﻿322 
  W. 
  D. 
  MacMillan 
  — 
  Mathematics 
  of 
  Isostasy. 
  

  

  Now 
  the 
  resources 
  of 
  mathematics 
  are 
  such 
  that 
  

   geodetic 
  observations 
  can 
  be 
  reduced 
  in 
  perfect 
  mathe- 
  

   matical 
  consistency 
  to 
  any 
  other 
  datum 
  surface 
  than 
  this 
  

   ideal 
  one, 
  either 
  above 
  it 
  at 
  any 
  selected 
  horizon 
  up 
  to 
  

   the 
  top 
  of 
  the 
  highest 
  mountain 
  or 
  beyond, 
  or 
  below 
  it 
  

   down 
  to 
  the 
  deepest 
  "deep" 
  of 
  the 
  ocean 
  or 
  beyond. 
  

   My 
  suggestions 
  did 
  not 
  relate 
  to 
  the 
  consistency 
  or 
  cor- 
  

   rectness 
  of 
  such 
  reduction 
  but 
  to 
  the 
  use 
  of 
  such 
  reduc- 
  

   tions 
  in 
  deriving 
  results 
  that 
  involved 
  natural 
  factors. 
  

   If 
  the 
  value 
  derived 
  from 
  reduction 
  to 
  artificial 
  datum 
  

   surfaces 
  are 
  used 
  as 
  though 
  they 
  represent 
  natural 
  

   values 
  actually 
  involved 
  in 
  isostatic 
  readjustments, 
  they 
  

   are 
  liable 
  to 
  lead 
  to 
  error. 
  This 
  I 
  endeavored 
  to 
  illus- 
  

   trate 
  concretely. 
  There 
  ought 
  to 
  be 
  no 
  difficulty 
  in 
  dis- 
  

   tinguishing 
  between 
  the 
  legitimacy 
  of 
  using 
  a 
  convenient 
  

   base 
  like 
  the 
  sea-level 
  in 
  reducing 
  observations 
  in 
  a 
  

   strictly 
  consistent 
  way, 
  and 
  the 
  danger 
  of 
  using 
  the 
  

   numerical 
  results 
  of 
  such 
  reduction 
  in 
  applications 
  that 
  

   have 
  natural 
  relations 
  to 
  the 
  true 
  base 
  of 
  isostatic 
  ad- 
  

   justment 
  in 
  the 
  earth. 
  It 
  should 
  be 
  obvious 
  on 
  the 
  mere 
  

   statement 
  that 
  the 
  reduction 
  of 
  observations 
  to 
  the 
  hori- 
  

   zon 
  that 
  would 
  become 
  a 
  real 
  surface, 
  if 
  perfect 
  isostasy 
  

   were 
  attained, 
  is 
  safest 
  and 
  best. 
  

  

  Dr. 
  Barrell 
  devotes 
  seven 
  pages 
  to 
  my 
  statement 
  that 
  

   the 
  geodetic 
  data 
  are 
  insufficient 
  to 
  demonstrate 
  mathe- 
  

   matically 
  the 
  depth 
  of 
  compensation. 
  This 
  was 
  not 
  

   elaborately 
  discussed 
  in 
  my 
  paper 
  and 
  I 
  do 
  not 
  think 
  it 
  

   needs 
  elaborate 
  discussion 
  here 
  after 
  what 
  has 
  already 
  

   been 
  said 
  about 
  the 
  capabilities 
  and 
  the 
  limitations 
  of 
  

   mathematics. 
  The 
  truth 
  about 
  the 
  depth 
  of 
  compensa- 
  

   tion 
  was 
  not 
  under 
  discussion 
  but 
  the 
  mathematical 
  proof 
  

   or 
  lack 
  of 
  proof. 
  The 
  very 
  fact 
  that 
  Hayford, 
  after 
  

   having 
  used 
  four 
  hypotheses 
  by 
  way 
  of 
  trial, 
  and 
  having 
  

   found 
  results 
  running 
  37, 
  76, 
  109, 
  and 
  178 
  miles 
  respec- 
  

   tively, 
  explicitly 
  declared 
  that 
  the 
  data 
  were 
  insufficient 
  

   to 
  decide 
  which 
  of 
  these 
  was 
  the 
  true 
  depth, 
  is 
  sufficient 
  

   evidence 
  that 
  nothing 
  like 
  a 
  conclusive 
  depth 
  of 
  compen- 
  

   sation 
  has 
  yet 
  been 
  reached, 
  even 
  with 
  mathematics 
  

   supplemented 
  by 
  other 
  resources. 
  It 
  is 
  idle 
  to 
  hope 
  that 
  

   mathematical 
  manipulation 
  of 
  geodetic 
  observations 
  

   alone 
  can 
  ever 
  demonstrate 
  such 
  depth. 
  It 
  may 
  greatly 
  

   aid 
  in 
  bringing 
  out 
  the 
  full 
  meaning 
  of 
  data 
  and 
  in 
  the 
  

   precise 
  application 
  of 
  interpretations 
  and 
  hypotheses. 
  

   It 
  may 
  thus 
  give 
  a 
  force 
  not 
  otherwise 
  attained 
  to 
  those 
  

  

  