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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]]. The earliest recorded ''fangcheng'' procedure is similar to what we now call [[Gaussian elimination]].
The ''fangcheng'' procedure was popular in ancient [[China]] and 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 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 |author=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'' ==
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#[[Yang Hui]]'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 [all forms of] measurement, 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
== Contents of the chapter titled ''Fangcheng'' ==
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=== 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 {{sfrac|9
** Mid-quality rice straw each produces {{sfrac|4
** Low-quality rice straw each produces {{sfrac|2
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".
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