Systematic Treatise on Arithmetic, Vol. 8

Taking the square root of this value yields 9, which is the difference between the base and the height. ○ Previously, we doubled the square of the hypotenuse; now, subtract the square of the base-height difference (9 squared equals 81) from that amount. This leaves a remainder of 3,600 The original text contains a likely clerical error, written as 3,900, but the subsequent steps confirm 3,600 was intended.. Taking the square root of this results in 60, which is the sum of the base and the height.

Calculations for the Base and Hypotenuse

○ Combining the base and hypotenuse results in a sum of 80. If we take the square of the height (which is 1,200) In this example, the square of the height (股) is given as 1,200, meaning the height itself is approximately 34.64. and divide it by this sum, we obtain 15. This value is the base-hypotenuse difference.
○ Conversely, if we take that base-hypotenuse difference of 15 and use it to divide the square of the height (1,200), we obtain 80, which is the sum of the base and the hypotenuse.

Calculations for the Height and Hypotenuse

Adding the height and hypotenuse together results in a sum of 80. By dividing the square of the base (which is 720) Here the square of the base (勾) is 720, meaning the base is approximately 26.83. by this sum of 80, we obtain 9. This is the height-hypotenuse difference.
○ That is to say, if we take the height-hypotenuse difference of 9 and use it to divide the square of the base (720), we obtain 80, which is the sum of the height and the hypotenuse.

Advanced Relationships of the Sides

○ Squaring the sum of the base and height (60) results in 3,600. From this, subtract the square of the hypotenuse (which is 1,700) Using the Pythagorean theorem (720 + 1200), the square of the hypotenuse here is actually 1,920. The text's 1,700 likely represents a different specific problem or a rounding common in woodblock manuals.; the remaining 1,900 serves as the dividend [實, shi]. Dividing this by the difference of the hypotenuse differences (30) yields 63, which is the sum of the hypotenuse differences.
○ Dividing the aforementioned dividend by the sum of the hypotenuse differences will conversely yield the difference of the hypotenuse differences.
○ Squaring the difference of the base and height (9) results in 81. Subtracting this from the square of the hypotenuse results in 1,900, which serves as the dividend for further operations...