General Source of Computational Methods: Volume VI

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The lower Side Method Lian fa: a subsidiary divisor used in the extraction of roots, representing the 'side' of the square expansion 60 OCR reads "16", but mathematical context for the root 35 suggests 60, derived from doubling the initial quotient of 30 is merged with the second quotient 5, making a total of 60. Both are matched with the upper quotient 5. Call out "five times six"

to subtract from the dividend 300. Then call "five times five" to subtract 25 paces. This results in a width of 35 paces, which matches the question.

$\bigcirc$ If the question asks for the length, arrange the calculation as follows: $\bigcirc$ The upper quotient of 15 the difference between length and width is multiplied by the deficit 50 to get 750.

Add this to the previous area for a total of 2,500 paces. Then call out "five times five" referring to the root 50 to subtract from the dividend 2,500. This exhausts the amount and yields the length, matching the question.

Problem: The Triangular Field

Now there is a triangular field Gui tian: literally "scepter field," an isosceles or right-angled triangle with an area of 126 paces. The width is 9 paces less than the length. What are the length and width?

Method: Double the field area to get 252 paces as the dividend Shi. Place the deficit of 9 paces as the "Vertical Square" Zong fang: a constant representing the difference between sides in quadratic equations on the right. $\bigcirc$

Place the first quotient of 10. $\bigcirc$ In the lower method, also place 10 above the Vertical Square's 9 paces, making a total of 19 paces. Combined with the upper quotient of 10, subtract 190 from the dividend. The remainder is 62 paces. $\bigcirc$ Separately, take the first quotient of the lower method 10 and double it to make 20. $\bigcirc$ Place the second

quotient of 2 paces on the left. $\bigcirc$ In the lower method, also place 2 paces and add it to the Vertical Square 9 paces, making a total of 11 paces. Both are combined with... The text cuts off here, but the calculation $2 \times (20 + 11) = 62$ would complete the extraction.