Systematic Treatise on Arithmetic, Vol. 5

The dividend original: 實, shi. The total area from which the root is being extracted. is a certain amount; the remaining dividend is a certain amount. Then, take the [first digit of the] root and multiply it by two operating on itself to obtain a certain amount, which serves as the side-divisor original: 廉法, lian fa. In geometric terms, these are the two rectangular "wings" added to the side of the initial square during the expansion process..

〇 Continue the process:

Place the second digit of the quotient The root being calculated. in the next position above the first digit to get a certain amount. For the lower method The auxiliary counter used to track the calculation's place value., also place this continuing quotient of a certain amount as the corner-divisor original: 隅法, yu fa. The small square that fills the gap between the two side-divisors to complete a larger square..

〇 At the position of the doubled root, combine all these parts; multiply them by the continuing quotient and subtract the result from the dividend until it is exhausted. This yields the value of one side of the square.

〇 If there is a remainder, repeat the previous steps to find the next digit. The "corner-divisor" is the corner of the two "side-divisors" shaped like a carpenter's square; this corner is the small square.

If the number [of the area] is insufficient [to form a whole number], use the "designation" method. 〇 What is meant by "designate"? If the remaining dividend of a certain amount does not divide evenly, take the obtained square root of a certain amount, double it, and add one. The total sum of a certain amount corresponds to the value of a side that is one unit larger. Because the original number was insufficient, we use this to designate the remainder This refers to an ancient Chinese method of expressing a remainder as a fraction, where the denominator is $2n + 1$ (the difference between $n^2$ and $(n+1)^2$)..

Calculating the area of a circle with a remainder is done in this same way. 〇 However, the methods for cubes and spheres are different.

If you wish to verify the result—as when measuring a square field—multiply the side of the square by itself to see if it matches the area. 〇 If there was originally a remainder below the side of the square, multiply the side by itself and then add the remainder to find the total area.