﻿424 
  JOURNAIv 
  OF 
  THE 
  WASHINGTON 
  ACADEMY 
  OF 
  SCIENCES 
  VOL. 
  12, 
  NO. 
  19 
  

  

  number 
  of 
  decimals 
  at 
  intervals 
  of 
  1" 
  for 
  the 
  first 
  and 
  last 
  degree 
  of 
  

   the 
  quadrant. 
  Detailed 
  information 
  in 
  regard 
  to 
  these 
  and 
  other 
  

   tables 
  is 
  given 
  by 
  Andoyer, 
  Glaisher/ 
  and 
  Horsburgh.^ 
  

  

  The 
  fundamental 
  values 
  required 
  for 
  the 
  evaluation 
  of 
  the 
  func- 
  

   tions 
  tabulated 
  in 
  the 
  accompanying 
  tables 
  were 
  obtained 
  by 
  the 
  

   substitution 
  of 
  

  

  X 
  = 
  0.04848 
  13681 
  10953 
  59935 
  89914 
  10235 
  79479 
  760 
  

   in 
  the 
  series, 
  thus 
  obtaining 
  values 
  of 
  sin 
  d 
  and 
  cos 
  6 
  for 
  10,000". 
  

   The 
  values 
  for 
  1", 
  10", 
  100", 
  and 
  1000" 
  were 
  then 
  easily 
  obtained 
  

   from 
  the 
  preceding 
  computations 
  of 
  x'/n! 
  by 
  making 
  the 
  appropriate 
  

   displacements 
  in 
  the 
  decimal 
  point. 
  With 
  these 
  values 
  as 
  a 
  basis, 
  

   new 
  values 
  were 
  computed 
  at 
  equal 
  intervals 
  of 
  the 
  argument 
  by 
  

   means 
  of 
  repeated 
  applications 
  of 
  the 
  formulae 
  for 
  sin 
  {x 
  + 
  y) 
  and 
  

   cos 
  {x 
  -\- 
  y). 
  Both 
  functions 
  were 
  computed 
  at 
  the 
  same 
  time 
  with 
  a 
  

   "Millionaire" 
  computing 
  machine. 
  Each 
  computation 
  was 
  verified 
  

   independently. 
  We 
  have 
  for 
  example. 
  

  

  Sin 
  {x 
  ^ 
  y) 
  = 
  sin 
  x 
  cos 
  y 
  ='= 
  cos 
  x 
  sin 
  y. 
  

  

  By 
  addition 
  of 
  the 
  last 
  to 
  the 
  preceding 
  term 
  of 
  this 
  equation 
  we 
  

   obtain 
  sin 
  {x 
  -\- 
  y), 
  a 
  new 
  quantity; 
  similarly 
  by 
  subtraction, 
  we 
  

   obtain 
  sin 
  {x 
  — 
  y), 
  2i 
  quantity 
  previously 
  computed. 
  Apart 
  from 
  

   checks 
  provided 
  by 
  each 
  subsequent 
  interpolation 
  at 
  equal 
  intervals, 
  

   comparisons 
  were 
  of 
  course 
  made 
  at 
  15°, 
  30°, 
  and 
  45° 
  by 
  means 
  of 
  

   the 
  known 
  values^ 
  of 
  ■\/2, 
  V3, 
  and 
  \/6. 
  All 
  of 
  the 
  computations 
  

   were 
  carried 
  to 
  35 
  places 
  of 
  decimals. 
  

  

  Andoyer's 
  table 
  1, 
  volume 
  1, 
  contains 
  expansions 
  in 
  series 
  for 
  each 
  

   of 
  the 
  six 
  trigonometric 
  functions, 
  the 
  variable 
  angle 
  being 
  written, 
  

   X 
  ir/2- 
  The 
  coefficients 
  in 
  these 
  series 
  are 
  tabulated 
  to 
  24 
  places 
  of 
  

   decimals. 
  In 
  table 
  3, 
  volume 
  1, 
  each 
  function 
  is 
  tabulated 
  to 
  17 
  

   places 
  of 
  decimals 
  at 
  intervals 
  of 
  9'. 
  Tables 
  I 
  and 
  II 
  may 
  be 
  useful 
  

   in 
  connection 
  with 
  these 
  tables 
  and 
  the 
  tables 
  of 
  Peters 
  previously 
  

   mentioned. 
  A 
  careful 
  comparison 
  of 
  our 
  values 
  with 
  the 
  correspond- 
  

   ing 
  values 
  given 
  by 
  Peters 
  and 
  Andoyer 
  revealed 
  no 
  errors 
  in 
  any 
  of 
  

   the 
  computations. 
  The 
  plus 
  signs 
  following 
  Andoyer's 
  tabulations 
  

   were 
  also 
  found 
  to 
  be 
  given 
  correctly. 
  

  

  * 
  See 
  article. 
  Table, 
  mathematical, 
  in 
  Encyclopedia 
  Brilannica. 
  

  

  * 
  HoRSBURGH, 
  E. 
  M. 
  Modern 
  instruments 
  and 
  methods 
  of 
  calculation, 
  a 
  handbook 
  of 
  the 
  

   Napier 
  tercentenary 
  exhibition. 
  

  

  ' 
  « 
  Bookman, 
  J. 
  M. 
  Square-root 
  notes. 
  Math. 
  Mag. 
  1: 
  207-8. 
  1882^. 
  Martin, 
  A. 
  

   Extraction 
  of 
  the 
  square 
  roots 
  by 
  series. 
  Math. 
  Mag. 
  1: 
  164-5. 
  1882-4. 
  

  

  