Systematic Treatise on Arithmetic, Vol. 3

Further examination reveals that, as shown in the previous diagram, if we take one square field 坵: a classifier for fields or plots of land measuring 40 steps on each side, it occupies an area of 1,600 square steps. The four surrounding "arc and sagitta" segments hushi: circular segments, literally "arc and arrow" occupy an area of 768 square steps. Combined, the total area of the circular field is 2,368 square steps. This is exactly 16 steps more than expected. Why is it more? 〇 This is because the side was first multiplied by itself resulting in 1,600 steps; for every hundred steps, there is one extra step, which accounts for the 16 extra steps The author is noting a discrepancy between different methods of calculation, likely comparing the "sum of parts" method to a direct circular area formula using Pi ≈ 3.

Examination of the Sagitta and Comparison with the Circle Diagram

Steps 〇 Further examination shows that an "arc and sagitta" field occupies three-quarters of a rectangular field.

Suppose there is an "arc and sagitta" field where the chord length is 40 steps and the sagitta width shi: the height of the segment, perpendicular to the chord is 8 steps. What is the diameter of the circle?

Answer: It is now corrected to a diameter of 56 steps. This question is posed to distinguish between the "void and solid" original: 虛實; refers to estimated vs. actual or theoretical values numbers of various large and small segments.

Method: Take the chord length and halve it to get 20. Multiply this by itself to get 400. Divide this by the sagitta of 8 steps to get 50. Add the sagitta of 8 steps to get 58 steps in total. However, this is 2 steps more than the diameter in the previous diagram.

Now, by changing these numbers, we are looking at a precise semi-circle. Because the chord is long and the sagitta is short, the calculated values are inconsistent and cannot be used as a standard.