Chapter 6 Chapter 3 Mathematics - "Nine Chapters on Arithmetic" and "Nine Chapters on Arithmetic Notes", "Nine Chapters on Mathematics", "Siyuan Yujian"
Until the middle of Ming Dynasty, in many branches of mathematics, China had always been far ahead in the world.Legend has it that as early as the time of the Yellow Emperor, Lishou created numerical symbols and calculation methods. "Historical Records" records that Dayu has used,,,,,,,,,,,,,,,,,Judging from the unearthed cultural relics, there are numbers in the pottery inscriptions unearthed from the Banpo site in Xi'an in the third and fourth millennium BC and the Erlitou site in Yanshi, Henan in the second millennium BC; and through the study of the numbers in the oracle bone inscriptions of the Shang Dynasty, we It was found that the germination of the positional value system already existed in it, and a decimal number system was formed.The decimal notation method is an indelible contribution made by the ancient people of our country to the people of the world.
In my country's ancient times and the Xia, Shang and Zhou dynasties, some specialized skills and knowledge were often in the hands of a small number of people and passed down from generation to generation.At that time, specialists in mathematics and astronomy were called "Chouren".In the Spring and Autumn Period, the Zhou family declined, the government moved downwards, and the children of the domain scattered, private schools began to rise, and mathematics knowledge gradually became popular.At the same time, the development of production also promoted the improvement of mathematical knowledge and calculation skills.We know from ancient documents that the nine-nine multiplication table was common sense at that time, and the concept of fractions and fractional operations had also been formed, especially for large-scale civil engineering projects such as building a capital city, which undoubtedly required more complex mathematical operations.It can be seen that in the Spring and Autumn Period, my country's mathematics has reached a relatively high level, but there are no special mathematics works handed down from generation to generation.
From the Warring States Period to the Western Han Dynasty, the ancient mathematics system represented by "Nine Chapters of Arithmetic" was established in our country.At the turn of the Spring and Autumn Period and the Warring States Period, Chinese society completed the transformation of production relations, and people's enthusiasm for production was greatly enhanced. Water conservancy was built, land was reclaimed, and farming techniques were improved. At the same time, handicrafts and commerce were further developed.In these activities, mathematical knowledge has been widely applied, providing new impetus for the development of mathematics.For example, in "Kao Gong Ji", there are many technical terms for right angles, obtuse angles, acute angles, and fractions.At this time, the ideological circle was also extremely active, with the rise of various schools of thought and contention among a hundred schools of thought, which not only achieved fruitful results in social and cultural aspects, but also promoted the research on the laws of thinking.Among them, the Mohists and famous scholars paid special attention to logical reasoning and rational speculation, and some propositions they put forward had profound mathematical connotations.The definitions of mathematical concepts such as circle, flat, and end (point) in the "Mo Jing" are already very rigorous.It should be said that at this time the conditions for summarizing and sorting out mathematical knowledge have been met.According to "Zhou Li", the mathematics education received by the "scholar" class at that time was called "Nine Numbers". "Nine numbers" refers to the division of mathematics into nine subdivisions.When Zheng Xuan of the Eastern Han Dynasty commented on "Zhou Li", he quoted Zheng Zhongzhi as saying: Nine Numbers are "Fang Tian, Corn, Difference, Shao Guang, Shang Gong, Even Losing, Equation, Winning Insufficiency, and Side Points." The titles are basically the same, except that the ninth is sidelines instead of Pythagorean shares. "Zhou Li" is generally considered to be written in the Warring States Period. Therefore, at the latest in the Warring States Period, mathematical works developed from Jiushu may have existed. At the end of 1983, the "Shushu Shu" unearthed from the tomb of the Western Han Dynasty in Zhangjiashan, Jiangling, Hubei Province, is the earliest mathematical work seen in my country so far. Many of its contents are similar to "Nine Chapters of Mathematical Sciences", and some titles and calculation problems are even completely consistent.Due to the simplicity of the text, it is generally believed that it is probably a pre-Qin work or recorded from a pre-Qin work.
The "Nine Chapters of Arithmetic", written in the Han Dynasty, is a summary and sublimation of mathematical knowledge from the pre-Qin to the Western Han Dynasty. It has achieved world-leading achievements in many aspects of mathematics at that time, and established a mathematical system centered on calculation in ancient China.In addition, the "Zhou Bi Suan Jing" we mentioned earlier is also regarded as an important mathematical work by later generations because of its mathematical content.
After the publication of "Nine Chapters of Arithmetic", there have been two climaxes in the study of ancient mathematics in my country. The first time was during the Wei, Jin, Southern and Northern Dynasties.A major feature of mathematics works in this period is the annotations for "Nine Chapters on Suanshu" and "Zhoubi Suanjing".According to the records of "Sui Shu·Jing Ji Zhi", there are eight kinds of works that only annotate "Nine Chapters of Arithmetic".During the Wei, Jin, Southern and Northern Dynasties, due to continuous wars and harsh political struggles, some people adopted the policy of calmness and restraint to deal with the chaotic and complicated society, so the style of talking was prevalent.In the field of thought, the dominant position of Confucianism was weakened and replaced by metaphysics based on "Book of Changes" and "Laozi".Metaphysics tries to demonstrate through abstract speculation that behind the real world there is an ontology that produces and dominates the phenomenal world, that is, the origin and fundamental laws of the world.Correspondingly, mathematicians have also begun to pay attention to the research of mathematical theory, trying to base the previously accumulated mathematical knowledge on the basis of inevitability. Chapter Arithmetic Notes is a typical representative.During this period, some new mathematical works also appeared, which made up for the content not covered in "Nine Chapters on Arithmetic", and created a new branch of mathematical research.Among them is Liu Hui's "Sea Island Calculus".The book was originally attached to the "Nine Chapters of Arithmetic Notes", and later generations made it an independent book.The book discusses the multi-difference technique developed from the method of measuring the height of the sun in "Zhou Bi Suan Jing", and it is named because the first topic in the book is to measure the height of an island. "Sunzi Suanjing", written around 400 A.D., describes the calculation and counting system, the rules of multiplication and division, fractions and square root, etc.Among them, the most famous one is the problem of "I don't know the number of things". The solution suggested in the book was extended to the solution of a congruence group by later generations. Since this topic was first proposed in this book, it is called "Sun Tzu's theorem" by historians. "Zhang Qiujian Suanjing", written by Zhang Qiujian in the 5th century A.D., the main achievements are the application of the greatest common divisor and the least common multiple, arithmetic series, open band from square and indefinite equation, etc., the famous "hundred chicken problem" It is from this book.Zhen Luan of the Northern Zhou Dynasty wrote three mathematical works: "Five Cao Suan Jing", "Five Classics of Numeracy", and "Shu Shu Ji Yi". "Wu Cao Suan Jing" is a practical arithmetic book written for local administrative officials, which contains the germination of decimal decimals; Three large-number bases and 14 algorithms were proposed, reflecting the historical situation of improving computing tools at that time.There are still many mathematical works in this period that have not been handed down to today. The more famous ones are "Xiahouyang Suanjing", "Zishu" by Zu Chongzhi, and "Three Class Numbers" by Dong Quan. The first two were included in the " "Ten Books of Calculus", the latter was also a textbook in the Tang Dynasty. "Zishu" has a very high achievement in mathematics. In the book, the ratio of pi is accurate to between 3.1415926 and 3.1415927. At the same time, there are major breakthroughs in the problem of spherical volume and the problem of the square root of the positive and negative coefficients of the quadratic equation.
The rulers of the Sui and Tang Dynasties set up a school of mathematics in Guozijian, and set up a subject of mathematics in the imperial examinations. In the Tang Dynasty, ten important mathematics works from the Han and Tang Dynasties were also sorted out and annotated as textbooks in the school of mathematics.The ten works are "Zhoubi Suanjing", "Nine Chapters Suanshu", "Sea Island Suanjing", "Sun Tzu Suanjing", "Xia Houyang Suanjing", "Zhang Qiujian Suanjing", "Five Cao Suanjing" ", "Five Classics of Arithmetic", "Zhushu" and "Jigu Suanjing".Among them, "Jigu Suanjing" was written by Wang Xiaotong in the early Tang Dynasty. Most of the 20 problems in the book are solved by higher-order equations.After the mid-Tang Dynasty, due to the great development of industry and commerce, people's requirements for simplifying the calculation process were more urgent, so many works on practical mathematics appeared, such as "Algorithm" by Long Yiyi, "One Algorithm" by Jiang Ben, Chen Congyun, etc. "Deyi Suan Jing" and so on.However, none of these works have been handed down to today. Only Han Yan's calculation book was named "Xiahouyang Suanjing" and added to the "Ten Books of Suanjing" due to the loss of the original "Xiahouyang Suanjing". pass down.The book records quite a lot of quick calculation methods, and promotes decimals.Generally speaking, the achievements of mathematics research in the Tang Dynasty were not high.Except for the quadratic interpolation method by Yixing et al., there is no major breakthrough.The reason is that the rulers of the Tang Dynasty did not pay enough attention to mathematics, and the social status of those who studied mathematics was very low, which was far lower than that of those who studied Confucian classics and poems and songs.However, the collation of ancient arithmetic books in the Tang Dynasty and the popularization of arithmetic knowledge laid the foundation for the development of mathematics in the Song and Yuan Dynasties.
During the Song and Yuan Dynasties, the society was relatively stable, the economy developed steadily, especially the development of industry and commerce, and the thirst for practical mathematical knowledge created conditions for the development of mathematics.At that time, many "shortcuts" and "gejue" appeared to help people quickly master various calculation methods.In addition, during this period, printing was widely used, and movable type printing was invented, which promoted the publication of mathematical works.In the seventh year of Song Yuanfeng (1084 A.D.), the secretary province published ten arithmetic scriptures as school textbooks. This is the first time that printed arithmetic books appeared in our country.At that time, most of the mathematical works written by mathematicians could be published soon after they were completed.Mathematical works have been widely circulated by means of printing.In this context, another climax was set off in Song and Yuan mathematics research. Especially in the second half of the 13th century, a group of outstanding mathematicians emerged, such as Qin Jiushao, Li Ye, Yang Hui, and Zhu Shijie. Brilliance can be said to be a peak stage in the development of ancient Chinese mathematics. In the first half of the 11th century, the publication of Jia Xian's "Nine Chapters of the Yellow Emperor's Computing Classics and Fine Grass" marked a leap in algebra for my country's algorithm system. Constitutional Triangle) and Zengcheng Kai method surpassed other nations for centuries.At about the same time, Shen Kuo pioneered the gap accumulation technique in "Mengxi Bi Tan", created a new branch of high-order arithmetic series summation, and also proposed an approximate formula for the arc length of a bow."Yi Gu Ji" written by Jiang Zhou used quadratic equations to solve various relationship problems of circles, and also made contributions to the development of Tian Yuan Shu. In the 12th century, Liu Yi's "Yigu Yuanyuan" introduced the negative coefficient equation again, and created the method of formulating Yiji and Minusong, which Yang Hui in the Southern Song Dynasty called it "Shiguanqiangu".During the Southern Song Dynasty, due to the long-term political confrontation between the North and the South in the Song, Liao, Song, and Jin Dynasties, mathematics research also formed two centers in the North and the South.The South Center is represented by Qin Jiushao and Yang Hui, and its main research objects are numerical solutions of higher-order equations, congruence solutions, and improved multiplication and division methods.Qin Jiushao's book is "Nine Chapters of Shushu", in which there are two important achievements that have attracted worldwide attention. One is the first systematic solution to the solution of a congruence group, and the other is a complete method for finding positive roots of higher-order equations.There are many works by Yang Hui, the main ones are "Detailed Explanation of Nine Chapters Algorithm", "Multiplication and Division, Changing the Basics", "Multiplication and Division Method of Field Mu Comparison", and "Continuing the Algorithm of Picking Oddities from Ancient Times".The latter three are Yang Hui's late works, and later generations are collectively called "Yang Hui's Algorithm".In Yang Hui's works, many calculation problems and algorithms in various mathematical works that have been lost have been collected, such as Jia Xian's "Increasing Multiplication and Opening Method" and "Original Diagram of Prescription Method", etc., and in the second order etc. High achievements have been made in the difference series and the simple arithmetic of multiplication and division.The North Mathematics Center is represented by Li Ye, and its main research object is Tianyuanshu and its solutions for high-degree equations.On the basis of predecessors, Li Ye systematically summarized the Tianyuan technique.His "Yi Gu Yan Duan" and "Che Yuan Hai Jing" are the earliest extant works about Tian Yuan Shu. The former is an introductory work written for beginners of Tian Yuan Shu; In order to comprehensively and systematically introduce the theory and algorithm of Tianyuanshu, the rich geometric content and the tendency of deductive reasoning are rare in ancient mathematics works.After Yuan unified China, the exchange of mathematics between North and South became a matter of course.Zhu Shijie lived in this environment.He has two books, "Enlightenment of Mathematics" is a mathematics enlightenment reader, including all aspects of mathematics at that time, from the method of multiplication and division to the root of multiplication, Tianyuan technique, and summation of high-order arithmetic series; "Yuan Yujian" is Zhu Shijie's famous work, which introduces the arrangement and solution of two-, three-, and four-dimensional high-order equations, and has made a major breakthrough in the problem of summation of high-order arithmetic series. Both achievements were earlier than Hundreds of years in the West, it has become a representative work of the peak of Song and Yuan mathematics.
Since the Ming Dynasty, Chinese classical mathematics began to decline, and there were many mathematical works at that time, but they were far less creative than the Song and Yuan calculation books.However, there are still two far-reaching events in mathematics during this period.One is the compilation of the "Yongle Dadian" in 1408, which classified and copied the mathematical works before the Ming Dynasty, and many mathematical works have been passed down to this day.The other thing is that with the completion of the simple calculation algorithm, the bead algorithm has also been developed and popularized.For example, Wu Jing's "Nine Chapters of Algorithmic Comparisons", Wang Wensu's "Treasures of Ancient and Modern Mathematics", Zhu Zaiyu (yu Yu)'s "New Book of Mathematics", etc., in addition to introducing calculation methods, they all mentioned abacus.In particular, "Algorithm Tongzong" written by Cheng Dawei in 1592 A.D., which systematically introduced the use of abacus, was once popular and widely spread.
At the end of the Ming Dynasty, the introduction of Western mathematics started a new stage of the integration of Chinese and Western mathematics.There were a lot of mathematical works in the Qing Dynasty. According to some preliminary statistics, there were more than 600 Chinese mathematicians with more than a thousand works.However, on an overall level, it has lagged behind the West.
"Nine Chapters of Suanshu", also known as "Nine Chapters of Suanjing" and "Nine Chapters of Suanjing" in Tang and Song Dynasties, is the most important mathematics classic in ancient China.According to the preface of Liu Hui's "Nine Chapters of Arithmetic" in the Wei and Jin Dynasties, mathematicians Zhang Cang and Geng Shouchang in the Western Han Dynasty made additions, deletions and additions to the book on the basis of the remnants of the book burned by Qin Shihuang.Modern researchers believe that "Nine Chapters of Arithmetic" was not written by one person for one lifetime, but the crystallization of the hard work of several generations. The final book was written between the end of the Western Han Dynasty and the early Eastern Han Dynasty.In China, the book has been directly used as a textbook for mathematics education for more than a thousand years; it has also influenced foreign countries, and North Korea and Japan have used it as a textbook.
"Nine Chapters of Arithmetic" collects 246 application questions, which are divided into nine chapters according to the nature and category of the questions.The order and content of each chapter are: 1. Fangtian, which is about the calculation of land area, including the calculation of the surface area of rectangle, triangle, trapezoid, circle, ring, bow and truncated sphere.Since fractions are used in the calculation of area, this chapter also systematically describes the calculation of fractions. 2. Corn, talking about the problem of proportion, especially how to exchange various grains in proportion. 3. Decay is the problem of proportional distribution of goods or taxes according to grades. 4. Shaoguang is to find the length of one side of a geometric object by knowing the area and volume, and it talks about the methods of square root and cube root derived from the calculation of fields. 5. Commercial work, including the calculation of volume in various projects, and the reasonable arrangement of labor. 6. Equity loss is a matter of calculating how to apportion taxes and dispatch migrant workers in a reasonable proportion according to conditions such as population size, price level, and distance. 7. Insufficiency of profit is about solving the problem of profit and loss in arithmetic, and it is called "sufficiency of profit".The issue of scale is also dealt with in this chapter. 8. Equations, mainly about linear equations. 9. Pythagorean, is about the application of the Pythagorean theorem in various measurements and geometric calculations.
The main part of "Nine Chapters of Arithmetic" adopts the form of application questions governed by algorithms, that is, a few example questions are listed first, and then abstract arithmetic texts are given. At this time, the example questions generally only have titles and answers; The specific technical text, and then list the sample questions. At this time, the sample questions generally include titles, answers and specific technical texts.In the nine chapters, there are nearly a hundred universal formulas and solutions, which have already included a considerable part of the content of mathematics in primary and secondary schools.For example, in the four arithmetic operations of fractions, proportion, area and volume, square root, cube root, positive and negative numbers, linear equations, quadratic equations, Pythagorean theorem, etc., there are relatively complete and detailed descriptions in the book.
For the calculation of fractions, our country has carried out in-depth research very early. The astronomical calculations in "Zhou Bi Suan Jing" already have quite complex fraction calculations, but because the division work is not done well, the calculations are more complicated.In "Nine Chapters of Arithmetic", a complete set of fraction calculation algorithms including reduction, general fraction, and four arithmetic operations are given.For example, the book uses the technique of "comparative subtraction" to find the greatest common divisor, pointing out that if the numerator and denominator can be divisible by 2, they should be divided by 2 first; The number equal to the minuend is the greatest common divisor.This method is basically the same as the method of rolling and dividing in modern arithmetic, and at that time, except for our ancestors, only the Greeks knew this method. "Nine Chapters of Arithmetic" is the first book in the world to systematically describe the arithmetic of fractions. Similar works appeared in India as late as the 7th century AD, while modern fractional algorithms were gradually formed in Europe after the 15th century.
The equation technique in the chapter on equations in "Nine Chapters of Arithmetic", that is, the solution of linear equations, can be said to be the most outstanding achievement in this classic.Since ancient China used counting chips to represent various numbers, the book uses the separation coefficient method to represent equations, which is equivalent to the current matrix.In solving equations, the method it uses is called "direct division method", which is basically the same as the current general method of addition, subtraction and elimination. It is the earliest complete solution method of linear equations in the world.In Europe, it was not until the 17th century that Leibniz proposed the complete solution of linear equations, more than 15 centuries later than in our country.Negative numbers will appear in the process of shifting items in column equations, merging similar items (profit and loss) and eliminating elements. In this part of "Nine Chapters of Arithmetic", the concept of negative numbers is introduced for the first time, and the rules of addition and subtraction of positive and negative numbers are proposed, and the multiplication and division of positive and negative numbers are carried out in actual operations.In the history of world mathematics, this is the first time to break through the range of positive numbers and expand the concept of number system.
"Today there are (several people) buying things together, (each) person pays (money) eight, and the profit is three; (each person) pays (money) seven, which is less than four. Ask the number of people and the price of the goods?" This is the "Nine Chapters" An application problem in the surplus and deficiency chapter of "Arithmetic".The book creatively applies two assumptions to solve such problems.Suppose the number of people is x, and the price is y; each person pays a, and the surplus is b; each person pays a, and the shortage is b, then there is the following equation: =(ab+ab)/(aa).
This method is called "profit-deficit technique", and it is used in the book to solve profit-loss problems and some mathematical miscellaneous problems. The "Insufficiency Technique" was introduced to Arabia around the 9th century AD, and it was called the "Chinese Algorithm". The same method did not first appear in the West until the 13th century in the works of the Italian mathematician Fibonacci.
The close combination of form and number is an important feature of "Nine Chapters on Arithmetic".In the Pythagorean chapter, geometrical problems are solved according to the Pythagorean theorem, and solutions to problems such as the area and volume of geometric figures and the measurement of "height, depth, width, and distance" are proposed, reflecting the development of measurement mathematics and map surveying and mapping at that time. s level.When computing area and volume problems, many square root calculations are encountered.In the Shaoguang chapter of "Nine Chapters of Arithmetic", the methods of square extraction and cubic extraction are given. They are basically the same as the current extraction methods, and they are the earliest square extraction procedures in the world.What needs to be pointed out is that using a counting chip to list several layers to perform square root and cube root operations is equivalent to listing a quadratic or cubic digital equation, that is, using different upper and lower layers to represent the coefficients of each term of an equation .Pythagorean chapter has a prospecting problem and it comes down to the open band from the square, that is, to solve the quadratic equation.Later, solving the positive roots of higher-degree equations was called "square root", which became the most developed field in ancient Chinese mathematics.
The mathematical achievements of "Nine Chapters of Arithmetic" are comprehensive and outstanding, which established its lofty position in ancient Chinese mathematics, and it is no exaggeration to call it the head of Chinese arithmetic.The book had an extremely profound influence on subsequent mathematical works.In terms of content, the nine parts of "Nine Chapters on Arithmetic" determined the basic framework of ancient Chinese mathematics and formed the characteristics of ancient Chinese mathematics centered on calculation; most of the 246 questions in the nine chapters came from the actual needs of people's production and life , creating a style in which mathematical theory is closely related to practice; the whole book does not contain any content of number mysticism, embodies a simple materialist view, and sets an example for future mathematical works.In terms of the structure of the book, "Nine Chapters on Arithmetic" generally has three parts: "questions", "answers" and "skills". This method of unifying questions with techniques has gradually formed a basic form of ancient Chinese mathematics works. After "Nine Chapters of Arithmetic", ancient Chinese mathematics works mainly adopt two modes, one is to write new works based on this book as a model, and the other is to make annotations for this book.
When it comes to annotating "Nine Chapters of Sumension", one cannot fail to mention Liu Hui.Liu Hui was a very outstanding mathematician in ancient my country, who lived in the 3rd century AD.Since "Nine Chapters of Arithmetic" was produced earlier, and it was not written by a single person, it also has its own shortcomings.For example, the text is simple, some content is not abstract, and only solutions and answers are given to the questions, lacking necessary explanations and proofs, etc.Liu Hui has repeatedly studied "Nine Chapters of Arithmetic" since he was a child. Later, he collected the research results of his predecessors and integrated his own mathematical experience into the book "Nine Chapters of Arithmetic", which is a comprehensive review of "Nine Chapters of Arithmetic". explanation and justification.
Liu Hui is the founder of ancient Chinese mathematical theory.He said in the preface of "Nine Chapters of Arithmetic": "Things are deduced from each other, and each has its own (what) to return. Therefore, although the branches are divided but have the same stem, the knowledge comes from one end." It means that there are many mathematics. The problems, seemingly different, have a common root in theory.This kind of logical reasoning thought was extremely important in the development of ancient Chinese mathematics.When explaining and demonstrating mathematical problems, he believes that "analyze with words and disintegrate with diagrams", which requires both language exposition and visual proof combined with graphics.This method of combining numbers and graphs is a unique method proved by ancient Chinese mathematics.
Another great contribution of Liu Huizhu in the history of mathematics is to use the "circle cutting technique" to obtain the value of pi.For a long time in ancient China, the value of π was 3.Liu Hui thinks that this is only the ratio of the perimeter to the diameter of a regular hexagon inscribed in a circle, which is incorrect.He first determined that the area of a regular polygon inscribed in a circle is smaller than the area of a circle, and after doubling the number of sides several times, the area increases accordingly. The more sides there are, the closer the area of a regular polygon inscribed in a circle is to the area of a circle.He wrote: "If you cut more thinly, you will lose less. If you cut again and again until you can't cut it, you will fit into the circle and nothing will be lost." This sentence reflects Liu Hui's limit thinking.Starting from calculating the area of the regular hexagon inscribed in a circle, Liu Hui successively calculated the area of the regular twelve, twenty-four, forty-eight... one hundred and ninety-two polygons inscribed in a circle, and obtained a pi approximation of 3.14.It is said that he was still not satisfied, and continued to calculate the value of π=3927/1250 (equivalent to 3.1416), which was the best value of pi in the world at that time.Theoretically speaking, the method in Liu Hui's note can be used to calculate pi very accurately.Researchers generally believe that Zu Chongzhi of the Southern Dynasty used Liu Hui's method to accurately measure pi to eight significant figures in "Zhu Shu", that is, between 3.1415926 and 3.1415927.Liu Hui's method laid the foundation for my country's pi calculation to lead the world for thousands of years.
There are many other aspects of Liu Huizhu's contribution to mathematics.For example, it develops the concept of rate in "Nine Chapters of Arithmetic", defines rate as "every number that is related to each other is called rate", that is, numbers and numbers are related to each other as rate, and discusses the nature of rate, and uses rate theory to discuss "Nine Most of Chapter Arithmetic.The book believes that Jinyoushu is a common method, and the solutions to many problems in the nine chapters can be attributed to this technique.Now there is a technique called proportional method, knowing the three items in the proportional formula to find the fourth item, such as a:b=c:d, knowing a, c, b, then d=bc/a.This method also existed in ancient India (the three-rate method), but the relevant records were later than "Nine Chapters of Arithmetic". In the 16th century, this method was introduced to Europe by the Arabs, and it was widely used in business, known as the golden rule.In addition, Liu Hui's annotations have unique insights into such issues as finding the area of an arc field, the volume of a cone, the volume of a sphere, decimal fractions, and solving equations, as well as the discussion of the properties of fractions and the definition of positive and negative numbers.
Liu Hui's annotations corrected the mistakes in "Nine Chapters of Mathematical Sciences", developed the mathematical theory in it, and enriched and perfected the mathematical system of "Nine Chapters of Mathematical Sciences". important part.
"Nine Chapters of Shushu" written by Qin Jiushao in the Southern Song Dynasty is an important mathematics work in ancient China and one of the masterpieces of the climax of mathematics in the Song and Yuan Dynasties.
Qin Jiushao (approximately AD 1202-approximately 1261) was named Daogu, and he called himself a native of Lujun (now Qufu and Yanzhou, Shandong). He was born in Anyue County, Puzhou (now Anyue County, Sichuan). astronomer.He was the leader of the volunteer army at the age of 18, but later experienced twists and turns in his official career.Qin Jiushao was smart and eager to learn since he was a child, and had a wide range of interests.He has in-depth research on astronomy, rhythm, arithmetic, construction, etc. As for games, horse bowing, kicking, swordsmanship, etc., he is also very familiar with it.It can be said that Qin Jiushao is a rare generalist, which will not be said to have no influence on his future great achievements in mathematics by learning from others' strong points and understanding by analogy.In 1247 AD, Qin Jiushao summed up his experience in mathematics research for many years and wrote "Nine Chapters of Shu Shu".
"Nine Chapters of Shushu", also known as "Nine Chapters of Shushu", "Shushu Dalu", "Mathematics Daxie", "Nine Chapters of Mathematics", etc., the title of "Nine Chapters of Shushu" only appeared in the late Ming Dynasty.The original title of the book has not yet been determined.The book was not published immediately after it was completed, and only manuscripts circulated. In the Ming Dynasty, it was classified and compiled into "Yongle Dadian", and in the Qing Dynasty, it was copied from "Yongle Dadian" and included in "Siku Quanshu".There is another copy that was handed down from Wenyuan Pavilion in the Ming Dynasty. After Shen Qinpei and Song Jingchang collected annotations and collated it, it was engraved into Yijiatang Series by Shanghai Yu Songnian in the 22nd year of Qing Daoguang (1842 A.D.) , is the most popular version.
"Nine Chapters of Shushu" contains 81 questions, which are divided into nine categories, and each category has nine questions.The nine major categories are: 1. Dayan category, which describes "Dayan seeks a technique" and uses it to solve various practical problems; 2. Astronomical category, related to mathematical problems such as calendar formulation, astronomical calculation, and calculation of rainfall and snowfall. The Tianchi Basin is the earliest rain gauge in the world; 3. The field category is about the calculation of the area of various shapes of fields, reflecting the activities of the people in the south of the Yangtze River to reclaim the sea and the lake; 4. The outlook category discusses the Pythagorean measurement , involving measuring the distance of mountains, waters, cities, towers, and enemy troops, and the restoration of historic sites; 5. Taxation and service, which is about the calculation of land tax and household tax, reflecting the actual situation of taxation in the Southern Song Dynasty; 6. Money and grain , about grain transshipment and warehouse volume, designed the mathematical calculation problems caused by the chaos caused by the increase in the number of utensils in various places in the Southern Song Dynasty and the increase in land rent. Exchange situation; 7. Construction category, which solves mathematical calculations in engineering construction, among which the question of calculating and clearing the platform is the earliest existing design drawing of the observatory in the world; Issues in the supply of military supplies, which were related to the fierce wars between the Song, Jin and Song and Yuan Dynasties at that time, had the purpose of serving the war, which is relatively rare in ancient calculation books; There are many historical materials on commodity transactions and related policies in the Southern Song Dynasty.
Judging from the classification and content of "Nine Chapters of Shushu", it is obvious that it was deeply influenced by "Nine Chapters of Arithmetic", closely related to real life, and reflected the social economy, culture, politics, science and technology at that time. One side of various activities.Qin Jiushao once said in the preface of this book: Mathematics "can pass through the gods and obey life if it is large, and can manage world affairs if it is small, and it can be compared with all things", which raised mathematics to a very high position.However, he connected mathematics with the origin of the world, and believed that "number and Tao are not two books", which also shows that he was influenced by Neo-Confucianism and Xiangmathematics at that time.
"Da Yan Qiu Yi Shu" is Qin Jiushao's most proud masterpiece and a great achievement of ancient Chinese mathematics, so it is very appropriate for Qin Jiushao to put it at the top of the book.As early as the 4th century AD, such a question was raised in "Sun Tzu's Mathematical Classics". In today's words, there is a number, and when divided by 3, the remainder is 2; when divided by 5, the remainder is 3; when divided by 7, the remainder is 2. Find this number.This is the famous "Grandson problem", and it is also a problem of a congruence group.In the ancient Chinese calendar, the problem of understanding the congruence formula group is also encountered when calculating the cumulative year of the Shangyuan.For this kind of problem, the solution given in "Sun Tzu's Mathematical Classics" is too simple, and there is no systematic algorithm in the calendar, and it is even mistaken for the solution of linear equations.It was not until Qin Jiushao's "Nine Chapters of Shushu" that this kind of problem was systematically solved for the first time.The key to Qin Jiushao's method is to use "odd numbers" and "fixed numbers" to divide and divide and a whole set of calculation procedures to find the "multiplication rate" that meets the requirements.Because the calculation of the "multiplication rate" requires repeated divisions until the final remainder is 1, so Qin Jiushao called it "the art of seeking one".In Qin Jiushao's problem, the data can be integers, fractions, or decimals, and he has given the corresponding resolution procedures.In short, Qin Jiushao systematically solved the problem of a congruence group for the first time in the world, and the calculation steps are quite rigorous.After more than 500 years, talents such as Euler and Gauss in Europe have carried out relatively in-depth research on the simultaneous first degree congruence formula. After the "Da Yan Qiu Yi Shu" was introduced to the West, it attracted great attention from European scholars.Western mathematics historians call this theorem the "Chinese remainder theorem", and the famous German mathematics historian Cantor praised the Chinese mathematician who discovered this algorithm as "the luckiest genius".
In the second category of the book - the ninth category, Qin Jiushao used many mathematical methods since "Nine Chapters of Arithmetic", and made creative developments, the most important of which is the positive and negative formula for finding positive roots of higher-order equations.In ancient my country, solving general high-order numerical equations was called "square extraction". The methods of square extraction and cubic extraction have been recorded in "Nine Chapters of Arithmetic". They are "open band from square" and "open band from cube", because they are all derived from the methods of square extraction and cubic extraction.The art of prescribing prescriptions made great progress in the Song Dynasty.First of all, Jia Xian created the "increasing multiplication and opening method". Through the method of multiplication and addition, the positive root of a high-order equation can be found. In the 12th century, Liu Yi introduced the square root of negative coefficients. The coefficients of the equation can be positive or negative, and the restriction that the coefficients of the equation are only allowed to be positive integers was canceled.In the Southern Song Dynasty, Qin Jiushao proposed the "Positive and Negative Prediction Method" in "Nine Chapters of Shushu", which is a complete set of procedures for gradually finding the positive roots of higher-order equations by multiplying and adding.In Qin Jiushao's method, there is no restriction except that the "real" (constant term) is often negative due to the need for calculation. It is a general solution method for any high-order equation, which is basically the same as the current method for finding the positive root of a high-order digital equation. .The modern algorithm was proposed by the Italian Ruffini in 1804 and the British Horner in 1819, which is the well-known Ruffini-Horner method, more than 600 years later than Qin Jiushao.Qin Jiushao also made use of Liu Hui's idea of continuing to calculate "micro-numbers" with the square root. When the square root reaches the irrational root, he uses decimals as the approximate value of the irrational root. This is also the earliest contribution in the history of world mathematics.
The mathematical achievements of "Nine Chapters of Shushu" are also manifested in more aspects.In terms of equations, that is, the solution of linear equations, it uses the method of mutual multiplication and elimination, that is, the coefficients of the x items of the two equations are multiplied by each equation, and the purpose of eliminating the x items can be achieved by one subtraction.This method eliminates the trouble of continuous subtraction by direct division, which is exactly the same as the method commonly used by people today.The book also carried forward the techniques of prediction in "Nine Chapters of Arithmetic" and "Sea Island Mathematical Classics", and made many inventions on the problems of Pythagorean shares and heavy differences.What is particularly worth mentioning is the "triclinic quadrature formula", that is, the formula for calculating the area by using the three sides of a triangle. It was independently invented with the Western Heron's formula, but it coincides with each other.In addition, "Nine Chapters of Shushu" has special treatises on natural numbers, fractions, decimals, and negative numbers, and has been developed. It is an important material for the study of ancient Chinese notation.
The mathematical achievements of "Nine Chapters of Shu Shu" far surpassed the previous mathematical works. Only in terms of the solution of first-order congruence equations and the numerical solution of higher-order equations, it has represented the mainstream and highest level of mathematics development in the medieval world. Level is a glorious page in the history of Chinese mathematics.
The "Siyuan Yujian" written by Zhu Shijie in the Yuan Dynasty is another masterpiece of the climax of mathematics in the Song and Yuan Dynasties, and has an important position in the history of ancient Chinese mathematics.
Zhu Shijie (year of birth and death unknown), styled Hanqing, named himself Songting, lived in Yanshan, which is near today's Beijing.After the Yuan Dynasty unified China, the confrontation between the North and the South ended.Zhu Shijie has traveled around the country for more than 20 years, conducting mathematics research and engaging in mathematics education activities at the same time.Through long-term and extensive travel, he has a deep understanding of the achievements of the North and South Institutes of Mathematics, and has become a famous scholar who is the head of the two mathematics centers.When Zhu Shijie traveled to Yangzhou, more and more people came to learn from him from all directions.In order to meet the requirements of the students, he began to write books for the students to use.In 1299 AD, he wrote "Enlightenment of Mathematics", which was engraved and printed by Zhao Yuanzhen. 1303年，《四元玉鉴》完成，也由赵元镇刊印。
《四元玉鉴》全书三卷，共24门，288问。书首先给出四种图；古今开方会要之图，给出了增乘开方法的图示和九层八次方的贾宪三角；四元自乘演段之图、五和自乘演段之图、五较自乘演段之图则是图示处理几何问题时立方程的各个步骤。四图之后是假令四草，给出了一气混元、两仪化元、三才运元、四象会元四个例题，分别阐述天元术、二元术、三元术、四元术的解题模式。这些图和例题都是为了举例发凡，是统御全书的纲纪。在全书其他各问中，朱世杰没有再记出任何一题的算草。这种写作形式在中国古代数学著作中是一种独特的创造。之后是各卷内容。上卷六门：1.直段求源，关于勾、股、弦的计算问题；2.混积问元，田亩面积问题；3.端匹互隐，有关绫、罗等纺织品的各种计算；4.廪粟回求，谷物容积问题；5.商功修筑，工程建筑问题；6.和分索引，关于分数的各种运算。中卷10门：1.如意混和，把性质不同的问题混和以增加问题难度；2.方圆交错，有关方、圆的混合问题；3.三率究圆，以古率π=3、微率157/50、密率22/7计算有关圆与球的问题；4.明积演段，与勾股形(直角三角形)有关的各种计算；5.勾股测望，用勾股定理及相似勾股形测算距离；6.或问歌彖〔tuan〕，以诗歌形式给出的问题；7.茭草形段，垛积问题；8.箭积交参，关于方箭、圆箭的垛积问题；9.拨换截田，截割田亩的面积问题；10.如象招数，招差术问题。下卷八门：1.果垛叠藏，垛积问题；2.锁套吞容，相互交错的图形的面积计算；3.方程正负，线性方程问题；4.杂范类会，是各种杂题；5.两仪合辙，关于勾股及面积的二元二次方程组；6.左右逢元，关于勾股及面积的二元高次(三次以上)方程组；7.三才变通，关于勾股问题的三元方程组；8.四象朝元，关于勾股问题的四元方程组。
在《四元玉鉴》中，几乎所有问题都与方程或方程组有关，其中主要记载了朱世杰的伟大创造——四元术。我们知道，用解方程的方法解决实际问题，一般来说都需要两个步骤。首先是列出含有未知数的方程，然后才是解方程求出它的根来。列方程，古代称“造术”，这对于今天具备初等数学知识的人来说是轻车熟路，然而在天元术未出现以前，却并不简单。当时数学家们列方程只有借助文字叙述，非常复杂。金元之际，北方出现了一批有关天元术的著作，李冶的《测圆海镜》是现存最早系统论述天元术的著作。所谓“天元术”，实际上是列方程的一种代数方法。天元术中“列天元一某某”，就是“设x为某某”的意思，方法是在筹算的一次项旁写上“元”字，或在常数项旁写上“太”字。天元术的出现解决了一元高次方程的列方程问题。据记载，李德载的《两仪群英集臻》和刘大鉴的《乾坤括囊》分别对二元术和三元术作了研究，但他们的著作都没有流传下来。流传至今并将其发展成四元术的是朱世杰的《四元玉鉴》。四元术用天、地、人、物四元表示四元高次方程组。它是在常数项右侧记一“太”字，天、地、人、物四元和它们的乘幂的系数分别列在“太”字的下、左、右、上，相邻二未知数和它们的乘幂的积的系数，记入相应的两行相交的位置上，不相邻的几个未知数的积的系数，记入相应的夹缝中。这实际上是多元高次方程组的分离系数表示法。朱世杰还创造出一套完整的消未知数方法，称为四元消法。通过逐次消元，最后得到只含一元的方程式，然后用增乘开方法求正根。虽然由于受到筹算的局限，朱世杰只达到四元高次方程，但这一成果却在世界上长期处于领先地位。直到18世纪法国数学家别朱才系统叙述了高次方程组消元法问题。
垛积招差术，即高阶等差级数求和，是《四元玉鉴》中的另一项重大成就。它们主要被记载于茭草形段、如象招差、果垛叠藏三门中。关于垛积的研究，最早的要算是沈括，在《梦溪笔谈》中，他为计算用酒坛堆积的长方台的酒坛数，提出了一个新的计算公式——隙积术，其后杨辉又给出了三角垛、方垛、果子垛等公式，但这些公式实际上可以看成沈括隙积术的特例。到了朱世杰，垛积术的研究出现了全新的局面。《四元玉鉴》中的垛积公式共有三大类：1.三角形，包括茭草垛、三角垛(或称茭草落一形垛)、三角撒星形垛(或称三角落一形垛)、三角撒星更落一形垛；2.岚峰形，包括四角垛、岚峰形垛、三角岚峰形垛(或称岚峰更落一形垛)；3.值钱形(垛积物的价格逐层递增或递减)，包括茭草值钱正垛、茭草值钱反垛、三角值钱正垛、三角值钱反垛、四角值钱正垛、四角值钱反垛。三类中，三角形垛积公式是最基本的。由于朱世杰在书前的贾宪三角中增加了平行于两斜边的连线，再加上他用“落一”、“更落一”表示几种三角垛积的关系，所以，人们认为朱世杰已掌握了一般三角形垛的求和公式。同样道理，朱世杰也掌握一般岚峰形垛的求和公式，而第三类公式可以从前两类公式推导而出。
《四元玉鉴》中的招差问题和垛积问题互为表里，也是该书最精彩的部分之一。在朱世杰以前，招差问题是独立发展的一门知识，它和我国古代历法中计算天体运行有着密切关系。公元206年，刘洪在《乾象历》中首次提出用一次内插法计算月亮的变速运动，隋初刘焯《皇极历》中使用了二次内插法，到元代郭守敬等人已采用三次差分的内插法原理计算日月五星的运动。而朱世杰则将垛积和招差联系起来，在世界上第一次给出了包括四次差的内插公式。书中明确指出，公式中的各项系数是三角垛的积。由于朱世杰已经掌握了三角形垛的构造规律，所以一般认为他已得到任意高次的内插法公式。在欧洲，直到17世纪格列高里、牛顿等人才取得同样的结果。
除了上述成就外，朱世杰的创造性工作还表现在几乎全书的每一门中。例如，他突破了有理式的限制，开始讨论无理方程。又如，在几何学上，他在传统的勾股和体积、面积的计算的基础上，进一步研究了勾股形和圆形内各几何元素的关系，使得几何研究的对象由图形整体深入到图形内部，体现了数学思想的进步。
《四元玉鉴》写成的时候，社会上对算学十分尊崇，所以受到重视。明代以后，该书被人们所忽视，到了几乎失传的地步。清朝嘉庆年间，阮元在浙江访得《四元玉鉴》抄本，送交四库馆，后来何元锡将抄本刊印。该书重新刊印后，许多数学家对它进行过研究，其中以罗士琳的《四元玉鉴细草》影响最大，以后的许多版本都源于此书。
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