Zhang Heng
Mathematics

For centuries the Chinese approximated pi as 3; Liu Xin (d. CE 23) made the first known Chinese attempt at a more accurate calculation of 3.154, but there is no record detailing the method he used to obtain this figure. In his work around 130, Zhang Heng compared the celestial circle to the diameter of the earth, proportioning the former as 736 and the latter as 232, thus calculating pi as 3.1724. In Zhang's day, the ratio 4:3 was given for the area of a square to the area of its inscribed circle and the volume of a cube and volume of the inscribed sphere should also be 4:3. In formula, with D as diameter and V as volume, D:V = 16:9 or V= 9 16 {\displaystyle {\tfrac {9}{16}}} D; Zhang realized that the value for diameter in this formula was inaccurate, noting the discrepancy as the value taken for the ratio. Zhang then attempted to remedy this by amending the formula with an additional 1 16 {\displaystyle {\tfrac {1}{16}}} D, hence V= 9 16 {\displaystyle {\tfrac {9}{16}}} D + 1 16 {\displaystyle {\tfrac {1}{16}}} D = 5 8 {\displaystyle {\tfrac {5}{8}}} D. With the ratio of the volume of the cube to the inscribed sphere at 8:5, the implied ratio of the area of the square to the circle is √ 8:√ 5. From this formula, Zhang calculated pi as the square root of 10 (or approximately 3.162). In the 3rd century, Liu Hui made the calculation more accurate with his π algorithm, which allowed him to obtain the value 3.14159. Later, Zu Chongzhi (429–500) approximated pi as 355 113 {\displaystyle {\tfrac {355}{113}}} or 3.141592, the most accurate calculation for pi the ancient Chinese would achieve.
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