Systematic Treatise on Arithmetic, Vol. 3

The nine divisions are all marked with the character fen fen: a traditional unit of length, approximately 1/10th of an inch or 3.2 mm. Every five feet constitutes one "step" bu: a traditional unit of distance based on a double-pace; these are marked in sequence up to thirty steps, or may stop just below forty steps. Apply clear oil to the bamboo tape original: 篾 (mie). Thin, flexible strips of bamboo used as a measuring line.; this ensures that even if it becomes soiled with mud, it can be easily washed clean.

○ There is also a later design that uses only a cross-frame with a groove carved inside. In this version, the "thunder-head" likely referring to the central pivot or axle does not pass through the center; instead, a circular wooden disc is used to turn the tape. Although this prevents the tape from coming loose, the act of turning causes the tape to rub and scrape against itself, leading to very rapid wear and tear. Overall, this is inferior to the previous "cart-style" design. In that design, the tape rests upon the cross-frame and the frame itself rotates; the tape remains stationary relative to the frame, making it much more durable.

The "Fixed Rules for Squares and Circles" are summarized in the first nine diagrams here; the remaining applications of the Right-Triangle Theorem 勾股 (gougu): the Chinese equivalent of the Pythagorean theorem, literally "base and height" are detailed later.

---

Circumference Three, Diameter One

When the circumference is three and the diameter is one, the ratio is insufficient. To find the diameter from the circumference, divide by three original: 三歸 (sangui). An ancient division method using a counting board.. When the diameter is one, the circumference is three plus a small remainder. To find the circumference from the diameter, multiply by three.

---

Square Five, Diagonal Seven

When the side of a square is five, the diagonal is seven plus a small remainder. To find the diagonal, multiply the side by seven and divide by five. When the diagonal is seven, the side of the square is five but slightly insufficient. To find the side, multiply the diagonal by five and divide by seven.

---

Hexagon Six, Side Seven

To find the chord original: 弦 (xian). Here referring to the horizontal base or altitude of the geometric figure shown. from the diameter, divide by six and multiply by seven. To find the diameter from the chord, divide by seven and multiply by six.