Systematic Treatise on Arithmetic, Volume 3

Page 5

Suppose there is a rectangular field zhitian: literally "straight field," meaning a rectangle 60 steps long and 32 steps wide. What are the area and the taxable area respectively?

Answer: The area is 1,920 square steps. The tax is 8 mu. original: 稅八畝. In this period, 240 square steps equaled 1 mu. Thus, 1,920 divided by 240 equals exactly 8 mu.

Rectangular Fields

Length: 60 steps Area: 1,920 steps Width: 32 steps

The method states: Set the length at 60 steps and multiply it by the width of 32 steps. The resulting product is 1,920 square steps, which serves as the shi: the dividend. Divide this by the standard mu conversion factor of 240 square steps mufa: the rule of the mu to match the inquiry.

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Suppose there is a circular field with a diameter jing: the diameter of 56 steps and a circumference zhou: the perimeter of 168 steps. What is the area?

Answer: 2,352 square steps.

Circular Fields

Circumference: 168 steps Inner Diameter: 56 steps

The method states: Regarding the diameter—
Set the diameter of 56 steps and multiply it by itself 56 x 56 = 3,136 to get 3,136 steps.
Multiply this by 3 3,136 x 3 = 9,408. This uses the ancient Chinese approximation of Pi as 3. to get 9,408 steps.
Then, perform "division by four" sigui: a division mnemonic used on the abacus to divide by 4 to get 2,352 steps.

○ Alternatively:
Take the circumference of 168 and multiply it by the diameter of 56 steps, then divide the result by four; this also yields the result (168 x 56) / 4 = 2,352.

○ Alternatively:
Multiply the circumference by itself and divide the result by twelve; this also matches the inquiry (168 x 168) / 12 = 2,352.

Vocabulary: Rectangular field, Circular field, Diameter, Circumference, Area, Mu conversion rule, Division by four, Systematic Treatise on Arithmetic.