﻿242 
  J. 
  M. 
  Blake 
  — 
  Plotting 
  Crystal 
  Zones 
  on 
  the 
  Sphere. 
  

  

  facilitate 
  this 
  plotting 
  if 
  we 
  prepare 
  a 
  suitable 
  drawing 
  board. 
  

   This 
  need 
  not 
  exceed 
  twenty 
  inches 
  square. 
  A 
  hard-wood 
  peg 
  

   is 
  inserted 
  in 
  its 
  center 
  and 
  is 
  cut 
  off 
  flush 
  with 
  the 
  surface 
  of 
  

   the 
  board. 
  From 
  this 
  center 
  a 
  circle 
  is 
  drawn, 
  and 
  is 
  divided 
  

   into 
  degrees. 
  The 
  graduation 
  marks 
  should 
  extend 
  sufficiently 
  

   outward 
  so 
  as 
  not 
  to 
  be 
  covered 
  by 
  the 
  square 
  of 
  the 
  drawing 
  

   paper. 
  There 
  should 
  be 
  two 
  sets 
  of 
  numbers 
  in 
  reverse 
  order 
  

   both 
  on 
  the 
  ring 
  and 
  on 
  the 
  board. 
  The 
  graduation 
  marks 
  on 
  

   the 
  board 
  correspond 
  to 
  the 
  divisions 
  on 
  the 
  ring 
  of 
  the 
  

   sphere. 
  

  

  We 
  next 
  mark 
  on 
  a 
  paper 
  strip 
  a 
  series 
  of 
  natural 
  tangents 
  

   up 
  to 
  eighty 
  degrees. 
  This 
  scale 
  of 
  tangents 
  can 
  be 
  one 
  cal- 
  

   culated 
  for 
  a 
  two-inch 
  radius. 
  A 
  shorter 
  radius 
  is 
  sometimes 
  

   used, 
  but 
  requires 
  more 
  exactness 
  in 
  drawing. 
  We 
  pin 
  this 
  

   paper 
  strip 
  by 
  its 
  zero 
  point 
  to 
  the 
  center 
  of 
  the 
  drawing 
  

   board. 
  The 
  graduated 
  edge 
  of 
  the 
  strip 
  should 
  be 
  radial. 
  A 
  

   bit 
  of 
  paper 
  is 
  pasted 
  to 
  the 
  strip 
  to 
  receive 
  the 
  pin 
  through 
  

   the 
  zero 
  point 
  of 
  the 
  scale 
  as 
  this 
  will 
  bring 
  about 
  the 
  radial 
  

   position 
  of 
  the 
  scale 
  edge. 
  

  

  We 
  can 
  now 
  go 
  on 
  and 
  plot 
  the 
  position 
  of 
  the 
  normal 
  in- 
  

   tersection 
  points 
  on 
  paper, 
  and 
  we 
  thus 
  make 
  the 
  gnomonic 
  

   projection. 
  The 
  equatorial 
  zone 
  will 
  appear 
  only 
  as 
  radial 
  

   marks. 
  The 
  radius 
  of 
  the 
  tangent 
  scale 
  can 
  be 
  shown 
  by 
  a 
  

   circle 
  drawn 
  about 
  the 
  center. 
  

  

  The 
  normal 
  lines 
  used 
  in 
  projecting 
  start 
  from 
  the 
  center 
  of 
  

   the 
  sphere. 
  When 
  we 
  use 
  the 
  projection 
  for 
  our 
  purposes 
  we 
  

   look 
  down 
  upon 
  it 
  from 
  the 
  outside 
  as 
  we 
  would 
  upon 
  a 
  

   crystal. 
  If 
  we 
  also 
  project 
  the 
  back 
  planes, 
  these 
  should 
  

   appear 
  as 
  though 
  we 
  looked 
  through 
  the 
  crystal 
  and 
  should 
  be 
  

   in 
  reverse. 
  

  

  When 
  we 
  have 
  made 
  our 
  tangent 
  plane 
  projection 
  from 
  the 
  

   proper 
  view-point, 
  the 
  indices 
  of 
  the 
  planes 
  can 
  be 
  obtained 
  

   by 
  inspection 
  without 
  calculating 
  them. 
  

  

  It 
  will 
  be 
  gathered 
  from 
  these 
  articles 
  that 
  there 
  has 
  been 
  a 
  

   constant 
  aim 
  throughout 
  to 
  penetrate 
  an 
  almost 
  untrodden 
  

   field, 
  which 
  we 
  know 
  instinctively 
  must 
  lie 
  just 
  beyond 
  our 
  

   reach. 
  It 
  has 
  also 
  been 
  the 
  aim 
  to 
  improve 
  the 
  means 
  for 
  

   securing 
  as 
  reliable 
  data 
  as 
  possible 
  to 
  be 
  used 
  in 
  making 
  

   further 
  advances 
  in 
  the 
  desired 
  direction. 
  The 
  present 
  article 
  

   treats 
  of 
  a 
  means 
  of 
  making 
  the 
  gnomonic 
  projection, 
  and 
  

   gives 
  a 
  few 
  illustrations 
  of 
  its 
  application 
  to 
  crystal 
  work. 
  

   There 
  remain 
  at 
  least 
  two 
  other 
  methods 
  to 
  be 
  described. 
  

   These 
  latter 
  are 
  needed 
  to 
  place 
  the 
  whole 
  series 
  on 
  a 
  complete 
  

   and 
  practical 
  working 
  basis. 
  We 
  should 
  then 
  be 
  able 
  to 
  

   describe 
  a 
  crystal 
  with 
  little 
  dependence 
  upon 
  algebraic 
  

   methods, 
  and 
  the 
  results 
  of 
  our 
  work 
  should 
  pave 
  the 
  way 
  for 
  

   really 
  effective 
  mathematical 
  treatment 
  in 
  the 
  future. 
  

  

  New 
  Haven, 
  Conn., 
  Jan. 
  1917. 
  

  

  