Introduction to Mathematical Studies: Volume Three

Now, suppose there is a square field with an area of 3,136 square paces. What is the length of one side?

The answer is: 56 paces.

The Method says: Place the area of the field as the Dividend Shi: The total value from which the root is extracted. ◯ Estimate the dividend to determine the First Quotient Chu shang: The first digit of the result of 50 paces and place it on the left. ◯ For the Lower Method Xia fa: A divisor used in the extraction process, also place 50 paces on the right. Multiply the left and right together—five times five—and subtract from the Dividend 2,500 paces. The remaining area is 636 paces.
◯ Regarding the Lower Method of 50 paces, double it to get 100 paces This creates the 'Side-strip' or 'Lian' divisor. ◯ Next, determine the Second Quotient of 6 paces on the left, placed below the first quotient of 50. Also place 6 paces on the right in the Corner Position Yu wei: The place value that completes the geometric square beneath the doubled value of 100. Together, these total 106 paces. Multiply both of these by the Second Quotient of 6 paces. One times six allows you to subtract from the Dividend 600 paces. ◯ Then, multiply the 6 on the left against the 6 on the right—six times six—to subtract from the Dividend 36 paces. The calculation is now exactly exhausted.

Now, suppose there is a square field with an area of 207,936 square paces. What is the length of one side?

This problem introduces a larger number to demonstrate how the root extraction process repeats across multiple place values (hundreds, tens, and units).

Vocabulary used in this text:

• Square field area original: 方田積, fang tian ji: The total surface area of a square.
• Side of the square original: 平一面, ping yi mian: The length of one side (the square root).
• Dividend original: 實, shi: The number being operated upon.
• First Quotient original: 初商, chu shang: The first digit found in the root.
• Second Quotient original: 次商, ci shang: The second digit found in the root.
• Lower Method original: 下法, xia fa: A temporary divisor used to calculate the root.
• Doubled Square original: 倍方, bei fang: The process of doubling the previous root to find the next divisor.
• Corner Position original: 隅位, yu wei: The placeholder for the "corner" square in the geometric visualization of the root.
• Introduction to Mathematical Studies original: 算學啟蒙, Suanxue Qimeng: The title of this 1299 AD textbook.
• Zhu Shijie original: 朱世傑: The author and one of the greatest mathematicians of the Yuan Dynasty.