Editing Zhang Heng (section)

===Mathematics===
{{see also|Chronology of computation of π|Chinese mathematics|History of geometry#Chinese geometry|label 4=Chinese geometry}}

For centuries the Chinese approximated [[pi]] as 3; [[Liu Xin (scholar)|Liu Xin]] (d. CE&nbsp;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.<ref name="needham 1986 volume 3 99 100">Needham (1986), Volume 3, 99–100.</ref><ref name="arndt haenel lischka 2001 176"/> In his work around 130,<ref name="needham 1986 volume 3 100">Needham (1986), Volume 3, 100.</ref> 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.<ref name="berggren borwein 2004 27">Berggren, Borwein & Borwein (2004), 27.</ref> 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<sup>2</sup>:3<sup>2</sup>.<ref name="berggren borwein 2004 27"/> In formula, with D as diameter and V as volume, D<sup>3</sup>:V = 16:9 or V=<math>\tfrac{9}{16}</math>D<sup>3</sup>; Zhang realized that the value for diameter in this formula was inaccurate, noting the discrepancy as the value taken for the ratio.<ref name="arndt haenel lischka 2001 176">Arndt and Haenel (2001), 176.</ref><ref name="berggren borwein 2004 27"/> Zhang then attempted to remedy this by amending the formula with an additional <math>\tfrac{1}{16}</math>D<sup>3</sup>, hence V=<math>\tfrac{9}{16}</math>D<sup>3</sup> + <math>\tfrac{1}{16}</math>D<sup>3</sup> = <math>\tfrac{5}{8}</math>D<sup>3</sup>.<ref name="berggren borwein 2004 27"/> 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 √<span style = "text-decoration:overline">8</span>:√<span style = "text-decoration:overline">5</span>.<ref name="berggren borwein 2004 27"/><ref name="arndt haenel lischka 2001 177">Arndt and Haenel (2001), 177.</ref> From this formula, Zhang calculated pi as the [[square root]] of 10 (or approximately 3.162).<ref name="yan 2007 128">Yan (2007), 128.</ref><ref name="crespigny 2007 1050"/><ref name="berggren borwein 2004 27"/><ref name="arndt haenel lischka 2001 177"/><ref name="wilson 2001 16">Wilson (2001), 16.</ref> In the 3rd&nbsp;century, [[Liu Hui]] made the calculation more accurate with his [[Liu Hui's π algorithm|π algorithm]], which allowed him to obtain the value 3.14159.<ref name="needham 1986 volume 3 100 101">Needham (1986), Volume 3, 100–101.</ref> Later, [[Zu Chongzhi]] (429–500) [[Milü|approximated pi]] as <math>\tfrac{355}{113}</math> or 3.141592, the most accurate calculation for pi the ancient Chinese would achieve.<ref>Berggren, Borwein & Borwein (2004), 20 & 24–26.</ref>