﻿COUNTING 
  THE 
  STAES 
  SEARES 
  189 
  

  

  The 
  rapid 
  increase 
  in 
  the 
  numbers 
  of 
  stars 
  with 
  increasing 
  magnitude 
  

   recalls 
  the 
  old 
  problem 
  of 
  the 
  cost 
  of 
  shoeing 
  the 
  horse, 
  with 
  a 
  penny 
  

   for 
  the 
  first 
  nail, 
  two 
  for 
  the 
  second, 
  four 
  for 
  the 
  third, 
  and 
  so 
  on. 
  

   Doubling 
  the 
  cost 
  for 
  each 
  successive 
  nail 
  runs 
  the 
  total 
  into 
  an 
  incred- 
  

   ible 
  sum; 
  but 
  with 
  the 
  stars, 
  as 
  shown 
  by 
  Table 
  II, 
  the 
  numbers, 
  

   on 
  the 
  whole, 
  are 
  rather 
  more 
  than 
  doubled 
  each 
  time 
  an 
  additional 
  

   magnitude 
  is 
  counted. 
  No 
  wonder 
  the 
  total 
  is 
  great.' 
  

  

  To 
  illustrate 
  further 
  the 
  meaning 
  of 
  Table 
  II, 
  imagine 
  a 
  small 
  

   stellar 
  system 
  in 
  which 
  the 
  individual 
  stars 
  are 
  candles, 
  all 
  alike 
  and 
  

   equally 
  spaced, 
  we 
  ourselves 
  being 
  at 
  the 
  center 
  of 
  the 
  system. 
  With 
  

   the 
  eye 
  alone 
  we 
  should 
  be 
  unable 
  to 
  see 
  candles 
  beyond 
  a 
  certain 
  

   distance, 
  because 
  the 
  light 
  reaching 
  the 
  eye 
  would 
  be 
  too 
  faint 
  to 
  

   produce 
  a 
  visual 
  sensation. 
  A 
  telescope, 
  however, 
  would 
  bring 
  some 
  

   of 
  them 
  into 
  view; 
  and 
  for 
  the 
  purpose 
  let 
  us 
  choose 
  an 
  instrument 
  just 
  

   powerful 
  enough 
  to 
  reveal 
  candles 
  exactly 
  one 
  magnitude 
  fainter 
  than 
  

   the 
  faintest 
  seen 
  without 
  the 
  telescope. 
  The 
  relation 
  between 
  inten- 
  

   sity 
  and 
  brightness 
  which 
  defines 
  the 
  unit 
  of 
  magnitude 
  tells 
  us 
  that 
  

   such 
  a 
  telescope 
  would 
  penetrate 
  about 
  one 
  and 
  six- 
  tenths 
  times 
  

   farther 
  into 
  space 
  than 
  the 
  unaided 
  eye. 
  Now 
  let 
  us 
  count 
  all 
  the 
  

   candles 
  visible 
  from 
  our 
  central 
  station, 
  both 
  with 
  and 
  without 
  the 
  

   telescope. 
  The 
  numbers 
  will 
  be 
  those 
  contained 
  in 
  the 
  two 
  spheres 
  

   whose 
  radii 
  are 
  to 
  each 
  other 
  as 
  1 
  to 
  1.6, 
  and, 
  since 
  the 
  candles 
  are 
  

   everywhere 
  equally 
  spaced, 
  their 
  ratio 
  will 
  be 
  equal 
  to 
  that 
  of 
  the 
  

   volumes 
  of 
  the 
  two 
  spheres, 
  or 
  very 
  nearly 
  4 
  to 
  1. 
  Under 
  the 
  condi- 
  

   tions 
  supposed, 
  we 
  must 
  therefore 
  expect 
  that 
  extending 
  the 
  counts 
  

   of 
  candles 
  by 
  one 
  magnitude 
  would 
  multiply 
  the 
  number 
  visible 
  by 
  4. 
  

  

  Now, 
  since 
  the 
  star 
  ratios 
  of 
  Table 
  II 
  nowhere 
  equal 
  this 
  theoretical 
  

   value 
  and, 
  for 
  the 
  most 
  part, 
  are 
  far 
  below 
  it, 
  there 
  must 
  be 
  some 
  

   essential 
  difference 
  between 
  the 
  real 
  stellar 
  system 
  and 
  the 
  miniature 
  

   system 
  of 
  candles. 
  Candles, 
  to 
  be 
  sure, 
  are 
  not 
  stars; 
  but 
  for 
  the 
  

   moment 
  that 
  is 
  not 
  an 
  essential 
  difference. 
  Stars, 
  on 
  the 
  other 
  hand, 
  

   may 
  not 
  all 
  be 
  of 
  the 
  same 
  candlepower, 
  as 
  the 
  candles 
  are. 
  In 
  fact, 
  

   they 
  are 
  not; 
  but 
  it 
  can 
  be 
  shown 
  that 
  this 
  also 
  is 
  not 
  the 
  explanation. 
  

   Again, 
  some 
  of 
  the 
  distant 
  stars 
  may 
  be 
  hidden 
  by 
  haze 
  and 
  dust 
  

   scattered 
  throughout 
  space. 
  This 
  certainly 
  would 
  reduce 
  the 
  ratios 
  

   of 
  the 
  numbers 
  counted 
  and 
  actually 
  may 
  have 
  some 
  effect 
  on 
  their 
  

   values 
  ; 
  but 
  the 
  presence 
  of 
  absorbing 
  material 
  seems 
  at 
  most 
  to 
  be 
  a 
  

   local 
  phenomenon, 
  and 
  can 
  not 
  be 
  the 
  complete 
  explanation. 
  The 
  

   only 
  other 
  significant 
  factor 
  is 
  a 
  possible 
  lack 
  of 
  uniformity 
  in 
  the 
  

   spacing 
  of 
  the 
  stars, 
  and 
  this 
  indeed 
  is 
  where 
  the 
  difference 
  lies. 
  

   Uniform 
  spacing 
  means 
  a 
  factor 
  of 
  4 
  ; 
  but 
  if 
  the 
  stars 
  should 
  thin 
  out 
  

   with 
  increasing 
  distance 
  from 
  our 
  station 
  in 
  space, 
  the 
  numbers 
  of 
  

   faint 
  stars 
  would 
  be 
  less 
  than 
  we 
  should 
  otherwise 
  find, 
  and 
  the 
  ratios 
  

   from 
  magnitude 
  to 
  magnitude 
  would 
  necessarily 
  be 
  less 
  than 
  4. 
  The 
  

   converse 
  is 
  equally 
  true, 
  and 
  since 
  in 
  the 
  stellar 
  system 
  the 
  increase 
  

  

  