Galley Division

Up until about the 18th century, one of the most common methods of division taught was the galley method. The algorithm is very similar to modern long division (after all, how many different ways can you really see how much of one thing fits into another?) but is seemingly more space intensive and, in my opinion, less easy to follow. The method is essentially to remove multiples of the divisor from the dividend, counting the amount removed as we go, until there is no longer enough left over to fit a full multiple.

As a quick introduction, let’s do an example with some numbers. We begin by writing the dividend with a solid line next to it and writing the divisor beneath the dividend, lining up the leftmost digits. For example, if we were calculating 4892385 / 362 we would line up the problem as follows:

4892385 |
362          |

We can see that 362 divides into 489 exactly once, so we write a one on the other side of the bar and then calculate the difference between 489 and 362 * 1, writing the answer above the original dividend, whose first three numbers we now cross out. We also cross out our original divisor and rewrite it one space to the right. Our work would now look like this (periods are for formatting purposes, which wouldn’t be an issue if we were writing this on a paper):

127               |
4892385      | 1
3622             |
..36               |

Now we are dividing 362 into 1272, which goes in three times. We do 1272 – 3 * 362 = 186 and repeat the process stated above. We continue repeating the steps until the divisor does not fit in to any of the remaining dividend. Once all of that is done, our work will look like this:

…..13        |
..18571     |
1276367   |
4892385   | 13514
3622222   |
..36666     |
….333       |

We can therefore see that 4892385 / 362 = 13514 with a remainder of 317. That wasn’t so bad at all! Comparatively, we can write out the same problem using long division as follows:

………__    13514
362 | 4892385
…….-362
……..1272
…….-1086
……….1863
………-1810
………….538
…………-362
………….1765
…………-1448
……………317

Notice that many of the numbers (indeed, all of the remainders after subtraction is finished) appear in both methods. Galley division is much more compact, but long division looks much less cluttered. While galley division writes the divisor many times and does not right the subtrahend at each step, long division writes the divisor once but writes each subtrahend we come across. The underlying algorithm for both is the same: go from highest place value to lowest subtracting as many multiples of the divisor from the dividend as possible, keeping track of the amount as you go. Since the algorithm is the same, it should go without saying that the answer they arrive at is the same.

Now, if you’re anything like me, you’re probably thinking: what does that have to do with a galley? Well, in the words of Dr. Lamb, “If you draw a boat around it when you’re finished, it looks like a boat!” The image below shows what the problem can look like if you choose to make your work look pretty at the end instead of recognizing that math is pretty in its own right and doesn’t need flowery decorations.

Image by Francis Rolt-Wheeler via Wikimedia commons.

Perhaps the biggest flaw I see in galley division is that it is very cluttered and it can be hard to follow each step since one number can end up being written across multiple lines. By contrast, in long division each step can obviously be seen down and to the right of the preceding one. One strategy that has been used to combat this lack of clarity in the galley method is performing each step of the division and erasing any unnecessary numbers before moving to the next step. I think this is a good compromise. After all, if your work isn’t going to be followable you may as well not show it at all.

Like most really old things, we’re not too sure on the particulars of the galley method’s origin. The method is thought to have originated somewhere in China or the Middle East around 400 CE and was probably designed for a sand abacus or abax, which was a table covered in sand where stones and other objects would be used as counters and columns of objects were used as place value markers. The galley method is no longer taught in Western schools, but is still taught in northern Africa and the Middle East and is probably equally as misunderstood and hated by the general public there as long division is here due to a focus on teaching algorithms rather than actual underlying mathematics. Galley division and long division lack an intuitiveness, in my opinion, that seems to be inherent in other methods of division, such as the bubble method (which consists of pulling out “bubbles,” aka multiples, of numbers and adding them together to get the quotient) or the Ancient Egyptian method (which relies on building up the quotient from powers of two). Even though the underlying mathematics are more or less the same, additive methods seem much more intuitive since adding is simpler than subtracting. However, galley division was used for centuries upon centuries before being replaced, so perhaps it’s not as hard as I’m making it out to be.

The Wikipedia page on galley division is woefully sparse, but in truth only so much can be said about one really old algorithm and Wikipedia covers most of it. For more information on performing the method, check out http://www.math.wichita.edu/history/activities/arith-act.html#worse or http://en.wikipedia.org/wiki/Galley_division

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