***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.