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Knowledge of Chinese mathematics before 254 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. The dating of the use of certain mathematical methods in Chinese history is problematic and disputed[citation needed]

In early times the focus was on astronomy and perfecting the calendar and not on establishing the proof. Axiomic proof was the strength of ancient Greek mathematician; ancient Chinese mathematicians excelled at place value decimal device computation, algorithm development and algebra, the weakness of their Greek counterparts.

Early Chinese mathematics

Visual proof for the (3, 4, 5) triangle as in the Zhou Bi Suan Jing 500–200 BC.
Oracle bone script decimal
counting rod place value decimal

Simple mathematics on Oracle bone script date back to the Shang Dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the Yi Jing, which greatly influenced written literature during the Zhou Dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching contained elements of binary numbers.

Mathematics was one of the Liù Yì (六艺) or Six Arts, students were required to master during the Zhou Dynasty (1122–256 BC). Learning them all perfectly was required to be a perfect gentleman, or in the Chinese sense, a "Renaissance Man". Six Arts have their roots in the Confucian philosophy.

The oldest existent work on geometry in China comes from the philosophical Mohist canon of c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point.[1] Much like Euclid's first and third definitions and Plato's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it."[2] Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved.[2] It stated that two lines of equal length will always finish at the same place,[2] while providing definitions for the comparison of lengths and for parallels,[3] along with principles of space and bounded space.[4] It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.[5] The book provided definitions for circumference, diameter, and radius, along with the definition of volume.[6]

Qin mathematics

Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars.

Knowledge of this period must be carefully determined by their civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were incredible feats of human engineering. Emperor Qin Shihuang（秦始皇）ordered many men to build large, lifesize statues for the palace tomb along with various other temples and shrines. The shape of the tomb is designed with geometric skills of architecture. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion.

Han mathematics

Further information: Science and technology of the Han Dynasty § Mathematics and astronomy
The Nine Chapters on the Mathematical Art.

In the Han Dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called chousuan,consisted of only nine symbols, a blank space on the counting board stood for zero. The mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used. Zhang also applied mathematics in his work in astronomy.

Suan shu shu

The Suàn shù shū (writings on reckoning) is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1984 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. Some linguistic hints point back to the Qin dynasty.

In an example of an elementary mathematics in the Suàn shù shū, the square root is approximated by using an "excess and deficiency" method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."[7]

The Nine Chapters on the Mathematical Art

The Nine Chapters on the Mathematical Art is a Chinese mathematics book, its oldest archeological date being 179 AD (traditionally dated 1000 BC), but perhaps as early as 300–200 BC. Although the author(s) are unknown, they made a huge contribution in the eastern world. The methods were made for everyday life and gradually taught advanced methods. It also contains evidence of the Gaussian elimination and Cramer's Rule for system of linear equations.

It was one of the most influential of all Chinese mathematical books and it is composed of some 246 problems. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns.[8] A date of about 300 BC would appear reasonable, thus placing it in close competition with another treatise, the Jiu zhang suanshu, composed about 250 BC, that is, shortly before the Han dynasty (202 BC). Almost as old at the Chou Pei, and perhaps the most influential of all Chinese mathematical books, was the Jiuzhang suanshu, or Nine Chapters on the Mathematical Art. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. Chapter eight of the Nine chapters is significant for its solution of problems of simultaneous linear equations, using both positive and negative numbers. The earliest known magic squares appeared in China.[9] The Chinese were especially fond of patterns, as a natural outcome of arranging counting rods in rows on counting board to carry out computation; hence,it is not surprising that the first record (of ancient but unknown origin) of a magic square appeared there. The concern for such patterns led the author of the Nine Chapters to solve the system of simultaneous linear equations by placing the coefficients and constant terms of the linear equations into a matrix and performing column reducing operations on the matrix to reduce it to a triangular form represented by the equations 36z = 99, 5y + z = 24, and 3x + 2y + z = 39 from which the values of z, y, and x are successively found with ease. The last problem in the chapter involves four equations in five unknowns, and the topic of indeterminate equations was to remain a favorite among Oriental peoples.

Mathematics in the period of disunity

Lui Hui's Survey of sea island
Sunzi algorithm for division 400 AD
al Khwarizmi division in 9th century

In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium.[10] He was the first Chinese mathematician to calculate π=3.1416 with his π algorithm. He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the integral and the differential calculus during the 3rd century CE.

fraction interpolation for pi

In the fourth century, another influential mathematician named Zu Chongzhi, introduced the Da Ming Li. This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in Book of Sui, we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927). He also used He Chengtian's interpolation method for approximating irrational number with fraction in his astronomy and mathematical works, he obtained as a good fraction approximate for pi; Along with his son, Zu Geng, Zu Chongzhi used the Cavalieri Method to find an accurate solution for calculating the volume of the sphere. His work, Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that Zhui Shu contains the formulas and methods for linear, matrix algebra, algorithm for calculating the value of π, formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics.

A mathematical manual called "Sunzi mathematical classic" dated around 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. The earliest record of multiplication and division algorithm using Hindu Arabic numerals was in writing by Al Khwarizmi in early 9th century. Khwarizmi's step by step division algorithm was completely identical to Sunzi division algorithm described in Sunzi mathematical classic four centuries earlier.[11] Khwarizmi's work was translated in to Latin in the 13th century and spread to the west, the division algorithm later evolved into Galley division.

Tang mathematics

By the Tang Dynasty study of mathematics was fairly standard in the great schools. Wang Xiaotong was a great mathematician in the beginning of the Tang Dynasty, and he wrote a book: Jigu suanjing (Continuation of Ancient Mathematics), in which cubic equations appear for the first time[12]

The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of gNam-ri srong btsan, who died in 630.[13][14][15][16][17][18][19][20][21][22]

The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang Dynasty.[23] However,early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics.[24] I-Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known.[23]

Song and Yuan mathematics

Northern Song Dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule.[25]

Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD
Li Ye's inscribed circle in triangle

Li Zhi on the other hand, investigated on a form of algebraic geometry. His book; Ce Hai Yuan Jing revolutionized the idea of inscribing a circle into triangles, which could be calculated using equations with the Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations.

The high point of this era came with Zhu Shijie's two books Suanxue qimeng and the Siyuan yujian. In one case he reportedly gave a method equivalent to Gauss's pivotal condensation.

Qin Jiushao (c. 1202–1261) was the first to introduce the zero symbol into Chinese mathematics from India.[26] One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations.

Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa (详解九章算法), although it was described earlier around 1100 by Jia Xian.[27] Although the Introduction to Computational Studies (算学启蒙) written by Zhu Shijie (fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics.[28]

Precious Mirror of the Four Elements

Si-yüan yü-jian《四元玉鑒》, or Precious Mirror of the Four Elements, was written by Chu Shi-jie in 1303 AD and it 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. The Ssy-yüan yü-chien 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.Cite error: A <ref> tag is missing the closing </ref> (see the help page). He also worked with 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 calendarical science and astronomical calculations.[23] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[23] Victor J. Katz writes that in Shen's formula "technique of intersecting circles", he created an approximation of the arc of a circle s by s = c + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc.[29] 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).[30] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[23][31] Along with a later 17th century Chinese illustration of Guo's mathematical proofs, Needham states that:

Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, 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).[32]

Later developments

However after the overthrow of the Yuan Dynasty China became suspicious of knowledge it used. The Ming Dynasty turned away from math and physics in favor of botany and pharmacology.

At this period, the abacus which first appeared in Song dynasty now overtook the counting rods and became the preferred computing device. Zhu Zaiyu, Prince of Zheng who invented the equal temperament used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy.

However, this switching from counting rods to abacus to gain speed in calculation was at a high cost, causing the stagnation and decline of Chinese mathematics. The pattern rich layout of counting rod numerals on counting board inspired many Chinese inventions in mathematics,such as cross multiply principe of fractions, method for solving linear equations. The pattern rich counting rods inspired Japanese mathematician to invent the concept of matrix. In Ming dynasty, mathematicians were fascinated with perfecting algorithms for abacus, many mathematical works devoted to abacus mathematics appeared in this period, at the expense of new ideas creation.

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).[33]

A revival of math in China began in the late nineteenth century, Joseph Edkins, Alexander Wylie and Li Shanlan translated works on astronomy, algebra and differential-integral calculus into Chinese, published by London Missionary Press in Shanghai.

Mathematical texts

Zhou Dynasty

Zhoubi Suanjing" c. 1000 BCE-100 CE -Astronomical theories, and computation techniques -Proof of the Pythagorean theorem (Shang Gao Theorem) -Fractional computations -Pythagorean theorem for astronomical purposes

Nine Chapters of Mathematical Arts1000 BCE? – 50 CE -ch.1, computational algorithm, area of plane figures, GCF, LCD -ch.2, proportions -ch.3, proportions -ch.4, square, cube roots, finding unknowns -ch.5, volume and usage of pi as 3 -ch.6, proportions -ch,7, interdeterminate equations -ch.8, Gaussian elimination and matrices -ch.9, Pythagorean theorem (Gougu Theorem)

See also

Footnotes and references

  •  This article incorporates text from Americanized Encyclopaedia britannica: rev. and amended A dictionary of arts, sciences and literature, to which is added biographies of living subjects. 96 colored maps and numerous illustrations, Volume 9, a publication from 1890, now in the public domain in the United States.
  •  This article incorporates text from The home encyclopædia: compiled and revised to date from the leading encyclopædias, Volume 18, a publication from 1895, now in the public domain in the United States.
  •  This article incorporates text from Americanized Encyclopedia britannica, revised and amended: A dictionary of arts, sciences and literature; to which is added biographies of livings subjects ..., a publication from 1890, now in the public domain in the United States.
  •  This article incorporates text from The encyclopædia britannica: a dictionary of arts, sciences, literature and general information, Volume 26, by Hugh Chisholm, a publication from 1911, now in the public domain in the United States.
  •  This article incorporates text from The Encyclopaedia Britannica: a dictionary of arts, sciences, and general literature, Volume 23, by Thomas Spencer Baynes, a publication from 1888, now in the public domain in the United States.
  •  This article incorporates text from The Encyclopedia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26, by Hugh Chisholm, a publication from 1911, now in the public domain in the United States.
  •  This article incorporates text from The Encyclopædia britannica: a dictionary of arts, sciences, and general literature ; the R.S. Peale reprint, with new maps and original American articles, Volume 23, by William Harrison De Puy, a publication from 1893, now in the public domain in the United States.
  •  This article incorporates text from The Life of the Buddha and the early history of his order: derived from Tibetan works in the Bkah-hgyur and Bstan-hgyur followed by notices on the early history of Tibet and Khoten, by Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio, a publication from 1907, now in the public domain in the United States.
  •  This article incorporates text from The life of the Buddha: and the early history of his order, by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio, a publication from 1884, now in the public domain in the United States.
  1. ^ Needham, Volume 3, 91.
  2. ^ a b c Needham, Volume 3, 92.
  3. ^ Needham, Volume 3, 92-93.
  4. ^ Needham, Volume 3, 93.
  5. ^ Needham, Volume 3, 93-94.
  6. ^ Needham, Volume 3, 94.
  7. ^ Dauben, p 210.
  8. ^ Boyer, 1991, "Chinese Math, China and India"
  9. ^ Boyer, 1991, "Magic Square, China and India"
  10. ^ Frank J. Swetz: The Sea Island Mathematical Manual, Surveying and Mathematics in Ancient China 4.2 Chinese Surveying Accomplishments, A Comparative Retrospection p63 The Pennsylvania State University Press, 1992 ISBN 0-271-00799-0
  11. ^ Lam Lay Yong, The Development of Hindu Arabic and Traditional Chinese Arithematic, Chinese Science, 13(1996) 35–54 [sic?]
  12. ^ Yoshio Mikami, Mathematics in China and Japan,p53
  13. ^ Americanized Encyclopaedia britannica: rev. and amended A dictionary of arts, sciences and literature, to which is added biographies of living subjects. 96 colored maps and numerous illustrations, Volume 9. Belford-Clarke co. 1890. p. 5826. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)Americanized Encyclopaedia Britannica: Rev. and Amended A Dictionary of Arts, Sciences and Literature, to which is Added Biographies of Living Subjects. 96 Colored Maps and Numerous Illustrations
  14. ^ The home encyclopædia: compiled and revised to date from the leading encyclopædias, Volume 18. Educational publishing co. 1895. p. 5826. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)The Home Encyclopædia: Compiled and Revised to Date from the Leading Encyclopædias
  15. ^ Americanized Encyclopedia britannica, revised and amended: A dictionary of arts, sciences and literature; to which is added biographies of livings subjects ... The "Examiner". 1890. p. 5826. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)Volume 9 of Americanized Encyclopedia Britannica, Revised and Amended: A Dictionary of Arts, Sciences and Literature; to which is Added Biographies of Livings Subjects
  16. ^ Hugh Chisholm, ed. (1911). The encyclopædia britannica: a dictionary of arts, sciences, literature and general information, Volume 26 (11 ed.). At the University press. p. 926. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)The Encyclopædia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, Hugh Chisholm
  17. ^ Thomas Spencer Baynes, ed. (1888). The Encyclopaedia Britannica: a dictionary of arts, sciences, and general literature, Volume 23 (9 ed.). C. Scribner's sons. p. 345. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, and General Literature, Thomas Spencer Baynes
  18. ^ Hugh Chisholm (1911). The Encyclopedia Britannica: a dictionary of arts, sciences, literature and general information, Volume 26 (11 ed.). The Encyclopedia Britannica Co. p. 926. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)The Encyclopedia Britannica: A Dictionary of Arts, Sciences, Literature and General Information, Hugh Chisholm
  19. ^ William Harrison De Puy (1893). The Encyclopædia britannica: a dictionary of arts, sciences, and general literature ; the R.S. Peale reprint, with new maps and original American articles, Volume 23 (9 ed.). Werner Co. p. 345. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)The Encyclopædia Britannica: A Dictionary of Arts, Sciences, and General Literature ; the R.S. Peale Reprint, with New Maps and Original American Articles, William Harrison De Puy
  20. ^ Translated by William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio (1907). The Life of the Buddha and the early history of his order: derived from Tibetan works in the Bkah-hgyur and Bstan-hgyur followed by notices on the early history of Tibet and Khoten. K. Paul, Trench, Trübner. p. 211. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help); More than one of |pages= and |page= specified (help)CS1 maint: multiple names: authors list (link)
  21. ^ William Woodville Rockhill, Ernst Leumann, Bunyiu Nanjio (1884). The life of the Buddha: and the early history of his order. Trübner & co. p. 211. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help); More than one of |pages= and |page= specified (help)CS1 maint: multiple names: authors list (link)
  22. ^ The Life of the Biddha and the Early History of His Order Derived from Tibetan Works in the Bkah-hgyur and Bstan-khoten. Taylor & Francis. p. 211. Retrieved 2011-7-1. {{cite book}}: Check date values in: |accessdate= (help)
  23. ^ a b c d e Needham, Volume 3, 109.
  24. ^ Needham, Volume 3, 108-109.
  25. ^ Martzloff, 142
  26. ^ Needham, Volume 3, 43.
  27. ^ Needham, Volume 3, 134–137.
  28. ^ Needham, Volume 3, 46.
  29. ^ Katz, 308.
  30. ^ Restivo, 32.
  31. ^ Gauchet, 151.
  32. ^ Needham, Volume 3, 109–110.
  33. ^ Needham, Volume 3, 110.

Sources

  • Boyer, C. B. (1989). A History of Mathematics. rev. by Uta C. Merzbach (2nd ed.). New York: Wiley,. ISBN 0-471-09763-2.{{cite book}}: CS1 maint: extra punctuation (link) (1991 pbk ed. ISBN 0-471-54397-7)
  • Dauben, Joseph W. (2007). "Chinese Mathematics". In Victor J. Katz (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 9780691114859.
  • Martzloff, Jean-Claude (1996). A History of Chinese Mathematics. Springer. ISBN 3-540-33782-2.
  • Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd.
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