Fangcheng (sometimes written as fang-cheng or fang cheng) () is the title of the eighth chapter of the Chinese mathematical classic

Jiuzhang suanshu ''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 2nd century CE. This book is one of the earliest sur ...

(The Nine Chapters on the Mathematical Art) composed by several generations of scholars who flourished during the period from the 10th to the 2nd century BC. This text is one of the earliest surviving mathematical texts from China. Several historians of Chinese mathematics have observed that the term ''fangcheng'' is not easy to translate exactly. However, as a first approximation it has been translated as " rectangular arrays" or "square arrays". The term is also used to refer to a particular procedure for solving a certain class of problems discussed in Chapter 8 of The Nine Chapters book. The procedure referred to by the term ''fangcheng'' and explained in the eighth chapter of The Nine Chapters, is essentially a procedure to find the solution of systems of ''n'' equations in ''n'' unknowns and is equivalent to certain similar procedures in modern

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

. The earliest recorded ''fangcheng'' procedure is similar to what we now call

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 ...

. The ''fangcheng'' procedure was popular in ancient China and was transmitted to

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 north ...

. It is possible that this procedure was transmitted to

Europe Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...

also and served as precursors of the modern theory of

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

, and

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

s. It is well known that there was not much work on linear algebra in

Greece Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders with ...

prior to

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

's studies of elimination and determinants, beginning in 1678. Moreover, Leibniz was a

Sinophile A Sinophile is a person who demonstrates a strong interest for China, Chinese culture, Chinese language, Chinese history, and/or Chinese people. Those with professional training and practice in the study of China are referred to as Sinol ...

and was interested in the translations of such Chinese texts as were available to him.

On the meaning of ''fangcheng''

There is no ambiguity in the meaning of the first character ''fang''. It means "rectangle" or "square." But different interpretations are given to the second character ''cheng'': #The earliest extant commentary, 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 o ...

, dated 263 CE defines ''cheng'' as "measures," citing the non-mathematical term ''kecheng'', which means "collecting taxes according to tax rates." Liu then defines ''fangcheng'' as a "rectangle of measures." The term ''kecheng'', however, is not a mathematical term and it appears nowhere else in the Nine Chapters. Outside of mathematics, ''kecheng'' is a term most commonly used for collecting taxes. #Li Ji's "Nine Chapters on the Mathematical Arts: Pronunciations and Meanings" also glosses ''cheng'' as "measure," again using a nonmathematical term, ''kelü'', commonly used for taxation. This is how Li Ji defines ''fangcheng'': "''Fang'' means

n the N, or n, is the fourteenth Letter (alphabet), letter in the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is English alphabet# ...

left and right. ''Cheng'' means terms of a ratio. Terms of a ratio

left and right, combining together numerous objects, therefore tis called a "rectangular array"." #

Yang Hui Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theor ...

's "Nine Chapters on the Mathematical Arts with Detailed Explanations" defines ''cheng'' as a general term for measuring weight, height, and length. Detailed Explanations states: What is called "rectangular" (''fang'') is the shape of the numbers; "measure" (''cheng'') is the general term for ll forms ofmeasurement, also a method for equating weights, lengths, and volumes, especially referring to measuring clearly and distinctly the greater and lesser. Since the end of the 19th century, in Chinese mathematical literature the term ''fangcheng'' has been used to denote an "equation." However, as already been noted, the traditional meaning of the term is very different from "equation."

Contents of the chapter titled ''Fangcheng''

The eighth chapter titled ''Fangcheng'' of the ''Nine Chapters'' book contains 18 problems. (There are a total of 288 problems in the whole book.) Each of these 18 problems reduces to a problem of solving a system of simultaneous linear equations. Except for one problem, namely Problem 13, all the problems are determinate in the sense that the number of unknowns is same as the number of equations. There are problems involving 2, 3, 4 and 5 unknowns. The table below shows how many unknowns are there in the various problems: The presentations of all the 18 problems (except Problem 1 and Problem 3) follow a common pattern: *First the problem is stated. *Then the answer to the problem is given. *Finally the method of obtaining the answer is indicated.

On Problem 1

* Problem: ** 3 bundles of high-quality rice straws, 2 bundles of mid-quality rice straws and 1 bundle of low-quality rice straw produce 39 units of rice ** 2 bundles of high-quality rice straws, 3 bundles of mid-quality rice straws and 1 bundle of low-quality rice straw produce 34 units of rice ** 1 bundles of high-quality rice straw, 2 bundles of mid-quality rice straws and 3 bundle of low-quality rice straws produce 26 units of rice ** Question: how many units of rice can high, mid and low quality rice straw produce respectively? * Solution: ** High-quality rice straw each produces 9 + 1/4 units of rice ** Mid-quality rice straw each produces 4 + 1/4 units of rice ** Low-quality rice straw each produces 2 + 3/4 units of rice The presentation of Problem 1 contains a description (not a crisp indication) of the procedure for obtaining the solution. The procedure has been referred to as ''fangcheng shu'', which means "''fangcheng'' procedure." The remaining problems all give the instruction "follow the ''fangcheng''" procedure sometimes followed by the instruction to use the "procedure for positive and negative numbers".

On Problem 3

There is also a special procedure, called "procedure for positive and negative numbers" (''zheng fu shu'') for handling negative numbers. This procedure is explained as part of the method for solving Problem 3.

On Problem 13

In the collection of these 18 problems Problem 13 is very special. In it there are 6 unknowns but only 5 equations and so Problem 13 is indeterminate and does not have a unique solution. This is the earliest known reference to a system of linear equations in which the number of unknowns exceeds the number of equations. As per a suggestion of Jean-Claude Martzloff, a historian of Chinese mathematics, Roger Hart has named this problem "the well problem."

On the meaning of ''fangcheng''

Contents of the chapter titled ''Fangcheng''

On Problem 1

On Problem 3

On Problem 13

References

Further reading