''The Nine Chapters on the Mathematical Art'' () is a Chinese

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from

China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...

, the first being ''

Suan shu shu The ''Book on Numbers and Computation'' (), or the ''Writings on Reckoning'', is one of the earliest known Chinese mathematical treatises. It was written during the early Western Han dynasty, sometime between 202 BC and 186 BC.Liu et al. (2003), ...

'' (202 BCE – 186 BCE) and '' Zhoubi Suanjing'' (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to

ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic pe ...

mathematicians, who tended to deduce propositions from an initial set of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s. Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by

Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...

in the 3rd century.

History

Original book

The full title of ''The Nine Chapters on the Mathematical Art'' appears on two

bronze Bronze is an alloy consisting primarily of copper, commonly with about 12–12.5% tin and often with the addition of other metals (including aluminium, manganese, nickel, or zinc) and sometimes non-metals, such as phosphorus, or metalloids suc ...

standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles.Needham, Volume 3, 24-25. Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when the Nine Chapters reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses

Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...

before

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

(1777–1855). There is also the

mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...

given in the treatise for the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

.Needham, Volume 3, 22. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of

Korea Korea ( ko, 한국, or , ) is a peninsular region in East Asia. Since 1945, it has been divided at or near the 38th parallel, with North Korea (Democratic People's Republic of Korea) comprising its northern half and South Korea (Republic ...

and

Japan Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the n ...

. Its influence on mathematical thought in China persisted until the

Qing Dynasty The Qing dynasty ( ), officially the Great Qing,, was a Manchu-led imperial dynasty of China and the last orthodox dynasty in Chinese history. It emerged from the Later Jin dynasty founded by the Jianzhou Jurchens, a Tungusic-speak ...

era.

wrote a very detailed commentary on this book in 263. He analyses the procedures of the Nine Chapters step by step, in a manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide

formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the seq ...

s in the

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

ean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians

Zhang Cang Zhang Cang 張蒼 (253—152 BC) was a Chinese military general, philosopher, and politician during the Western Han dynasty. He was the representative thinker of the Yin-Yang School, as well as a Confucian scholar, general, and prime-minister un ...

(

fl. ''Floruit'' (; abbreviated fl. or occasionally flor.; from Latin for "they flourished") denotes a date or period during which a person was known to have been alive or active. In English, the unabbreviated word may also be used as a noun indicatin ...

165 BCE - d. 142 BCE) and Geng Shouchang (fl. 75 BCE-49 BCE) (see

armillary sphere An armillary sphere (variations are known as spherical astrolabe, armilla, or armil) is a model of objects in the sky (on the celestial sphere), consisting of a spherical framework of rings, centered on Earth or the Sun, that represent lines of ...

) with the initial arrangement and commentary on the book, yet

Han Dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...

records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.Needham, Volume 3, 24. The ''Nine Chapters'' is an anonymous work, and its origins are not clear. Until recent years, there was no substantial evidence of related mathematical writing that might have preceded it, with the exception of mathematical work by those such as Jing Fang (78–37 BCE), Liu Xin (d. 23), and

Zhang Heng Zhang Heng (; AD 78–139), formerly romanized as Chang Heng, was a Chinese polymathic scientist and statesman who lived during the Han dynasty. Educated in the capital cities of Luoyang and Chang'an, he achieved success as an astronomer, mat ...

(78–139) and the geometry clauses of the ''

Mozi Mozi (; ; Latinized as Micius ; – ), original name Mo Di (), was a Chinese philosopher who founded the school of Mohism during the Hundred Schools of Thought period (the early portion of the Warring States period, –221 BCE). The ancie ...

'' of the 4th century BCE. This is no longer the case. The Suàn shù shū (算數書) or ''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 1983 when

archaeologist Archaeology or archeology is the scientific study of human activity through the recovery and analysis of material culture. The archaeological record consists of artifacts, architecture, biofacts or ecofacts, sites, and cultural landsca ...

s opened a tomb in

Hubei Hubei (; ; alternately Hupeh) is a landlocked province of the People's Republic of China, and is part of the Central China region. The name of the province means "north of the lake", referring to its position north of Dongting Lake. The p ...

province. It is among the corpus of texts known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western

Han dynasty The Han dynasty (, ; ) was an Dynasties in Chinese history, imperial dynasty of China (202 BC – 9 AD, 25–220 AD), established by Emperor Gaozu of Han, Liu Bang (Emperor Gao) and ruled by the House of Liu. The dynasty was preceded by th ...

. 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 ''Suàn shù shū'' 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. The '' Zhoubi Suanjing'', a mathematics and

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by

Cai Yong Cai Yong (Chinese: ; 132–192), courtesy name Bojie, was Chinese astronomer, calligrapher, historian, mathematician, musician, politician, and writer of the Eastern Han dynasty. He was well-versed in calligraphy, music, mathematics and astrono ...

Western translations

The title of the book has been translated in a wide variety of ways. In 1852, Alexander Wylie referred to it as ''Arithmetical Rules of the Nine Sections.'' With only a slight variation, the Japanese historian of mathematics Yoshio Mikami shortened the title to ''Arithmetic in Nine Sections.'' David Eugene Smith, in his ''History of Mathematics (Smith 1923)'', followed the convention used by Yoshio Mikami. Several years later,

George Sarton George Alfred Leon Sarton (; 31 August 1884 – 22 March 1956) was a Belgian-born American chemist and historian. He is considered the founder of the discipline of the history of science as an independent field of study. His most influential work ...

took note of the book, but only with limited attention and only mentioning the usage of red and black rods for positive and negative numbers. In 1959,

Joseph Needham Noel Joseph Terence Montgomery Needham (; 9 December 1900 – 24 March 1995) was a British biochemist, historian of science and sinologist known for his scientific research and writing on the history of Chinese science and technology, i ...

and

Wang Ling (historian) Wang Ling (王鈴; 王铃), 1917 or 1918–1994) was a Chinese historian and sinologist. He was known for his collaboration with Joseph Needham on the history of science and technology in China. Biography Wang Ling was born in Nantong, China ...

translated Jiu Zhang Suan shu as The Nine Chapters on the Mathematical Art for the first time. Later in 1994, Lam Lay Yong used this title in her overview of the book, as did other mathematicians including John N. Crossley and Anthony W.-C Lun in their translation of Li Yan and Du Shiran's ''Chinese Mathematics: A Concise History'' (Li and Du 1987). Afterwards, the name The Nine Chapters on the Mathematical Art stuck and became the standard English title for the book.

Contents of ''The Nine Chapters'' are as follows: # 方田 ''Fangtian'' - Bounding fields.

Area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

s of fields of various shapes, such as rectangles, triangles, trapezoids, and circles; manipulation of vulgar fractions. Liu Hui's commentary includes a method for calculation of π and the approximate value of 3.14159. # 粟米 ''Sumi'' - Millet and rice. Exchange of commodities at different rates; unit pricing; the Rule of Three for solving proportions, using fractions. # 衰分 ''Cuifen'' - Proportional distribution. Distribution of commodities and money at proportional rates; deriving arithmetic and geometric sums. # 少廣 ''Shaoguang'' - Reducing dimensions. Finding the diameter or side of a shape given its volume or area. Division by mixed

number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...

s; extraction of square and

cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. F ...

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

, perimeter and

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

. # 商功 ''Shanggong'' - Figuring for construction. Volumes of

solid Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structur ...

s of various shapes. # 均輸 ''Junshu'' - Equitable taxation. More advanced word problems on proportion, involving work, distances, and rates. # 盈不足 ''Yingbuzu'' - Excess and deficit. Linear problems (in two unknowns) solved using the principle known later in the West as the '' rule of false position''. # 方程 '' Fangcheng'' - The two-sided reference (i.e. Equations). Problems of agricultural yields and the sale of animals that lead to systems of linear equations, solved by a principle indistinguishable from the modern form of

.http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf # 勾股 ''Gougu'' - Base and altitude. Problems involving the principle known in the West as the

Major contributions

Real number system

''The Nine Chapters on the Mathematical Art'' does not discuss natural numbers, that is, positive integers and their operations, but they are widely used and written on the basis of natural numbers. Although it is not a book on fractions, the meaning, nature, and four operations of fractions are fully discussed. For example: combined division (addition), subtraction (subtraction), multiplication (multiplication), warp division (division), division (comparison size), reduction (simplified fraction), and bisector (average). The concept of negative numbers also appears in "Nine Chapters of Arithmetic". In order to cooperate with the algorithm of equations, the rules of addition and subtraction of positive and negative numbers are given. The subtraction is "divide by the same name, benefit by different names. The addition is "divide by different names, benefit from each other by the same name. Among them, "division" is subtraction, "benefit" is addition, and "no entry" means that there is no counter-party, but multiplication and division are not recorded. ''The Nine Chapters on the Mathematical Art'' gives a certain discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. It basically has the prototype of real number system.

Gou Gu (Pythagorean) Theorem

The geometric figures included in ''The Nine Chapters on the Mathematical Art'' are mostly straight and circular figures because of its focus on the applications onto the agricultural fields. In addition, due to the needs of civil architecture, ''The Nine Chapters on the Mathematical Art'' also discusses volumetric algorithms of linear and circular 3 dimensional solids. The arrangement of these volumetric algorithms ranges from simple to complex, forming a unique mathematical system. Regarding the direct application of the Gou Gu Theorem, which is precisely the Chinese version of the Pythagorean Theorem, the book divides it into four main categories: Gou Gu mutual seeking, Gou Gu integer, Gou Gu dual capacity, Gou Gu similar. Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including famously the triple 3,4,5. Gou Gu dual capacity discusses algorithms for calculating the areas of the inscribed rectangles and other polygons in the circle, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles.

Completing of squares and solutions of system of equations

The methods of completing the squares and cubes as well as solving simultaneous linear equations listed in ''The Nine Chapters on the Mathematical Art'' can be regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms in ''The Nine Chapters on the Mathematical Art'' are very detailed. Through these discussions, one can understand the achievements of the development of ancient Chinese mathematics. Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but also general quadratic and cubic equations. It is the basis for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics. The "equations" discussed in the Fang Cheng chapter are equivalent to today's simultaneous linear equations. The solution method called "Fang Cheng Shi" is best known today as Gaussian elimination. Among the eighteen problems listed in the Fang Cheng chapter, some are equivalent to simultaneous linear equations with two unknowns, some are equivalent to simultaneous linear equations with 3 unknowns, and the most complex example analyzes the solution to a system of linear equations with up to 5 unknowns.

Significance

The word "Jiu", or "9", means more than just a digit in ancient Chinese. In fact, since it is the largest digit, it often refers to something of a grand scale or a supreme authority. Further, the word "Zhang", or "Chapter", also has more connotations than simply being the "chapter". It may refer to a section, several parts of an article, or an entire treatise. Given these historical understanding of ancient Chinese, the book ''The Nine Chapters on the Mathematical Art'' is actually a somewhat mistranslation; it should really mean a grand book for mathematics. In this light, many scholars of the history of Chinese mathematics compare the significance of ''The Nine Chapters on the Mathematical Art'' on the development of Eastern mathematical traditions to that of Euclid's ''Elements'' on the Western mathematical traditions. However, the influence of ''The Nine Chapters on the Mathematical Art'' stops short at the advancement of modern mathematics due to its focus on practical problems and inductive proof methods as opposed to the deductive, axiomatic tradition that Euclid's ''Elements'' establishes. However, it is dismissive to say that ''The Nine Chapters on the Mathematical Art'' has no impact at all on modern mathematics. The style and structure of ''The Nine Chapters on the Mathematical Art'' can be best concluded as "problem, formula, and computation". This process of solving applied mathematical problems is now pretty much the standard approach in the field of applied mathematics.

Notable translations

*Abridged English translation: Yoshio Mikami: "Arithmetic in Nine Sections", in ''The Development of Mathematics in China and Japan'', 1913. *Highly Abridged English translation:

Florian Cajori Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools first ...

: "Arithmetic in Nine Sections", in ''A History of Mathematics,'' Second Edition, 1919 (possibly copied or paraphrased from Mikami). *Abridged English translation: Lam Lay Yong: ''Jiu Zhang Suanshu: An Overview'', Archive for History of Exact Sciences, Springer Verlag, 1994. *A full translation and study of the Nine Chapters and Liu Hui's commentary is available in Kangshen Shen, ''The Nine Chapters on the Mathematical Art'', Oxford University Press, 1999. *A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary by Karine Chemla and Shuchun Guo is ''Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires''. Paris: Dunod, 2004. . *German translation: Kurt Vogel, ''Neun Bücher Arithmetischer Technik'', Friedrich Vieweg und Sohn Braunsweig, 1968 *Russian translation: E. I Beriozkina, ''Математика в девяти книгах (Mathematika V Devyati Knigah)'', Moscow: Nauka, 1980.

Notes

References

*Needham, Joseph (1986). ''Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth''. Taipei: Caves Books, Ltd. *Straffin, Philip D. "Liu Hui and the First Golden Age of Chinese Mathematics", ''Mathematics Magazine'' (Volume 71, Number 3, 1998): 163–181. *

External links

Full text of the book
(Chinese) {{DEFAULTSORT:Nine Chapters On The Mathematical Art Nine Chapters on the Mathematical Art, The Nine Chapters on the Mathematical Art, The Han dynasty texts Han dynasty literature 2nd-century books