General Source of Computational Methods, Volume 3

The Ox Horn Shape

Ox Horn (niú jiǎo): A shape resembling a curved segment of a ring or a tapered crescent.

Borrow the inner curve to create a diagonal; combine it with the outer arc to form the internal area.

Method: Set down the middle length of 17.5 steps. Take the width of 8 steps and halve it to get 4 steps. Multiply these together to find the area original: "積" (jī) — the total area or accumulation. This matches the query.

Alternative Method: One may also measure the inner and outer curves and add them together. Halve that sum and multiply by the radius to obtain the area as well.

The Olive Shape

Olive shape (lǎn xíng): Geometrically known as a mandorla or vesica piscis, formed by the intersection of two circular arcs.

Suppose there is an olive shape with a middle length of 40 steps and a width of 16 steps. Ask: what is the area?

Method: Set down the length of 40 steps to serve as the chord chord (xián): the straight line connecting the ends of an arc. Take half the width, which is 8 steps, to serve as the sagitta sagitta (shǐ): literally the "arrow," the perpendicular distance from the center of the chord to the arc.
Add these to get 48 steps, then halve it to get 24 steps. Multiply this by the sagitta of 8 steps to get 192 steps. This is the area of a single arc-segment; double it to find the area of the olive shape. This matches the query.

Suppose there is a "three-width field" where the southern width is 26 steps, the northern width is 54 steps, and the middle width is 18... The text cuts off here and likely continues on the next page with the calculation for an irregular trapezoidal or curved field with three measurements.