Editing Chinese mathematics (section)

==Song and Yuan dynasties==
[[Northern Song dynasty]] mathematician [[Jia Xian]] developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule.{{sfn|Martzloff|1987|p=142}}
[[File:Yanghui_triangle.gif|thumb|Yang Hui triangle ([[Pascal's triangle]]) using rod numerals, as depicted in a publication of [[Zhu Shijie]] in 1303&nbsp;AD]]
Four outstanding mathematicians arose during the [[Song dynasty]] and [[Yuan dynasty]], particularly in the twelfth and thirteenth centuries: [[Yang Hui]], [[Qin Jiushao]], [[Li Zhi (mathematician)|Li Zhi]] (Li Ye), and [[Zhu Shijie]]. Yang Hui, Qin Jiushao, Zhu Shijie all used the [[Horner scheme|Horner]]-[[Ruffini's rule|Ruffini]] method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "[[Pascal's Triangle]]", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on a form of algebraic geometry based on [[tiān yuán shù]]. His book; [[Ceyuan haijing]] revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians. Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with [[Zhu Shijie]]'s two books ''[[Suanxue qimeng]]'' and the ''[[Jade Mirror of the Four Unknowns]]''. In one case he reportedly gave a method equivalent to [[Carl Friedrich Gauss|Gauss]]'s pivotal condensation.

[[Qin Jiushao]] ({{circa|1202}}{{snd}}1261) was the first to introduce the [[0 (number)|zero symbol]] into Chinese mathematics."{{sfn|Needham|1959|p=43}} Before this innovation, blank spaces were used instead of zeros in the system of [[counting rods]].{{sfn|Needham|1959|pp=62–63}} One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?"{{sfn|Mikami|1913|p=77}} Qin also solved a 10th order equation.{{sfn|Libbrecht|1973|p=211}}

[[Pascal's triangle]] was first illustrated in China by Yang Hui in his book ''Xiangjie Jiuzhang Suanfa'' (詳解九章算法), although it was described earlier around 1100 by [[Jia Xian]].{{sfn|Needham|1959|pp=134–137}} Although the ''Introduction to Computational Studies'' (算學啓蒙) written by [[Zhu Shijie]] ([[Floruit|fl.]] 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of [[Japanese mathematics]].{{sfn|Needham|1959|p=46}}

===Algebra===
====''Ceyuan haijing''====
{{main|Ceyuan haijing}}
[[File:圆城图式.jpg|right|thumb|273x273px|Li Ye's inscribed circle in triangle:'''Diagram of a round town''']]
[[File:Yang_Hui_magic_circle.svg|thumb|[[Yang Hui]]'s magic concentric circles – numbers on each circle and diameter (ignoring the middle 9) sum to 138]]
''[[Ceyuan haijing]]'' ({{zh|t=測圓海鏡|p= Cèyuán Hǎijìng}}), or ''Sea-Mirror of the Circle Measurements'', is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by [[Li Zhi (mathematician)|Li Zhi]] (or Li Ye) (1192–1272 AD). He used [[Tian yuan shu]] to convert intricated geometry problems into pure algebra problems. He then used ''fan fa'', or [[Horner's method]], to solve equations of degree as high as six, although he did not describe his method of solving equations.{{sfn|Boyer|1991|loc="China and India"|p=204}} "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His ''Ts'e-yuan hai-ching'' (''Sea-Mirror of the Circle Measurements'') includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275).

====''Jade Mirror of the Four Unknowns''====
[[File:Sixianghuiyuan.jpg|right|thumb|upright=1.1|Facsimile of the ''Jade Mirror of Four Unknowns'']]
The ''[[Jade Mirror of the Four Unknowns]]'' was written by [[Zhu Shijie]] in 1303&nbsp;AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of ''fan fa'', today called Horner's method, to solve these equations.{{sfn|Boyer|1991|loc="China and India"|p=203}}

There are many summation series equations given without proof in the ''Mirror''. A few of the summation series are:{{sfn|Boyer|1991|loc="China and India"|p=205}}

<math>1^2 + 2^2 + 3^2 + \cdots + n^2 = {n(n + 1)(2n + 1)\over 3!}</math>
<math>1 + 8 + 30 + 80 + \cdots + {n^2(n + 1)(n + 2)\over 3!} = {n(n + 1)(n + 2)(n + 3)(4n + 1)\over 5!}</math>

====''Mathematical Treatise in Nine Sections''====
The ''[[Mathematical Treatise in Nine Sections]]'', was written by the wealthy governor and minister [[Ch'in Chiu-shao]] ({{circa|1202}}{{snd}}{{circa|1261}}) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis.{{sfn|Boyer|1991|loc="China and India"|p=204}}

====Magic squares and magic circles====
The earliest known [[magic square]]s of order greater than three are attributed to [[Yang Hui]] (fl. ca. 1261–1275), who worked with magic squares of order as high as ten.{{sfn|Boyer|1991|loc="China and India"|pp=204–205}} "The same "Horner" device was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten." He also worked with [[Magic circle (mathematics)|magic circle]].

===Trigonometry===
The embryonic state of [[trigonometry]] in China slowly began to change and advance during the Song dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendar science and astronomical calculations.{{sfn|Needham|1959|pp=108–109}} The [[polymath]] and official [[Shen Kuo]] (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.{{sfn|Needham|1959|pp=108–109}} Joseph W. Dauben notes that in Shen's "technique of intersecting circles" formula, he creates an approximation of the arc of a circle ''s'' by ''s'' = ''c'' + 2''v''<sup>2</sup>/''d'', where ''d'' is the [[diameter]], ''v'' is the [[versine]], ''c'' is the length of the chord ''c'' subtending the arc.{{sfn|Dauben|2007|p=308}} Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for [[spherical trigonometry]] developed in the 13th century by the mathematician and astronomer [[Guo Shoujing]] (1231–1316).<ref name="restivo 322">{{Cite book |last=Restivo |first=Sal |title=Mathematics in Society and History: Sociological Inquiries |date=1992 |publisher=Dordrecht: Kluwer Academic Publishers |isbn=1-4020-0039-1 |pages=32}}.</ref> Gauchet and Needham state Guo used [[spherical trigonometry]] in his calculations to improve the [[Chinese calendar]] and [[Chinese astronomy|astronomy]].{{sfn|Needham|1959|pp=108–109}}<ref name="gauchet 1512">{{Cite journal |last=Gauchet |first=L. |date=1917 |title=Note sur la trigonométrie sphérique de Kouo Cheou-king |url=https://www.jstor.org/stable/4526535 |journal=T'oung Pao |volume=18 |issue=3 |pages=151–174 |doi=10.1163/156853217X00075 |jstor=4526535 |issn=0082-5433|lang=fr}}</ref> Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham writes:

{{blockquote|Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two [[meridian arc]]s, one of which passed through the [[summer solstice]] point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree).{{sfn|Needham|1959|pp=109–110}}}}

Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of ''[[Euclid's Elements]]'' by Chinese official and astronomer [[Xu Guangqi]] (1562–1633) and the Italian Jesuit [[Matteo Ricci]] (1552–1610).{{sfn|Needham|1959|pp=110}}