Song and Yuan mathematics Chinese mathematics

1 song , yuan mathematics

1.1 algebra

1.1.1 ceyuan haijing
1.1.2 jade mirror of 4 unknowns
1.1.3 mathematical treatise in 9 sections
1.1.4 magic squares , magic circles


1.2 trigonometry

song , yuan mathematics

northern song dynasty mathematician jia xian developed additive multiplicative method extraction of square root , cubic root implemented horner rule.

yang hui triangle (pascal s triangle) using rod numerals, depicted in publication of zhu shijie in 1303 ad

four outstanding mathematicians arose during song dynasty , yuan dynasty, particularly in twelfth , thirteenth centuries: yang hui, qin jiushao, li zhi (li ye), , zhu shijie. yang hui, qin jiushao, zhu shijie used horner-ruffini method 6 hundred years earlier solve types of simultaneous equations, roots, quadratic, cubic, , quartic equations. yang hui first person in history discover , prove pascal s triangle , along binomial proof (although earliest mention of pascal s triangle in china exists before eleventh century ad). li zhi on other hand, investigated on form of algebraic geometry based on tian yuan shu. book; ceyuan haijing revolutionized idea of inscribing circle triangles, turning geometry problem algebra instead of traditional method of using pythagorean theorem. guo shoujing of era worked on spherical trigonometry precise astronomical calculations. @ point of mathematical history, lot of modern western mathematics discovered chinese mathematicians. things grew quiet time until thirteenth century renaissance of chinese math. saw chinese mathematicians solving equations methods europe not know until eighteenth century. high point of era came zhu shijie s 2 books suanxue qimeng , siyuan yujian. in 1 case reportedly gave method equivalent gauss s pivotal condensation.

qin jiushao (c. 1202–1261) first introduce 0 symbol chinese mathematics. before innovation, blank spaces used instead of zeros in system of counting rods. 1 of important contribution of qin jiushao method of solving high order numerical equations. referring qin s solution of 4th order equation, yoshio mikami put it: can deny fact of horner s illustrious process being used in china @ least 6 long centuries earlier in europe? qin solved 10th order equation.

pascal s triangle first illustrated in china yang hui in book xiangjie jiuzhang suanfa (详解九章算法), although described earlier around 1100 jia xian. although introduction computational studies (算学启蒙) written zhu shijie (fl. 13th century) in 1299 contained nothing new in chinese algebra, had great impact on development of japanese mathematics.

algebra
ceyuan haijing


li ye s inscribed circle in triangle:diagram of round town



yang hui s magic circle

ceyuan haijing (pinyin: cèyuán hǎijìng) (chinese characters:測圓海鏡), or sea-mirror of circle measurements, collection of 692 formula , 170 problems related inscribed circle in triangle, written li zhi (or li ye) (1192–1272 ad). used tian yuan shu convert intricated geometry problems pure algebra problems. used fan fa, or horner s method, solve equations of degree high six, although did not describe method of solving equations. li chih (or li yeh, 1192–1279), mathematician of peking offered government post khublai khan in 1206, politely found excuse decline it. ts e-yuan hai-ching (sea-mirror of circle measurements) includes 170 problems dealing with[...]some of problems leading polynomial equations of sixth degree. although did not describe method of solution of equations, appears not different used chu shih-chieh , horner. others used horner method ch in chiu-shao (ca. 1202 – ca.1261) , yang hui (fl. ca. 1261–1275).

jade mirror of 4 unknowns


facsimile of zhu shijie s jade mirror of 4 unknowns

si-yüan yü-jian (《四元玉鑒》), or jade mirror of 4 unknowns, written zhu shijie in 1303 ad , marks peak in development of chinese algebra. 4 elements, called heaven, earth, man , matter, represented 4 unknown quantities in algebraic equations. deals simultaneous equations , equations of degrees high fourteen. author uses method of fan fa, today called horner s method, solve these equations.

the jade mirror opens diagram of arithmetic triangle (pascal s triangle) using round 0 symbol, chu shih-chieh denies credit it. similar triangle appears in yang hui s work, without 0 symbol.

there many summation series equations given without proof in precious mirror. few of summation series are:

1

2


+

2

2


+

3

2


+
⋯
+

n

2


=



n
(
n
+
1
)
(
2
n
+
1
)


3
!





{\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}}






1
+
8
+
30
+
80
+
⋯
+




n

2


(
n
+
1
)
(
n
+
2
)


3
!



=



n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
(
4
n
+
1
)


5
!





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



mathematical treatise in 9 sections

shu-shu chiu-chang, or mathematical treatise in 9 sections, written wealthy governor , minister ch in chiu-shao (ca. 1202 – ca. 1261 ad) , invention of method of solving simultaneous congruences, marks high point in chinese indeterminate analysis.

magic squares , magic circles

the earliest known magic squares of order greater 3 attributed yang hui (fl. ca. 1261–1275), worked magic squares of order high ten. worked magic circle.

trigonometry

the embryonic state of trigonometry in china began change , advance during song dynasty (960–1279), chinese mathematicians began express greater emphasis need of spherical trigonometry in calendarical science , astronomical calculations. polymath chinese scientist, mathematician , official shen kuo (1031–1095) used trigonometric functions solve mathematical problems of chords , arcs. victor j. katz writes in shen s formula technique of intersecting circles , created approximation of arc of circle s s = c + 2v/d, d diameter, v versine, c length of chord c subtending arc. sal restivo writes shen s work in lengths of arcs of circles provided basis spherical trigonometry developed in 13th century mathematician , astronomer guo shoujing (1231–1316). historians l. gauchet , joseph needham state, guo shoujing used spherical trigonometry in calculations improve calendar system , chinese astronomy. along later 17th-century chinese illustration of guo s mathematical proofs, needham states that:

guo used quadrangular spherical pyramid, basal quadrilateral of consisted of 1 equatorial , 1 ecliptic arc, 2 meridian arcs, 1 of passed through summer solstice point...by such methods able obtain du lü (degrees of equator corresponding degrees of ecliptic), ji cha (values of chords given ecliptic arcs), , cha lü (difference between chords of arcs differing 1 degree).