Five Unequal Sides

○ If one follows the ancient method, measuring the southern side along the diagonal Xie xian (斜弦): literally "slanted string," referring to the hypotenuse or a diagonal line yields a surplus of 0.012 steps compared to the vertical leg Gu (股): the longer leg of a right triangle. Measuring the eastern side along the diagonal yields a surplus of 0.02 steps compared to the vertical leg. The total calculated area would then exceed the true area by twenty-seven and 0.27 steps.

○ Having examined this now, one should use the cutting method Jie fa (截法): a technique of partitioning irregular shapes into standard rectangles and triangles to ensure all measurements are accurate. This demonstrates the errors in the previous ancient methods and allows students to easily understand the underlying logic. Whenever one encounters slanted boundaries, there must be "slanted steps" This refers to the physical distance along a slope or diagonal.; how could these possibly be multiplied as if they were "straight steps" Orthogonal measurements used for standard area calculation.? If the field is partitioned, errors will be avoided.

Five Unequal Sides

Suitable for the reference of Liu Juchun of Ningnan Prefecture, Haijie original: "海界寧南州柳聚純佩備考宜". This likely indicates the owner of the manuscript or the specific regional jurisdiction for which these surveying standards were intended.

Suppose there is a single field with five unequal sides. Partition it into one four-sided and one triangle. The diagonal length is 36 steps; the upper width is 15.3 steps and the lower width is 12.8 steps. The length of the triangular section is 22 steps and its width is 12 steps. Question: What is the total area?

Method: First, take the two widths of the four-sided section and combine them to get 28 steps. The text says 28 steps, though 15.3 + 12.8 is 28.1; this may reflect a simplification in the manual or a minor transcription variation. Halve this to get 14 steps. Multiply this by the length of 36 steps to obtain 504 square steps.

○ Next, take the triangle's length of 22 steps and multiply it by the width of 12 steps. Halve the result to obtain an area of 132 square steps.

Combining these two results yields a total area of 636 square steps, which matches the question.

Technical Terms:
Five Unequal Sides An irregular pentagon., Partition into To divide a shape for calculation., Four-sided Quadrilateral., Triangle, Diagonal/Hypotenuse, Vertical leg, Cutting method Partitioning., Area, Step A unit of length, approximately 5 feet; square steps are used for area., Tenths, Hundredths, Slant length, Width/Height, Halve it, Combined result, Matches the question.