Systematic Treatise on Arithmetic, Vol. 5 (1592) - Page 18

◯ The explanation says: If we take the area of 409 9 paces and add it to 40 paces 20 paces, then subtract the numerator 6, we still obtain 500 20 paces. Then we set the quotient as 20 paces; this is what is called "falling short," therefore it is designated as the remainder This note clarifies the fractional remainder logic from the previous page's calculation of the square root of 490.

Song for the Return-and-Divide Method for Square Roots with an Additional Dimension

Guichu pingfang daizong ge: A mnemonic for solving quadratic equations where the area and the difference between length and width are known

The method for square roots with an additional dimension is most marvelous:

Multiply the area by four without hesitation.

Square the "longitudinal excess" Zongduo: the amount by which the length exceeds the width and add it to the multiplied area,

Then apply the square root extraction method to divide it.

Furthermore, take this extracted root and add the "longitudinal excess" to it;

Half of this sum shall be established as the length.

If you ask how many paces the width is:

Simply take the length and subtract the original "longitudinal excess."

Now suppose there is a rectangular field with a total area of 1,750 paces. The length exceeds the width by 15 paces. We ask: what are the length and width respectively?

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

The mathematical logic provided in the song follows the formula for solving $x(x + b) = Area$. 1\. $4 \times Area = 7000$ 2\. $7000 + b^2$ (where $b=15$, so $225$) $= 7225$ 3\. $\sqrt{7225} = 85$ 4\. Length $= (85 + 15) / 2 = 50$ 5\. Width $= 50 - 15 = 35$