Editing History of mathematics (section)

== Chinese ==
{{Main|Chinese mathematics}}
{{further|Book on Numbers and Computation}}
{{see also|History of science#Chinese mathematics}}
[[File:Qinghuajian, Suan Biao.jpg|thumb|right|upright|The [[Tsinghua Bamboo Slips]], containing the world's earliest [[decimal]] multiplication table, dated 305 BC during the [[Warring States]] period]]
An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.<ref>{{Harv|Boyer|1991|loc="China and India" p. 201}}</ref> The oldest extant mathematical text from China is the ''[[Zhoubi Suanjing]]'' (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the [[Warring States Period]] appears reasonable.<ref name="Boyer 1991 loc=China and India p. 196">{{Harv|Boyer|1991|loc="China and India" p. 196}}</ref> However, the [[Tsinghua Bamboo Slips]], containing the earliest known [[decimal]] [[multiplication table]] (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.<ref name="Nature">{{cite journal | url =http://www.nature.com/news/ancient-times-table-hidden-in-chinese-bamboo-strips-1.14482| title =Ancient times table hidden in Chinese bamboo strips | journal =Nature |first=Jane|last=Qiu|author-link=Jane Qiu|date=7 January 2014| access-date =15 September 2014| doi =10.1038/nature.2014.14482 | s2cid =130132289 | doi-access =free}}</ref>

[[File:Chounumerals.svg|thumb|left|[[Counting rod numerals]]]]
Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.<ref>{{Harvnb|Katz|2007|pp=194–99}}</ref> Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.<ref>{{Harv|Boyer|1991|loc="China and India" p. 198}}</ref> [[Counting rods|Rod numerals]] allowed the representation of numbers as large as desired and allowed calculations to be carried out on the ''[[suanpan|suan pan]]'', or Chinese abacus. The date of the invention of the ''suan pan'' is not certain, but the earliest written mention dates from AD 190, in [[Xu Yue (mathematician)|Xu Yue]]'s ''Supplementary Notes on the Art of Figures''.

The oldest extant work on geometry in China comes from the philosophical [[Mohism|Mohist]] canon {{circa|330 BC}}, compiled by the followers of [[Mozi]] (470–390 BC). The ''Mo Jing'' described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.<ref>{{Harv|Needham|Wang|1995|pp=91–92}}</ref> It also defined the concepts of [[circumference]], [[diameter]], [[radius]], and [[volume]].<ref>{{Harv|Needham|Wang|1995|p=94}}</ref>

[[File:九章算術.gif|thumb|upright|right|''[[The Nine Chapters on the Mathematical Art]]'', one of the earliest surviving mathematical texts from [[China]] (2nd century AD).]]
In 212 BC, the Emperor [[Qin Shi Huang]] commanded all books in the [[Qin Empire]] other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the [[Burning of books and burying of scholars|book burning]] of 212 BC, the [[Han dynasty]] (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is ''[[The Nine Chapters on the Mathematical Art]]'', the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for [[Chinese pagoda]] towers, engineering, [[surveying]], and includes material on [[right triangle]]s.<ref name="Boyer 1991 loc=China and India p. 196"/> It created mathematical proof for the [[Pythagorean theorem]],<ref>{{Harv|Needham|Wang|1995|p=22}}</ref> and a mathematical formula for [[Gaussian elimination]].<ref>{{Harv|Straffin|1998|p=164}}</ref> The treatise also provides values of [[Pi|π]],<ref name="Boyer 1991 loc=China and India p. 196"/> which Chinese mathematicians originally approximated as 3 until [[Liu Xin (scholar)|Liu Xin]] (d. 23 AD) provided a figure of 3.1457 and subsequently [[Zhang Heng]] (78–139) approximated pi as 3.1724,<ref>{{Harv|Needham|Wang|1995|pp=99–100}}</ref> as well as 3.162 by taking the [[square root]] of 10.<ref>{{Harv|Berggren|Borwein|Borwein|2004|p=27}}</ref><ref>{{Harv|de Crespigny|2007|p=1050}}</ref> [[Liu Hui]] commented on the ''Nine Chapters'' in the 3rd century AD and [[Liu Hui's π algorithm|gave a value of π]] accurate to 5 decimal places (i.e. 3.14159).<ref name="Boyer 1991 loc=China and India p. 202">{{Harv|Boyer|1991|loc="China and India" p. 202}}</ref><ref>{{Harv|Needham|Wang|1995|pp=100–01}}</ref> Though more of a matter of computational stamina than theoretical insight, in the 5th century AD [[Zu Chongzhi]] computed [[Milü|the value of π]] to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.<ref name="Boyer 1991 loc=China and India p. 202"/><ref>{{Harv|Berggren|Borwein|Borwein|2004|pp=20, 24–26}}</ref> He also established a method which would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]].<ref>{{cite book
|title=Calculus: Early Transcendentals
|edition=3
|first1=Dennis G.
|last1=Zill
|first2=Scott
|last2=Wright
|first3=Warren S.
|last3=Wright
|publisher=Jones & Bartlett Learning
|year=2009
|isbn=978-0-7637-5995-7
|page=xxvii
|url=https://books.google.com/books?id=R3Hk4Uhb1Z0C}} [https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of p. 27]
</ref>

The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the [[Song dynasty]] (960–1279), with the development of Chinese algebra. The most important text from that period is the ''[[Jade Mirror of the Four Unknowns|Precious Mirror of the Four Elements]]'' by [[Zhu Shijie]] (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to [[Horner's method]].<ref name="Boyer 1991 loc=China and India p. 202"/> The ''Precious Mirror'' also contains a diagram of [[Pascal's triangle]] with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.<ref name="Boyer 1991 loc=China and India p. 205">{{Harv|Boyer|1991|loc="China and India" p. 205}}</ref> The Chinese also made use of the complex combinatorial diagram known as the [[magic square]] and [[Magic circle (mathematics)|magic circles]], described in ancient times and perfected by [[Yang Hui]] (AD 1238–1298).<ref name="Boyer 1991 loc=China and India p. 205" />

Even after European mathematics began to flourish during the [[Renaissance]], European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. [[Jesuit]] missionaries such as [[Matteo Ricci]] carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.<ref name="Boyer 1991 loc=China and India p. 205"/>

[[Japanese mathematics]], [[Korean numerals|Korean mathematics]], and [[Vietnamese numerals|Vietnamese mathematics]] are traditionally viewed as stemming from Chinese mathematics and belonging to the [[Confucian]]-based [[East Asian cultural sphere]].<ref>{{Harv|Volkov|2009|pp=153–56}}</ref> Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's [[Ming dynasty]] (1368–1644).<ref>{{Harv|Volkov|2009|pp=154–55}}</ref> For instance, although Vietnamese mathematical treatises were written in either [[Chinese characters|Chinese]] or the native Vietnamese [[Chữ Nôm]] script, all of them followed the Chinese format of presenting a collection of problems with [[algorithm]]s for solving them, followed by numerical answers.<ref>{{Harv|Volkov|2009|pp=156–57}}</ref> Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of [[History of astronomy|mathematicians and astronomers]], whereas in Japan it was more prevalent in the realm of [[private school]]s.<ref>{{Harv|Volkov|2009|p=155}}</ref>