Newly Compiled Systematic Treatise on Computing Methods, Volume Six

Compiled by Cheng Dawei (styled Rusi, pen name Binqu) of Xin’an Reviewed by Qin Guanghui (styled Baoyuan) of Kuaiji, Kuang Tingrui of Yuhang, and Weng Hong (styled Zaonan) of Jixi

Chapter Four: Reducing the Width

This chapter deals with instances where a field’s length original: 縱, zong is increased while its width original: 廣, guang is decreased; hence it is titled "Reducing the Width" original: 少廣, Shao Guang. This is a classical category in Chinese mathematics, dating back to the Han Dynasty, primarily dealing with finding the side of a shape when the area is known—effectively division and square roots.. It follows the principle of reversing the calculations used in field measurement. One uses the "Square Method" to divide the area original: 積, ji to find the side of a square, and the "Circle Method" to divide the area to find the dimensions of a circle. The essential steps for extracting square roots original: 開平方, kai pingfang and circular roots are often perceived as complex and tedious for beginners. Therefore, I have provided a simplified and clear explanation in the notes below.

Mnemonic for Determining the Quotient in Square Root Extraction


For one hundred, [the root is] ten, fixed without doubt.	For one thousand, [the root is] thirty with a remainder.
For nine thousand, [the root is] ninety-nine, close to one hundred.	For ten thousand, it is always reckoned as one hundred.
Once the first quotient is found, double it to form the Lian original: 廉. In the traditional Chinese method of root extraction, the Lian or "joint" represents the rectangular sections added to the square to expand it..	The next quotient is called the Yu original: 隅. The Yu or "corner" represents the small square that completes the larger square during the extraction process.; combine it with the Lian to divide the remaining area.