Systematic Treatise on Arithmetic, Vol. 5

[...]five? thousand. Furthermore, match the left six against the placed six. Say: six times six, and subtract from the Dividend Shi: The number being divided or from which a root is extracted thirty-six paces. The calculation is exactly completed and matches the question.

Suppose there are 1,461 square bricks. If you wish to arrange them into a large square, what is the length of one side?

The Method says: Place the total area of the bricks as the Dividend. ◯ Estimate the First Quotient Chu shang: The first digit of the result of 30 bricks in the left position. Also place 30 on the right as the Square Method Fang fa: The divisor representing the primary square area. Multiplying the left and right together—three times three—subtract from the Dividend 900 bricks. ◯ The remaining Dividend is 561 bricks. ◯

Then, take the Square Method and double it The text says "tenfold," but the gloss clarifies "60," which is double the initial 30; this reflects an abacus technique where the operator shifts decimal places while doubling the root to make 60, which serves as the Side Method Lian fa: A divisor representing the expansion strips of the square. ◯

Estimate the Second Quotient Ci shang of 8 in the left position, below the first quotient of 30. Also, place 8 below the Side Method of 60 on the right to serve as the Corner Method Yu fa: The small corner piece that completes the new, larger square. Together, these total 68. Both are multiplied by the upper quotient of 8.

Say: six times eight, and subtract from the Dividend 480. Then say: eight times eight, and subtract from the Dividend 64. The remaining Dividend is 17—it does not divide evenly. Now, double the quotient of 38 and add one brick more to get a total of 77. Use the 17 as the numerator Literally "name the 17," creating a fraction where 17 is the remainder and 77 is the denominator. Why is it "named"? By taking the remainder of 17 and placing it over the divisor of 77, the side of the square is found to be 38.

Vocabulary used in this section: Square Bricks (方磚), Square Root/Square (平方), Dividend (實), Quotient/Estimate (商), Square Method (方法), Side Method (廉法), Corner Method (隅法), Systematic Treatise on Mathematics (算法統宗), Cheng Dawei (程大位).