Systematic Treatise on Arithmetic, Volume 6

Page 10

...the result is the length. If you wish to verify the result, take the obtained width as the dividend (shi); separately, take the obtained quotient [the width] and add the longitudinal excess [the difference between length and width] to it. Multiply these together to see if they match the original area.

Suppose there is a field with an area of 1,750 square paces. It is only stated that the length exceeds the width by 15 paces. Question:

Answer: The length is 50 paces. The width is 35 paces.

The method says: Set the area [1,750] as the dividend (shi) Shi: The constant or "target" number in Chinese division and root extraction, placed in the center of the counting board. Take the excess of 15 paces as the longitudinal (zong) Zong: The difference between length and width; in modern algebra, this is the coefficient 'a' in the equation x² + ax = Area and list it in the lower position. Divide it using the "Square Root Method with an Additional Dimension" Pingfang dai zong fa: A specialized method for finding the sides of a rectangle, similar to solving a quadratic equation.

First, determine the initial quotient This is the first digit of the width, in the tens place to be 30, and place it in the left position. In the lower method, also place 30 and add it to the longitudinal value [15], which gives a total of 45. These values—the upper quotient and the lower divisor—now "call out" to each other [multiply]:

The 3 on the left corresponds to the 4 on the right: "Three fours are twelve," so subtract 120 Representing 1,200 from the dividend of 1,750 from the dividend.
Next, the 3 on the left corresponds to the 5 on the right: "Three fives are fifteen," so subtract 15 meaning 150 from the dividend.

Separately, take the initial quotient of 30 from the lower method and double it to make 60; add the longitudinal excess of 15 to this, resulting in a total of 75. Next, determine the second quotient to be 5 and place it in the left position...