Editing History of geometry (section)

===''The Nine Chapters on the Mathematical Art''===
''[[The Nine Chapters on the Mathematical Art]]'', the title of which first appeared by 179&nbsp;AD on a bronze inscription, was edited and commented on by the 3rd century mathematician [[Liu Hui]] from the Kingdom of [[Cao Wei]]. This book included many problems where geometry was applied, such as finding surface areas for squares and circles, the volumes of solids in various three-dimensional shapes, and included the use of the [[Pythagorean theorem]]. The book provided illustrated proof for the Pythagorean theorem,<ref name="needham volume 3 22">Needham, Volume 3, 22.</ref> contained a written dialogue between of the earlier [[Duke of Zhou]] and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical [[gnomon]], the circle and square, as well as measurements of heights and distances.<ref name="needham volume 3 21">Needham, Volume 3, 21.</ref> The editor Liu Hui listed pi as 3.141014 by using a 192 sided [[polygon]], and then calculated pi as 3.14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporary [[Wang Fan]], a mathematician and astronomer from [[Eastern Wu]], would render pi as 3.1555 by using <sup>142</sup>⁄<sub>45</sub>.<ref name="needham volume 3 100">Needham, Volume 3, 100.</ref> Liu Hui also wrote of mathematical [[surveying]] to calculate distance measurements of depth, height, width, and surface area. In terms of solid geometry, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a [[tetrahedral]] wedge.<ref name="needham volume 3 98 99">Needham, Volume 3, 98&ndash;99.</ref> He also figured out that a wedge with [[trapezoid]] base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid.<ref name="needham volume 3 98 99"/> Furthermore, Liu Hui described [[Cavalieri's principle]] on volume, as well as [[Gaussian elimination]]. From the ''Nine Chapters'', it listed the following geometrical formulas that were known by the time of the Former Han dynasty (202 BCE&nbsp;&ndash; 9 CE).

'''Areas for the'''<ref name="needham volume 3 98">Needham, Volume 3, 98.</ref>
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*Square
*Rectangle
*Circle
*[[Isosceles triangle]]
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*[[Rhomboid]]
*[[Trapezoid]]
*Double trapezium
*Segment of a circle
*Annulus ('ring' between two concentric circles)
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'''Volumes for the'''<ref name="needham volume 3 98 99" />
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*Parallelepiped with two square surfaces
*Parallelepiped with no square surfaces
*Pyramid
*[[Frustum]] of pyramid with square base
*Frustum of pyramid with rectangular base of unequal sides
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*Cube
*[[Prism (geometry)|Prism]]
*Wedge with rectangular base and both sides sloping
*Wedge with trapezoid base and both sides sloping
*[[Tetrahedral]] wedge
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*Frustum of a wedge of the second type (used for applications in engineering)
*Cylinder
*Cone with circular base
*Frustum of a cone
*Sphere
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Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician [[Shen Kuo]] (1031–1095 CE), [[Yang Hui]] (1238–1298) who discovered [[Pascal's Triangle]], [[Xu Guangqi]] (1562–1633), and many others.