﻿Celestial 
  Bodies 
  in 
  relation 
  to 
  Attraction 
  of 
  Gravitation. 
  d27 
  

  

  11. 
  The 
  application 
  of 
  the 
  law 
  to 
  the 
  moon's 
  orbit 
  is 
  the 
  

   subject 
  o£ 
  the 
  fourth 
  proposition 
  of 
  the 
  third 
  book 
  of 
  the 
  

   * 
  Principia/ 
  and 
  is 
  of 
  so 
  much 
  importance 
  in 
  relation 
  to 
  my 
  

   paper 
  that 
  I 
  have 
  thought 
  it 
  well 
  to 
  give 
  an 
  abstract 
  of 
  the 
  

   text 
  from 
  Motte^s 
  classical 
  translation. 
  The 
  linear 
  quantities 
  

   in 
  the 
  original 
  are 
  expressed 
  in 
  Paris 
  feet, 
  but 
  are 
  here 
  given 
  

   in 
  English 
  measure 
  in 
  accordance 
  with 
  modern 
  usage 
  : 
  — 
  

  

  " 
  Let 
  us 
  assume 
  the 
  mean 
  distance 
  of 
  the 
  moon 
  to 
  be 
  

   GO 
  semi-diameters 
  of 
  the 
  earth, 
  and 
  if 
  we 
  imagine 
  the 
  moon 
  

   deprived 
  of 
  all 
  motion 
  to 
  be 
  let 
  go, 
  so 
  as 
  to 
  descend 
  towards 
  

   the 
  earth 
  with 
  the 
  impulse 
  of 
  all 
  that 
  force 
  b}' 
  which 
  it 
  is 
  

   retained 
  in 
  its 
  orbit, 
  it 
  will, 
  in 
  the 
  space 
  of 
  one 
  minute 
  of 
  

   time, 
  describe 
  in 
  its 
  fall 
  16 
  feet 
  1 
  inch. 
  For 
  the 
  versed 
  sine 
  

   of 
  that 
  arc 
  which 
  the 
  moon, 
  in 
  the 
  space 
  of 
  one 
  minute 
  of 
  

   time, 
  would 
  by 
  its 
  mean 
  motion 
  describe 
  at 
  the 
  distance 
  of 
  

   60 
  semi-diameters 
  of 
  the 
  earth 
  is 
  nearly 
  16 
  feet 
  1 
  inch, 
  

   ^"herefore 
  since 
  that 
  force 
  in 
  approaching 
  to 
  the 
  earth 
  in- 
  

   creases 
  in 
  the 
  reciprocal 
  duplicate 
  ])roportion 
  of 
  the 
  distance, 
  

   and, 
  upon 
  that 
  account, 
  at 
  the 
  surface 
  of 
  the 
  earth 
  60 
  x 
  60 
  times 
  

   greater 
  than 
  at 
  the 
  moon, 
  a 
  body 
  in 
  our 
  regions 
  falling 
  with 
  

   that 
  force 
  ought, 
  in 
  the 
  space 
  of 
  one 
  minute 
  of 
  time, 
  to 
  

   describe 
  60 
  x 
  60 
  x 
  16 
  feet 
  1 
  inch. 
  And 
  with 
  this 
  very 
  force 
  

   we 
  actually 
  find 
  that 
  bodies 
  here 
  upon 
  earth 
  do 
  really 
  descend. 
  

   And 
  therefore 
  the 
  force 
  by 
  which 
  the 
  moon 
  is 
  retained 
  in 
  its 
  

   orbit 
  becomes, 
  at 
  the 
  very 
  surface 
  of 
  the 
  earth, 
  equal 
  to 
  the 
  

   force 
  of 
  gravity 
  which 
  we 
  observe 
  in 
  bodies 
  there. 
  And 
  

   therefore 
  the 
  force 
  by 
  which 
  the 
  moon 
  is 
  retained 
  in 
  its 
  orbit 
  

   is 
  that 
  very 
  same 
  force 
  which 
  we 
  commonlv 
  call 
  gravity.'^ 
  

  

  12. 
  I 
  have 
  already 
  demonstrated 
  that 
  the 
  moon's 
  orbital 
  

   velocity 
  is 
  BS'IGO 
  miles 
  per 
  minute, 
  and 
  that 
  a 
  body 
  would 
  

   describe 
  an 
  arc 
  of 
  4*936 
  miles 
  of 
  the 
  earth's 
  circumference 
  

   in 
  one 
  second, 
  during 
  which 
  time 
  it 
  would 
  fall 
  through 
  the 
  

   versed 
  sine 
  of 
  that 
  arc 
  =193 
  inches. 
  Now 
  as 
  the 
  versed 
  

   sines 
  of 
  arcs 
  of 
  the 
  same 
  leugth 
  are 
  as 
  the 
  radii 
  inversely 
  

  

  from 
  the 
  centre, 
  we 
  have 
  ^ 
  ^ 
  . 
  ^ 
  = 
  3'21 
  inches 
  as 
  the 
  

  

  versed 
  sine 
  of 
  4' 
  936 
  miles 
  of 
  the 
  moon's 
  orbit. 
  Again, 
  

  

  as 
  versed 
  sines 
  are 
  as 
  the 
  squares 
  of 
  their 
  arcs, 
  we 
  have 
  

  

  38-160 
  miles 
  ^ 
  -^o 
  o 
  ^i 
  i 
  ^ 
  .- 
  c 
  l 
  i 
  - 
  ^ 
  ^-u 
  

  

  . 
  , 
  . 
  - 
  — 
  Tj 
  — 
  r<3-x.r21, 
  or 
  nearlv 
  10 
  leet 
  1 
  men. 
  as 
  the 
  

   •4*930 
  miles 
  

  

  versed 
  sine 
  of 
  38*160 
  miles 
  through 
  which 
  the 
  moon 
  falls 
  

  

  towards 
  the 
  earth 
  during 
  one 
  minute 
  of 
  time 
  in 
  accordance 
  

  

  with 
  Newton's 
  demonstration. 
  

  

  13. 
  The 
  assumption 
  by 
  Newton 
  of 
  the 
  ^^hole 
  number 
  60 
  as 
  

   the 
  distance 
  of 
  the 
  moon 
  in 
  semi-diameters 
  of 
  the 
  earth 
  in 
  

   combination 
  with 
  the 
  same 
  even 
  number 
  in 
  seconds 
  and 
  

   minutes 
  of 
  time 
  greatly 
  facilitated 
  the 
  demonstration 
  of 
  the 
  

  

  