Galley or Scratch Method of DivisionDate: 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. |
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