Editing Chinese mathematics (section)

===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}}