﻿Fundamental 
  Laws 
  of 
  Matter 
  and 
  Energy. 
  709 
  

  

  inconsistency 
  in 
  the 
  equations 
  for 
  the 
  momentum 
  and 
  the 
  

   energy 
  of 
  a 
  beam 
  of 
  radiation. 
  The 
  momentum 
  of 
  the 
  beam 
  

   of 
  mass 
  m 
  we 
  have 
  given 
  in 
  equation 
  (5) 
  as 
  

  

  M 
  = 
  mV. 
  

  

  From 
  our 
  assumption 
  that 
  the 
  energy 
  of 
  the 
  beam 
  is 
  simply 
  

   the 
  kinetic 
  energy 
  of 
  the 
  moving 
  mass 
  m, 
  we 
  might 
  expect 
  

   from 
  our 
  knowledge 
  of 
  elementary 
  mechanics 
  to 
  find 
  for 
  the 
  

   energy 
  the 
  equation 
  

  

  E 
  = 
  l/2mV 
  2 
  ; 
  

  

  whereas 
  in 
  fact 
  we 
  find 
  from 
  equations 
  (4) 
  and 
  (5) 
  that 
  

  

  E 
  = 
  mV 
  2 
  (9) 
  

  

  We 
  shall 
  see, 
  however, 
  in 
  the 
  next 
  section 
  that 
  this 
  comparison 
  

   of 
  equations 
  (5) 
  and 
  (9) 
  instead 
  of 
  demolishing 
  our 
  theory 
  

   actually 
  furnishes 
  a 
  remarkably 
  satisfactory 
  argument 
  in 
  its 
  

   favour. 
  

  

  Non- 
  Newtonian 
  Mechanics. 
  

  

  One 
  of 
  the 
  interesting 
  branches 
  of 
  modern 
  mathematics 
  has 
  

   grown 
  out 
  of 
  the 
  study 
  of 
  those 
  geometries 
  which 
  would 
  

   result 
  from 
  the 
  change 
  of 
  one 
  or 
  more 
  of 
  the 
  axioms 
  of 
  Euclid. 
  

   These 
  non-Euclidian 
  geometries 
  present 
  the 
  properties 
  of 
  

   purely 
  imaginary 
  kinds 
  of 
  space 
  and 
  are 
  therefore 
  so 
  far 
  

   mere 
  exercises 
  in 
  logic, 
  without 
  any 
  physical 
  significance. 
  

   But 
  their 
  investigation 
  was 
  doubtless 
  prompted 
  in 
  some 
  cases 
  

   by 
  the 
  belief 
  that 
  experiment 
  itself 
  may 
  sometime 
  show 
  that 
  

   there 
  are 
  deviations 
  from 
  the 
  ordinary 
  laws 
  of 
  space 
  when 
  

   these 
  laws 
  are 
  subjected 
  to 
  tests 
  of 
  a 
  different 
  order 
  from 
  

   those 
  of 
  common 
  mensuration. 
  Indeed 
  it 
  is 
  not 
  unlikely 
  that 
  

   Euclidian 
  geometry 
  may 
  prove 
  inadequate 
  when 
  we 
  are 
  able 
  

   to 
  subject 
  to 
  an 
  accurate 
  metric 
  investigation 
  the 
  vast 
  stretches 
  

   of 
  interstellar 
  space 
  or 
  the 
  minute 
  regions 
  which 
  we 
  believe 
  to 
  

   be 
  encompassed 
  within 
  an 
  atom 
  or 
  an 
  electron. 
  

  

  The 
  science 
  of 
  mechanics, 
  like 
  geometry, 
  has 
  been 
  built 
  up 
  

   from 
  a 
  set 
  of 
  simple 
  axioms, 
  which 
  were 
  laid 
  down 
  by 
  Newton. 
  

   But 
  the 
  conclusions 
  of 
  the 
  previous 
  section 
  lead 
  us 
  to 
  modify 
  

   one 
  of 
  these 
  axioms 
  and 
  thus 
  lay 
  the 
  foundation 
  of 
  a 
  system 
  

   of 
  non-Newtonian 
  mechanics. 
  

  

  The 
  axiom 
  which 
  we 
  must 
  surrender 
  is 
  the 
  one 
  which 
  states 
  

   that 
  the 
  mass 
  of 
  a 
  body 
  is 
  independent 
  of 
  its 
  velocity. 
  We 
  

   have 
  concluded 
  that 
  mass 
  is 
  proportional 
  to 
  content 
  of 
  energy. 
  

   When 
  a 
  body 
  is 
  set 
  in 
  motion 
  it 
  gains 
  kinetic 
  energy 
  and 
  

   therefore 
  its 
  mass 
  must 
  change 
  with 
  its 
  velocity. 
  In 
  place 
  of 
  

  

  