One of the earliest and most influential pieces of mathematical literature from ancient China is the Chiu-chang Suan Shu or The Nine chapters on the Mathematical Art. This treatise is a complication of centuries of mathematical knowledge that have been conveyed through the generations by many Chinese figures that have worked in mathematics or in the applications of mathematics. A few prominent figures are known in modern knowledge as contributors to the Chiu-chang Suan Shu but many of the authors have been lost in time. The The Nine Chapters provides a foundation that supported the growth of Chinese and Eastern Mathematics.
The Nine Chapters is divided into nine chapters or sections. Each of these sections covers several problems that related to a central topic. Solutions are provided to the reader as well as methods to find the solutions. In all 246 problems are presented and worked out in the The Nine Chapters. The presentational structure of the Nine Chapters is foundational in the communication of mathematics in China and similar methods are used to this day. Each problem is presented with background information and details about the objective of the problem. Following this section is an explanation of the methods used to solve the problem. Finally, the solution is given. This structure of presenting the problem, explaining the methods, and showing the solution is a model that other mathematical books drew from. The question and answer form shows that the inductive method was used to some extent to show that one application of the problem could be used on similar problems. Although the Nine Chapters shows practical methods for solving common problems, it also serves as a compendium of the concepts and methods that where present in ancient China.
Historical records cannot determine the exact date of when The Nine Chapters were written or who the original authors were. It is known, however, that the development of the manuscript began sometime during or preceding the Zhou Dynasty. The Nine Chapters were then added to throughout the Qin Dynasty and into the Han Dynasty. The development of the Chiu-chang Suan Shu spanned man y centuries, starting circa 1100 BCE and ending around 220 CE. Many of the oldest copies of the Nine Chapters have been lost and intentionally destroyed. In the year 213 BCE, by decree of the despotic Emperor of the Qin Dynasty, Shih Hoang-ti, all books were to be burned.
Many important works were lost forever, including astronomy and mathematical texts, as well as the Chiu-chang Suan Shu. Shortly after the death of Shih Hoang-ti, the corrupt regime collapsed and the development of science and culture was no longer prevented or destroyed Fortunately the burned books of the Chiu-chang Suan Shu were not the only copies that were created. Wooden block carvings, bamboo records and manuscripts found in other areas of China preserved the content from destruction. Several authors have contributed to the creation and organization of the Chiu-chang Suan Shu over the centuries but three notable figures are recognized by history.
The most noted contributor to The Nine Chapters was by a commentator, Liu Hui. Born in the Wei Kingdom in the Shansi province of China around the year 220 CE, Liu Hui lived and worked during a time of great upheaval in ancient China. Hui's childhood occurred in the shadow of a major social and political change. The Han Dynasty, which had controlled China for over four hundred years, had collapsed and out of the political change, the top three generals in the now former Emperor's military had recently set up kingdoms across the country. This time of history in China was known as the Period of the Three Kingdoms. The birth and childhood of Liu Hui is thought to have coincided with the eve of the Period of the Three Kingdoms.
Liu Hui commented on The Nine Chapters and provided documentation that is the key source of the history of Nine Chapters. From Hui's accounts it is known that copies of the manuscript were destroyed by the tyrant Shih Hoang-ti. Hui also credited other figures in history as having contributed to The Nine Chapters. In addition to commenting and editing The Nine Chapters, Hui contributed an algorithm that attempts to find the value of the constant, πand he began to work on fundamental ideas behind Integral and Differential Calculus.
After the collapse of Shih Hoang-ti's government the Han Dynasty came into power. It was during this tim that the arguably the most important figure in the history of The Nine Chapters came into view. Zheng Cang was a marquis and statesman under the Han Dynasty born in. Working for several years as a head minister. Prior to his involvement with the Han Dynasty, he served in similar positions in the now defunct Ch'in monarchy. During the transition of power, Cang fought alongside the Han. While being heavily involved in politics, Cang also had over one hundred wives and possibly several hundred children. While the birth date of Zheng Cang is not known, it is believed that Cang might have lived to be hundred years of age, dying in 152 BCE. Shortly after the fall of the Hoang-ti's government, Cang gathered what he could of the remaining manuscripts and complied them into one of the first forms of The Nine Chapters. Cang's work was key in the resurrecting of the ancient chinese mathematics. By reconstructing the documents, Cang was able to preserve a part of Eastern Mathematical history.
Zhang Heng was a mathematician and political figure during his lifetime. Heng was born in 78 CE and died in 139 CE. He served in the high ranks of the government as well as many other achievements. Heng is credited for inventing one of the first seismographs that was able to detect earthquakes from over 300 miles away. Heng also created a hydraulically powered equatorial armillary sphere. This ancient star chart helped chinese astronomers create more accurate celestial maps and aided in the calendar reform that was happening in China at the time. As for contributions to the The Nine Chapters, Heng worked on approximations of πamong other things.
Chapter VIII of The Nine Chapters goes by the name 'Rectangular Arrays'. In this chapter 18 problems are presented that related to the concepts that are known in modern times as Linear Algebra and positive and negative quantities. The topic of interest is the applications or linear algebra in The Nine Chapters. Problem number one of chapter eight begins as follows:
Top-grade ears of rice, one bundle, medium grade ears of rice, two bundles, low grade ears of rice, one bundle, makes 39 dou*.
Top-grade ears of rice two bundles, medium grade ears or rice three bundles, low grade ears of rice one bundle makes 34 dou.
Top-grade ears if rice one bundle, medium grade ears of rice two bundles, low grade ears of rice three bundles makes 26 dou.
*dou, a measurement of rice by volume.
Using modern techniques this is written as:
The modern notation is functionally the same as the Ancient Chinese methods but instead of numeric characters, the Ancient Chinese used counting rods, probably pieces of wood or bamboo, arranged in a meaningful way, to represent the system of equations. The next step is to place the system of linear equations into an augmented matrix.
Next by using elementary row operations, the matrix can be changed into reduced row form. The following is a result of the row operations.
This gives the following set or linear equations:
Solving for each coefficient results in the following:
This Ancient chinese method can be easily converted to modern techniques.
The Nine Chapters on the Mathematical Art are one of the foundations of mathematics in Asia. Through the centuries many devoted contributors complied, preserved and saved the manuscripts from the passing of time and destruction from civil unrest. The Nine Chapters provide a place for the mathematical knowledge from Ancient China to be saved in a central piece of work. The Nine Chapters on the Mathematical Art acted as the keystone for the development or Mathematics in China.
Sources:
Li, Yen. Chinese mathematics a concise history. Trans. John N. Crossley and Anthony W. Lun. Oxford [England]: Clarendon, 1987. Print.
"Liu_Hui biography." Welcome to the Turnbull Server. Web. 09 Dec. 2009. .
Mikami, Yoshio. Mathematics in China and Japan. New York: Chelsea Company, 1913. Print.
"Zhang_Heng biography." Welcome to the Turnbull Server. Web. 09 Dec. 2009. .
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