What are the famous ancient Chinese mathematical works?

Mathematics in ancient China, like astronomy and many other sciences and technologies, also made extremely brilliant achievements. It is no exaggeration to say that until the middle of the Ming Dynasty, China was in the leading position in many branches of mathematics. Many ancient Chinese mathematicians had written many famous mathematical works. Many achievements of world significance have survived precisely because of these ancient arithmetic books. These ancient Chinese mathematical masterpieces are a rich treasure trove for understanding ancient mathematical achievements.

For example, the earliest mathematical works now known, the Zhou Thigh Calculations and the Nine Chapters of the Mathematical Art, both of which date from before and after the A.D. Era, are about two thousand years old now. To be able to make mathematical books from two thousand years ago circulate to the present is a remarkable achievement in itself.

In the beginning, people learned and passed on their knowledge of mathematics to the next generation by copying. It was not until the Northern Song Dynasty, with the development of printing, that printed math books began to appear, which is probably the earliest appearance of printed math works in the world. Now in the Beijing Library, Shanghai Library, Peking University Library, the heirloom of the Southern Song Dynasty, "Zhou Thigh Calculator", "nine chapters of arithmetic," and other five mathematical books, but also a valuable cultural relics to be treasured.

From the Han and Tang dynasties to the Song and Yuan dynasties, famous books of arithmetic have appeared in all generations: either they annotated the existing books of arithmetic with traditional Chinese methods, and put forward their own new algorithms in the process of annotating; or they wrote a new book to create new sayings and establish new ideas. In these ancient books of arithmetic handed down in the cohesion of successive generations of mathematicians of the fruits of their labor, they are the generations of mathematicians *** with the valuable legacy left behind.

The Ten Books of Arithmetic refers to the ten famous mathematical works of the Han and Tang Dynasties for more than a thousand years, which used to be the textbooks of the Arithmetic Section of the State Children's Prison (the mathematical section of the school established by the state) in the Sui and Tang Dynasties. The names of the ten books of arithmetic are: Zhou Thighs Arithmetic, Nine Chapters of Arithmetic, Hai Dao Arithmetic, Wu Cao Arithmetic, Sun Zi Arithmetic, Xia Hou Yang Arithmetic, Zhang Qiu Jian Arithmetic, Wu Jing Arithmetic, Zhi Gu Arithmetic, and Shu Jiao (Suffixing Art).

These ten books of arithmetic, with the "Zhou Thigh arithmetic scripture" for the earliest, do not know who its author is, according to the evidence, it was written in the late Western Han Dynasty (the first century BC) when the age of the book is no later than. The book is not only a mathematical work, but also an astronomical work about a school of astronomy at that time, the "Gaitian theory". As far as the mathematical content is concerned, the book contains astronomical calculations using the hook and strand theorem, as well as more complicated fractional calculations. Of course, it cannot be said that both of these algorithms were not mastered until the first century B.C., but it simply means that the Zhou Thigh Calculator is the earliest of the now-known sources.

A comprehensive and complete account of all aspects of ancient mathematics is the Nine Chapters of Arithmetic, which is the most important of the ten books of arithmetic. Its influence on the later development of ancient Chinese mathematics was just as profound as that of the Geometrical Principles of Euclid (c. 330-275 BC) in ancient Greece on Western mathematics. In China, it was used directly as a textbook for math education for over a thousand centuries. It also influenced foreign countries, and was used as a textbook in both Korea and Japan.

The Nine Chapters of Arithmetic, and it is not known who the actual author was, only that Zhang Cang (201-152 BC), Geng Shouchang, and other famous mathematicians in the early Western Han Dynasty, had added to and deleted from it. In 1984, the Book of Arithmetic was unearthed in the early Western Han Dynasty tomb of Zhangjiashan in Jiangling, Hubei Province, and was projected to have been written more than one and a half centuries earlier than the Book of Nine Chapters, and its content was very similar to that of the Book of Nine Chapters, with some of the questions and problems being similar to those of the Book of Arithmetic. Nine chapters of arithmetic" arithmetic questions and sentences are basically the same, can be seen in the two books have some inheritance relationship. It can be said that "Nine Chapters of the Mathematical Art" was gradually formed over a long period of time after many revisions, although some of its algorithms may have been in place before the Western Han Dynasty. As reflected in the title of the book, the whole book *** is divided into nine chapters, a *** collection of two hundred and forty-six mathematical problems, together with the solution of each problem, is divided into nine categories, each category is considered a chapter.

From the point of view of mathematical achievements, the first thing that should be mentioned is that: the book recorded the world's most advanced fractions at the time of the four operations and proportionality algorithms. The book also contains algorithms for solving various area and volume problems as well as various problems of measurement using the hook-and-square theorem. The most important achievements in the Nine Chapters on Arithmetic are in the area of algebra, where the methods of squaring and cubing are documented, and where numerical solutions to general quadratic equations (with non-negative coefficients in the first term) are based on them. There is also a whole chapter on the solution of associative quadratic equations, which is essentially the same as the method now taught in secondary schools. This is more than fifteen hundred years ahead of similar algorithms in Europe. In the same chapter, for the first time in the history of world mathematics, the concept of negative numbers and the rules for adding and subtracting positive and negative numbers are also recorded.

The Nine Chapters of the Algorithm not only occupies an important position in the history of Chinese mathematics, but its influence also extends far beyond its borders. In the European Middle Ages, some of the algorithms in the Nine Chapters, such as fractions and proportions, may have been introduced to India and then to Europe through Arabia. Another example is "surplus and deficit" (can also be regarded as a kind of one-time interpolation), in the Arab and early European mathematical works, was called "Chinese algorithm". Now, as one of the world's leading scientific works, the Nine Chapters of Arithmetic has been translated and published in many languages.

The third of the Ten Books of Arithmetic is the Haidao Shuijing, which was written by Liu Hui (c. 225-c. 295) during the Three Kingdoms period. This book is all about using markers to make two, three, and most complexly, four measurements to solve various measurement math problems. This mathematics of measurement was the mathematical basis of the very advanced cartography of ancient China. In addition, Liu Hui's work on the annotations of the Nine Chapters of the Mathematical Art is also very famous. Generally speaking, these annotations can be regarded as mathematical proofs of several algorithms in the Nine Chapters of the Mathematical Art. Liu Hui's "Circle Cutting" pioneered an important method of calculating pi in ancient China (see page 98 of this book), and he also applied the concept of limit for the first time to the solution of mathematical problems.

The rest of the Ten Books of Arithmetic also contain some achievements of world significance. For example, Sun Tzu's problem of "not knowing the number of things" (the solution of one congruent equation, see page 106 of this book), and Zhang Qiujian's problem of "a hundred chickens" (indeterminate equations) in the book of arithmetic are all relatively famous. And the method of solving cubic equations in the "Searching for Ancient Arithmetic Scriptures", especially the method of making cubic equations by geometrical methods described therein, is also very distinctive.

The Suffixed Art is the work of Zu Chongzhi, a famous mathematician in the period of North and South Dynasties. Unfortunately, this book was lost around the 10th century AD in the Tang and Song dynasties. When the Song people published the Ten Books of Arithmetic, they used another book of arithmetic found at that time, the Recorded Remains of Mathematical Art, to fill the book. Zu Chongzhi's famous work on the calculation of pi (accurate to the sixth decimal place) is recorded in the Sui Shu - Ruling and Calendar Zhi (see page 101 of this book).

Mathematical terms used in the Ten Books of Arithmetic, such as numerator, denominator, open square, open cube, positive, negative, equation, and so on, have been used up to the present day, some of them nearly 2,000 years old.

Ancient Chinese mathematics, after more than a thousand years of development from the Han to the Tang Dynasty, has formed a more complete system. On this basis, there were new developments in the Song-Yuan period (10th to 14th centuries A.D.). Song and Yuan mathematics, from the point of view of its rapid development, the number of mathematical works appeared and the high achievements, can be said to be the most glorious page in the history of ancient Chinese mathematics.

In particular, in the second half of the thirteenth century A.D., four famous mathematicians, Qin Jiushao (1202-1261), Li Ye (1192-1279), Yang Hui, and Zhu Shijie, appeared in just a few decades. The so-called Song-Yuan Algorithms refer to the mathematical works of these four masters that have been passed down to the present, including:

Nine Chapters of the Book of Numbers by Qin Jiushao (1247 A.D.);

Measurement of the Circular Sea Mirror by Li Ye (1248 A.D.) and Yigu yuanduan (1259 A.D.);

Detailed Explanation of the Algorithms of the Nine Chapters by Yang Hui (1261 A.D.), The Algorithms for Daily Use (1261 A.D.), and Algorithms for Daily Use (1262 AD), and Yang Hui's Algorithms (1274-1275 AD),

Ju Shijie's Enlightenment of Arithmetic (1299 AD), and Siyuan Yujian (1303 AD).

The Nine Chapters of the Book of Numbers focuses on two important achievements: the numerical solution of higher equations and the solution of one congruent equation (see pp. 119 and 110 of this book, respectively). Some problems in the book require the solution of equations of ten times, and some problems have as many as one hundred and eighty answers. The Measuring Circle and Sea Mirror and the Yigu Yan Duan tell of another achievement of Song-Yuan mathematics: tianyuan jutsu (equations by algebraic methods, see p. 121 of this book); they also tell of the relationship between the segments of a right-angled triangle and the inner circle, a unique geometry in ancient Chinese mathematics. Yang Hui's work describes another important aspect of Song and Yuan mathematics: practical mathematics and various shortcut algorithms. This was a new direction that emerged in response to the socio-economic development of the time and created the conditions for the creation of the bead abacus. Zhu Shijie's Enlightenment of Arithmetic is worthy of being a textbook of enlightenment at that time, from the shallow to the deep, step by step, until the more advanced content of mathematics at that time. The Four Elements of Yujian records two other achievements of Song-Yuan mathematics: the Four Elements Technique (for solving higher-order systems of equations, see page 123 of this book) and the Higher-Order Equivalent Difference Series, and the Higher Recruitment Difference Method (see page 131 of this book).

These achievements in Song-Yuan arithmetic books are compared with their Western counterparts: the numerical solution of higher-order equations preceded Horner's (1786-1837) method by more than 500 years, the quaternion technique by more than 400 years before that of Bezeau (1730-1783),1 and the higher-order solving method by more than 400 years before that of Newton (1642-1727) and others. -1727) and others by nearly four hundred years.

The brilliant achievements recorded in the Song and Yuan arithmetic books prove once again that, until the middle of the Ming Dynasty, many aspects of Chinese science and technology were in a far advanced position.

After the Song and Yuan dynasties, there were also many books on arithmetic in the Ming and Qing dynasties. For example, in the Ming Dynasty, there is a famous book of arithmetic, "Algorithm Unification". This is a popular book about the bead abacus. After the Qing Dynasty, although there are also many books on arithmetic, but not many achievements as significant as those contained in the Ten Books of Arithmetic and the Song and Yuan books on arithmetic. Especially in the late Ming and early Qing dynasties, there were a lot of books that introduced Western mathematics. This reflects the gradual backwardness of China's science and technology after the development of Western capitalism into the modern scientific period, but also reflects the gradual integration of Chinese mathematics into the world's mathematical development of the general trend of a process.

The history of the development of Chinese mathematics shows that Chinese mathematics has made outstanding contributions to the development of world mathematics, but it is only in modern times that it has gradually fallen behind. We are convinced that, with hard work, Chinese mathematics will be able to catch up with the rest of the world.