Volume Three

Warning: The right edge of the first column is partially cut off in the original manuscript.

Why is it that we now subtract two steps? This is because half of the chord length xian: the straight line segment joining the ends of an arc is 20 steps. Within these twenty steps, there is one extra step for every ten; therefore, we subtract two steps. 〇 Alternatively, if the chord length is 40 and the sagitta shi: the "arrow," or the height of the segment from the center of the chord to the arc is 20, how do we find the diameter of the circle? Take the chord of 40 and halve it to get 20. Multiply this by itself to get 400. Divide this result by the sagitta of 20 to get 20. Add the sagitta of 20 to this, and you obtain the diameter The formula used here is $Diameter = \frac{(Chord/2)^2}{Sagitta} + Sagitta$. 〇 This is the method for calculating a flat semi-circular field; it is effective and will result in no further errors.

The "Gui" Shape

Gui: A ritual jade tablet with a pointed top. In Chinese mathematics, this term is used to describe an isosceles triangle.

This is half of a "shuttle" shape original: 梭 suo; a rhombus or diamond-like shape

Suppose there is a Gui-shaped field with a center length of 60 steps and a base width of 32 steps. What is its area?
Answer: 960 square steps.
Method: Take the length of 60 steps and multiply it by the width of 32 steps to get 1,920 steps. Halve this to obtain the area ji: the product or accumulated area of 960 steps. This matches the inquiry. The Gui shape is exactly half of a rectangular field zhitian: literally "straight field", which is why the method of halving the product is used. A shuttle shape is simply two Gui shapes joined together.

Suppose there is a triangular field where each side is 14 steps. What is the resulting area?