General Source of Computational Methods, Volume 12

The method says: Set the hypotenuse 弦 (Xian): the diagonal side of a right triangle 45 feet and multiply it by itself To "multiply by itself" (zi cheng) is to square the number. to get 2,025. Within this sum, there is one base squared and one height squared. Separately, take the base 勾 (Gou): the shorter horizontal leg of a right triangle and multiply it by itself to get 729 feet. Subtract these two numbers; the remainder is 1,296 feet, which serves as the dividend 實 (Shi): the area or number from which a root is extracted. This is the square of the height.

Resolve this using the method for extracting the square root 開平方法 (Kai ping fang fa). The first quotient 初商 (Chu shang): the first digit of the root is 30; place this on the left side. Also place 30 on the right side as the fixed divisor 方法 (Fang fa): the initial divisor used in the extraction process.

Align the 3 on the left with the 3 on the right and call out "three times three"; subtract 900 from the dividend. The remaining dividend is 396 feet. Separately, take the lower first quotient of 30 and double it to make 60; this is the side-term divisor 廉法 (Lian fa): literally "edge method," representing the rectangular areas added when expanding a square.

The second quotient 次商 (Ci shang) is 6 feet; place it on the left following the 30. Also place 60 on the right, and next to the side-term divisor 60, place the corner-term divisor 隅法 (Yu fa): the final small square unit needed to complete the larger square. Align the 6 on the left with the 6 on the right and call out "six times sixty"; subtract 360 from the dividend.

Then, align the 6 on the left with the 6 on the right again and call out "six times six"; subtract 36 feet from the dividend. The dividend is now exhausted exactly. The resulting length of the height is 36 feet, which fits the question.

Now there is a height of 36 feet and a hypotenuse of 45 feet; ask what the width of the base is?

The page concludes with a list of the mathematical terms used in these operations, which are essential for understanding the traditional Chinese algebraic system.

Vocabulary: Base, Height, Hypotenuse, General Source of Computational Methods, Square Root Extraction Method, Side-term Divisor, Corner-term Divisor, Dividend, Answer, Fixed Divisor.