The History

• Early algorithms for division up until the 16th century were done by using the Galley method, also known as the scratch method. The reason for this method being called “Galley” is because Galley is a type of boat, and when you solve using this method the pattern creates a boat. The term scratch had to do with the fact that when solving a division problem, you are crossing out the values to be replaced with new ones in the process.

The origin

Research shows that the method is believed to come from the early Hindu or Chinese origins. The reason math researches do not know exactly where the Galley method came from, is because the Hindus method of division was identical to the Chinese method of division and they were both unknowing aware of how similar their process of long division is. Around the 4th century is the predicted time where the method was found in India, and the earliest it was found in China was in the 3rd century in Sunzi suanjing ( a math manual) written by Sun Zi. Another version of this method was also said to be from the Arabs, where famous mathematician Al-Khwarizmi used a version of this method in one of his writing around 825 AD. Eventually, this method made its way to Europe where it was the most known way to solve division up until the 1600.

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How the Method Works

The purpose of the Galley method is 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. In other words, multiply each digit of the divisor by the quotient digit one at a time and subtract it from the dividend, crossing out digits and replacing them above with the result of the subtraction.

For example, 65284÷594 a problem given in Treviso arithmetic (1478)

The answer is 109 with a remainder of 538.

Let us take look at a simpler examples with steps on how to obtain the quotient. This example is from An Introduction to the History of Mathematics (6th Ed) page 291. 9414÷37

1. Write the divisor below the dividend. Then 37 goes into 94 two times, so we can write the 2 on the right side, as shown in the picture.
2. Take the 2 and multiply it by the 3 in the #37. Which equals 6 and then take the 9 in the #9413 and subtract it by the 6 which will equal 3, and now we have used the numbers 9 and 3, so we can scratch it off and write the number we just solved for (3) above the #9. Now we will do the same thing, take the 2 multiply it by the 7 in #37 which equals 14. Now the #34 is formed on the diagonal, so 34-14=20 which is our new number we have to write in. We can scratch off 7 from #37, and 3,4 from #34, and we will write the number 2 (from 20) above the scratched 3 and the number 0 (from 20) above the scratched 4.

3. The resulting division is now 2013. Now we need to write the divisor on the bottom again, so under the crossed out 37, write a new number 3 under the old 7 and and a new number 7 next to the old 7 to create a diagonal 37. So we start back to step one, 37 goes into 201, five times, so write a #5 on the right side. 5x3=15, 15-20=5, so we can scratch off the 3,2, and 0 and write the new number 5 above the scratched 0. 5x7=35, 51-35=16, so we can scratch off 7,5,1 and write the new number 16, where the 1 will go above the scratched 5 and the 6 above the scratched 1.

4. The resulting division in now 163. Write the divisor on the bottom, which is technically being moved one place to the right diagonally. 37 goes into 163 four times, so write a #4 on the right side. 4x3=12, 16-12=4, so scratch off 3,1,6 and write the new number 4 above the scratched 6. Next, 4x7=28, 43-28=15, scratch off 7,4,3 and write 1 above the scratched 4 and 5 above the scratched 3.

5. We know we can stop here because if we add a new 37 on the bottom, 37 does not go into 15; therefore the quotient is 254 with a remainder of 15.

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The Hindu method of division was similar to galley method except the math was done on sand, and the numbers were erased not scratched off. The Chinese method used rod-numerals, and they would change the counting rods which is similar idea to erasing the number. The Galley and Hindu method both placed the divisor under the dividend starting from the beginning of the dividend, and the quotient on the right side like the examples shown above.

However, the Chinese method places the divisor under the dividend depending on if the divisor is greater than or less than the dividend, and the quotient is placed above the dividend. Where to place the quotient depends on what place the divisor is. For example, if the divisor is in the 10th place, then the quotient will be paced in the 10th place above the dividend. This example is from the article On the Chinese Origin of the Galley Method

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Since 256 is less than 751, the divisor needs to be shifted two places to the left because 256 is not dividable by 751. Also since the divisor is in the 100th place, the quotient will be in the 100th placed. The steps after that are similar to galley method.

This website has a more detailed explanation of the scratch method and the erase method that the Hindus and Chinese used.

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Citations

Blog, P. (1970, January 1). Some notes on Division, and its history. Some Notes on Division, and Its History. Retrieved October 12, 2022, from https://pballew.blogspot.com/2019/11/some-notes-on-division-and-its-history.html?m=1

Eves, H. W. (1992). 8.12 The Gelosia and Galley Algorithms. In An introduction to the history of Mathematics (pp. 290–291). essay, Saunders College Publishing.

Lam Lay-Yong. (1966). On the Chinese Origin of the Galley Method of Arithmetical Division. The British Journal for the History of Science, 3(1), 66–69. http://www.jstor.org/stable/4025103