Systematic Treatise on Arithmetic, Vol. 5

Next to the side-strip method 20 steps is the position called the "corner method" (yufa); together they total 28 steps. Multiply this by the second quotient on the left, which is 8 steps. 2 times 8 [20 x 8] removes 160 steps from the dividend (shi). Furthermore, the 8 on the left corresponds to the 8 on the right; 8 times 8 [8 x 8] removes 64 steps from the dividend. This accounts for the area exactly. The side of the square is found to be 18 steps. Squaring this result [18 x 18] returns the original area. The method above clarifies the names of the Square, Side-Strip, and Corner.

Suppose there is a Go board A traditional board for the game of Weiqi (Go), which is composed of a grid of 19 by 19 lines, creating 361 intersection points. with 361 pieces. The question is: how many pieces are on each side?

The method says: Place the total number of pieces as the dividend (shi). The number 361. ○ Estimate the first quotient of 10 steps to the left of the dividend. ○ For the lower method, also place 10 steps to the right of the dividend. The left and right correspond to each other: 1 times 1 [10 x 10] removes 100 pieces from the dividend, leaving a remainder of 261 pieces. ○ Then, double the lower method value of 10 to get 20. This "side-strip" value prepares for the next digit of the root. ○ Estimate the second quotient of 9 pieces to the left. Following the first quotient of 10, also place 9 pieces to the right, following the doubled value of 20. Together, these total 29. Both are multiplied by the second quotient of 9 on the left: 2 times 9 [20 x 9] removes 180 pieces from the dividend. Then, the left 9 corresponds to the right 9: 9 times 9 [9 x 9] removes 81 pieces from the dividend. 180 + 81 = 261, which clears the remaining dividend exactly.

Side-strip method (lianfa): The middle step of square root extraction where you double the current root to find the next digit.
Corner method (yufa): The final small square area (the square of the new digit) added to the side-strips to complete the geometric expansion.
Side of the square (fangmian): The resulting square root.
Go board (qipan): A standard 19x19 grid, used here as a practical example of a square area.
Dividend (shi): The total area or number from which the root is being extracted.
First quotient (chushang): The first digit of the square root (in the tens place).
Second quotient (cishang): The second digit of the square root (in the units place).
Extracting the square root (kaipingfang): The algebraic process demonstrated here through geometric partitioning.