Editing History of geometry (section)

==Chinese geometry==
{{see also|Chinese mathematics}}
[[Image:九章算術.gif|thumb|right|220px|The ''[[Nine Chapters on the Mathematical Art]]'', first compiled in 179 AD, with added commentary in the 3rd century by [[Liu Hui]]]]
[[Image:Sea island survey.jpg|thumb|right|220px|''[[Haidao Suanjing]]'', Liu Hui, 3rd century]]

The first definitive work (or at least oldest existent) on geometry in China was the ''[[Mo Jing]]'', the [[Mohist]] canon of the early philosopher [[Mozi]] (470–390 BC). It was compiled years after his death by his followers around the year 330 BC.<ref name="needham volume 3 91"/> Although the ''Mo Jing'' is the oldest existent book on geometry in China, there is the possibility that even older written material existed. However, due to the infamous [[Burning of books and burying of scholars|Burning of the Books]] in a political maneuver by the [[Qin dynasty]] ruler [[Qin Shihuang]] (r. 221–210 BC), multitudes of written literature created before his time were purged. In addition, the ''Mo Jing'' presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.

The ''Mo Jing'' described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point.<ref name="needham volume 3 91">Needham, Volume 3, 91.</ref> Much like [[Euclid]]'s first and third definitions and [[Plato]]'s 'beginning of a line', the ''Mo Jing'' stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it."<ref name="needham volume 3 92">Needham, Volume 3, 92.</ref> Similar to the [[atomist]]s of [[Democritus]], the ''Mo Jing'' stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved.<ref name="needham volume 3 92"/> It stated that two lines of equal length will always finish at the same place,<ref name="needham volume 3 92"/> while providing definitions for the ''comparison of lengths'' and for ''parallels'',<ref name="needham volume 3 92 93">Needham, Volume 3, 92-93.</ref> along with principles of space and bounded space.<ref name="needham volume 3 93">Needham, Volume 3, 93.</ref> It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.<ref name="needham volume 3 93 94">Needham, Volume 3, 93-94.</ref> The book provided definitions for circumference, diameter, and radius, along with the definition of volume.<ref name="needham volume 3 94">Needham, Volume 3, 94.</ref>

The [[Han dynasty]] (202 BC&nbsp;– 220 AD) period of China witnessed a new flourishing of mathematics. One of the oldest Chinese mathematical texts to present [[geometric progression]]s was the ''[[Suàn shù shū]]'' of 186 BC, during the Western Han era. The mathematician, inventor, and astronomer [[Zhang Heng]] (78–139 AD) used geometrical formulas to solve mathematical problems. Although rough estimates for [[pi]] ([[Pi|π]]) were given in the ''[[Zhou Li]]'' (compiled in the 2nd century BC),<ref name="needham volume 3 99">Needham, Volume 3, 99.</ref> it was Zhang Heng who was the first to make a concerted effort at creating a more accurate formula for pi. Zhang Heng approximated pi as 730/232 (or approx 3.1466), although he used another formula of pi in finding a spherical volume, using the square root of 10 (or approx 3.162) instead. [[Zu Chongzhi]] (429–500 AD) improved the accuracy of the approximation of pi to between 3.1415926 and 3.1415927, with [[Milü|<sup>355</sup>⁄<sub>113</sub>]] (密率, Milü, detailed approximation) and [[Proof that 22/7 exceeds π|<sup>22</sup>⁄<sub>7</sub>]] (约率, Yuelü, rough approximation) being the other notable approximation.<ref name="needham volume 3 101">Needham, Volume 3, 101.</ref> In comparison to later works, the formula for pi given by the French mathematician [[Franciscus Vieta]] (1540–1603) fell halfway between Zu's approximations.

===''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>
{{col-begin}}
{{col-4}}
*Square
*Rectangle
*Circle
*[[Isosceles triangle]]
{{col-4}}
*[[Rhomboid]]
*[[Trapezoid]]
*Double trapezium
*Segment of a circle
*Annulus ('ring' between two concentric circles)
{{col-4}}
{{col-4}}
{{col-end}}

'''Volumes for the'''<ref name="needham volume 3 98 99" />
{{col-begin}}
{{col-4}}
*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
{{col-4}}
*Cube
*[[Prism (geometry)|Prism]]
*Wedge with rectangular base and both sides sloping
*Wedge with trapezoid base and both sides sloping
*[[Tetrahedral]] wedge
{{col-4}}
*Frustum of a wedge of the second type (used for applications in engineering)
*Cylinder
*Cone with circular base
*Frustum of a cone
*Sphere
{{col-4}}
{{col-end}}
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.