⚠The image is a high-contrast woodblock print; while some strokes are slightly broken due to ink quality, the characters remain legible.

Systematic Compendium of Mathematical Methods, Volume 10, Part 2

The source image is a high-contrast woodblock print; while some strokes are slightly broken due to ink quality, the characters remain legible.

Definitions of the Right-Angle Triangle: Thirteen Derivative Terms

In traditional Chinese mathematics, the right-angled triangle is understood through the relationship of its three sides: the "base" (gou), the "altitude" (gu), and the "hypotenuse" (xian).


Base Horizontal is called base	Base-Hypotenuse Difference Subtracting base from hypotenuse	Base-Hypotenuse Sum Adding base and hypotenuse	Altitude Vertical is called altitude	Altitude-Hypotenuse Sum Adding altitude and hypotenuse	Hypotenuse Diagonal is called hypotenuse	Hypotenuse-Sum Sum Adding hypotenuse to the sum of base and altitude	Hypotenuse-Difference Difference Subtracting the difference of base and altitude from the hypotenuse
Base-Altitude Difference Subtracting base and altitude	Base-Altitude Sum Adding base and altitude		Altitude-Hypotenuse Difference Subtracting altitude from hypotenuse		Hypotenuse-Difference Sum Adding hypotenuse to the difference of base and altitude	Hypotenuse-Sum Difference Subtracting the sum of base and altitude from the hypotenuse

Explanatory Definitions and Discourse on the Right-Angle Triangle

original: Gougu Lunshuo Shiyi (勾股論說釋義)

Volume 12 of the Newly Compiled Direct Guide to the Systematic Compendium of Mathematical Methods

Note that while the header identifies this as Volume 10, the internal title lists it as Volume 12. Such discrepancies are common in different woodblock editions of this 16th-century bestseller.

Compiled by Cheng Dawei Styled Rusi, from Binqu, Xi’an Revised by his great-grandson Guang Shen (styled Suting and Peiwei); Reviewed by You Hong (styled Mei’an and Wufu)

Chapter 9: Base and Altitude

The term "Gougu" (勾股) refers to the two legs of a right-angled triangle. Gou represents the shorter leg (base) and Gu represents the longer leg (altitude).

The horizontal width is called the base (gou), the vertical length is called the altitude (gu), and the diagonal connecting the two corners is called the hypotenuse (xian). This chapter uses the base and altitude to find the diagonal hypotenuse; the base and hypotenuse to find the length of the altitude; and the altitude and hypotenuse to find the width of the base. Finding inscribed squares or circles within these triangles, determining the height of mountains, the depth of valleys, the width of cities, or the distance of roads—all can be known through this method.

### Diagram of the Base and Altitude

The diagonal is the hypotenuse
The long side is the altitude
The short side is the base

The shape of the right-angle triangle is exactly like that of a carpenter’s square original: quchi (曲尺). The base is the short arm original: zhi (咫), an ancient measurement of about 8 inches, and the altitude is the long arm original: chi (尺), a standard foot. The diagonal extending from the tip of the short arm to the end of the long arm is the hypotenuse.

For example, if the base is 3 feet and the altitude is 4 feet, t...