Song for the Square Root Method with an Additional Dimension

Including the Return-and-Divide Method Jianshang chu: A division technique used in square root extraction to find the next digit of the root

The method says: Set the area Ji: The total product or area of 1,750 50 The gloss "50" here refers to the final length calculated later and multiply it by four to obtain 7,000. Separately, take the longitudinal excess Zongduo: the amount by which the length exceeds the width 15 paces and multiply it by itself i.e., square it to obtain 225 paces. Add these two figures together to get a total of 7,225 paces, which serves as the dividend Shi: The "solid" number from which the root will be extracted. Use the square root extraction method to divide it.

○ Estimate the quotient 80 on the left and also place 80 on the right. Call out to each other between left and right: "eight times eight" to subtract from the dividend 6,400 paces 80 x 80 = 6,400. The remaining dividend is 825 paces.

○ Take the lower divisor 80 and double it to get 160, which becomes the divisor Fa: The number used to divide the remaining area. Use this to divide the remainder. Call out "where there is a five, advance 5" 5 and place it next to the initial quotient of 80, making a total of 85 paces.

○ For the lower divisor, also place the 5 beneath the 160, making a total of 165 paces. Match the 8 and 5 on the left against the 6 on the right This describes the alignment of numbers on a counting board or abacus. Call out "five times six" to subtract 300 from the dividend 5 x 60 = 300. Then match the 5 on the left against the 5 on the right, calling "five times five" to subtract 25 paces. The dividend is now exactly exhausted.

○ The quotient on the left is 85 paces. This represents the sum of the length and width added together. Add the longitudinal excess of 15 paces to this for a total of 100 paces. Halve this amount to get 50 paces. From within this, subtract the longitudinal excess of 15 paces; the remaining 35 paces is the width.

The mathematics here follows the formula: 1\. $(4 \times \text{Area}) + (\text{Difference})^2 = (\text{Length} + \text{Width})^2$ 2\. $\sqrt{7225} = 85$ (which is $L+W$) 3\. $(L+W + \text{Difference}) / 2 = \text{Length}$ 4\. $\text{Length} - \text{Difference} = \text{Width}$

Daizong kaipingfang fa: Square root extraction with an additional dimension (solving $x^2 + ax = b$)
Zhiji: To set or establish the area/product
Shi: The dividend or the "solid" number being operated upon
Yueshang: To estimate or approximate the quotient
Kuo: The width of a rectangle
Zongduo: The "longitudinal excess" or difference between length and width