Systematic Treatise on Arithmetic, Vol. 5

Page Nine

Nine: The rule is established because the compasses cannot reach it; the others follow this example. This likely refers to a specific mnemonic or layout rule on the abacus for handling multi-digit roots.

Now suppose there is a square field with an area of 71,824 paces. What is the length of one side?

The Method says: Place the area of the square field as the Dividend Shi: The number from which the root is to be extracted. Divide it using the Square Root Extraction Method. ◯ Place the First Quotient Chu shang: The first digit of the root, representing the hundreds place of 200 in the left position. ◯ Also place 200 in the right position as the Square Method Fang fa: The primary divisor for the square area. ◯ Align the 2 on the left with the 2 on the right and call out the calculation:

Two times two—subtract 40,000 from the Dividend. The remaining Dividend is 31,824 paces. Then, take the Square Method of 200 and double it to make 400, which becomes the Side Method Lian fa: A divisor representing the "strips" or rectangles added to the sides of the square. ◯ Place the Second Quotient Ci shang: The second digit of the root, representing the tens place of 60 in the left position, below the first quotient of 200. ◯ Also place 60 below the Side Method of 400.

In the right position, this becomes the Corner Method Yu fa: The divisor representing the small corner square required to complete the larger square. The total combined divisor is 460. Both are called out and divided against the second quotient of 60. First, align the 6 on the left with the 4 on the right. Call out: four times six—subtract 24,000 from the area. Then, align the 6 on the left with the 6 on the right. Call out: six times six—subtract 3,600 from the area.

The remaining Dividend is 4,224 paces. Then, take the second quotient of 60 from the right position, double it to add 60 to the 400 below, making a total of... The text cuts off here, but the process would continue to find the third digit, 8, to complete the root 268.