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Les fonts primàries són El Desenvolupament de les Matemàtiques a Xina i Japó de Mikami i Matemàtiques Xineses, una Historia Precisa de Li Yan i Du Shiran. Veure la bibliografia a baix.

1. Anotació numèrica, computació aritmètica, recompte de barres.

a) Notació Decimal Tradicional -un símbol per cadascún dels números 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 100, 1000, i 10000. Exemple: 2034 s´escriuria amb els símbols del 2,1000,3,10,4, significant 2 vegades 1000 per 3 vegades 10 més 4. Retorns als orígens de l´escriptura Xinesa.

b) Els càlculs es feien utilitzant una petita taula de bambú on es comptaven barres. Les posicions de les barres van donar lloc i valor al sistema decimal, i també es feien per enregistrar les dades a llarg plaç. El número 0 es representava amb un espai. S´ordenaven d´esquerra a dreta com la numeració Aràbica. Parlem del 400 aC o anterior.

c) Suma: Les barres de recompte per sumar dos números es posaven a sota, amb un número sobre l´altre. Els dígits es sumaven d´esquerra a dreta amb la resta col·locada on fos necessari. Les restes es feien de manera similar.

d) Multiplicació: les taules de multiplicació fins al número 9 es memoritzaven. Les multiplicacions llargues es feien com ara nosaltres amb l´avantatge de les barres físiques. Les divisions llargues de manera anàloga als algoritmes actuals, potser més proper al "mètode de la Galera"

2. Zhoubi suanjing,
The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) (c. 100 B.C.E.-c. 100 C.E.). Descriu una de les teories celestials. Abans de la dinastia Han (206 B.C.E -220 C.E.) o anterior. El llibre fou cremat l´any 213 B.C.E. Estats i usos del Teorema de Pitàgoes per a la investigació, l´astronomia, etc.Els càlculs inclouen les fraccions comuns.

3. Jiuzhang Suanshu,
The Nine Chapters on the Mathematical Art (c. 100 B.C.E.-50 C.E.)
Recull matemàtic dels inicis de la dinastia Han. 246 problemes en 9 capítols. Un dels llibres de matemàtica Chinese més longívol i influent. Alguns comentaris.

Ch 1, Field measurement: systematic discussion of algorithms using counting rods for common fractions including alg. for GCD, LCM; areas of plane figures, square, rectangle, triangle, trapezoid, circle, circle segment, sphere segment, annulus -- some accurate, some approximations.
Ch 2,3,6 on proportions, Cereals, Proportional distribution, Fair taxes.
Ch 4, What width?: given area or volume find sides. Describes usual algorithms for square and cube roots but takes advantage of computations with counting rods
Ch 5, Construction consultations: volumes of cube, rectangular parallelepiped, prism frustums, pyramid, triangular pyramid, tetrahedron, cylinder, cone, and conic frustum, sphere -some approximations, some use pi=3
Ch 7, Excess and deficients: false position and double false position
Ch 8, Rectangular arrays: Gives elimination algorithm for solving systems of three or more simultaneous linear equations. Involves use of negative numbers (red reds for pos numbers, black for neg numbers). Rules for signed numbers.
Ch 9, Right triangles: applications of Pythagorean theorem and similar triangles, solves quadratic equations with modification of square root algorithm, only equations of the form x^2 + a x = b, with a and b positive.

4. Sun Zi (c. 250? C.E.)
Va escriure el seu manual de matemàtiques. Inclou "Chinese remainder problem" o "problem of the Master Sun": find n so that upon division by 3 you get a remainder of 2, upon division by 5 you get a remainder of 3, and upon division by 7 you get a remainder of 2. His solution: Take 140, 63, 30, add to get 233, subtract 210 to get 23.

5. Liu Hui (c. 263 C.E.)
Commentary on the Nine Chapters
Approximates pi by approximating circles polygons, doubling the number of sides to get better approximations. From 96 and 192 sided polygons, he approximates pi as 3.141014 and suggested 3.14 as a practical approx.
States principle of exhaustion for circles
Suggests Calvalieri's principle to find accurate volume of cylinder
Haidao suanjing (Sea Island Mathematical Manual). Originally appendix to commentary on Ch 9 of the Nine Chapters. Includes nine surveying problems involving indirect observations.

6. Zhang Qiujian (c. 450?)
Wrote his mathematical manual. Includes formula for summing an arithmetic sequence. Also an undetermined system of two linear equations in three unknowns, the "hundred fowls problem"

7. Zu Chongzhi (429-500) Astronomer, mathematician, engineer.
Collected together earlier astronomical writings. Made own astronomical observations. Recommended new calendar.
Determined pi to 7 digits: 3.1415926. Recommended use 355/113 for close approx. and 22/7 for rough approx.
With father carried out Liu Hui's suggestion for volume of sphere to get accurate formula for volume of a sphere.

8. Liu Zhuo (544-610) Astronomer
Introduced quadratic interpolation (second order difference method).

9. Wang Xiaotong (fl. 625) Mathematician and astronomer.
Wrote Xugu suanjing (Continuation of Ancient Mathematics) of 22 problems. Solved cubic equations by generalization of algorithm for cube root.

10. Translations of Indian mathematical works.
By 600 C.E., 3 works, since lost. Levensita, Indian astronomer working at State Observatory, translated two more texts, one of which described angle measurement (360 degrees) and a table of sines for angles from 0 to 90 degrees in 24 steps (3 3/4 degree) increments.
Hindu decimal numerals also introduced, but not adopted.

11. Yi Xing (683-727) tangent table.

12. Jia Xian (c. 1050)
Written work lost. Streamlined extraction of square and cube roots, extended method to higher-degree roots using binomial coefficients.

13. Qin Jiushao (c. 1202 - c. 1261)
Shiushu jiuzhang, Mathemtaical Treatise in Nine Sections, 81 problems of applied math similar to the Nine Chapters. Solution of some higher-degree (up to 10th) equations. Systematic treatment of indeterminate simultaneous linear congruences (Chinese remainder theorem). Euclidean algorithm for GCD.

14. Li Chih (a.k.a. Li Yeh) (1192-1279)
Ceyuan haijing, Sea Mirror of Circle Measurements, 12 chapters, 170 problems on right triangles and circles inscribed within or circumscribed about them. Yigu yanduan, New Steps in Computation, geometric problems solved by algebra.

15. Yang Hui (fl. c. 1261-1275)
Wrote sevral books. Explains Jiu Xian's methods for solving higher-degree root extractions. Magic squares of order up through 10.

16. Guo Shoujing (1231-1316).
Shou shi li, Works and Days Calendar. Higher-order differences (i.e., higher-order interpolation).

17. Zhu Shijie (fl. 1280-1303)
Suan xue qi meng, Introduction to Mathematical Studies, and Siyuan yujian, Precious Mirror of the Four Elements. Solves some higher degree polynomial equations in several unknowns. Sums some finite series including (1) the sum of n^2 and (2) the sum of n(n+1)(n+2)/6. Discusses binomial coefficients. Uses zero digit.

(Rest of outline yet to write.)

Cronologia dels Matemàtics i Obres Matemàtiques

Els primers textes tradicionals
Es van poder fer per l´acumulació gradual de material de les centúries anteriors. Estan datats segons les dates de finalització dels manuscrits.

Suan shu shu, A Book on Arithmetic (c. 180 B.C.E.). A book of bamboo strips found in 1984 near Jiangling in Hubei province.

Zhoubi suanjing, The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven (c. 100 B.C.E.-c. 100 C.E.)

Jiuzhang suanshu, Nine Chapters of the Mathematical Art (c. 100 B.C.E.-50 C.E.)

El desenvolupament dels matemàtics

Zhang Heng (78-139)
Ling xian (Spiritual Constitution of the Universe)

Liu Hong (fl. 178-187)
o Qian xiang li (Calendrical Science Based on the Celestial Appearances) (178-187)

Wang Fan (217-257)

Sun Zi (c. 250?)
o Sunzi suanjing (Master Sun's Mathematical Manual)

Zhao Shuang (Jun Qing) (c. 260)
o Zhoubi suanjing zhu (Commentary on the `Zhoubi Suanjing')

Liu Hui (c. 263)
o Jiushang suanshu zhu (Commentary on the `Nine Chapters of the Mathematical Art')
o Haidao suanjing (Sea Island Mathematical Manual)

Xiahou Yang (c. 350?)
o Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual)

Zhang Qiujian (c. 450?)
o Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual)

Zu Chongzhi (Wenyuan) (429-500)
o Da ming li (Da Ming Calendar) (462)
o Zhui shu (Method of Interpolation)
o Jiuzhang shu yi zhu (Commentary on the Methods and Essence of the `Nine Chapters')
o Chong cha zhu (Commentary on Double Differences)

Zu Geng
o Zhui shu (Method of Interpolation)

Zhen Luan (Shuzun) (fl. 566)
o Tian he li (Tian He Calendar) (462)
o Wucao suanjing (Mathematical Manual of the Five Government Departments)
o Wujing suanshu (Arithmetic in Five Classics)
o Shushu juji (Memoir on some Traditions of Mathematical Art)

Liu Zhuo (544-610)
o Huang ji li (Imperial Standard Calendar) (600)

Wang Xiaotong (fl. 625)
o Xugu suanjing (Continuation of Ancient Mathematics)

Li Chunfeng (fl. 664)
o Edited the Shibu suanjing (Ten Books of Mathematical Classics) .
This collection included the Jiuzhang suanshu (Nine Chapters of the Mathematical Art), Haidao suanjing (Sea Island Mathematical Manual), Sunzi suanjing (Master Sun's Mathematical Manual), Wucao suanjing (Mathematical Manual of the Five Government Departments), Wujing suanshu (Arithmetic in Five Classics), Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual), Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual), Zhui shu (Method of Interpolation), and Xugu suanjing (Continuation of Ancient Mathematics).

Yi Xing (683-727)
o Da yan li (Da Yan Calendar) (727)

Levensita (fl. 718)
o Jiu zhi li (Catching Nines Calendar) (718) translated from an Indian work
Three century interlude
followed by
The zenith of mathematical development

Jia Xian (c. 1050)
o Jia Xian suanjing (Jia Xian's Mathematical Manual)

Shen Kuo (1031-1095)
o Meng qu bi tan (Dream Pool Essays)

Li Zhi (Li Ye) (Jingzhai) (1192-1279)
o Ceyuan haijing (Sea Mirror of Circle Measurements) (1248)
o Yigu yanduan (New Steps in Computation) (1259)

Liu Yi (fl. c. 1225)
o Yigu genyuan (Discussion of the Old Sources)

Qin Jiushao (Daogu) (c. 1202-c. 1261)
o Shiushu jiuzhang (Mathemtaical Treatise in Nine Sections) (1247)

Guo Shoujing (1231-1316)

Yang Hui (Qianguang) (fl. 1261-1275)
o Xiangjie jiuzhang suanfa (A Detailed Analysis of the Mathematical Methods in the `Nine Chapters') (1261)
o Riyong suan fa (Computing Methods for Daily Use) (1262)
o Yang Hui suan fa (Yang Hui's Methods of Computation) (1274)
o Fasuan qu yong ben mo (Alpha and Omega of a Selection on the Applications of Arithmetical Methods (1274)
o Xugu zhaiqi suan fa (Continuation of Ancient Mathematical Methods for Elucidating the Strange [Properties of Numbers]) (1275)
o Jiuzhang suan fa zuan lei (Reclassification of the Mathematical Methods in the `Nine Chapters')
o Tian mu bi lei cheng chu jie fa (Practical Rules of Arithmetic for Surveying) (1275)
o Cheng chu tong bian suan bao (Precious Reckoner for Variations of Multiplication and Division)

Wang Xun (1235-1281)

Shou shi li (Works and Days Calendar), Guo Shoujing, Wang Xun, and others. (1280).

Zhu Shijie (Hanqing, Songting) (fl. 1280-1303)
o Suan xue qi meng (Introduction to Mathematical Studies) (1299) There is a Japanese edition of 1658.
o Siyuan yujian (Precious Mirror of the Four Elements) (1303)

L´ocàs dels matemàtics

Sha keshi (fl. 1321)
o He fang tong yi (On the Prevention of River Flooding) (1321)

Ding Ju (fl. 1355)
o Ding ju suan fa (Ding Ju's Arithmetical Methods) (1355)

He Pingzi (fl. 1373)
o Xiangming suan fa (Explanations of Arithmetic) (1373)

Liu Shilong (fl. 1424)
o Jiu zhang tong ming suanfa (Methods of Calculation in the `Nine Chapters') (1424)

Wu Jing (fl. 1450)
o Jiu zhang suan fa bi lei da quan (Complete Description of the `Nine Chapters' on Arithmetical Techniques) (1450)

Wang Wensu (fl. 1524)
o Suan xue baojian (Precious Mirror of Mathematics) (1524)

Xu Xinlu
o Pan zhu suan fa (Method of Calculating on an Abacus) (1573)

Ke Sangquin (fl. 1578)
o Shu xue tong gui (Rules of mathematics) (1578)

Matteo Ricci (1552-1610)

Niccolo Longobardi (1559-1654)

Xu Guangqi (Zixian, Xuanhu) (1562-1633)

Li Zhizao (Zhenzhi) (1565-1630)
Introduction of Western Mathematics

Cheng Dawei (Rusi, Binqu)(fl. 1592)
o Suan fa tong zong (Systematic Treatise on Arithmetic) (1592) Reprinted in Japan in 1675.
o Zhi zhi uan fa tong zong (Postscript to the Systematic Treatise on Arithmetic) (1592)
o Suan fa zuan yao (Highlights of Calculation Methods (1598)

Huang Longyi (fl. 1604)
o Suan fa ji nan (Directory of Calculation Methods) (1604)

Johann Terrenz Schreck (1576-1630)
o Da ce (Complete Surveying)
o Ge tu ba xian biao (Tables of Trigonometric Functions) (1631)
o Ce tian yue shuo (Brief Description of the Measurement of the Heavens)

Li Tianjing (1579-1659)
o Traducció dels sis primers Llibres dels Elements d´Euclides, Matteo Ricci and Xu Guangqi (1607)

Giulio Aleni (1582-1649)
o Ji he yao fa (Essentials of Geometry)

Johann Adam Schall von Bell (1591-1666)

Giacomo Rho (1593-1638)
o Chou suan (Napier's Bones) (1628)
o Ce liang quan yi (Complete Theory of Surveying) (1631)
o Bi li gui jie (Manual for proportional dividers) (1631)

Tong wen suan zhi (Treatise on European Arithmetic) an edited translation of Clavius's Epitome of Practical Arithmetic, Matteo Ricci and Li Zhizao (1631)

Chong zhen li shu (Chong Zhen Reign Treatise on Astronomy and Calendrical Science). (1631-1634).
A collection of 137 books in five submissions edited by Xu Guanqi and Li Tianjing with support of many others. It included Aleni's Ji he yao fa (Essentials of Geometry); Terrenz's Da ce (Complete Surveying), Ge tu ba xian biao (Tables of Trigonometric Functions), and Ce tian yue shuo (Brief Description of the Measurement of the Heavens); and Rho's Ce liang quan yi (Complete Theory of Surveying), Bi li gui jie (Manual for proportional dividers), and Chou suan (Napier's Bones).

Jean Nicolas Smogulecki (1611-1656)

Xi yang xin fa li shu (Treatise on Astronomy and Calendrical Science According to the New Western Methods).
A collection of 100 books in 17 volumes emended by Schall von Bell from the Chong zhen li shu (Chong Zhen Reign Treatise on Astronomy and Calendrical Science) (1645)

Xue Fengzuo (d. 1680)

Fang Zhongtong (1633-1698)

Li xue hui tong (Understanding Calendar Making) (1652-1654)
A collection of books published by Smogulecki and Xue Fengzuo. Included are
o Bi li si xian xin biao (New Tables for Four Logarithmic Trigonometric Functions)
o Bi li dui shu biao (Logarithm Tables with Explanations)
o San jiaofa (Essentials of Trigonometry) (1653)

Tian bu zhen yuan (True Course of Celestial Motions) (1653)
A collection of books written by Smogulecki and Xue Fengzuo. Includes
o San jiao suan fa (Method of Trigonometrical Calculations)

Mei Wending (Dingjiu, Wu'an) (1633-1721)
o See Mei Juecheng, Mei shi congshu jiyao (Collected Works of the Mei Family) for publication of Mei Wending's written comments on mathematics

Shu li jing yun (Collected Basic Principles of Mathematics) (1723).
Supervised by Emperor Kang Xi (Aixinjueluo) (1654-1722), edited by Mei Juecheng, Chen Houyao, He Guozong, Ming Antu, Mei Wending, and others.

Mei Juecheng
o In 1761, Mei Juecheng complied Mei Wending's written commentaries into the Mei shi congshu jiyao (Collected Works of the Mei Family). It included several works on mathematics: Bisuan (Pen Calculations), Chou suan (Napier's bones), Du suan shi li (Proportional Dividers), Shao guang shi yi (Supplement to `What Width'), Fang cheng lun (Theory of Rectangular Arrays), Gougu ju yu (Right-angled Triangles), Jihe tong jie (Explanations in Geometry), Ping san jiao ju yao (Elements of Plane Trigonometry), Fang yuan mi ji (Squares and Circles, Cubes and Spheres), Jihe bu bian (Supplement to Geometry), Hu san jiao ju yao (Elements of Spherical Trigonometry), Huan zhong shu chi (Geodesy), and Qiandu celiang (Surveying Solids).
Mathematics under the "Closed Door" Policy

Chen Shiren (1676-1722)
o Shao guang bu yi (Supplement to `What Width')

Ming Antu (d. 1765)
o Suanjing shishu (Ten Mathematical Manuals) (1773)
o Ge yuan mi lu jie fa (Quick Method for Determining Close Ratios in Circle Division) (1774)

Jiao Xun (1763-1820)
o Da yan qiu yi shu (Technique for Finding 1 by the Great Extension)

Ruan Yuan (1764-1849)

Wang Lai (Xiaoying, Hengzhai) (1768-1813)
o Hengzhai suanxue (Hengzhai's Mathematics)
o Hengahai yi shu (Unpublished Works of Hengzhai) (1834, edited by Xia Xie)

Chou ren zhuan (Biographies of Mathematicians and Astronomers) (1795-1799). Edited by Ruan Yuan.

Li Huang (d. 1811)
o Jiuzhang suanshu xi cao tu shuo (Careful Explanation of the `Nine Chapters on the Mathematical Art')
o Haidao suanjing xi cao tu shuo (Careful Explanation of the `Sea Island Mathematical Manual')
o Xu gu suanjing kao zhu (Commentary on the `Continuation of Ancient Mathematical Methods for Elucidating the Strange [Properties of Numbers]')

Li Rui (Shangzhi, Sixiang) (1773-1817)
o Li shi suan xue yi shu (Collected Mathematical Works of Li Rui)

Luo Tengfeng
o Yi you lu (Records of the Art of Learning) (1815)

Xiang Mingda (1789-1850)
o Xiang shu yi yuan (The Source of Series) (1888, edited by Dai Xu)

Luo Shilin (1789-1853)
o Siyuan yujian xicao (Commentary on the `Precious Mirror of the Four Elements') (1836)

Dong Youcheng (Fangli) (1791-1823)
o Ge yuan mi lu tu jie (Explanation for the `Determination of Close Ratios in Circle Division')

Gu Guanjuang (1799-1862)
o Zhoubi suanjing xiao kan ji (A Textual Criticism of the `Zhoubi Suanjing')

Shen Qinpei (fl. 1829)
o Siyuan yujian xicao (Commentary on the `Precious Mirror of the Four Elements') (1829)

Zhang Dunren (fl. 1831)
o Qiu yi suan shu (Techniques of Finding 1) (1831)

Dai Xu (1805-1860)
o Dai shu jian fa (Concise Technique of Logarithms) (1846)
International Mathematics in China

Li Shanlan (Renshu, Qiuren) (1811-1882)
o Duo ji bi lei (Sums of Piles of Various Types)
o Fang yuan chan you (Explanation of the Square and the Circle)
o Hu shi qi mi (Unveiling the Secrets of Arc and Sagitta)
o Dui shu tan yuan (Seeking the Source of Logarithms)
o Several translations of Western mathematics (1852-1866)

Hua Hengfan (Ruo Ting) (1833-1902)
o Xingsu xuan suan (Mathematical Papers form the Xing Su Study)
o Several translations of Western mathematics (1868-1886)

Shi Richun
o Qiu yi shu zhi (Path to the Technique of Finding 1) (1873)

Huang Zongxian
o Qiu yi shu tong jie (Explanation of the Technique of Finding 1) (1873)

REFERÈNCIES SELECTES

BIBLIOGRAFIA

David E. Joyce
Department of Mathematics and Computer Science
Clark University

NOTES

La Societat Chinesa de Matemàtiques

Encara que parlem de 1935 com la data de fundació de la Societat Matemàtica Xinesa, hi ha una raó per a dir 1929. La raó és que la Societat Xinesa de les Cièncias Matemàtiques fou fundada el 1929 i va portar a terme les mateixes funcions que la Societat Matemàtica Xinesa fins el 1936. La fundació de la Societat Matemàtica Xinesa el juliol de 1935 va comportar el tancament de la Societat Xinese de les Ciències Matemàtiques a l´any següent.

El primer President de la Societat era Hu T-F que va establir les bases de la Societat el 1935. Ocupà la presidència fins el 1948, veient travessar la Societat per un període molt difícil. Al principi la Societat va prosperar i el 1936 va començar la publicació de dos diaris, un d´investigació i una edició popular. Els aconteixements polítics van portar grans dificultats. El juliol de 1937 la invasió japonesa derrota als chinesos i a finals d´any Japó ja había conquerit Shanghai i Nanking. El 1938 el païs fou dividit entre la part controlada per Japó i la Xina Lliure que es va negar a sometre´s. La Societat Matemàtica Xina ja no era operativa tot i persistir d´alguna manera ja que de 1938 a 1945 no va poder funcionar com a Societat.

Japó es rendeix el 1945 però els problemes de Xina no acaben pel país ja que va patir una guerra civil durant quatre anys entre Nacionalistes i Comunistes. Els intents van ser refundar la Societat Matemàtica Xina i el 1948 va iniciar l´activitat de nou per donar suport als matemàtics chinesos. Hu T-F, el primer president va acabar el seu mandat el 1948. La victòria comunista va portar a la creació de la República Popular Xina l´1 d´octubre de 1949. La Societat Matemàtica Xina es va expandir ràpidament sota la direcció del president Hua -nomenat el 1951- i va exercir fins el 1983. Actualment:

... és una societat professional que compren milers de membres i que publica prop de deu diaris matemàtics i varies series de llibres.

Il·lustració del Zhou Pei Suan Jing. El Llibre data del periode dela Dinastia Zhou (1046 BCE - 700 BCE). Consisteix en una col·lecció anònima de 246 problemes descoberts per Duke of Zhou i el seu astròlog Shang Gao.

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