Liu Hui (born around 250 AD), Zu Chongzhi (born in 429 AD), Zuxuan (son of Zu Chongzhi), (died in 784 AD), (born in the Northern Wei Dynasty), Qin (born in 1208) and Guo Shoujing (born in123/kloc-).

Born in 249), Jia Xian (Northern Song Dynasty), Yang Hui (Southern Song Dynasty), Zhao Shuang (Wu from the end of the Eastern Han Dynasty to the Three Kingdoms Period), Wang Xun (born in 1235), Xu Guangqi (born in 1562), (born in 1633), Xue.

(1) Ten Classic Books on Computing

The Ten Calculations Classic is a compilation of ten mathematical works that have appeared in China since the Han and Tang Dynasties. Mathematics was set up in national universities in the Tang Dynasty, with ten mathematics books as teaching materials. These ten calculation classics are: Calculation Classics, Nine Chapters Arithmetic, Calculation Classics of Sun Zi, Calculation Classics of Wu Cao, Calculation Classics of Xiahou Yang, Calculation Classics of Zhang Qiu, Calculation Classics of Island, Arithmetic of Five Classics and Composition and Compilation of Ancient Calculation Classics. Among them, Composition by Zu Chongzhi and his son has been lost, and the other nine books are introduced as follows.

First, "Week Fast Shu Jing"

The Book of Zhou Zhi ·suan, which is now circulating, was written in the first century BC and is a work on mathematics and astronomy. People have noticed this throughout the ages, such as Zhao Shuang, Li and other works, most of which are related to astronomical calculation. There are three kinds of people who talk about celestial bodies in China since ancient times, that is, covering the sky, announcing the night and clouding the sky. The origin of Gaitian theory is very early, and Zhouyi ·suan Jing can be called Gaitian theory. Those who cover the sky, as the name implies, call the sky like a hat and the ground like chess. The sun, the moon and the stars run on the sky and live in the mainland. This mathematical work is a book in which astronomers use triangulation to measure the distance between celestial bodies and explain the four poles and seasons. Its contents include the method of learning mathematics, the measurement of Pythagorean theorem, and the calculation of height, depth, distance and proximity with fractions.

Second, "Nine Chapters of Arithmetic"

Nine Chapters Arithmetic is a famous mathematical monograph in ancient China. It inherited the development of pre-Qin mathematics, was added or deleted by many scholars in the Han Dynasty, and finally finalized in the second half of the first century. Most mathematicians in later generations began to study and study mathematics from "Nine Chapters of Arithmetic". Many scholars have also done annotation work, such as "263" by Liu Wei and "656" by Li. Their notes have been circulated with the original book, which was officially published by the state as a mathematics textbook for official schools in the Tang and Song Dynasties.

The book Nine Chapters Arithmetic marks the formation of China's ancient mathematical system, which has the following characteristics:

1. The whole book is expressed in the form of application questions. * * * There are 246 directional questions, most of which are of practical significance. It can be called China ancient applied mathematics system.

2. Its second characteristic is that it pays attention to algorithm. The book consists of three parts: direction, answer and skill, with "skill" as the main content, which is a practical and applicable algorithm.

3. Taking calculation as a calculation tool, the technique can be called calculation and calculation method.

"Nine Chapters Arithmetic" divides 246 questions and 202 "skills" into nine chapters, including square field, millet, declining score, less generalization, business work, average loss, surplus and deficiency, equations, Pythagorean and so on.

Nine Chapters Arithmetic is a world-famous work, and its outstanding achievements include: fractional operation, proportional problem, double solution, calculation of area and volume, solution of linear equation, introduction of negative number and its algorithm, square root, square root and solution of general quadratic equation, etc.

Third, Sun Tzu's Art of War

The book was written in the 4th and 5th centuries, and the author's life and time of writing are unknown. It has three volumes, * * *. In terms of volume, the algorithms of counting, multiplication, division, square root and fraction with counting are described in detail. There are 64 questions in the last two volumes, which belong to practical problems in daily life. There are 67 "skills" in the book, which is the main content.

Question 26 of Volume 2 is the famous "Things are Unknown", which is related to the compilation of astronomical calendar and needs to be solved mathematically once. The algorithm to solve this problem developed into a "big swallow with one skill" in the Song Dynasty. Volume 3 1 is the ancestor of "chickens and rabbits in the same cage" in later generations, and later spread to Japan, becoming "crane turtle counting".

Fourth, "Five Grass Calculations"

Zhen Luan wrote it in the Northern Zhou Dynasty. Cao was an ancient official department, that is, the commercial management department of governments at all levels. "Wu Cao Shu Jing" is a practical mathematics manual for five types of Cao officials, written in the form of problem sets, with a total of 67 questions. These include:

Cao Tian: Calculation of field area;

Soldiers and Cao Cao: army configuration calculation, supply transportation, etc.

Counting grass: calculation of trade exchange;

Storehouse: grain tax, grain cellar volume calculation, etc.

Jin Cao: Calculation of Silk Trade.

Five, "Xiahou Yang Suanjing"

The original book has been lost, and the Book of Xiahou Yangshu, which was engraved in the ninth year of Yuanfeng in Beijing (1084), is a calculation book in the middle Tang Dynasty. The book has 83 mathematical problems in three volumes, citing popular algorithms such as multiplication, division and agile method to solve the application problems in daily life, and also saving quite a lot of mathematical historical materials.

Six, "Zhang Qiujian suan jing"

Neither the author nor the year of writing can be verified, and now it is considered to be a mathematics book in the middle of the fifth century. The book has 92 questions in three volumes, many of which are very useful. In addition to the content in Nine Chapters of Arithmetic, the book also contains arithmetic progression, quadratic equation and indefinite equation. The "hundred chickens problem" of indefinite equation is a typical example of indefinite equation in later generations.

Seven, "island computing"

The Archipelago was written by Liu Wei, a mathematician in the Three Kingdoms period. Is a book about gravity difference technology, gravity difference technology is a method to measure the depth of islands, cities, mountains and wells. Liu Weiyuan attached this part of the book to Nine Chapters of Arithmetic and Pythagoras. As an independent work, the early Tang Dynasty was named after the method of measuring islands in the first chapter, so it was named "Island Calculation". Surveying in the book is the mathematical basis of ancient cartography.

Eight, "Five Classics Arithmetic"

Liu Wei believes that "there are nine numbers in Zhou Gong's rites", and his mathematical works are from Confucian classics.

The Five Classics Arithmetic written by Zhen Luan in the Northern Zhou Dynasty consists of two volumes, namely, Zhen Jing, The Book of Songs, Shu Jing, Li Jing, Chunqiu and other ancient Confucian classics, especially the questions about astronomy and calendar, which are listed in the form of problem sets and answered one by one. This is not only a reference for learning Confucian classics, but also a reference for learning.

IX. Compilation of Ancient Calculations

About the beginning of the seventh century, Wang Kaotong, a mathematician in the Tang Dynasty, wrote twenty questions, most of which were about the technical solution of cubic algebraic equations. The first question is about astronomy. The algorithm belongs to arithmetic. The other nineteen paragraphs are all about the calculation of volume and length. Most of these geometric problems come from civil engineering problems, and the algorithm is the' technical' solution of algebraic equations.

(See Wang Xiaotong, an ancient mathematician in China) The Composition written by Zu Chongzhi and his son has been lost.

X. "Composition"

Composed by Zu Chongzhi and his son, it has been lost.

(2) Other famous ancient mathematical works.

I. Shu Shu's Nine Chapters

Written by Qin, a mathematician in the Southern Song Dynasty in China, 1247 was originally named "Introduction to Mathematical Methods" and was renamed "Nine Chapters" in the late Ming Dynasty. It was included in Yongle Dadian and Sikuquanshu respectively, and Shu Shu Jiu Zhang listed 8 1 questions, which were divided into nine categories with nine questions in each category. These nine categories are as follows: The book has the following characteristics:

1, classified according to problems: the composition talks about mathematics, but also involves natural phenomena and social life.

2. Complete reservation counting method and expression. Natural numbers, fractions, decimals and negative numbers are all discussed in special articles.

3. Summarize the "Large Diffraction-One Technique" and make the solution of a congruence group programmed, which is more than 500 years earlier than the similar method founded by western Gauss.

4. Equation solving has complete calculation steps, which can be used to solve rational roots or irrational roots of any equation, more than 500 years earlier than the similar method of Horner in Britain.

5. The triclinic product formula listed in the book is the same as the Greek Helen formula.

Counting Nine Chapters is the inheritance and development of Nine Chapters Arithmetic, and the book also contains the main mathematical achievements of the Song Dynasty.

Second, the "circle sea mirror"

Written by Ye Li in the Jin and Yuan Dynasties in China, written on 1248. The book is divided into 12 volumes and 170 questions, which is the representative work of discussing Rong Yuan and Tianyuan in China.

The issues discussed in the book are:

1, Pythagorean solution: known Pythagorean method inscribed circle, diameter of tangent circle and so on.

2. The theory of celestial bodies is systematically summarized, which is equivalent to the modern equation theory, and the word algebra begins to evolve into symbolic algebra.

3. Ye Li opposes the tendency to mystify mathematics, such as the positive root of higher-order equations and the algorithm of polynomials. He believes that mathematics comes from the objective world. In the preface, he said that his book originated from the theories of Dong Yuan and Jiu Rong.

The later scholars' evaluation of "Measuring the Round Sea Mirror" is "a treasure book of China's mathematics".

Third, "Siyuan Jade Mirror"

This book was written by Zhu Shijie, a mathematician of Yuan Dynasty in China, and was written in 1303, with three volumes, 24 subjects and 288 questions. This paper mainly discusses the elimination method of higher-order equation, the summation of higher-order arithmetic progression and the method of higher-order interpolation. This book is an important work of China from celestial art to Quaternary art.

Zhu Shijie gave the triangular crib formula of higher-order arithmetic progression sum:

And the formula of recruiting difference, which is more than 400 years earlier than the west.

V. "geometrical features"

Ancient Greek mathematics was written by Euclid (around 300 BC), when there were thirteen volumes. It is a model of establishing mathematical deduction system by axiomatic method and the crystallization of Greek mathematical achievements, methods and thoughts at that time. It has been popular for more than 2000 years since it came out. It has been translated and revised many times. Since the first printed version of 1482 was published, there have been more than 1000 different versions. The earliest Chinese translation of China was 1607 (Wanli period of Ming Dynasty), which was jointly translated by matteo. Matteo Ricci and Xu Guangqi, translated into the first six volumes of fifteen volumes. 250 years later, he and Li He translated the last nine volumes 1857. The Elements of Geometry is a classic work in the historical era of the world history of mathematics, which has great historical value and practical significance and has a great influence on mathematics science and mathematics education in China.

Six, abacus and "unified family of algorithms"

Before the introduction of western mathematics in Ming Dynasty, the greatest achievement was the invention of abacus, and the most important mathematical work was Cheng Dawei's Arithmetic Unity (1592). Before the popularization of electronic computers, abacus was welcomed by the masses for hundreds of years because of its simple structure, low price and rapid calculation, and it is still popular today. The name "abacus calculation" has appeared in the book Numerology, which may be the bud of abacus calculation in later generations. Unfortunately, this book is too simple to describe in detail.

China has always attached importance to computing instruments, and it is natural to develop from computing to abacus. There is a metaphor of "abacus beads" in Tao's Record of Dropping Farms in Ming Dynasty (1366): "abacus beads, when you talk, move." In the Ming Dynasty, Jason Wu (from Renhe County, Hangzhou) wrote "Nine Chapters Arithmetic Comparison" at 1450, "No abacus, no mistakes"; This is the earliest mathematical work that mentioned abacus calculation, such as "It is not difficult to multiply, divide, add and subtract with abacus".

The book about abacus calculation that can really be tested was written by Ke's "Mathematical Passage" (1578), which contains thirteen truss abacus calculations, exactly the same as now, and the formula for calculating songs. The unified school of arithmetic who went to Cheng Dawei expounded the system and usage of abacus, and it was completely mature.

Cheng Dawei, 1533, from Xin 'an. 1592 compiled Directing at Arithmetic Unification School (hereinafter referred to as Arithmetic Unification School). In the 21st year of Wanli (1593), Jianjiang (that is, Zhejiang) made a preface, which was widely circulated.

The algorithm unified team is rich in content, but there is no new creation except abacus and songs. Basically, it is a book to sort out the works of predecessors. Moreover, some important parts such as higher-order equations and multivariate higher-order equations are omitted.

According to legend, at the end of the Ming Dynasty, Japanese Maori returned to China to study mathematics and brought back arithmetic unity. His book, The Book of Reduction Calculations.

(1622) and his disciple Hiroyuki Yoshida (1598- 1672) both described the abacus calculation method, but abacus calculation had already flowed into Japan before arithmetic was popularized.

The Japanese abacus is called "Ten Exposed Plate", and the beads are changed from round to diamond (longitudinal section), and the two beads on the beam become one bead. Now this kind of abacus is used in Northeast China, which is much smaller than Shanhaiguan's abacus, narrow and long (common 7×38cm), up to 27 trusses.

There was also an "abacus" in the Greek era, but it was different from the current abacus. It is the most primitive counting method to carve many straight lines or horizontal lines on a plate and put stones or wooden nails on the lines for counting. At the same time, the sand table used for drawing geometric figures or counting is also called abacus, which was later converted into Latin abacus or abacus and English abacus.

Rome improved this abacus. Carve a groove on the disk, put the beads in the groove, or take them away, and then embed the beads in the metal groove, which can move up and down, but can't be taken away. The Romans didn't know the notation of the value system, so they had to carve letters on the grooves of the abacus to indicate units. On the other hand, they use 12 to input the score, and add a small groove on the abacus to represent the score, which is very troublesome.

Westerners 1999 have no multiplication formula, but Chinese characters are written in one word and one sound, which is fluent and fluent, and foreign languages are in one word and one sound, which is inconvenient for formula.

Roman abacus is made of copper, which is not conducive to popularization, and it is very heavy, unlike China's abacus, which is made of bamboo, light and cheap. Due to these shortcomings, the Roman abacus was gradually eliminated and finally became a museum exhibit. Europeans returned to the old road of "counting board" for placing stones.

Ancient Russians also had an abacus. A number of curved battens were placed horizontally in a wooden frame, with ten beads on each batten, one for each bead and two for each bead. Unlike China, where there were five beads on each bead and one bead on each bead, the calculation speed was greatly limited.