Galley or Scratch Method of Division

Date: 12/08/2002 at 14:59:04
From: Julie
Subject: Division in the 1400s

In the 1400's what was the method for performing division?

Date: 12/08/2002 at 23:00:04
From: Doctor Peterson
Subject: Re: Division in the 1400s

Hi, Julie.

According to D. E. Smith's History of Mathematics,

  By far the most common plan in use before 1600 is known as the
  galley, batello, or scratch, method and seems to be of Hindu
  origin.

After showing just part of an example (which shows a pile of crossed-
out digits), he says

  The method is by no means as difficult as it seems at first sight,
  and in general it uses fewer figures than our common plan. Maximus
  Planudes (c. 1340) throws some light upon its early history,
  saying that it is "very difficult to perform on paper, with ink,
  but it naturally lends itself to the sand abacus. The necessity
  for erasing certain numbers and writing others in their places
  gives rise to much confusion where ink is used, but on the sand
  table it  is easy to erase numbers with the fingers and to write
  others in their places."

If you need to see the details, I can try figure out how it is done, 
and to work out a way to explain it.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/09/2002 at 13:33:08
From: Julie
Subject: Division in the 1400s

Hi,

I really appreciate your answer. I would like to see how it is done in 
practice.  This is for a school project and I would like to understand 
it.

Date: 12/09/2002 at 16:59:51
From: Doctor Peterson
Subject: Re: Division in the 1400s

Hi, Julie.

I was hoping you'd ask - I spent some time last night studying the 
explanation in my book so I could be sure I understood it.

The tricky part is that there are several examples that seem to be 
variants on the method; all that Smith shows for those is the end 
result, and I can't see how the same method could give the numbers 
they used. But the rest of the examples do agree, so I can tell you 
the method pretty cleanly.

I can actually give you three variants: one using cross-outs (the 
standard form used on paper), one using erasure (apparently the 
original form, used on a sand board) and one without either (the form 
a printer would have preferred). I'll do it in that order, for the 
problem 3945 / 27.

First, we write the dividend, followed by a bar, after which we will 
write the quotient. Then below the left side of the dividend, we 
write the divisor:

    3945 |
    27

Now, as in the modern algorithm, we divide 39 by 27. The quotient is 
1, so we write that in the answer at the right. Now here's what's 
different: we multiply each digit of the divisor by the quotient 
digit, 1, ONE AT A TIME, and subtract it from the dividend, crossing 
out digits and replacing them with the result of our subtraction. 
First, we subtract 2 from 3:

    1
    /945 | 1
    /7

Note that I crossed out the 3 and the 2, which have been used up, and 
wrote the difference, 1, at the top of that column. (When you do this 
on paper, of course, you will still see the numbers you've crossed 
off. You may want to write this out and cross digits off on your own 
copy, so you can see what's happening.) Essentially, what I have done 
is to subtract 2000 from 3945, and I have 1945 left. So the digits on 
top, taken together, represent what is left of the dividend, and we 
are going to be subtracting from that at each step.

So now we do the same with the 7, subtracting 7*1=7 from 19; we don't 
have to touch the 1 (though we might have), but can cross off the 9 
and write the difference, 2, above it:

    12
    //45 | 1
    //

So far we've subtracted 2000 and 700 from 3945, leaving 3945-2700 = 
1245. That takes care of the hundreds digit in the quotient.

Now we write the divisor on the bottom again, shifted one place to the 
right, always putting each digit on the bottom of the appropriate 
column:

    12
    //45 | 1
    //7
     2

Now, how many times does 27 go into 124, the number on the top above 
these digits? Let's try 4; we'll subtract 2*4=8 from 12, and then 7*4 
= 28 from the 44 we have left after that:

     4
    //
    //45 | 14
    //7
     /

     1
     /
    //6
    ///5 | 14
    ///
     /

So now we've subtracted 400 times 27 from the 1945 we had left, and we 
know that 3945 - 140*27 = 165.

One more step; we'll guess a 6, then subtract 6*2 and 6*7 from the 
respective columns:

     1
     /
    //6
    ///5 | 146
    ///7
     /2

     /
     /4
    ///
    ///5 | 146
    ///7
     //

     /
     //
    ///3
    //// | 146
    ////
     //

So the quotient is 146, with a remainder of 3.

One thing I am not sure of is what they would do if they guessed a 
digit wrong - with all the crossing out, it would seem hard to back up 
and try again. If they guessed too small, they would subtract and have 
too large a remainder; it would be no trouble to subtract the divisor 
an extra time, cross off the wrong digit in the quotient and increase 
it by one, repeating again as needed. If they guessed too small, they 
would probably have to add the divisor back to the dividend pile 
(however many digits they had changed before discovering the problem) 
and try again. But none of the examples I've seen show this being 
done.

Here's what I would have done if I had guessed 3 for the second digit:

     6
    //
    //45 | 13
    //7
     /

     4
     /
    //3
    ///5 | 13
    ///
     /

Since the 43 on the top is still greater than 27, I would subtract 27 
again without shifting it to the next place:

     4
     /
    //3     4
    ///5 | 1/
    ///
     /7
     2

     2
     /
     /
    //3     4
    ///5 | 1/
    ///
     /7
     /

     1
     /
     /
     /6
    ///     4
    ///5 | 1/
    ///
     //
     /

Now we're where we would have been if we had guessed right.

Now, the erasure method, used on a sand table (or black- or white-
board today) would work the same, but we would entirely erase each 
digit as it is used, so they don't pile up:

    3945 |
    27

    1945 | 1  <--- I erased the 3 and 2, and replaced with 3-2
     7

    1245 | 1  <--- I erased the 9 and 7, and replaced with 9-7

    1245 | 14 <--- I wrote the next digit in the quotient
     27

     445 | 14 <--- I subtracted 2*4 from 12
      7

     165 | 14 <--- I subtracted 7*4 from 44

     165 | 146 <-- I wrote in the next digit
      27

      45 | 146 <-- I subtracted 2*6 from 16
       7

       3 | 146 <-- I subtracted 7*6 from 45

The book gives no examples of this (since it wasn't printed) but 
this seems to be what they must have done. The hard part is that it 
could be easy to lose track of where you are, since there is no 
evidence as to what digit you last worked on.

Finally, you can do the same thing without erasing or crossing out, 
just remembering that only the top digit on each column is current. 
You have to write a zero to remove the top digit entirely:

     0
     10
    044
    1263
    3945 | 146
    2777
     22

This method was used in some books.

Incidentally, the name "galley method" comes from seeing the final 
picture as looking like a ship, with tall sails and a low keel.

If you are less concerned than they were about wasting paper, you can 
line up the work at each step, rather than draping each number over 
the existing pile, so it makes more sense:

      03
     04
     16
    04
    12
    3945 | 146
    27
     27
      27

Having done all that, I searched the Web and found this explanation 
of the method:

 Things Could be Worse! - Carey Eskridge-Lybarger 
 http://www.math.wichita.edu/history/activities/arith-act.html#worse 

Let me know if you need any more help. It's fun learning these old 
methods.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/10/2002 at 12:51:24
From: Julie
Subject: Thank you (1400's division)

Hmmm, now I know why they came up with the "current" long 
division method .... I think most people would have given 
up on that galley (scratch) method :-)

I apprecaite the help.