Newly Compiled Direct Guide to the Systematic Treatise on Arithmetic, Volume Three

Xin’an refers to modern-day Huangshan, Anhui province. Cheng Dawei was a successful merchant whose practical experience informed this work.

Corrected by great-great-grandson Cheng Guangshen (styled Peizhang, nicknamed Suting)
Consulted by Yunzhai, Fang, Hong, and Sheng
The inclusion of family members in the credits highlights the hereditary nature of mathematical knowledge and the prestige of the Cheng family lineage in the Qing dynasty editions.

Chapter One: Square Fields (Fangtian)

original: 方田 (fāngtián). This is the traditional first chapter of classical Chinese mathematical texts, focusing on calculating the areas of various shapes of land.

This chapter uses the shapes and boundaries of fields to determine the total area in units of mu and bu 1 mu (畝) was approximately 614 square meters during the Ming, composed of 240 square bu (步).. By using the breadth and length, one can find the area of shapes such as:
square rectangular isosceles triangular diamond-shaped trapezoidal and slanted.
Using the circumference and diameter, one can calculate the area of circular fields, "bowl-shaped" concave or semi-spherical depressions fields, and ring-shaped fields.

〇 Note: The shapes of fields are extremely numerous, and it is difficult to record them all in detail. Those who study this art must not be rigid or stubborn; rather, they must be adaptable when on-site. One must "cut from the surplus to fill the void" original: 截盈補虛 (jié yíng bǔ xū). A fundamental principle in Chinese geometry where complex shapes are mentally or physically rearranged into simpler rectangles to calculate area., making pointed or irregular sections fit into larger, standard shapes to conform to the mathematical formulas.

However, for the pointed parts of a field, one should first extract standard shapes such as square rectangular base altitude original: 勾股 (gōugǔ), referring to the legs of a right-angled triangle. triangular or diamond shapes. Calculate the area of any remaining side pieces separately, combine them into a single total, and then apply the methods of multiplication and division. Using the methods of Extracting the Square Root original: 開平 (kāipíng) found in the "Deduction of Breadth" original: 少廣 (shǎoguǎng), the name of the fourth chapter of the Nine Chapters on the Mathematical Art. chapter to return the figures to their original state is the most precise of techniques.