Fangcheng (mathematics): Difference between revisions

'''Fangcheng''' (sometimes written as '''fang-cheng''' or '''fang cheng''') ({{zh|c=方程|p=fāng chéng}}) is the title of the eighth Chapter chapter of the [[Chinese mathematics|Chinese mathematical]] classic [[Jiuzhang suanshu]] (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 BCEBC. 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.<ref name="Hist01">{{cite book |last1author=Jean-Clause Martzloff |title=A History of Chinese Mathematics |url=https://archive.org/details/historychinesema00mart_058 |url-access=limited |date=2006 |publisher=Springer |page=[https://archive.org/details/historychinesema00mart_058/page/n268 250]}}</ref><ref name="Hart01">{{cite book |last1author=Roger Hart |title=The Chinese Roots of Linear Algebra |date=2011 |publisher=The Johns Hopkins University Press |url=https://muse.jhu.edu/chapter/322683 |accessdate=6 December 2016}}</ref> However, as a first approximation it has been translated as "[[Matrix (mathematics)|rectangular arrays]]" or "square arrays".<ref name="Hist01" /> The term is also used to refer to a particular procedure for solving a certain class of problems discussed in the Chapter 8 of the The Nine Chapters book.<ref name="Hart01" />

The procedure referred to by the term ''fangcheng'' and explained in the eighth Chapterchapter of The Nine Chapters, is essentially a procedure to find the solution of systems of ''n'' equations in ''n'' unknowns and it is equivalent to certain similar procedures in modern [[linear algebra]]. The earliest recorded ''fangcheng'' procedure is similar to what we now call [[Gaussian elimination]].

The ''fangcheng'' procedure was popular in ancient [[China]] and it was transmitted to [[Japan]]. It is possible that this procedure was transmitted to [[Europe]] also and served as precursors of the modern theory of [[Matrix (mathematics)|matrices]], [[Gaussian elimination]], and [[determinant]]s.<ref name="Hart02" /> It is well known that there was not much work on linear algebra in [[Greece]] or [[Europe]] prior to [[Gottfried Leibniz]]’s's studies of [[Elimination theory|elimination]] and [[Determinant|determinants]], beginning in 1678. Moreover, Leibniz was a [[Sinophile]] and was interested in the translations of such Chinese texts as were available to him.<ref name="Hart02">{{cite book |last1author=Roger Hart|title=The Chinese Roots of Linear Algebra |date=2011 |publisher=The Johns Hopkins University Press |url=https://muse.jhu.edu/chapter/322679 |accessdate=6 December 2016}}</ref>

== On the meaning of ''fangcheng'' ==

There is no ambiguity in the meaning of the first character ''fang''. It means “rectangle”"rectangle" or “square"square.”" But different interpretations are given to the second character ''cheng'':<ref name="Hart01" />

#The earliest extant commentary, by [[Liu Hui]], dated 263 CE defines ''cheng'' as "measures,”" citing the non-mathematical term ''kecheng'', which means “collecting"collecting taxes according to tax rates.”" Liu then definesdefines ''fangcheng'' as a “rectangle"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’sJi'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 [on the] left and right. ''Cheng'' means terms of a ratio. Terms of a ratio [on the] left and right, combining together numerous objects, therefore [it] is called a "rectangular array"."

#[[Yang Hui]]’s'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”"rectangular" (''fang'') is the shape of the numbers; “measure”"measure" (''cheng'') is the general term for [all forms of] measurement, also a method for equating weights, lengths, and volumes, especially referring to measuring clearly and distinctly the greater and lesser.

Interestingly, sinceSince 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:

! Number of unknowns<br /> in the problem !! Number of equations<br /> in the problem !! Serial numbers of problems !! Number of problems || Determinacy

| || || Total || 18

The presentations of all the 18 problems (except Problem 1 and Problem 3) follow a common pattern:

=== On Problem 1 ===

=== On Problem 3 ===

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

=== On Problem 13 ===

== References ==

*{{cite journal|last1author=Christine Andrews-Larson|title=Roots of Linear Algebra: An Historical Exploration of Linear Systems|journal=PRIMUS|date=2015|volume=25|issue=6|pages=507 - 528507–528|urldoi=http://dx.doi.org/10.1080/10511970.2015.1027975|accessdates2cid=6 December 2016122250602}}

*{{cite book|last1author=Kangshen Shen, |author2=John N. Crossley, |author3=Anthony Wah-Cheung Lun, Hui Liu|title=The Nine Chapters on the Mathematical Art: Companion and Commentary|date=1999|publisher=Oxford University Press|isbn=9780198539360978-0-19-853936-0|pages=386 - 440386–440|url=https://books.google.co.incom/books?id=eiTJHRGTG6YC&pg=PA1&lpg=PA1&dqq=nine+chapters+of+mathematical+art&sourcepg=bl&ots=i5RaRayT5L&sig=tO2daShj5Ey0TlOw5YD_Dl5LIGk&hl=en&sa=X&ved=0ahUKEwiJtOejvOLQAhWsIMAKHSjHDrY4FBDoAQgeMAE#v=onepage&q=nine%20chapters%20of%20mathematical%20art&f=falsePA1|accessdate=7 December 2016}}