stereograph 
  ic 
  projection 
  which 
  involves 
  the 
  use 
  of 
  

   the 
  tangent 
  of 
  the 
  half 
  angle. 
  By 
  such 
  a 
  substitution 
  we 
  lose 
  

   some 
  important 
  advantages 
  which 
  can 
  follow 
  the 
  use 
  of 
  the 
  

   gnomonic 
  projection. 
  The 
  objection 
  alluded 
  to 
  can 
  be 
  over- 
  

   come 
  in 
  the 
  case 
  of 
  the 
  orthorhombic 
  and 
  oblique 
  species 
  by 
  

   taking 
  care 
  in 
  selecting 
  the 
  proper 
  projecting 
  point. 
  

  

  There 
  is 
  a 
  certain 
  position 
  on 
  a 
  crystal 
  which 
  gives 
  the 
  most 
  

   compact 
  projection 
  of 
  the 
  planes. 
  This 
  position 
  will 
  furnish 
  

   the 
  least 
  complicated 
  set 
  of 
  indices. 
  A 
  projection 
  made 
  at 
  

   right 
  angles 
  to 
  this, 
  will 
  probably 
  show 
  the 
  planes 
  less 
  compact 
  

   but 
  still 
  within 
  reasonable 
  bounds. 
  A 
  third 
  position 
  at 
  right 
  

   angles 
  to 
  the 
  other 
  two 
  is 
  apt 
  to 
  give 
  a 
  very 
  widespread 
  pro- 
  

   jection 
  of 
  the 
  planes. 
  This 
  third 
  position 
  is 
  to 
  be 
  avoided 
  as 
  a 
  

   rule. 
  If 
  the 
  drawing 
  were 
  made 
  from 
  this 
  third 
  view-point 
  

   some 
  of 
  the 
  planes 
  would 
  be 
  seriously 
  foreshortened. 
  

  

  Fig. 
  3 
  gives 
  three 
  projections 
  of 
  humite, 
  type 
  I, 
  an 
  ortho- 
  

   rhombic 
  species. 
  It 
  shows 
  a 
  remarkable 
  segregation 
  of 
  planes 
  

   in 
  projection 
  1 
  at 
  right 
  angles 
  to 
  the 
  c 
  axis. 
  The 
  projection 
  

   shown 
  in 
  2 
  is 
  also 
  made 
  at 
  right 
  angles 
  to 
  the 
  c 
  axis 
  and 
  is 
  also 
  

   at 
  right 
  angles 
  to 
  the 
  1 
  position. 
  The 
  projection 
  on 
  the 
  c 
  

   plane 
  given 
  in 
  3 
  is 
  made 
  at 
  right 
  angles 
  to 
  the 
  1 
  and 
  2 
  posi- 
  

   tions. 
  It 
  is, 
  on 
  the 
  contrary, 
  very 
  diffuse, 
  and 
  for 
  this 
  reason 
  

   the 
  drawing 
  3 
  has 
  been 
  made 
  to 
  include 
  only 
  one 
  quadrant. 
  

   The 
  radiating 
  short 
  lines 
  in 
  1 
  and 
  2 
  represent 
  the 
  normals 
  of 
  

   the 
  prism 
  planes 
  according 
  as 
  we 
  adopt 
  one 
  or 
  the 
  other 
  posi- 
  

   tion. 
  The 
  humite 
  series 
  of 
  minerals 
  are 
  unique 
  in 
  having 
  

   forms 
  bounded 
  by 
  a 
  multiplicity 
  of 
  planes. 
  

  

  Oblique 
  crystals 
  may 
  give 
  several 
  projections 
  in 
  which 
  the 
  

   planes 
  are 
  not 
  widely 
  scattered. 
  Fig. 
  4 
  represents 
  a 
  triclinic 
  

   crystal 
  of 
  albite. 
  It 
  is 
  selected 
  to 
  show 
  an 
  inclined 
  crystal 
  

   in 
  which 
  the 
  projection 
  can 
  be 
  made 
  from 
  several 
  view-points 
  

   without 
  wide-spreading 
  the 
  plane 
  positions. 
  The 
  upper 
  cut 
  

   represents 
  a 
  crystal 
  laid 
  flat 
  on 
  the 
  o 
  plane 
  and 
  the 
  numbered 
  

   lines 
  pointing 
  towards 
  its 
  center 
  give 
  the 
  direction 
  of 
  each 
  pro- 
  

   jection. 
  The 
  middle 
  figures 
  show 
  the 
  crystal 
  in 
  several 
  posi- 
  

   tions 
  while 
  being 
  rotated 
  on 
  the 
  b 
  axis. 
  The 
  favored 
  positions 
  

   are 
  selected 
  with 
  reference 
  to 
  giving 
  an 
  edge-on 
  position 
  to 
  

   certain 
  planes. 
  The 
  lower 
  figure 
  shows 
  the 
  several 
  projections 
  

   corresponding 
  to 
  the 
  different 
  positions. 
  Radial 
  lines 
  at 
  the 
  

   margins 
  indicate 
  what 
  are 
  temporarily 
  the 
  prismatic 
  zones. 
  

   The 
  triclinic 
  character 
  of 
  the 
  crystal 
  is 
  shown 
  in 
  measurable 
  

   quantity 
  on 
  the 
  original 
  drawings 
  which 
  were 
  made 
  to 
  a 
  two 
  

   inch 
  radius, 
  but 
  is 
  not 
  plainly 
  shown 
  on 
  the 
  greatly 
  reduced 
  

   scale. 
  

  

  Plotting 
  the 
  Planes 
  on 
  the 
  Sphere. 
  — 
  The 
  planes 
  of 
  a 
  single 
  

   selected 
  zone 
  are 
  first 
  plotted 
  on 
  a 
  great 
  circle 
  drawn 
  on 
  the 
  

   sphere 
  with 
  a 
  pencil 
  guided 
  by 
  the 
  equatorial 
  ring. 
  If 
  we 
  

   have 
  a 
  crystal 
  with 
  two 
  rectangular 
  axes, 
  a 
  second 
  great 
  circle 
  

   can 
  be 
  drawn 
  at 
  once 
  at 
  right 
  angles 
  to 
  the 
  first 
  circle 
  at 
  the 
  

  

  