﻿cniTTESDEK.] 
  THE 
  SAK 
  JUAN 
  DISTEICT. 
  3C5 
  

  

  tbe 
  office, 
  I 
  should 
  generally 
  have 
  omitted 
  the 
  auxiliary 
  sketches. 
  

   The 
  accuracy 
  was 
  such 
  that, 
  starting 
  from 
  the 
  eastern 
  edge 
  of 
  the 
  sheet 
  

   with 
  rather 
  badly-conditioned 
  points 
  for 
  location, 
  and 
  running 
  by 
  the 
  

   plane-table 
  alone 
  to 
  the 
  southwestern 
  corner 
  of 
  the 
  sheet, 
  the 
  maximum 
  

   error 
  was 
  slightly 
  over 
  a 
  mile, 
  a 
  really 
  small 
  amount 
  when 
  distributed 
  

   through 
  fiity 
  stations 
  and 
  carried 
  over 
  a 
  distance 
  of 
  more 
  than 
  a 
  

   hundred 
  miles, 
  and 
  when 
  copied 
  by 
  small 
  areas 
  utterly 
  inappreciable 
  

   in 
  the 
  drainage. 
  

  

  The 
  district 
  through 
  which 
  this 
  work 
  was 
  carried 
  is 
  well 
  suited 
  in 
  

   every 
  way 
  to 
  this 
  system 
  of 
  working. 
  But 
  I 
  would 
  not 
  by 
  any 
  means 
  

   recommended 
  it 
  in 
  a 
  very 
  level 
  or 
  in 
  a 
  mountainous 
  district, 
  in 
  the 
  

   first 
  for 
  lack 
  of 
  near 
  " 
  points," 
  and 
  in 
  the 
  second 
  for 
  the 
  multitudinous 
  

   detail 
  and 
  advantage 
  of 
  distant 
  sketches. 
  In 
  1871, 
  while 
  learning 
  the 
  

   use 
  of 
  the 
  plane-table, 
  the 
  great 
  difficulty 
  to 
  beginners 
  in 
  immediately 
  

   assuming 
  the 
  correct 
  point 
  from 
  a 
  given 
  triangle 
  of 
  error 
  attracted 
  in 
  

   our 
  party 
  considerable 
  attention, 
  and 
  many 
  attempts 
  were 
  made 
  at 
  the 
  

   determination 
  of 
  a 
  graphical 
  method 
  for 
  the 
  solution. 
  One 
  member 
  of 
  

   the 
  force, 
  Mr. 
  D. 
  H. 
  Pierpont, 
  advanced 
  a 
  method 
  which 
  I 
  ha^e 
  often 
  

   employed 
  during 
  the 
  season 
  to 
  facilitate 
  locations, 
  and 
  always 
  found 
  it 
  

   of 
  great 
  assistance. 
  

  

  Since 
  the 
  time 
  of 
  our 
  working 
  together 
  Mr. 
  Pierpont 
  has 
  died, 
  and 
  I 
  

   am 
  indebted 
  to 
  his 
  family 
  for 
  the 
  demonstration 
  given 
  below, 
  which 
  I 
  

   hope 
  may 
  be 
  found 
  useful, 
  at 
  least, 
  by 
  some 
  who, 
  like 
  hira, 
  are 
  beginners 
  

   in 
  the 
  use 
  of 
  this 
  chief 
  of 
  topographical 
  instruments, 
  or 
  by 
  older 
  hands 
  

   in 
  reconnaissances 
  where 
  it 
  is 
  impossible 
  to 
  assume 
  at 
  first 
  sight 
  a 
  close 
  

   approximation 
  to 
  the 
  true 
  position, 
  or 
  where 
  it 
  is 
  impossible 
  to 
  obtain 
  

   well-conditioned 
  points 
  for 
  location. 
  

  

  The 
  problem 
  of 
  the 
  three 
  points 
  offers 
  five 
  cases, 
  depending 
  on 
  the 
  

   relation 
  of 
  the 
  point 
  sought 
  to 
  the 
  three 
  given 
  points. 
  The 
  cases 
  may 
  

   be 
  stated 
  as 
  follows 
  : 
  • 
  

  

  Case 
  1. 
  When 
  the 
  required 
  point 
  is 
  within 
  the 
  triangle 
  formed 
  hj 
  the 
  three 
  

   ■givin 
  points. 
  

  

  Case 
  2. 
  When 
  the 
  required 
  point 
  is 
  ivithout 
  the 
  triangle 
  hut 
  tcithin 
  the 
  

   circle 
  of 
  the 
  three 
  given 
  points. 
  

  

  Case 
  3. 
  When 
  the 
  required 
  point 
  is 
  without 
  the 
  circle 
  and 
  the 
  central 
  point 
  

   is 
  nearest. 
  

  

  Case 
  4. 
  When 
  the 
  required 
  point 
  is 
  without 
  the 
  circle 
  and 
  tlie 
  central 
  point 
  

   is 
  the 
  most 
  distant. 
  

  

  Case 
  5. 
  When 
  the 
  required 
  point 
  is 
  on 
  the 
  circle 
  of 
  the 
  three 
  given 
  points. 
  

  

  If 
  the 
  table 
  be 
  put 
  in 
  approximate 
  position 
  and 
  lines 
  be 
  drawn 
  through 
  

   three 
  plotted 
  points 
  from 
  the 
  direction 
  of 
  their 
  respective 
  signals, 
  they 
  

   will 
  unite 
  in 
  forming 
  a 
  triangle 
  of 
  error, 
  each 
  angle 
  of 
  which 
  will 
  be 
  on 
  

   a 
  great 
  circle 
  through 
  the 
  required 
  point 
  and 
  two 
  of 
  the 
  given 
  points. 
  

  

  These 
  three 
  great 
  circles, 
  each 
  containing 
  the 
  required 
  point, 
  they 
  

   must 
  intersect 
  in 
  this 
  point. 
  

  

  The 
  absolute 
  construction 
  of 
  these 
  circles 
  would 
  be 
  impossible 
  in 
  the 
  

   field. 
  It 
  then 
  remains 
  to 
  find 
  some 
  means 
  of 
  quick 
  determination 
  of 
  

   • 
  sufficient 
  arcs 
  of 
  these 
  circles 
  to 
  satisfy 
  the 
  needs 
  of 
  the 
  location. 
  

  

  If, 
  after 
  determining 
  the 
  triangle 
  of 
  error 
  by 
  resection, 
  the 
  table 
  be 
  

   moved 
  slightly 
  by 
  the 
  tangent-screw 
  and 
  lines 
  be 
  drawn 
  as 
  before, 
  a 
  new 
  

   triangle 
  of 
  error 
  will 
  be 
  found, 
  which, 
  since 
  the 
  angles 
  are 
  necessarily 
  

   equal, 
  will 
  be 
  exactly 
  similar 
  to 
  the 
  first. 
  The 
  similar 
  angles 
  of 
  these 
  

   triangles 
  will 
  in 
  each 
  case 
  be 
  on 
  the 
  same 
  great 
  circle, 
  and 
  if 
  these 
  two 
  

   angles 
  be 
  joined 
  by 
  a 
  right 
  line, 
  this 
  line 
  will 
  be 
  the 
  chord 
  of 
  a 
  great 
  

   circle 
  of 
  the 
  known 
  and 
  required 
  points, 
  and 
  within 
  a 
  small 
  arc 
  will 
  

   sensibly 
  coincide 
  with 
  the 
  circle 
  itself. 
  If 
  the 
  other 
  similar 
  angles 
  be 
  

  

  