﻿202 
  ANNUAL 
  REPORT 
  SMITHSONIAN 
  INSTITUTION, 
  1927 
  

  

  tion." 
  These 
  laws 
  gave 
  a 
  complete 
  answer 
  to 
  the 
  question 
  how 
  the 
  

   planets 
  moved 
  round 
  the 
  sun 
  (elliptical 
  orbit, 
  equal 
  areas 
  described 
  

   by 
  the 
  radius 
  vector 
  in 
  equal 
  periods, 
  relation 
  between 
  semi-major 
  

   axis 
  and 
  period 
  of 
  revolution). 
  But 
  these 
  rules 
  do 
  not 
  satisfy 
  the 
  

   requirement 
  of 
  causality. 
  The 
  three 
  rules 
  are 
  logically 
  independent 
  

   of 
  one 
  another, 
  and 
  show 
  no 
  sign 
  of 
  any 
  interconnection. 
  The 
  third 
  

   law 
  can 
  not 
  be 
  extended 
  numerically 
  as 
  its 
  stands, 
  from 
  the 
  sun 
  

   to 
  another 
  central 
  body; 
  there 
  is, 
  for 
  instance, 
  no 
  relation 
  between 
  

   a 
  planet's 
  period 
  of 
  revolution 
  round 
  the 
  sun 
  and 
  the 
  period 
  of 
  

   revolution 
  of 
  a 
  moon 
  round 
  its 
  planet. 
  

  

  But 
  the 
  principal 
  thing 
  is 
  that 
  these 
  laws 
  have 
  reference 
  to 
  motion 
  

   as 
  a 
  whole, 
  and 
  not 
  to 
  the 
  question 
  how 
  there 
  is 
  developed 
  from 
  one 
  

   condition 
  of 
  motion 
  of 
  a 
  system 
  that 
  which 
  immediately 
  follows 
  it 
  

   in 
  time. 
  They 
  are, 
  in 
  our 
  phraseology 
  of 
  to-day, 
  integral 
  laws, 
  

   and 
  not 
  differential 
  laws. 
  

  

  The 
  differential 
  law 
  is 
  the 
  form 
  which 
  alone 
  entirely 
  satisfies 
  the 
  

   modern 
  physicist's 
  requirement 
  of 
  causality. 
  The 
  clear 
  conception 
  

   of 
  the 
  differential 
  law 
  is 
  one 
  of 
  the 
  greatest 
  of 
  Newton's 
  intellectual 
  

   achievements. 
  What 
  was 
  needed 
  was 
  not 
  only 
  the 
  idea 
  but 
  a 
  formal 
  

   mathematical 
  method 
  which 
  was, 
  indeed, 
  extant 
  in 
  rudiment 
  but 
  had 
  

   still 
  to 
  gain 
  a 
  systematic 
  shape. 
  This 
  also 
  Newton 
  found 
  in 
  the 
  

   differential 
  and 
  integral 
  calculus. 
  It 
  is 
  unnecessary 
  to 
  consider 
  

   whether 
  Leibnitz 
  arrived 
  at 
  these 
  same 
  mathematical 
  methods 
  inde- 
  

   pendently 
  of 
  Newton 
  or 
  not; 
  in 
  any 
  case, 
  their 
  development 
  was 
  a 
  

   necessity 
  for 
  Newton, 
  as 
  they 
  were 
  required 
  in 
  order 
  to 
  give 
  Newton 
  

   the 
  means 
  of 
  expressing 
  his 
  thought. 
  

  

  THE 
  STEP 
  FROM 
  GALILEO 
  TO 
  NEWTON 
  

  

  Galileo 
  had 
  already 
  made 
  a 
  significant 
  first 
  step 
  in 
  the 
  recognition 
  

   of 
  the 
  law 
  of 
  motion. 
  He 
  discovered 
  the 
  law 
  of 
  inertia 
  and 
  the 
  law 
  

   of 
  free 
  falling 
  in 
  the 
  earth's 
  field 
  of 
  gravitation 
  : 
  A 
  mass 
  (or, 
  more 
  

   accurately, 
  a 
  material 
  point) 
  uninfluenced 
  by 
  other 
  masses 
  moves 
  

   uniformily 
  in 
  a 
  straight 
  line; 
  the 
  vertical 
  velocity 
  of 
  a 
  free 
  body 
  

   increases 
  in 
  the 
  field 
  of 
  gravity 
  in 
  proportion 
  to 
  the 
  time. 
  It 
  may 
  

   seem 
  to 
  us 
  to-day 
  to 
  be 
  only 
  a 
  small 
  step 
  from 
  Galileo's 
  observations 
  

   to 
  Newton's 
  laws 
  of 
  motion. 
  But 
  it 
  has 
  to 
  be 
  observed 
  that 
  the 
  two 
  

   propositions 
  above, 
  in 
  the 
  form 
  in 
  which 
  they 
  are 
  given, 
  relate 
  

   to 
  motion 
  as 
  a 
  whole, 
  while 
  Newton's 
  law 
  of 
  motion 
  gives 
  an 
  answer 
  

   to 
  the 
  question 
  : 
  How 
  does 
  the 
  condition 
  of 
  motion 
  of 
  a 
  point-mass 
  

   change 
  in 
  an 
  infinitely 
  small 
  period 
  under 
  the 
  influence 
  of 
  an 
  external 
  

   force 
  ? 
  Only 
  after 
  proceeding 
  to 
  consider 
  the 
  phenomenon 
  during 
  an 
  

  

  - 
  Everyone 
  knows 
  to-day 
  what 
  gigantic 
  efforts 
  were 
  needed 
  to 
  discover 
  these 
  laws 
  from 
  

   the 
  empirically 
  ascertained 
  orbits 
  of 
  the 
  planets. 
  But 
  few 
  reflect 
  on 
  the 
  genius 
  of 
  the 
  

   method 
  by 
  which 
  Kepler 
  ascertained 
  the 
  true 
  orbits 
  from 
  the 
  apparent 
  ones 
  ; 
  i. 
  e., 
  their 
  

   directions 
  as 
  observed 
  from 
  the 
  earth. 
  

  

  