Nine Chapters on Mathematical Art, Vol. 1

Whether the compass is large or small, the realization lies in the relationship between the arc and the chord. However, what has been passed down through generations regarding this method...

...no one has been willing to examine it precisely. Scholars follow the ancients and practice their errors, and because there are no clear proofs...

...to distinguish them, this is difficult. Generally, all objects are either round or square in form. The ratio of square and round...

...is truly evident in the near, so even if it is distant, it can be known. Speaking from this, its...

...application is broad. I respectfully note that I have verified the circle and recreated the dense ratio. I fear the method was established in vain, and the numbers...

...are obscure and difficult to explain, so I have set them out for inspection. I have carefully detailed the notes. To cut...

...six arcs to make twelve arcs. The procedure says: Take a circle with a diameter of 2 chi. Halve it...

...to be 1 chi, which is the side of the six arcs inside the circle. Let the radius of 1 chi be the chord, and 5 cun inches be the gou shorter leg of a right triangle. Seek the gu longer leg of a right triangle for it. Using the gou power of...

...25 cun, subtract it from the chord power to find the remainder of 75 cun. Extract the square root, down to...

...miao ten-thousandths and hu hundred-thousandths. Also, use the remainder as a divisor to find the decimal fractions. The decimal fractions have no name; one knows to take...

...the remainder as the numerator and the divisor as the denominator, reducing the denominator to five parts of a hu and two. Thus...

...the two results in a gu of 8 cun, 6 fen, 6 li, 3 hao, 5 si, 9 hu, 2...

...Subtract this from the radius, leaving 1 cun, 3 fen, 3 li, 9 hao, 7 si, 4...

...hu, 3/5 of a hu. Call this the small gou. The small gou...

...is the 5 cun gou, the half-side of the arc, and it is also called the small gu. Seek the chord for it. Its power is 267,949,193,445 hu. Add the full fractions together.