Quick Answer: What Is The Nine Chapters?
The Nine Chapters (九章算术, Jiǔzhāng Suànshù) is a Chinese mathematical classic that presents problems and methods on topics such as land measurement, grain exchange and solving for several quantities. The transmitted text has a commentary by Liu Hui dated 263 CE and further commentary from the Tang period.[1][4]
- The book is conventionally counted as containing 246 problems.[1]
- Its fangcheng (方程, fāngchéng) chapter includes an array-based procedure for a problem about grain.[2][3]
- The commentary helps explain how procedures work.[5]
The problems give something to calculate. The commentary provides a separate layer for understanding the methods.
Suppose three bundles of high-grade grain, two of medium-grade grain and one of low-grade grain yield a known total. How much does a single bundle of each grade yield? The classic uses several mixtures to make that question answerable.
How the Book Organizes Its Problems
The opening chapter covers areas and arithmetic with fractions. Later sections treat grain ratios, proportional distribution, extracting roots and measuring volumes. A chapter toward the end deals with arrays for finding unknown quantities. The last concerns right triangles.[1]
Those subjects explain why fractions and roots, surveying and arithmetic can all appear in the same book.
The Three Grades of Grain
In the opening fangcheng problem, groups of high-, medium- and low-grade bundles are listed as 3, 2 and 1, then 2, 3 and 1, then 1, 2 and 3. Their total yields are 39, 34 and 26 dou, a historical unit of volume. Each group gives one relationship involving the same three unknown yields.[2]
The procedure sets entries out in columns. The array helps explain why counting rods were useful for calculations involving several quantities at once.[3]
What the Elimination Step Does
Compare the first group with the second. Both have one low-grade bundle, so their totals differ by five dou entirely because the first has one more high-grade bundle and one fewer medium-grade bundle. This is a modern explanation of cancellation using the book’s data. The historical instructions organize and transform the columns to remove unknown quantities in turn.[2][3]
The yields per bundle are 9¼, 4¼ and 2¾ dou, in descending grade order. Substituting them back into the first group gives 27¾ + 8½ + 2¾ = 39. Other problems in the chapter supply material for understanding signed quantities.[2]
What Liu Hui’s Commentary Adds
The older problems and procedures survive with later explanations. Liu Hui’s commentary examines why particular transformations or relations work. Later commentary offers another layer. Separating the layers helps place the classic next to other mathematical records, including excavated manuscripts, the later tianyuan method and the different tradition of numerical squares.[4][5][6]
FAQs About The Nine Chapters
Did Liu Hui write the entire work?
He wrote a commentary on the older work. Later editors and commentators contributed another layer to the transmitted text.[4][5]
Why does the grain problem use three mixtures?
Each mixture gives a relationship involving the same three unknown yields. Comparing the relationships makes it possible to eliminate quantities and solve for each one.[2][3]
Was the original notation the same as modern algebra?
The instructions use a historical arrangement of numbers in columns. Modern equations can express the same relationships for comparison.[2]
Источники
[1] MacTutor, University of St Andrews. Nine Chapters on the Mathematical Art
[2] Chinese Text Project. The Nine Chapters, Fang Cheng
[3] Shidian Classics. The Nine Chapters, volume eight
[4] Mathematical Association of America. Nine Chapters on the Art of Mathematics
[5] Karine Chemla. Reading instructions of the past
[6] Karine Chemla. How excavated manuscripts and transmitted canons shed light on each other