General Source of Computational Methods: Volume 13

Page 4

Extracting square roots reveals the hypotenuse;	These methods have been established since antiquity.
○ To find the height from the base and hypotenuse,	Square each side respectively.
Subtract the smaller from the larger; the remainder is the dividend;	To find the height, extract the square root of this dividend.
Finding the base from the hypotenuse and height follows this same rule;	Mathematicians must memorize this well and never forget.

Method for Finding the Hypotenuse from the Base and Height
Rule: Set down the base 勾 (Gou): the shorter leg of a right triangle multiplied by itself original: 自乘 (zi cheng); to square a number and the height 股 (Gu): the longer leg of a right triangle multiplied by itself. Combine these two numbers and use the method of extracting the square root to resolve it; this yields the value of the hypotenuse 弦 (Xian): the diagonal or hypotenuse. The base squared and the height squared combined equal the hypotenuse squared; therefore, using the square root method yields the length of the diagonal hypotenuse.

Method for Finding the Height from the Base and Hypotenuse
Rule: Set down the hypotenuse squared. From this, subtract the base squared. Use the method of extracting the square root on the remainder; this yields the length of the height. Within the square of the hypotenuse, there is both a base-squared and a height-squared value. By subtracting the base squared, the remainder is the height squared; therefore, extracting the square root yields the length of the height.

Method for Finding the Base from the Height and Hypotenuse
Rule: Set down the hypotenuse squared. From this, subtract the height squared. Use the method of extracting the square root on the remainder; this yields the width of the base 濶 (Kuo): width or breadth. The square of the hypotenuse contains the squares of both the base and the height. By subtracting the height squared, the remainder is the base squared; therefore, extracting the square root yields the width of the base.