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 A page ofThe Nine Chapters on the Mathematical Art (1820 edition) | |||||||||||||||||
| Traditional Chinese | 九章算術 | ||||||||||||||||
| Simplified Chinese | 九章算术 | ||||||||||||||||
| Literal meaning | nine chapters on arithmetic | ||||||||||||||||
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The Nine Chapters on the Mathematical Art is a Chinesemathematics 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 fromChina, the others being theSuan shu shu (202 BCE – 186 BCE) andZhoubi 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 toancient Greek mathematicians, who tended to deduce propositions from an initial set ofaxioms.
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 byLiu Hui in the 3rd century.
The book was later included in the early Tang collection, theTen Computational Canons.
The full title ofThe Nine Chapters on the Mathematical Art appears on twobronze standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles.[1]
The title is also mentioned in volume 24 of theBook of the Later Han as one of the books studied by Ma Xu (馬續). Based on the known dates of his younger brother Ma Rong (馬融) this places the date of composition no later than 93 CE.[2]
Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time whenThe 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 usesGaussian elimination beforeCarl Friedrich Gauss (1777–1855).[3] There is also themathematical proof given in the treatise for thePythagorean theorem.[4] The influence ofThe Nine Chapters greatly assisted the development of ancient mathematics in the regions ofKorea andJapan. Its influence on mathematical thought in China persisted until theQing dynasty era.
Liu Hui wrote a detailed commentary in 263. He analyses the procedures ofTheNine 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 provideformal proofs in theEuclidean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematiciansZhang Cang (fl. 165 BCE – d. 142 BCE) and Geng Shouchang (fl. 75 BCE – 49 BCE) (seearmillary sphere) with the initial arrangement and commentary on the book, yetHan dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century[5]
TheNine 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 asJing Fang (78–37 BCE),Liu Xin (d. 23), andZhang Heng (78–139) and thegeometry clauses of theMozi of the 4th century BCE. This is no longer the case. TheSuàn shù shū (算數書) orWritings on Reckonings 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 whenarchaeologists opened a tomb inHubei province. It is among the corpus of texts known as theZhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the WesternHan dynasty. While its relationship toThe Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of theSuàn shù shū is however much less systematic thanThe Nine Chapters; and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. TheZhoubi Suanjing, a mathematics andastronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE byCai Yong.
The title of the book has been translated in a wide variety of ways.
In 1852,Alexander Wylie referred to it asArithmetical Rules of the Nine Sections.
With only a slight variation, the Japanese historian of mathematicsYoshio Mikami shortened the title toArithmetic in Nine Sections.[6]
David Eugene Smith, in hisHistory of Mathematics (Smith 1923), followed the convention used byYoshio Mikami.
Several years later,George Sarton 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 andWang Ling (historian) translatedJiu Zhang Suan shu asThe 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 includingJohn N. Crossley and Anthony W.-C Lun in their translation of Li Yan and Du Shiran'sChinese Mathematics: A Concise History (Li and Du 1987).[6]
Afterwards, the nameThe Nine Chapters on the Mathematical Art stuck and became the standard English title for the book.
Contents ofThe Nine Chapters are as follows:
| chapter | contents |
|---|---|
| 方田Fangtian Bounding fields | Areas 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 ofπ and the approximate value of 3.14159.[7] |
| 粟米Sumi Millet and rice | Exchange of commodities at different rates; unit pricing; theRule 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 mixednumbers;extraction of square andcube roots;diameter ofsphere,perimeter anddiameter ofcircle. |
| 商功Shanggong Figuring for construction | Volumes ofsolids 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 therule of false position. |
| 方程Fangcheng The two-sided reference (i.e. Equations) | Problems of agricultural yields and the sale of animals that lead tosystems of linear equations, solved by a principle indistinguishable from the modern form ofGaussian elimination.[8] |
| 勾股Gougu Base and altitude | Problems involving the principle known in the West as thePythagorean theorem. |
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).[9]
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.[9]
The Nine Chapters on the Mathematical Art gives a certain discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. Generally speaking, it has the prototype of the real number system used in modern mathematics.
The geometric figures included inThe 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.[9]
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.
The methods of completing the squares and cubes as well as solving simultaneous linear equations listed inThe Nine Chapters on the Mathematical Art can be regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms inThe Nine Chapters on the Mathematical Art are very detailed. Through these discussions, one can understand the achievements of the development of ancient Chinese mathematics.[9]
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.[9]
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.[9]
The wordjiu, 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 wordzhang, 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.[10]
In this light, many scholars of the history of Chinese mathematics compare the significance ofThe Nine Chapters on the Mathematical Art on the development of Eastern mathematical traditions to that of Euclid'sElements on the Western mathematical traditions.[11][12] However, the influence ofThe Nine Chapters on the Mathematical Art stops short[according to whom?] 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'sElements establishes.
However, it is dismissive to say thatThe Nine Chapters on the Mathematical Art has no impact at all on modern mathematics. The style and structure ofThe Nine Chapters on the Mathematical Art can be best concluded as "problem, formula, and computation".[13] This process of solving applied mathematical problems can now be considered the standard approach in the field of applied mathematics.[citation needed]
{{citation}}: CS1 maint: work parameter with ISBN (link)
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