The Nine Chapters on the Mathematical Art
Encyclopedia
The Nine Chapters on the Mathematical Art is a Chinese mathematics
book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE. This book is one of the earliest surviving mathematical texts from China
, the first being Suàn shù shū
(202 BCE – 186 BCE) and Zhou Bi Suan Jing (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
mathematicians, who tended to deduce propositions from an initial set of axiom
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
in the 3rd century.
standard measures dated to 179 CE, yet there is speculation that the same book existed beforehand only under different titles.
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
before Carl Friedrich Gauss
(1777–1855). There is also the mathematical proof given in the treatise for the Pythagorean theorem
. There are also features of ancient Western mathematics that are not found in ancient China. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea
and Japan
. Its influence on mathematical thought in China persisted until the Qing Dynasty
era.
Liu Hui
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 proofs in the Euclid
ean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl.
165 BCE - d. 142 BCE) and Geng Shouchang (fl. 75 BCE-49 BCE) (see armillary sphere
) with the initial arrangement and commentary on the book, yet Han Dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.
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, spare mathematical work by those such as Jing Fang
(78–37 BCE), Liu Xin
(d. 23), and Zhang Heng
(78–139) and the geometry clauses of the Mozi
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 archaeologists opened a tomb in Hubei
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
. 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. The Zhou Bi Suan Jing, a mathematics and astronomy
text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE. This book is one of the earliest surviving mathematical texts from China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, the first being Suàn shù shū
Suàn shù shu
The Suàn shù shū , or the Writings on Reckoning , is one of the earliest known Chinese mathematical treatises...
(202 BCE – 186 BCE) and Zhou Bi Suan Jing (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 Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
mathematicians, who tended to deduce propositions from an initial set of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...
in the 3rd century.
History
The full title of The Nine Chapters on the Mathematical Art appears on two bronzeBronze
Bronze is a metal alloy consisting primarily of copper, usually with tin as the main additive. It is hard and brittle, and it was particularly significant in antiquity, so much so that the Bronze Age was named after the metal...
standard measures dated to 179 CE, yet there is speculation that the same book existed beforehand only under different titles.
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
Gaussian elimination
In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
before Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
(1777–1855). There is also the mathematical proof given in the treatise for the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
. There are also features of ancient Western mathematics that are not found in ancient China. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea
Korea
Korea ) is an East Asian geographic region that is currently divided into two separate sovereign states — North Korea and South Korea. Located on the Korean Peninsula, Korea is bordered by the People's Republic of China to the northwest, Russia to the northeast, and is separated from Japan to the...
and Japan
Japan
Japan is an island nation in East Asia. Located in the Pacific Ocean, it lies to the east of the Sea of Japan, China, North Korea, South Korea and Russia, stretching from the Sea of Okhotsk in the north to the East China Sea and Taiwan in the south...
. Its influence on mathematical thought in China persisted until the Qing Dynasty
Qing Dynasty
The Qing Dynasty was the last dynasty of China, ruling from 1644 to 1912 with a brief, abortive restoration in 1917. It was preceded by the Ming Dynasty and followed by the Republic of China....
era.
Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...
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 proofs in the Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
ean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl.
Floruit
Floruit , abbreviated fl. , is a Latin verb meaning "flourished", denoting the period of time during which something was active...
165 BCE - d. 142 BCE) and Geng Shouchang (fl. 75 BCE-49 BCE) (see armillary sphere
Armillary sphere
An armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
) with the initial arrangement and commentary on the book, yet Han Dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.
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, spare mathematical work by those such as Jing Fang
Jing Fang
Jing Fang , born Li Fang , courtesy name Junming , was a Chinese music theorist, mathematician and astrologer. Born in present-day Puyang, Henan during the Han Dynasty , he was the first to notice how closely a succession of 53 just fifths approximates 31 octaves...
(78–37 BCE), Liu Xin
Liu Xin
Liu Xin , later changed name to Liu Xiu , courtesy name Zijun , was a Chinese astronomer, historian, and editor during the Xin Dynasty . He was the son of Confucian scholar Liu Xiang and an associate of other prominent thinkers such as the philosopher Huan Tan...
(d. 23), and Zhang Heng
Zhang Heng
Zhang Heng was a Chinese astronomer, mathematician, inventor, geographer, cartographer, artist, poet, statesman, and literary scholar from Nanyang, Henan. He lived during the Eastern Han Dynasty of China. He was educated in the capital cities of Luoyang and Chang'an, and began his career as a...
(78–139) and the geometry clauses of the Mozi
Mozi
Mozi |Lat.]] as Micius, ca. 470 BC – ca. 391 BC), original name Mo Di , was a Chinese philosopher during the Hundred Schools of Thought period . Born in Tengzhou, Shandong Province, China, he founded the school of Mohism, and argued strongly against Confucianism and Daoism...
of the 4th century BCE. This is no longer the case. The Suàn shù shū
Suàn shù shu
The Suàn shù shū , or the Writings on Reckoning , is one of the earliest known Chinese mathematical treatises...
(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 archaeologists opened a tomb in Hubei
Hubei
' Hupeh) is a province in Central China. The name of the province means "north of the lake", referring to its position north of Lake Dongting...
province. It is among the corpus of texts known as the Zhangjiashan Han bamboo texts
Zhangjiashan Han bamboo texts
The Zhangjiashan Han bamboo texts are ancient Han Dynasty Chinese written works dated 196–186 BCE. They were discovered in 1983 by archaeologists excavating tomb no. 247 at Mount Zhangjia of Jiangling County, Hubei Province . The tomb was built for an early Western Han era official who had...
. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western Han dynasty
Han Dynasty
The Han Dynasty was the second imperial dynasty of China, preceded by the Qin Dynasty and succeeded by the Three Kingdoms . It was founded by the rebel leader Liu Bang, known posthumously as Emperor Gaozu of Han. It was briefly interrupted by the Xin Dynasty of the former regent Wang Mang...
. 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. The Zhou Bi Suan Jing, a mathematics and astronomy
Chinese astronomy
Astronomy in China has a very long history, with historians considering that "they [the Chinese] were the most persistent and accurate observers of celestial phenomena anywhere in the world before the Arabs."...
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
Cai Yong was a Chinese scholar of the Eastern Han Dynasty. He was well-versed in calligraphy, music, mathematics and astronomy. One of his daughters is the famous Cai Wenji.-Early life:...
.
Table of contents
Contents of the Nine Chapters are as follows:- 方田 Fangtian - Rectangular fields. AreaAreaArea is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
s of fields of various shapes; manipulation of vulgar fractions. - 粟米 Sumi - Millet and rice. Exchange of commodities at different rates; pricing.
- 衰分 Cuifen - Proportional distribution. Distribution of commodities and money at proportional rates.
- 少廣 Shaoguang - The lesser breadth. Division by mixed numberNumberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s; extraction of square and cube roots; dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s, area and volumeVolumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
of circleCircleA circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
and sphereSphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. - 商功 Shanggong - Consultations on works. Volumes of solidSolidSolid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
s of various shapes. - 均輸 Junshu - Equitable taxation. More advanced problems on proportion.
- 盈不足 Yingbuzu - Excess and deficit. Linear problems solved using the principle known later in the West as the rule of false position.
- 方程 Fangcheng - The rectangular array. Systems of linear equations, solved by a principle similar to Gaussian eliminationGaussian eliminationIn linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations. It can also be used to find the rank of a matrix, to calculate the determinant of a matrix, and to calculate the inverse of an invertible square matrix...
. - 勾股 Gougu - Base and altitude. Problems involving the principle known in the West as the Pythagorean theoremPythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
.
Accessibility
- Abridged English translation: Florian CajoriFlorian CajoriFlorian Cajori was one of the most celebrated historians of mathematics in his day.- Biography :...
: Arithmetic in Nine Sections, 1893. - Abridged English translation: Lam Lay Yong: Jiu Zhang Suanshu, An Overview, Archive for History of Exact Science, 1994 Springer Verlag.
- A full translation and study of the Nine Chapters and Liu Hui's commentary is available in SHEN Kangshen "The Nine Chapters on the Mathematical Art" Oxford 1999. ISBN 0-19-853936-3
- A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary is Chemla, Karine, and Shuchun Guo. 2004. Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod. ISBN 978-2100495894.
- German translation: Kurt Vogel: Neun Bücher Arithmetischer Technik, Friedrich Vieweg und Sohn Braunsweig 1968
- Russian translation: E. I Brerezkina: Mathematika V Devyati Knigah, Moscow Science Press 1980.
- Japanese translation.
See also
- The Sea Island Mathematical ManualThe Sea Island Mathematical ManualThe Sea Island Mathematical Manual or Haidao suanjing was written by the Chinese mathematician Liu Hui of the Three Kingdoms era as an extension of chapter 9 of The Nine Chapters on the Mathematical Art....
- History of mathematicsHistory of mathematicsThe area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
- History of geometry
External links
- Full text of the book (Chinese)