Cheng Dawei was born in a small business, smart and studious since childhood, especially fond of mathematics, and often spent a lot of money on books to do calculations. When he was about 20 years old, he took advantage of his business trip to invite him to visit Wu Chu, visit famous teachers all over the world, and get to know "those who are proficient in mathematics, those who are diligent and tireless". He lives in a small county and attaches great importance to land survey. He once created a "surveyor" and drew pictures handed down from generation to generation. After the age of 40, Cheng Dawei was tired of traveling abroad, so he "went back to floating on the water for more than 20 years". He carefully studied ancient books, explained their meanings, tried to make them into laws, learned from the strengths of various schools, and combined with his own experience, and finally wrote the Arithmetic Unification School (formerly known as the Arithmetic Unification School) in the twentieth year of Wanli (1592), with a volume of 17. In the following six years (1598), the book was deleted, its essence was revealed, and it was written into four volumes and published in Xiuning.
The first and second volumes are the basic knowledge used in the whole book. Volumes 3 to 12 are a compilation of solutions to various application problems, and each volume is basically titled with the chapter names of nine chapters on arithmetic. Volumes 13 to 16 are "difficult problems". In fact, the algorithm is very simple, just expressing the conditions in poetry. More obscure; Volume 17 is Miscellaneous Law. All kinds of problems in the book are calculated by abacus, and a set of concise and fluent abacus formulas and prescriptions used by Cheng Dawei have been used to this day. This book systematically summarizes China's bead algorithm and becomes a relatively complete bead calculation book. Its publication and wide spread marked the transition from pre-calculation to abacus calculation in the history of Chinese mathematics, so Cheng Dawei himself was called "the master of abacus calculation".
Xu Guangqi, a thinker in the late Ming Dynasty, once pointed out that there were two reasons for the backwardness of mathematics in the Ming Dynasty. One is that "Confucianism of fame and reason is the truth of the world", and the other is that "the fallacies of numbers are the truth of God." Cheng Dawei, as a mathematician, is different from other "Confucian scholars of name and reason". He attaches great importance to practical work and the application of mathematics. His "unity of arithmetic" can be "popular all over the world" and make "a person who holds a plan in the sea should also raise a series at home", which is inseparable from its practicality.
Attach importance to the application of mathematics
Cheng Dawei believes that mathematics has a wide range of uses. He said: "Far away, the world is vast and the mountains and rivers are vast; The demand of the court and the army is large, and the daily expenses of people's livelihood are small. " Wu Jisui also quoted what he said in the preface of Arithmetic Unity: "More is better than less, but is it worse?" In Cheng Dawei's view, mathematics is indispensable to society and life. In his book "Arithmetic Unity", he wrote in the form of poetry: "The six arts in the world are the foundation of human beings; Knowing books and algorithms is like fainting in a dark room. " This is in sharp contrast with the attitude of Neo-Confucianists who opposed practical knowledge and despised mathematics at that time. At that time, the popular stereotyped scholar selection system was based on the proposition of four books, five classics and eight essays, which guided intellectuals away from natural science and seriously bound their thoughts. For the sake of fame and fortune, many scholars are immersed in Confucian classics and only talk about feudal ethics such as the Three Cardinals and the Five Permanents. Where will they care about practical science and technology such as mathematics? Cheng Dawei, on the other hand, was able to break through the shackles of Confucianism and devote himself to writing arithmetic unification after middle age. In order to solve the practical problems urgently needed by the society at that time, this spirit is very valuable.
Not only that, Cheng Dawei also dared to point out the disadvantages of the times, and wrote directly from the book, exposing the fooling of corrupt officials to the people from a mathematical point of view. This idea is embodied in the third volume of "On Mu Fa". The article said: "In the nine years of Wanli, the imperial edict was clear, and the general book of our city (Xiuning) was good at changing the wooden method. The site is divided into four grades, upper 190, middle 220, lower 260 and lower 300. ..... Compared with the former sages, it is two hundred and forty steps per mu. It is also appropriate to take advantage of the fertile soil in Japan to levy it, and to compete with it. Don't change the mu method easily, no matter how much rice and wheat there are. The first book has its shortcomings, and the city is followed. How can I put it? I remember this and regard it as a smart and confused cloud. " Obviously, it is absurd to determine the unit of cultivated land by "land fertility" and "severity of expropriation" Its purpose is nothing more than to fish in troubled waters and blackmail the people. Between the lines of this passage, an upright mathematician showed deep sympathy for the people.
Throughout the whole book "Algorithm Classics", the author attaches great importance to the application of mathematics. Most of the 595 questions are applied questions closely related to people's lives. There are some pure mathematical problems in prescription and inspection, which are also preparations for practical problems. In the application problems, including field survey, transportation, material distribution, volume calculation, tax trade, engineering technology and so on. The topic classification basically follows the nine-chapter arithmetic, but it is obviously different from the nine-chapter arithmetic in style, that is, the basic knowledge needed for learning the whole book is listed first, including algorithm outline, large numbers, decimals, weights and measures, field measurement system, abacus positioning method and four abacus operation formulas. This makes the book not only rich in content, but also convenient for self-study, and it is a good introduction to mathematics.
Improved bead algorithm
Another characteristic of algorithm unification is that most problems are abacus calculation, which also reflects the author's spirit of paying attention to application. The abacus is simple in structure, low in price and easy to carry. Compared with calculation, abacus calculation is more convenient and faster. But at that time, the abacus calculation method was not perfect, and some formulas were not smooth enough, so Cheng Dawei made great efforts to improve the abacus calculation algorithm and abacus calculation formula. In order to distinguish between multiplication and division formulas, he clearly stipulated in the first book that "99 composite number" is called decimal above, large number below, and "Drunken Song" is called decimal above and decimal below. For example, "684 18" is a multiplication formula and "8674" is a division formula. The book records a complete collision formula, such as "once: I didn't see anything except 9/kloc-0, so I got up and went back"; "two returns, see two without dividing 92, get up and return to two" and so on. In the sixth and seventh volumes, Cheng Dawei also gave the method of calculating Kaiping and Cubic with abacus. Although it is not certain that this is his invention, this book is indeed one of the earliest ancient books that recorded this method. (Zhu Zaikun's "New Arithmetic Theory" was written earlier and published later than the arithmetic unification school, and it also has the method of opening and establishing abacus. ) The abacus positioning rule in the book should be attributed to Cheng Dawei, because it was not mentioned in the popular abacus books at that time. Although there is a positioning method in Jason Wu's "Nine Chapters of Algorithm Analogy", it is used for calculation. The Positioning Song in Arithmetic Unity completely describes the abacus positioning method for the first time:
"Several positioning methods are surprising, because they are all pushed down.
Addition and subtraction only need to identify the standard and return to the upper application.
The original number of dharma is inversely proportional to the upper number, and the zero before dharma is appropriate.
If the law is few, the real number will be reduced, and zero will be known before the law. "
Cheng Dawei attached great importance to the abacus formula. He believes that formulas are the basis of learning and using abacus, so they must be remembered. He repeatedly stressed: "First, we must be familiar with the nine songs, and second, we must recite the method of dividing songs." "Arithmetic learners must study hard and learn nine numbers from time to time."
Supplementary area formula
Among all kinds of practical problems solved by bead algorithm, the area problem is particularly noticeable. For the vast rural areas, field investigation is essential, so Cheng Dawei attaches great importance to the area problem. In the third volume "Square Field" of "Arithmetic Unity", he summed up a large number of area formulas combined with field measurement, compiled them into ballads and gave figures. There are more than 60 kinds of drawings in this volume, of which more than a dozen are basic, and others are composed of these drawings. Among these dozens of figures, there are some formulas in nine chapters of arithmetic, such as square field (square), straight field (rectangle), Keita field (triangle), evil field (trapezoid), circular field (circle) and arc field. , while others are not nine chapters of arithmetic. Cheng Dawei gave formulas respectively.
For the calculation results, Cheng Dawei not only requires as accurate as possible, but also advocates moderation according to specific conditions.
Cheng Dawei created the "truncation method" instead of the old method, just for the accuracy of the calculation results. He said: "When the skew is not equal, there must be an oblique step. How can you multiply in front? " ? If you cut it off, it will be correct. "For more complex graphics, it is not enough to only use the' cutting method', so Cheng Dawei adopted the method of' cutting surplus to fill vacancy'". He said: "There are many shapes of fields, which are difficult to carry. Scholars don't have to stick to the mud, but have to improvise, so they must cut profits and fill vacancies, be modest and reduce costs, and follow the rules. But Tanaka first took out the shapes of square, straight, Pythagoras, GUI and shuttle. , and then merge into one, and then multiply and divide, with Shao's prescription reduction, this is the essence. "But his requirements for accuracy are limited because he focuses on applications. He pointed out: "The world's habitual calculator is based on the square five oblique seven and the circumference three radial one, but I don't know that the square five oblique seven is odd and the radial one is odd. "This shows that he knows that there is a more accurate ratio, but he doesn't think it is necessary to use it because more numbers are difficult to collect", that is, there are many accurate data bits, and the calculation of mining is too complicated, which is often unnecessary in practical application.
Create a survey walker
In order to meet the needs of measuring fields at that time, Cheng Dawei also created a kind of measuring walker, which was painted with numbers and explained in detail in the book "Unified Clan Arithmetic"; This measuring tool is similar to the existing tape measure, and consists of a ring, a cross, a rotating shaft, a lock, a drill angle and a bamboo ruler (a ruler made of thin bamboo pieces) wrapped in the cross. This was a very advanced measuring tool at that time. Cheng Dawei was very proud of his invention. He wrote on the edge of the picture: "Qu Bin is good at manufacturing, and mathematics is good at official script."
Add arithmetic to poetry.
In addition to the content of the arithmetic unification school, we can also see from its text form that the author attaches importance to the application and popularization of mathematics. The words in the book are divided into three forms: narrative, poetry and rhyme, and words in charts. Poetry runs through the book and accounts for a considerable proportion. These poems are not only beautiful literary works, but also directly serve mathematics. For example, the song style of "Leave your head and take a ride" is a seven-line: "The method of taking a ride begins with two reasons, three or four or five times, and the standard is broken." A poem entitled "Xijiang Moon" written by Xia Zhang is used in the proposition: "A flock of sheep is one hundred and forty, and shearing sheep is not afraid of hard work. There are ewes in the flock, so it is better to cut two hairs first. A big sheep shears two kilograms of wool, one or two lambs, and one hundred and fifty kilograms is a seedling. How many should there be between mother and son? " These poems are easy to understand, vivid and interesting, so that readers can enjoy the beauty while learning abacus. Another example is the five laws used by Yinglizhang to make a proposition: "Take a pot of wine today and walk in the suburbs in spring. Double your friends and drink nine in the shop. I met three shops and drank all the wine in the pot. If you can learn, how can you know the original? " This poem is not only catchy, but also full of life. After reading the complete poem, it seems that there is an interesting picture of a spring outing with wine in front of you. This popular and vivid poem will undoubtedly interest readers. The book "Arithmetic Unification" contains arithmetic problems with poems, endowing mathematics books with literary color, and its effect of popularizing mathematics is obvious. While enjoying these poems happily, people began to understand mathematics. Arithmetic Tongzong became the most popular arithmetic book in Ming and Qing Dynasties, even surpassing the country, and was welcomed by people in Japan, Korea and Southeast Asia. Undoubtedly, its fascinating words are one of the reasons.