Before there were calculators and computers, this is the way square
roots had to be calculated:


Start with this. The 0s are in pairs and 1 digit goes over each pair
of 0's
   ________________
  V8.00 00 00 00 00

The largest integer whose square does not exceed 8 is 2.
So we put 2 above th 8 

   2.              
  v8.00 00 00 00 00

We square 2 getting 4. Place 4 under the 8, draw a line
under the 4, subtract and bring down the next pair of 0's

   2.              
  v8.00 00 00 00 00
   4
   4 00

Draw a vertical bar to the left of the bottom 4.
To the left of the bar, multiply the partial answer 2 by 2, getting 4
Annex a 0 to the 4 getting 40

    2.               
   v8.00 00 00 00 00
    4
 40|4 00 
   |

We ask "How many times does 40 go into 400?  We get 10, but we must use
a single digit, so we try 9.

    2. 9            
   v8.00 00 00 00 00
    4
 40|4 00

Now we must add the 9 that we are trying to the 40, getting 49,
so we write the 49 under the 40, multiply 9 by 49, getting 441,
and write it under the 4 00, and draw a line.
  
    2. 9             
   v8.00 00 00 00 00   
    4
 40|4 00
 49|4 41

Uh oh! 441 is larger than 400, so we must erase the 9, reduce it to an
8, and repeat the process with 8 instead of 9.  Add the 8 to the 40, getting 48,
put the 48 under the 40. Multiply 8 by 48 getting 384, and put that
under the 400 and draw a line 
  
    2. 8            
   v8.00 00 00 00 00   
    4
 40|4 00
 48|3 84

This time we can subtract, so 8 is the correct tenths digit.
We subtract getting 16 and bring down the next two 0's beside it:
 
    2. 8            
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
      16 00

Draw a vertical bar to the left of the bottom 16. To the left of the bar,
Ignoring the decimal of 2.8 and use only 28.  Multiply by 2, getting 56. 
Annex a 0 to the 56 getting 560 and write it left of the bar

    2. 8            
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
     |

We ask "How many times does 560 go into 1600"?  We get 2, so we try 2.
  
    2. 8  2         
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
     |

Now we must add the 2 that we are trying to the 560, getting 562,
so we write the 562 under the 560, multiply 2 by 562, getting 1124,
and write it under the 14 00, and draw a line.
 
    2. 8  2         
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24

We subtract getting 476 and bring down the next two 0's beside it:
   
    2. 8  2         
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
       4 76 00

Draw a vertical bar to the left of the bottom 47600.
To the left of the bar, multiply the partial answer 282 (ignoring the decimal)
by 2, getting 564 Annex a 0 to the 564 getting 5640.

    2. 8  2         
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24
  5640|4 76 00
      |

We ask "How many times does 5640 go into 47600"?  We get 8, so we try 8.
Now we must add the 8 that we are trying to the 5640, getting 5648,
so we write the 5648 under the 5640, multiply 8 by 5648, getting 45184,
and write it under the 47600, and draw a line.
    
    2. 8  2  8      
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
  5648|4 51 84

We subtract getting 2416 and bring down the next two 0's beside it:
   
    2. 8  2  8      
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
      |4 51 84 
         24 16 00

Draw a vertical bar to the left of the bottom 241600.
To the left of the bar, multiply the partial answer 2828 (ignoring the decimal)
by 2, getting 5656. Annex a 0 to the 5656 getting 56560
   
    2. 8  2  8      
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
      |4 51 84 
   56560|24 16 00
        |

We ask "How many times does 56560 go into 241600"?  We get 4, so we try 4.
Now we must add the 4 that we are trying to the 56560, getting 56564,
so we write the 56564 under the 56560, multiply 4 by 56564, getting 226256,
and write it under the 241600, and draw a line.
   
    2. 8  2  8  4   
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
      |4 51 84 
   56560|24 16 00
   56564|22 62 56

We subtract getting 15344 and bring down the next two 0's beside it:
   
    2. 8  2  8  4      
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
      |4 51 84 
   56560|24 16 00
   56564|22 62 56 
          1 53 44 00

Draw a vertical bar to the left of the bottom 1534400.
To the left of the bar, multiply the partial answer 28284 (ignoring the decimal)
by 2, getting 56568. Annex a 0 to the 56568 getting 565680  and write it left of
the bar:

    2. 8  2  8  4   
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
      |4 51 84 
   56560|24 16 00
   56564|22 62 56 
   565680|1 53 44 00
         |

We ask "How many times does 565680 go into 1534400"?  We get 2, so we try 2.
Now we must add the 2 that we are trying to the 565680, getting 565682,
so we write the 565682 under the 565680, multiply 2 by 565682, getting 1131364,
and write it under the 241600, and draw a line.
 
    2. 8  2  8  4  2
   v8.00 00 00 00 00
    4
 40|4 00
 48|3 84
  560|16 00
  562|11 24 
  5640|4 76 00
  5642|4 51 84 
   56560|24 16 00
   56564|22 62 56 
   565680|1 53 44 00
   565682|1 13 13 64
            40 30 36

Soce 403036 is more than half of 565682, we know to round the last
decimal up. So the square root of 8 to 5 decimals is 2.82843.

Edwin