﻿320 
  W. 
  D. 
  MacJIUlan 
  — 
  Mathematics 
  of 
  Isostasy. 
  

  

  I 
  think 
  the 
  discriminating 
  reader 
  will 
  see 
  that 
  there 
  is 
  

   nothing 
  whatever 
  in 
  these 
  passages 
  that 
  is 
  either 
  objec- 
  

   tionably 
  skeptical 
  or 
  destructive 
  respecting 
  the 
  truth 
  of 
  

   isostasy. 
  They 
  do 
  involve, 
  however, 
  not 
  merely 
  skepti- 
  

   cism 
  bnt 
  a 
  firm 
  conviction 
  that 
  the 
  isostasy 
  is 
  not 
  mathe- 
  

   matically 
  proved. 
  My 
  offense 
  then 
  is 
  that, 
  as 
  a 
  

   mathematician, 
  speaking 
  to 
  geologists 
  I 
  have 
  affirmed 
  

   that 
  certain 
  things 
  were 
  not 
  mathematically 
  proved, 
  not 
  

   that 
  they 
  are 
  not 
  true. 
  

  

  Now 
  the 
  main 
  purpose 
  of 
  my 
  paper 
  was 
  to 
  point 
  out 
  

   just 
  what 
  mathematics 
  had 
  done 
  or 
  could 
  do 
  in 
  the 
  isos- 
  

   tatic 
  inquiry 
  and 
  just 
  what 
  it 
  could 
  not 
  do. 
  I 
  very 
  

   naturally 
  assumed 
  that 
  such 
  a 
  discrimination 
  would 
  be 
  

   welcomed 
  by 
  all 
  students 
  of 
  the 
  subject 
  controlled 
  by 
  the 
  

   scientific 
  spirit, 
  as 
  I 
  understand 
  this 
  to 
  be 
  precisely 
  what 
  

   the 
  scientific 
  spirit 
  demands. 
  I 
  did 
  not 
  dream 
  that 
  it 
  

   would 
  be 
  offensive 
  to 
  any 
  scientific 
  student. 
  I 
  supposed 
  

   it 
  to 
  be 
  a 
  prerequisite 
  of 
  safe 
  procedure 
  to 
  employ 
  

   mathematics 
  just 
  so 
  far 
  as 
  it 
  really 
  may 
  be 
  employed 
  and 
  

   to 
  be 
  very 
  careful 
  to 
  distinguish 
  clearly 
  those 
  conclusions 
  

   which 
  the 
  mathematics 
  warrant 
  and 
  those 
  conclusions 
  

   which 
  they 
  do 
  not 
  warrant. 
  It 
  is 
  in 
  this 
  latter 
  respect, 
  

   mainly, 
  that 
  a 
  mathematician 
  can 
  be 
  of 
  service. 
  

  

  Xow 
  mathematics 
  has 
  wonderful 
  resources 
  in 
  setting 
  

   forth 
  the 
  relations 
  of 
  things. 
  "When 
  it 
  is 
  employed 
  in 
  a 
  

   physical 
  field 
  this 
  extraordinary 
  power 
  is 
  naturally 
  

   matched 
  by 
  a 
  complete 
  inability 
  to 
  discriminate, 
  so 
  far 
  

   as 
  objective 
  reality 
  is 
  concerned, 
  between 
  the 
  many 
  pos- 
  

   sible 
  solutions 
  which 
  satisfy 
  the 
  given 
  data. 
  From 
  a 
  

   mathematical 
  point 
  of 
  view 
  they 
  are 
  all 
  true, 
  so 
  that 
  it 
  is 
  

   necessary 
  to 
  go 
  outside 
  of 
  its 
  field 
  to 
  find 
  a 
  basis 
  of 
  dis- 
  

   crimination. 
  

  

  Now 
  these 
  wonderful 
  capacities 
  of 
  mathematics 
  set 
  

   over 
  against 
  its 
  necessary 
  limitations 
  are 
  matters 
  of 
  

   common 
  knowledge 
  among 
  mathematicians. 
  It 
  did 
  not 
  

   seem 
  to 
  me 
  more 
  than 
  necessary 
  to 
  state 
  that 
  particular 
  

   phase 
  of 
  them 
  that 
  related 
  to 
  the 
  subject 
  in 
  hand. 
  Dr. 
  

   Barrell, 
  however, 
  thinks 
  I 
  should 
  have 
  been 
  more 
  specific 
  

   and 
  demonstrative. 
  To 
  show 
  that 
  the 
  geodetic 
  observa- 
  

   tions 
  can 
  be 
  satisfied 
  mathematically 
  by 
  a 
  non-isostatic 
  

   solution 
  as 
  well 
  as 
  an 
  isostatic 
  solution, 
  let 
  us 
  suppose 
  

   that 
  there 
  are 
  n 
  geodetic 
  observations. 
  At 
  each 
  obser- 
  

   vation 
  three 
  things 
  are 
  determined, 
  two 
  deflections 
  and 
  

   one 
  intensity. 
  In 
  n 
  observations 
  there 
  are, 
  therefore, 
  3n 
  

   quantities 
  determined. 
  In 
  order 
  to 
  satisfy 
  these 
  n 
  obser- 
  

   vations 
  it 
  is 
  sufficient 
  to 
  pick 
  out 
  3n 
  points 
  in 
  the 
  interior 
  

  

  