Systematic Treatise on Arithmetic, Vol. 8

Now there is a base of 27 feet and a height of 36 feet; ask what the length of the diagonal hypotenuse is?

The method says: Set down the base 勾 (Gou): the shorter leg of a right triangle of 27 feet. Multiply it by itself original: 自乘 (zi cheng); to square a number to get 729 feet. Separately, take the height 股 (Gu): the longer leg of a right triangle of 36 feet and multiply it by itself to get 1,296 feet. Combine these two numbers to get 2,025 feet. Now these two combined form the square of the hypotenuse. Use this sum as the dividend 實 (Shi): the total area from which the side length or root is extracted, and resolve it using the method for extracting the square root 開平方 (Kai ping fang).

First, determine the first quotient 初商 (Chu shang): the first digit of the root, which is 40. Place this on the left. Also place 40 on the right as the fixed divisor 方法 (Fang fa): the divisor used for the first stage of extraction. Aligning the 4 on the left with the 4 on the right, we say "four times four" and subtract 1,600 from the dividend. The remaining dividend is 425. Next, take the first quotient on the left and multiply it by ten to make 80, which becomes the side-term divisor 廉法 (Lian fa): the portion of the divisor representing the rectangular areas in a geometric square extraction.

The second quotient 次商 (Ci shang) is 5. Place it on the left after the first quotient of 40. Also place 5 on the right, added to the side-term divisor of 80; this becomes the corner-term divisor 隅法 (Yu fa): the final unit added to the divisor to complete the square area. Align the 5 on the left with the 8 on the right; say "five times eight" and subtract 400 from the dividend. Then align the 5 on the left with the 5 on the right; say "five times five" and subtract the remaining 25 from the dividend. The dividend is now completely exhausted. Thus, we find the diagonal hypotenuse 弦 (Xian) is 45 feet.

Now there is a base of 27 feet and a hypotenuse of 45 feet; ask what the length of the height is?