Systematic Treatise on Arithmetic, Vol. 5

...six Referring to the multiplication steps of the previous calculation's final digit divides the Dividend by 242 paces, and again calling out "four times four," which divides the Dividend by 16 paces, exactly exhausting the amount. ◯ Thus, the quotient obtained at the top-left, 234 paces, is the length of one side of the square.

Suppose we have a square area of 490 paces. We wish to find the side of the square. Question: What is the length of each side?

The rules for Square Root Extraction by the Return-and-Divide Method Original: Guichu kai pingfang fa state: Place the total area of 490 as the Dividend Shi: the number to be divided in the center of the board. Seeing that the Dividend contains 400, set a quotient Shang: the estimated root of 20 to the left of the Dividend. ◯ Also place 20 paces at the lower right. Calling out between the left and right: "Two times two [is four]," which divides the Dividend by 400 paces. The remaining Dividend is 90 paces.

Immediately double the 20 paces on the right to get 40 paces, which serves as the Divisor Fa: the number used to divide the remainder. Perform the "Return-and-Divide" calculation. Call out the mnemonic: "Encountering eight, advance two." This abacus mnemonic indicates that within the remainder of 90, the divisor 40 goes in twice, as 2 times 4 is 8. Accordingly, place the 2 paces below the 40 on the right. Calling between the positions: "two [times forty is eighty]" and "two times two [is four]," these divide the Dividend by 4 paces The text omits the subtraction of the 80, focusing on the final units. A remainder of 6 paces is left unexhausted.

To designate the remainder using the "Direct Square" method, the rule says: Double the current quotient of 22 paces and add one more pace, giving a total of 45 paces. This shall be the Denominator Fenmu: the "mother" or divisor of the fraction. Designate the final result as: 22 paces and six parts of a pace divided into forty-five.