# The History of Mathematics in the Ancient China

It is common knowledge that mathematics is the science of quantitative relations and spatial forms of the real world. However, to be able to investigate these forms and relations in their pure form, it is necessary to separate them entirely from their content, putting it aside as irrelevant. The applications of mathematics are very diverse. In principle, the scope of the mathematical method is unlimited: all types of motion of matter can be studied mathematically.

As a result, with it being in close connection with the demands of natural science and technology, the stock of quantitative relations and spatial forms studied within the boundaries of mathematics is filled with more and more rich content. It is no secret that the science of mathematics dates back to ancient times, but in the different states, the rate of its development varied. This statement is especially true for the countries that spent certain periods of their existence in isolation, i.e. without maintaining a relationship with the others, which has provided for the development of unique culture and science. Among them is the ancient China, which, at the time, had almost no contacts with the European and Arabic countries that are well-known for their contribution to the development of mathematics. Thus, the purpose of this essay is to reveal the peculiarities of the history of this branch of science in China.

## Mathematics in China

### General Information

Mathematics has been developing in China since ancient times, more or less independently and reached its peak in the 14th century. After that, it was enriched by the western ideas and finding, mainly brought by the European missionaries, thus marking another era in the history of the Chinese science. As a result, it is best to focus the attention on the history of mathematics in the ancient China, up to the 7th century, as well as its most typical problems. In most cases, they are of the initial level, being common for the branch o science that has only stated its development. Thus, they relate to the concept of number, shape and its area, body and its volume, the formation of elementary number-theoretic concepts of the arithmetic mean, the greatest common divisor, the least common multiple, and the history of the Pythagorean Theorem.

The Chinese mathematicians possessed a highly developed technology of computation and demonstrated the interest of the general algebraic methods as described in a number of Chinese texts belonging to the ancient and medieval authors. These texts can be divided into the two groups. The first of them includes the collection consisting of the ten classic treatises on mathematics (The Ten Computational Canons). This work has formed the basis for the progress of mathematics in China until the 14th century. The texts included in it were written in the period between the 3rd and 6th centuries. All of them focus on the different topics while possessing certain common properties. In particular, all the texts are essentially anonymous, although some titles of the treatises contain the authors' names. Moreover, the problems presented in the Ten Computational Canons are mostly arithmetic and algebraic rather than geometric in their nature. For example, these works describe such concepts as inter alia and the methods for extracting the square and cube roots of integers. At the same time, they review some calendar issues and even the musical ones. The second group includes more recent works by the individual authors. They are unique in their nature and do not have the scope of the Ten Computational Canons, instead focusing on the individual problems.

### Periodization

At the same time, due to the lack of the contacts with the outside world, the works described above, as well as the Chinese mathematics in general are rather peculiar, which makes the periodization of its history a rather complex issue. However, in general, it is possible to define the four stages of its overall development, providing a qualitative picture. The first of them is the accumulation of mathematical knowledge and the creation of practical mathematics. Next, there is the period of elementary mathematics, i.e. the one that primarily operates with constants. The third stage is focused on the creation of the mathematical variables, being followed by the period of modern mathematics. In this regard, the ancient Chinese mathematics fits entirely in the second period of development, being the science of constant quantities, with the Chinese mathematicians making several striking discoveries. First of all, they have managed to develop the method of numerical solution of n-degree equations (an equivalent of Ruffini-Horner method). They have also developed methods for solving the systems of linear equations (similar to the one created by Gauss). Finally, they were able to calculate the number Pi as a ratio of a diagonal to the side of a square, which demonstrates their ability to keep up with the Western world in the terms of scientific discoveries.

At the same time, the detailed presentation of the history of the ancient Chinese mathematics requires special periodization with the involvement of the traditional Chinese history. In this regard, it is possible to divide it into the three periods. The first of them is the so-called deep antiquity, which embraces the period from the times of the legendary Yellow Emperor to the establishment of the Han dynasty. The second one (antiquity) lasted 100 BC to 600 AD and included the times of Han and Sui dynasties. The third period is the late antiquity (600 – 1367), i.e. Tang, Song and Yuan dynasties.

Thus, the first period in the history of the Chinese mathematics was a normal initial stage of development of science that is common for all ancient civilizations. This was the era of knowledge-building in connection with the inquiries of the economy and the appearance of the first specialized texts on the subject. The numbers were introduced, and the people started using different tools to simplify the process of calculation, including the compass (a drawing tool) and try square. These tools represented the order. During the first period, Chinese also started using calendars. In the middle of the first millennium (the start time of the iron smelting), there have been significant changes in all spheres of life in China. By that time, mathematics was considered to be an independent science, being known under the name of the art of computing. It has also become the subject of the study by the nobles , which has resulted in an increase in its popularity, marking the so-called golden age of this branch of science. The development of mathematics during this period is not studied well enough due to the fact that no texts of that time have survived. However, it is likely that they served as the basis for the Ten Computational Canons mentioned before. As a result, it is possible to obtain the information about the period of the formation of the Chinese mathematics from the individual fragments of this work, as well as the facts provided by the nonmathematical literature of China. The latter includes The Book of Changes, which is based on the sixty-four hexagrams. According to this book, mathematicians of the ancient China were working with the problems of combinatorics and were familiar with binary and ternary systems. It is also possible to include the treatises associated with the development of the dialectic in ancient China, as well as the logics, optics, dynamics and a number of definitions and axioms of geometry. Thus, mathematics was an integral part of the Chinese society.

The second period is associated with the Han Dynasty, the reign of which is divided into two halves: early or the western Han (up to the 1st century) and late or eastern Han (up to the beginning of the 3rd century). During this period, there was a division between the orthodox and unorthodox sciences. In particular, astronomy and mathematics were considered the official (i.e. orthodox) sciences. On the other hand, the part of medical science that relied on natural-philosophical ideas was considered orthodox while the other, which was based on magic, was unorthodox. The year 192 marked the beginning of the era of the Three Kingdoms. By this time, almost all mathematical treatises of The Ten Computational Canons were already written, but the collection was compiled at the beginning of the third period, which signified the creation of the original Chinese algebraic school. As a result, without the constant contacts with the West, the Chinese were able to create unique mathematics. However, after several centuries, the situation has changed significantly. The medieval era has become the period of decline of classical mathematics, providing the conditions for the development of the so-called folk methods of computation. There is a wide distribution of guidelines on the rules on the calculation by using the special tools, as well as the rhymed rhetorical rules. The first Western missionaries arrive in China, resulting in the emergence of the translations of the Elements by Euclid and the other Western literature. As a result, the work of mathematicians goes in the two following directions: a theoretical substantiation of the Western methods and processing and development of the old, traditional problems.

## Computing Techniques and Tools

Little is known about the computing technique of the ancient China, which is sometimes not mentioned at all. However, it is an essential complement to the overall picture of the development of mathematics in the ancient times. The counting technique of the Chinese was based on the decimal numeration, similar to the one in the West, but it used a positional principle. According to it, the same number could have different values depending on its location. In other words, it is a system that is universally used nowadays. It should be noted that the primary reason for its implementation in the ancient China was the system of Chinese hieroglyphs, which did not allow the use of the system similar to the Roman one. At the same time, in the era of the Sui-Tang (the 7th century), the Chinese developed yet another counting technique that was based on the use of the rods. Under this system, the numbers and figures were depicted as a combination of horizontal and vertical rods. A vertical row was used to refer to single digits, hundreds, tens of thousands, and so on while the horizontal one indicated the tens, thousands, hundreds of thousands, etc. The red-colored rods were used to denote positive numbers while the black ones – the negatives. Thus, the counting techniques were rather diverse.

Still, in practice, the calculations were usually carried out on the counting board named suanpan, which also used the positional principle, as well as the decimal numerical system, decimal. It has appeared around the 6th century BC. The modern type of countable unit was created much later, presumably in the 12th century. Suanpan was a rectangular frame in which the parallel wires or ropes (at least nine) were stretched, being the equivalent of the decimal places. It was also partitioned off into the two unequal parts. In a large compartment, each of wires housed five beads (ossicles) while in the smaller one, there were only two. Suanpan was manufactured in various sizes, which means that it was used widely by the people of many professions (merchants, officials, travelers, etc.). As a result, the Chinese have developed a sophisticated technique of work on such counting board. These methods allowed producing all of the four basic arithmetic operations, as well as extract square and cube roots. Additionally, the Chinese sources significantly complement the overall picture of the development of computational methods in antiquity, providing an insight into the problems of numerical systems, arithmetic of integers, and decimals. Thus, it is clear that the computational techniques were quite developed in the ancient China.

### Numbers, Algebra, and Geometry

When reviewing the history of mathematics in the ancient China, it is important to consider the algebraic way of transition from the integers to the rational numbers. However, the historical process that took place in China during the development of the concept of a number was rather general in its nature, being quite similar to the ones that and occurred in all ancient civilizations. It included such concepts as the common fractions, proportion and progression, as well as the problem of the division with a remainder. On the other hand, algebraic methods were characteristic of the Chinese mathematics. It should be noted that at the time, algebra was expounded verbally, without the use of symbols (Martzloff 82). Its primary problems were the linear systems, as well as the solution of equations of higher degrees of through the use of numerical methods. Finally, the geometry of the ancient China was directly connected to algebra, as the algebraic methods were used to solve geometric problems. These included measuring of areas and volumes, the application of Pythagorean Theorem, measurement of the circle and the sphere, and the determination of distances to inaccessible objects. In China, the traces of the application of mathematics can be found everywhere – the construction of pagodas, temples, burial chambers, dams, as well as mapping and astronomical calculations. At the same time, mathematics was permitted only as a science that is used by the state. In China, it was considered a compulsory subject at the state examination, which signifies its importance for the development of the country.

Conclusion

In conclusion, it is possible to say that the development of mathematics in ancient China up to the 7th century gave a strong push for its further improvement and application of methods developed in the future. The development of the group and decimal counting, as well as the positional principle of fixing the numbers, as well as the further invention of counting board (suanpan) for the calculations, has led to the emergence of the positional number system with decimals. In the terms of making estimates of ordinary and decimals, the Chinese mathematicians studied the decimal fractions, which are associated with the division procedure, as well as the extraction of roots, as well as the common fractions and number-theoretic problems. Such concepts as the arithmetic mean of two or more numbers, properties of arithmetic and geometric progression, as well as the negative numbers were well-known in China. The arithmetic of residues, the use of Pythagoras Theorem, and the wide array of works covering many issues of mathematical nature testify to the immense practice in the terms of solving of the decision-theoretic problems, which makes it possible to say that mathematics of the ancient China was very developed for its time.