Systematic Treatise on Arithmetic, Vol. 5

The Answer is: 456 Paces

The Method says: Place the area of the square as the Dividend Shi: The total area being solved. ◯ Estimate the First Quotient Chu shang: The hundreds-place digit of the root of 400 and place it in the left position. ◯ Also place 400 in the right position as the Square Method Fang fa: The primary divisor for the first stage. Multiply the upper quotient and this value together—four times four—and subtract from the Dividend 160,000. The remaining area is 47,936 paces.

◯ Next, take the Square Method of 400 and double it to make 800, which serves as the Side Method Lian fa: A divisor representing the "side strips" of the geometric square. ◯ Estimate the Second Quotient Ci shang: The tens-place digit of 50 on the left, below the first quotient of 400. ◯ Also place 50 on the right, below the Side Method of 800, to serve as the Corner Method Yu fa: The small corner square that completes the new section. Together, these total 850.

These are both multiplied by the second quotient of 50 to be subtracted from the dividend. First, match the left 5 against the right 8; five times eight subtracts from the Dividend 40,000. Next, match the left 5 against the right 5; five times five subtracts from the Dividend 2,500. The remaining area is 5,436 paces.

◯ Then, take the previous lower method’s second quotient of 50, double it, and add it to the Side Method to get a total of 900, which again becomes the Side Method. ◯ Again, estimate a quotient of 6 paces on the left, below the previous quotients of 450. ◯ Also place 6 paces on the right, shifting it one place value below the Side Method of 900. Together, these total 906.

Both of these are multiplied by the left’s third quotient of 6 paces to be subtracted. First, match the left 6 against the right 9; six times nine subtracts from the Dividend...

The text describes the process of calculating the square root of 207,936. The step-by-step subtraction mimics the physical removal of areas from a large square: first a large 400x400 block, then two 400x50 rectangles and a 50x50 corner, and so on.