Editing Chinese mathematics (section)

===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]].