Systematic Treatise on Arithmetic, Vol. 6

The method states: Set the area and double it to get 450 paces as the dividend (shi: the value to be divided). Use the length as the divisor (fa: the measure used to divide) and divide it to find the width, which is 10 paces 10 paces. Mathematically, if the area is 225 and length is 30, the width should be 15. The manuscript or transcription records "10 paces" here, which may be a clerical error.

If the question asks for the central length: double the area to use as the dividend and use the width as the divisor; the result is then obtained.

The two items above are called "Forgetting the Length or Losing the Short-side." The logic is the same as that for sectioning the area of a rectangular field.

Now suppose there is a right-angled triangular field original: 勾股田 (gou-gu tian) with a length of 30 paces and a width of 15 paces. If a section is cut from the apex the pointed corner or vertex at a length of 12 paces, what is the width at that point?

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Sectioning a Right-Angled Triangle

The principle is the same as for the triangular field original: 圭 (gui), referring to the isosceles triangular "tablet" shape.
Vertical side (gu) is 30 paces long
Horizontal side (gou) is 15 paces wide
The resulting width is 6 paces

The method states: Set the length of the cut 12 paces and multiply it by the horizontal width (gou) to get 180 as the dividend. Use the vertical length (gu) as the divisor and divide. Calculation: (12 × 15) ÷ 30 = 6.

Another method: Set the horizontal width (gou) as the dividend and the vertical length (gu) as the divisor. For every 1 pace of vertical length, the width increases by 0.5 5 tenths. Multiply this by the length of the cut to also obtain the result. Calculation: (15 ÷ 30) × 12 = 0.5 × 12 = 6.

Now suppose there is a slanted field original: 斜田 (xie tian), usually referring to a trapezoid with a southern width of 4 paces, a northern width of 12 paces, and a length of 32 paces. Now, if it is cut through the middle...

The diagram provided in the manuscript illustrates a right-angled triangle with a vertical line marking the point where the cut (section) is made. This visual aid helps the student understand similar triangles—the ratio of the height to the base remains constant regardless of where the triangle is sliced.