Systematic Treatise on Computational Methods, Volume 2

## Inverted and Upright Triangles

Inverted Triangle Gui (圭): a wedge-shaped ritual tablet, used here to denote a triangular geometric shape　Width: 8 steps Bu (步): a "step," the standard unit of measurement in Chinese surveying, roughly equivalent to 5 feet　Central length: 8 steps
Upright Triangle　Central length: 10 steps　Width: 12 steps

## The Three-Triangle Shape

Eastern diameter: 8 steps　Diagonal: 24 steps　Middle diagonal: 26 steps
Western diameter: 12 steps　Diagonal: 23 steps　Southern diameter: 3 steps

Partitioning the shape into two quantities	The Method states:	Partitioning the shape into three quantities	The Method states:
The Inverted Triangle below has a central length of 8 steps and a width of 8 steps. ○ The Upright Triangle above has a central length of 10 steps The OCR text here reads "12 steps," but the subsequent calculation uses "10 steps" to arrive at the correct sum. and a width of 12 steps. ○ Question: What is the total area? ○ Answer: The total area is 124 steps.	Set the Inverted Triangle's central length of 8; halve it Original: "半之". While halving 8 and multiplying by 8 should yield 32, the text records the result as 64, suggesting either a calculation error in the original woodblock or that the "width" refers to a semi-measurement. and multiply to get 64 steps. ○ Then take the Upright Triangle's central length of 10 and halve it to get 5. ○ Multiply by the width of 12 to get an area of 60 steps. ○ Combine these to get a total area of 124 steps.	The eastern diameter is 8 steps and the western diameter is 12 steps. ○ Additionally, the southern diameter is 3 steps. The diagonals are 24 steps, 26 steps, and 23 steps. ○ Answer: The total area is 295 steps.	Take the length of the middle diagonal between the east and west (26). Combine the two diameters 8 and 12 and halve the sum to get 10. Multiply these to get 260 steps. ○ Then take the diagonal of 23, halve it, and multiply by the 3 steps 11.5 multiplied by 3 is 34.5; the text rounds this up to 35 steps. to get 35 steps. ○ Combine these to get a total area of 295 steps.