Zhu Shijie (1249- 13 14), a native of Yanshan (present-day Beijing), was a mathematician and educator in the Yuan Dynasty, and engaged in mathematics education all his life. It has the reputation of "the greatest mathematician in the medieval world".
Zhu Shijie developed the "Quaternary Technique" on the basis of celestial sphere technique at that time, that is, he listed the polynomial equation of higher degree and its elimination method. In addition, he also created the "superposition method", that is, the summation method of high-order arithmetic progression, and the "trick", that is, the high-order interpolation method.
His main works are "Arithmetic Enlightenment" and "Meeting with Siyuan".
Chinese name: Zhu Shijie.
Alias: Han Qing,No. Songting.
Nationality: Yuan
Ethnic group: Han nationality
Birthplace: Yanshan (now Beijing)
Date of birth: 1249
Date of death: 13 14.
Occupation: Mathematician
Faith: science
Main achievements: creating "four elements"
Creating "Stacking Method" and "Calling for Difference"
Representative works: Arithmetic Enlightenment and Meeting with Siyuan.
outline
Zhu Shijie "traveled around the lake and sea as a famous mathematician for more than 20 years" and "gathered scholars by following the door" (Mo Ruo and Zu Yi: Introduction to the Meeting of Siyuan).
During the Song and Yuan Dynasties, there were "Qin, Yang Hui and Zhu Shijie" among the outstanding mathematicians in the heyday of mathematics in China, and Zhu Shijie was one of them. Zhu Shijie is a civilian mathematician and math educator. Zhu Shijie studied nine chapters of arithmetic diligently all his life, bypassing other algorithms and becoming a famous mathematician in Yuan Dynasty.
life experience
After the Yuan Dynasty unified China, Zhu Shijie traveled around the world as a mathematician for more than 20 years, and many people learned from him. When he arrived in Guangling (now Yangzhou), he "followed the door and gathered scholars". He fully inherited the mathematical achievements of predecessors, not only absorbed the celestial science in the north, but also absorbed the positive and negative formulas, various daily algorithms and popular songs in the south. On this basis, he conducted creative research, and for the purpose of summarizing and popularizing all kinds of mathematical knowledge at that time, he wrote "Arithmetic Enlightenment" (3 volumes) and "Siyuan Jade Mirror" (3 volumes), which were published in 65438+ successively. The Enlightenment of Arithmetic, from simple to deep, from one-digit multiplication to the latest mathematical achievement at that time-Tiandao, suddenly formed a complete system.
The book clearly puts forward the multiplication rule of positive and negative numbers, gives the concept and basic properties of reciprocal, summarizes some new multiplication formulas and radical operation rules, summarizes some quick formulas of multiplication and division, and applies the method of setting auxiliary unknowns to solving linear equations. The main content of Siyuan Encounter is quaternary technology, that is, the establishment and solution of multivariate higher order equations. Qin's numerical solution of higher-order equations and his celestial skills are all included.
During the Song and Yuan Dynasties, Zhu Shijie's works had special significance. If many mathematicians are compared to mountains, Zhu Shijie is the highest and most majestic mountain. Looking down on traditional mathematics from the height of Zhu Shijie's mathematical thought, there will be a feeling that "the stones of other mountains are dwarfed under the sky". .
The significance of Zhu Shijie's works is to sum up the mathematics of Song and Yuan Dynasties and make it reach a new height in theory. This is mainly manifested in the following three aspects. The first is equation theory. In terms of equations, Jiangzhou's staging method has prepared for the art of heaven, and has already had the idea of finding equivalent polynomials. Dong He is a pioneer of celestial art, but their equation derivation is still bound by geometric thinking. Ye Li basically got rid of this bondage and summed up a set of fixed celestial art procedures, which made celestial art enter a mature stage. Jia Xian gave the method of increasing, multiplying and opening when solving the equation, while Liu Yi found the positive root of the quartic equation with the positive and negative square roots. On this basis, Qin solved the numerical problem of higher-order equations. So far, the establishment and solution of one-dimensional higher-order equations have been realized. Linear equations have existed since ancient times, so they have the conditions for generating multivariate higher power equations. Li Dezai's binary art and Liu Dajian's ternary art have appeared one after another. Zhu Shijie's quaternary art is the summary and improvement of binary art and ternary art ... Now that the four elements have filled the upper, lower, left and right sides of constant terms, the development of equation theory here is obviously over. Judging from the types of equations, the equations before the emergence of celestial art are all integral equations.
From Dong Yuan to Ye Li, the fractional equation developed gradually. Zhu Shijie, on the other hand, broke through the limitation of rational number formula and began to deal with irrational number equation. Secondly, the study of advanced arithmetic progression. Shen Kuo's gap product technique pioneered the study of higher-order arithmetic progression, and Yang Hui gave a series of summation formulas of second-order arithmetic sequence including gap product technique. On this basis, Zhu Shijie studied the summation of second-order, third-order, fourth-order, and even fifth-order arithmetic progression in turn, thus discovering its laws and mastering the unified formula of triangular crib. He also found the internal relationship between superposition and interpolation, and gave the standard quartic interpolation formula by using superposition formula. The third is the study of geometry. Before the Song Dynasty, geometry research could not be separated from Pythagorean and area and volume. Jiangzhou's Yiji Valley also takes the area problem as the research object. Ye Li began to pay attention to the relationship between elements in the round city factor, and obtained some theorems, but they could not be extended to more general situations. Zhu Shijie not only summarized the previous pythagorean and quadrature theories, but also went further on the basis of Ye Li's thought, deeply studied the quantitative relationship between pythagorean and geometric elements in a circle, and found two important theorems-projective theorem and chord power theorem. In solid geometry, he also began to notice the relationship between elements in graphics. Zhu Shijie's work makes the object of geometry research go deep into the interior of the figure from the whole figure, which embodies the progress of mathematical thought.
Famous anecdote
/kloc-At the end of 0/3, the war-torn motherland was unified by the Yuan Dynasty, and the destroyed economy and culture flourished rapidly. For the sake of national prosperity and security, Mongolian rulers respect knowledge, select talents and push all sciences to a new height.
One day, by the scenic Slender West Lake in Yangzhou, a teacher came and hung a sign in front of his apartment, which read in big letters: "Teacher Zhu Songting of Yanshan specializes in teaching four elements." A few days later, Zhu Shijie's front door was crowded with people who were curious about knowledge. Just as Zhu Shijie was accepting students' registration, suddenly, a lot of shouting caught his attention.
I saw a semi-old Xu Niang dressed in satin and silver, chasing a young girl, beating and cursing: "You bitch, didn't you catch a lot of money?" Do you want to be a good family? I am afraid that you voted for the wrong baby and will not think about it in the next life. " The girl was beaten so badly that even the clothes inside her were torn. The girl curled up into a ball and asked her to fight, but she didn't go back with her. When Zhu Shijie saw that the road was rough, he came forward to ask. Xu Niang, a half-aged man, saw a nosy person and sneered, "Do you want to be keen? You give me fifty taels of silver, and the girl is yours! "
Zhu Shijie was furious at this. "Can't I pay fifty taels of silver?" . You can't run amok in broad daylight. Is there no king's law? "
Old Xu Niang said sarcastically, "You poor wretch, what are you talking about? Silver is the king. If you can pay fifty taels of silver, I won't fight. "
Zhu Shijie was so angry that he took out 50 taels of silver from his pocket, threw it in front of Xu Niang, picked up the girl and went back to his teaching place. It turned out that Xu Niang, who was half-aged, was the madam of a prostitute's family. The girl's father borrowed 10 silver from the madam, but he could not pay back the money due to natural disasters, so he had to sell his daughter to pay off his debts. I happened to meet Zhu Shijie today and let this girl out of her misery.
Later, under the careful instruction of Zhu Shijie, the girl also learned a lot of mathematics and became Zhu Shijie's right-hand man. In a few years, the two became husband and wife. Therefore, there is a saying among Yangzhou people: Zhu Hanqing taught and educated people in Yuan Dynasty. Saving people from fire and water, a big marriage.
Related works
Zhu Shijie has been engaged in mathematics research and education for a long time. He traveled around the world as a famous mathematician for more than 20 years, and many people came from all directions to study. Zhu Shijie's representative works in mathematics include "Arithmetic Enlightenment" (1299) and "Meeting with the Source" (1303). "Arithmetic Enlightenment" is a well-known mathematical masterpiece, which spread overseas and influenced the development of mathematics in Korea and Japan. "Thinking of the source meets" is another symbol of the peak of China's mathematics in the Song and Yuan Dynasties, among which the most outstanding mathematical creations are "thinking of the source" (the formulation and elimination of multivariate higher-order equations), "overlapping method" (the summation of higher-order arithmetic progression) and "seeking difference method" (the high-order interpolation method).
In mathematical science, Zhu Shijie comprehensively inherited the mathematical achievements of Qin, Yang Hui and developed them creatively. He wrote such famous works as Arithmetic Enlightenment, Meeting with Siyuan, which pushed the ancient mathematics in China to a new height and formed the highest peak of China's mathematics in the Song and Yuan Dynasties. The Enlightenment of Arithmetic was published by Zhu Shijie in Yuan Chengzong Dade for three years (1299). There are 20 books in three volumes, with 259 questions and corresponding answers. Since the multiplication and division method, this book has been talking about the highest achievement of mathematics development at that time, "Tianyuan Shu", which comprehensively introduced all aspects of mathematics at that time.
Its system is complete, the content is simple and easy to understand, and it is a very famous enlightenment reading. This book was later spread to Korea, Japan and other countries, and reprinted and annotated editions were published successively, which had a certain influence. Philip Burkart Meeting is a brilliant mathematical masterpiece. It is highly praised by researchers in the history of modern mathematics, and it is considered as the most important and greatest mathematical masterpiece among China's ancient mathematical scientific works. Meeting Siyuan was written in the seventh year of Dade (1303), with three volumes, 24 doors and 288 questions. This paper introduces Zhu Shijie's research and achievements in solving multivariate higher-order equations-quaternary method and calculating higher-order arithmetic progression-superposition method and differential method.
"Tianyuan Shu" means "Tianyuan is XXX", that is, XXX is X. However, when there is more than one unknown quantity, besides the unknown Tianyuan (X), it is necessary to set geographical elements (Y), humanistic elements (Z) and material elements (U), and then list the high-order binary, ternary or even quaternary equations before solving them. In Europe, the solution of linear equations began in16th century, and the study of high-order simultaneous equations was from 18 to19th century. Another great contribution of Zhu Shijie is the study of "piling". He studied a series of new summation problems of crib-shaped sequences, which were summarized as "triangular crib" formula. In fact, he obtained a systematic and universal solution to this kind of arbitrary high-order arithmetic progression summation problem. Zhu Shijie also introduced the triangle crib formula into the unique skill, pointing out that the coefficient in the unique skill formula is just the product of the triangle crib in turn, thus obtaining the unique skill formula with the fourth difference.
He also extended this formula to include any higher-order difference, which is the first time in the history of mathematics in the world, nearly four centuries earlier than the same achievement of Newton in Europe. Because of this, Zhu Shijie and his "Meet with Siyuan" enjoy a high reputation in the world. In modern Japan, France, the United States, Belgium and many countries in Asia, Europe and the United States, people introduce homesickness to their countries. Sutton, a famous American historian of science, commented on Zhu Shijie in this way: "(Zhu Shijie) is the most outstanding mathematical scientist of the Chinese nation, the era in which he lived, and the most outstanding throughout the ages." Meeting in Philip Burkart is the most important mathematical work of China and one of the most outstanding mathematical works in the Middle Ages. It is a rare treasure in the world mathematics treasure house. "Thus, scientists and their works in the Song and Yuan Dynasties played an inestimable role in the history of world mathematics.
Character contribution
Zhu Shijie's main contribution is to create a set of methods to eliminate unknowns, which is called quaternary elimination method. This method has been in the leading position in the world for a long time, and it was not until18th century that the French mathematician Bezot put forward the general solution of higher-order equations that Zhu Shijie was surpassed. Four-element jade mirror has two important achievements besides four-element method, namely, the general formula of high-order arithmetic progression summation and the formula of quartic equidistant interpolation, which are usually called difference method. This book represents the highest level of mathematics in the Song and Yuan Dynasties, and American science historian G. Sutton praised it as "the most important book in China's mathematics works and one of the outstanding mathematics works in the Middle Ages". Zhu Shijie is in the heyday of the development of traditional mathematics in China. At that time, the society "respected arithmetic and gradually popularized science", and mathematical works were widely circulated.
There are in-depth studies on solving multivariate higher-order equations, higher-order arithmetic progression summation, and higher-order interpolation methods. He is the author of three volumes: Arithmetic Enlightenment (1299) and Siyuan Jade Mirror (1303). In the latter, he discussed the solution, expression, operation and elimination of related polynomials of high-order four-element simultaneous equations.
It is another outstanding creation of mathematicians in Song and Yuan Dynasties to extend celestial sphere to higher-order simultaneous equations of binary, ternary and quaternary. What has been handed down to this day is Zhu Shijie's Meet with Siyuan, which systematically discusses this outstanding creation. Meeting in Philip Burkart was written on 1303. The book has 3 volumes, 24 subjects and 288 questions. This paper mainly discusses the solution of higher-order equation (which is also Zhu Shijie's greatest contribution), higher-order arithmetic progression summation and higher-order interpolation method. It is an important masterpiece that has been circulated so far and systematically discusses the four elements.
On the basis of the theory of heaven, Zhu Shijie established the theory of four-dimensional higher-order equation, and he put the constant term in the center (that is, "Tai"), and then "set Tianyuan in the bottom, the ground in the left, people in the right and things in the top". The four elements of "heaven, earth, people and things" represent the unknown (that is, equivalent to today's). If modern X, Y, Z and W are used to represent heaven, earth, people and things, then we can express Zhu Shijie's high-order multivariate equation. The equations expressed by the above two graphs "Quadratic Primary Support Formula" and "Quadratic Secondary Support Formula" are: x+y+z+w=0 respectively.
After listing the quaternary high-order equations in the above way, solve the equations simultaneously. The method is to solve the equations by elimination method. First, one variable is unknown, and polynomials composed of other elements are used as the coefficients of this unknown. Then the quartic equation with four variables is eliminated to form the quartic equation with three variables, then the univariate is changed into the quartic equation with two variables, and then the univariate is eliminated to obtain the Tianyuan open module with only one variable. Then find the positive root by multiplying the root. This is an important development of linear method for solving problems. In the west, the systematic study of multivariate equations will wait until16th century. Higher-order arithmetic progression summation and higher-order interpolation are also important contents that Siyuan meets. In many summation problems, formulas can be derived from a series of triangular superposition formulas. Zhu Shijie gave the formula when p = 1, 2,6 in the above formula. In addition, there are other higher-order arithmetic progression summation formulas. In terms of recruitment methods, Zhu Shijie has given the formula of recruitment, which is more than 400 years earlier than the West.
Sutton, a famous American historian of science, commented: "Zhu Shijie was an outstanding mathematician in his time and throughout the ages", and "Meeting Philip Burkart" is "the most important mathematical work of China and one of the most outstanding mathematical works in the whole Middle Ages." Zhu Shijie is not only an outstanding mathematician, but also a mathematics educator. He has traveled all over the world and taught students for more than 20 years. And personally wrote an introductory book on mathematics called "Arithmetic Enlightenment". Zhu Shijie put forward the method of solving Pythagorean formula with known chord sum and chord sum in the volume of Arithmetic Enlightenment, and supplemented the deficiency of Nine Chapters Arithmetic.
Historical evaluation
Mr. Zhu Songting of Yanshan was an outstanding mathematician in Yuan Dynasty. His Four Lessons from Yu Juan and Arithmetic Enlightenment are important milestones in the development of ancient mathematics in China and precious legacies of ancient mathematics in China. /kloc-In the middle of the third century, Zhu Shijie not only accepted the mathematical achievements of the north, but also absorbed the mathematical achievements of the south, especially various daily algorithms, business arithmetic and popular songs.
Zhu Shijie once "traveled around the world". In the preface of Muruo (an ancient mathematician), "Mr. Zhu Songting of Yanshan Mountain, as a famous mathematician, has traveled around the lake and sea for more than 20 years. There are more and more scholars all over the world, so Mr. Wang invented Nine Wonders Books, and named the three-volume book Siyuan Encounter after Shu. In the preface to Ancestor's Legacy, there is also a saying that Han Qing is a famous world hero, and Song Ting has his own number. "Liu Zhou is everywhere, revisiting Guangling, and gathering people after the door." After a long period of investigation and lectures, his two mathematical masterpieces, The Enlightenment of Arithmetic and The Encounter of Thinking Source, were finally published in Yangzhou 1299 and 1303. The formula of eliminating songs in Yang Hui's book was further developed in Zhu Shijie's Arithmetic Enlightenment.
In the Qing Dynasty, Luo Shilin thought: "Han Qing was the third best harmony between Qin Dynasty and Yuan Dynasty. There are pros and cons in the valley, and Tianyuan in Han Qing has ups and downs. There are all kinds of things, and the more the better, especially beyond Qin and Li. " Wang Jian, a mathematician in the Qing Dynasty, also said: "Mr. Zhu Songting is the director of Qin and Li, and has achieved great success." Zhu Shijie comprehensively inherited and creatively carried forward the mathematical achievements in Qin and Li's works, such as "Tian Mo" and "Forward-Inverse Method", and included various algorithms in Yang Hui's works that were closely related to social life at that time, and made new development.
related data
Before the Yuan Dynasty destroyed the Southern Song Dynasty, North-South exchanges, especially academic exchanges, were almost cut off. Mathematicians in the south know nothing about celestial science in the north, and mathematicians in the north are rarely influenced by the south.
From this point of view, in Zhu Shijie's works, there are not only the achievements of northern mathematics, such as solving higher-order equations and celestial skills, but also the achievements of southern mathematics, such as daily and commercial algorithms and various songs, which are all reflected in Yang Hui's works. They not only inherited the glorious legacy of ancient mathematics in China, but also developed creatively. Zhu Shijie's work, in a sense, can be regarded as the representative of mathematics in Song and Yuan Dynasties and the peak of the development of ancient computer systems. Even western bourgeois scholars can't deny this. George Sarton said: Zhu Shijie "belongs to the Han nationality, belongs to the era in which he lived, and is also an outstanding mathematician in all previous dynasties", and called "Meet with Siyuan" the most important mathematical work in China and one of the most outstanding mathematical works in the Middle Ages ". With his outstanding works, Zhu Shijie pushed ancient mathematics in China to a new height, added a new chapter to the glorious history of ancient mathematics in China, and formed the highest peak of mathematics development in China-Song Dynasty.
Yangzhou anecdote
From this, we know that Zhu Shijie was born in Beijing. /kloc-in the late third century, he traveled all over the country for more than 20 years as a famous mathematician. Zhu Shijie finally settled in Yangzhou, where he studied mathematics and gave lectures. He attracted many scholars to engage in academic exchanges. Yangzhou is located at the intersection of north and south, where various academic ideas are integrated; At that time, Yangzhou's printing industry was very developed and it was the national book publishing center. Two books reflecting Zhu Shijie's achievements in mathematics, Arithmetic Enlightenment and Thinking of the Source, were printed and published in Yangzhou in the third year of Yuan Dade (1299) and the seventh year of Yuan Dade (1303) respectively.
***3 volumes, divided into 20 subjects, including 259 math problems. At the beginning of the book, Zhu Shijie gave 18 commonly used mathematical songs and various commonly used mathematical constants, including: 99 pieces of multiplication, 99 pieces of division (exactly the same as the abacus calculation formula later), zero pieces of weight, counting rules, decimal method, metrological conversion, pi, plus or minus multiplication and division rules, and root. The text includes multiplication and division and its flexible algorithm, multiplication and division, celestial technology, solving linear equations, and higher-order arithmetic progression summation. The book covers almost all aspects of mathematics at that time, forming a relatively complete system, which can be said to be a good mathematics textbook. Luo Shilin, a scholar in Yangzhou in Qing Dynasty, said that "the enlightenment of arithmetic" was "as shallow as reality", and such comments were very pertinent.
"Meet with Siyuan" is the representative work of Zhu Shijie's carefully arranged research results for many years. The book is divided into 3 volumes, 24 subjects and 288 questions. All the problems in the book are related to solving equations or equations. Among them, there are four unknown 7 questions, three unknown 13 questions, two unknown 36 questions and one unknown 232 questions. The preface lists four kinds of five diagrams such as Jia Xian Triangle, and gives examples of solving celestial sphere, binary, ternary and quaternary techniques. The last three are the column methods and solutions of binary, ternary and quaternary higher-order equations respectively. The book's greatest contribution is the creation of the four-element elimination method, which solves the problem of multivariate higher-order equations. Another great achievement in the book is the systematic solution to the problems of higher-order arithmetic progression summation and higher-order differential method.
Before Zhu Shijie, there was a way to understand the equation in ancient Chinese mathematics-"Tianyuan Shu", which solved the equation by setting "Tianyuan as so-and-so", so-and-so as (x). Zhu Shijie not only inherited the celestial sphere technique, but also extended the solution of equations from binary and ternary to quaternary. When there is more than one unknown quantity, in addition to the unknown Tianyuan (X), we also set up soil element (Y), human element (Z) and matter element (U), and then list binary, ternary or even quaternary simultaneous equations and solve them. In Europe, the solution of simultaneous linear equations began in16th century, and the study of simultaneous equations of multiple degrees began in18th and19th century. Zhu Shijie's "celestial skills" were more than 400 years earlier than those in Europe.
Zhu Shijie's research on "stacking" actually obtained a general solution to the higher-order arithmetic progression summation problem. Since the Song Dynasty, there has been a study on the summation of higher-order arithmetic progression in China. There are overlapping problems in the works of Shen Kuo (103 1- 1095) and Yang Hui (126 1- 1275).
"Meet with Siyuan" is a brilliant mathematical masterpiece, a master of mathematics in the Song and Yuan Dynasties, and the highest-level mathematical work in ancient China. Researchers in the history of modern mathematics spoke highly of Philip Burkart's encounter. George Sarton, a famous expert in the history of science, said that "Meeting with Siyuan" is "one of China's most important mathematical works and one of the most outstanding mathematical works in the Middle Ages". Joseph Needham, who wrote History of Science and Technology in China, commented on the meeting between Zhu Shijie and Philip Burkart in this way: "His previous mathematicians failed to touch the mysterious truth contained in this extensive and profound work".
Unfortunately, after Zhu Shijie, there were no profound mathematical works in Yuan Dynasty, and there were few new mathematical works in Han, Tang, Song and Yuan Dynasties, and many of them were even lost. In the thirty-seventh year of Qianlong (1772), when the Siku Quanshu Library opened, many ancient mathematical classics were discovered, but Zhu Shijie's works were not discovered, so they were not compiled at first. 1799, Ruan Yuan, Li Rui and others didn't introduce Siyuan's meeting when they compiled A Family Biography of Mathematics. Soon after, Ruan Yuan inspected the book in Zhejiang, and immediately compiled it into Sikuquanshu, which was handed over to Li Rui for proofreading (unfinished) and later carved by He Yuanxi. This is the first reprint of Siyuan Meeting since the first edition of 1303. From 65438 to 0839, Luo Shilin, a scholar from Yangzhou, published a book "Siyuan Meets Fine Grass" after years of research, and Luo Shi made a fine grass on every question in the book "Siyuan Meets Fine Grass". Just like Luo Shilin's second edition of Meet with Siyuan, arithmetic enlightenment is still missing. Later, Luo Shilin "heard that North Korea took the Book of Poetry as the arithmetic topic", so he asked people to find a reprint engraved by Jin Shizhen, the governor of the whole state of North Korea in the seventeenth year of Shunzhi (1660) in Beijing. In this way, "Arithmetic Enlightenment" was reprinted in Yangzhou, which is the mother of the existing version of the book.
Zhu Shijie's two outstanding mathematical works in Yuan Dynasty were both completed and engraved in Yangzhou. After hundreds of years of loss, it was discovered, collated and annotated by Yangzhou scholars, and reprinted and published in Yangzhou. This shows that Yangzhou has a very important position in the history of mathematics development in China.