Zhoubi Suanjing, formerly known as Zhoubi, is one of the ten books of Suanjing. It is an ancient work on astronomy and mathematics, mainly illustrating theGaitian saidandQuarternary calendarLaw.In the early Tang Dynasty, it was defined asImperial CollegeMingsuan SectionOne of the teaching materials, so it was renamed "Zhou Bi Suan Jing".
The Zhoubi Suanjing is determined by the most simple and feasible methodAstronomical calendar, revealingSun, moon and starsThe operation rules of, includingSeasonal alternationClimate change includes the fact that there are poles in the north and the south, and the day and night are the same.Since then, mathematicians of all dynasties have taken the Zhoubi Suanjing as a reference, and on this basis, they have continued to innovate and develop.
Chinese name
Zhou Bi Suanjing
Foreign name
Chou Bi Suan Ching
Alias
Pericycle
Translator
Jiang Xiaoyuan/Xie Jun
press
Liaoning Education Press
ISBN
nine trillion and seven hundred and eighty-seven billion five hundred and thirty-eight million two hundred and twenty-two thousand four hundred and fifty-six[3]
The text of Zhoubi Suanjing is limited to the question and answer between Shang Gao and Zhou Gong, as ifbe published in book formIt is not difficult to determine the age.However, after the reign of Emperor Qianlong and Emperor Jiaqing, the study of textual research rose, and the suspicion of the ancient became increasingly popular. In modern times, almost all Chinese and foreign scholars had to accept the inference that not only Shang Gao was invented by later generations, but even Chen Zi was also invented by later generations.Therefore, it is no longer meaningful to regard Shanggao's Q&A as the text of Zhoubi Suanjing.Therefore, many scholars regard the words after Chen Zi's question and answer as a part of the full text of Zhoubi Suanjing and no longer distinguish them.As a result, people began to infer the age of the book from the contents of Zhoubi Suanjing.
Common methods can be divided into two categories:History of AstronomyExperts like to use modern astronomical means, according to some specialAstronomical phenomenonOr data, calculate the age when it should appear, and then determine the age of the book.For example, the Japanese scholar Naoda Zhongliang used thePolaris(North Pole Xuanji)North Celestial PoleIt is calculated that the book was written between the 5th and 7th century BC.
The other method is to infer the age of the book by comparing some contents of Zhoubi Suanjing with other historical books with relatively clear ages.Qian Baocong (1892-1974) made the following textual research on the age of the Zhoubi Suanjing in the Zhoubi Suanjing Textual Research: First, the Zhoubi Suanjing is mainly divided into two parts, the first is the question and answer of Shang Gao, and the second is the Chen Zi model;Second, because of the suspicion that Shang Gao is the hypocrisy of later generations, we believe that the text below Chen Zi is the main body of the Zhoubi Suanjing《Huainanzi Astronomical Training》From six aspects, this paper discusses that the characters below Chen Zi were written around 100 BC;Third, the meaning of "Zhou Bi" is based on Chen Zizhi's theory, and other explanations are also mentioned;Fourth, compare 24 names and their order with《Trinity calendar》Similarities and differences, referring toZhao ShuangNote that the original solar term length of 15 days is similar to the rough notation of Huainanzi;Fifth, the complexity and《Chapter Nine Arithmetic》Similar.He concluded that Zhoubi Suanjing was written around 100 BC.Under the influence of the trend of skepticism, there is also a tendency that deserves people's attention. That is, the lower limit of all the contents in Zhoubi Suanjing is used to determine the age of the book. C. Cullen can be regarded as a representative of this aspect.
Publication background
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GukriIt is believed that most of the previous scholars mistakenly tried to discover the age when Zhoubi Suanjing was completed as a whole, so their conclusions were obtained in a false illusion.He believes that this book was written by researchers with the same idealsproceedings。First, he investigated the internal structure of Zhoubi Suanjing, divided it into different chapters, and discussed the relationship between sections;Second, discuss the content related to each sectionExternal worldInformation and activities;Third, discuss possible issues related to each sectionHistorical environment。He disorganized the overall arrangement of Zhoubi Suanjing and divided them intoOuter PartAnd the inner part.The inner part is mainly based on the Chenzi model, with its lower limit in the first century AD.
In the discussion about the external environment, it was pointed out that as a royalBibliography，Ban Gu(32-92)《Hanshu · Yiwenzhi》There are "Xu Shang Arithmetic" and "Du Zhong Arithmetic" without "Zhou Bi Suan Jing";Gaitian saidIt was well known in the first century AD,Cai YongIn 180 AD, it was clearly listed as the three ancient schools in the Middle CircleCosmologyone of.The conclusion is that due to theHuntian saidThe Zhoubi Suanjing could not be written earlier than the first century BC, but it could not be written later than 200 AD.To judge the completion time of ancient Chinese scientific classics, we shouldScientific thinkingorKnowledge levelThe age reflected is the criterion of judgment, rather than a few words mixed by the descendants in the book as the condition of inference.Since early scientific classics are usually the crystallization of human knowledge, it may be more meaningful to understand the origination period and transmission context of scientific ideas than to simply determine the age of the book.
History of ScienceIt has been proved repeatedly that scientific truth, which seems to be very obvious, often experienced long-term doubt, or even resistance, in the early stage of human understanding of it.For example,precession of the equinoxesThe dispute between the preservation and abolition of the phenomenon in the Northern and Southern Dynasties is a typical example.Therefore, we try to identify the content of Zhoubi Suanjing as being between two ancient documents by taking the completeness or correctness of the content of Zhoubi Suanjing as an examplebe published in book formThe method that the age must be somewhere between the two is unreliable.However, it is not feasible to use theoretical calculation of some important data to determine the age of the book.Feng Ligui once collected 14 different views on the discussion of the time when Zhoubi Suanjing was written.Although there are great differences in the judgment of the time when Zhoubi Suanjing was written, almost all researchers have a common understanding that Zhoubi Suanjing was not written by one person at a time. It has passed through many dynasties to complete its length and structure.
Zhang HongzhaoThe formation of Zhoubi Suanjing was clearly divided into three periods: the first period was the question and answer of Shanggao;The second issue is Chen Zi's Q&A;The third issue is about the words after Chen Zi.This division is a fact that many people acquiesce.As Chen Fangzheng said when summarizing the previous discussions on the process of the completion of Zhoubi Suanjing, Zhoubi is not only not a personal work, nor even a work of a single nature, but may be the result of the accumulation of several related, similar but different theories that have emerged in different historical periods.Therefore, it is not appropriate to regard Zhoubi Suanjing as a self consistent system to express the theory of covering heaven, while ignoring its stratification nature.The formation of Zhoubi Suanjing can be divided into three periods.Specifically, one of the first volumes, the question and answer between Shanggao and Zhougong, should be the original text of Zhoubi Suanjing, which reflects the early Chinese ancient mathematicians represented by Shanggao's understanding of mathematics and its use.Shanggao's answer to Duke Zhou's question is intended to explain the problem of solving geometric surveyingMathematical methodHe did it.This method containsPythagorean theoremAnd the way of using moments.According to Shang Gao, these mathematical contents were already available when Dayu controlled the flood and should be credible.In the second period, Chen Zi's model was put forward, and its content was the question and answer between Chen Zi and Rongfang in Volume II, which had been formed in the Warring States Period.During this period, Chen Zi further developed Shanggao's method of using moments to measure the sun's heightWeightotherapy。It is also believable.What Chen Zi tried to solve in his Q&A was to further improve the more ambitiousSky surveyThe theory and practice of land measurement.The proposal of Chen Zi's model, in fact, is a key step towards the goal of transforming Shang Gao's Zhoubi into Gaitian's Zhoubi, and transforming a relatively simple mathematical work into a purely mathematical astronomy work.
From the third part of the first volume, it is the expansion and improvement of the theory of Gaitian.First, under the basic assumptions of Chenzi modelSeven weights and six roomsOfCosmic model, and give it every day in the form of textsunlightThe calculation method of running track makes the seven scale diagram a real movable form that can be operatedAstrolabe。On this basis, a new pattern of the shape of heaven and earth is further introduced, and the geography is giventhe five zonesDivision, explanation of the causes of cold and heat, orientation of sunrise and sunset, and establishment of the theory of covering the skyAstrometry, introduced the concept of de extreme, and produced a relatively complete《Quarternary calendar》Wait.Although these greatly enriched the theoretical connotation of Chen Zi's model, they also created some contradictions within Gaitian's theory system that could not be completely self consistent, which became the target of continuous criticism by later scholars.The formation of this part means the completion of Zhoubi Suanjing as a monograph on the theory of Gaitian.From the third volume of the first volume of Zhoubi Suanjing, there appeared a large number of "Shuyue", which formed a clear contrast with the writing style of Shanggao's Q&A and Chenzi's Q&A, reflecting from one side the fact that its formation period should be more recent than the previous two parts.
To sum up, the first part of Shanggao's Q&A in Zhoubi Suanjing was once an independent article of Zhoubi Suanjing. It should have been completed in the early Western Zhou Dynasty, about the 11th century BC.The mathematical theory and universe model in Chen Zi's Q&A were completed in about the 4th and 5th centuries BC.As a mathematical astronomy work to explain the theory of Gaitian, Zhoubi Suanjing is the improvement and expansion of Chen Zi's model from the third volume of the first volume. Some of the basic data and structures, such as the seven scale diagram and de extreme, should have been determined after Chen Zi's model was proposed. However, Chen Zi's hypothesis of a parallel plane heaven and earth model has been modified to some extent,And some new things have been added, such as the causes of cold and heat and the calendar. In a word, the third part of the Zhoubi Suanjing is shaped according to the relationship between Qian Baocong andLiu ChaoyangIt should not be later than 100 BC.
Pythagorean theorem
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First, the Zhoubi Suanjing clearly recordsPythagorean theoremThe formula of "If you want to get the most evil sun, take the lower part of the sun as the hook, the higher part of the sun as the share, multiply the higher part by the lower part, and then divide the square to get rid of it." (Zhoubi Suanjing, Volume I, II)
The proof of Pythagorean Theorem is in Volume I of Zhoubi Suanjing——
Former Duke Zhou asked YuShanggaoHe said, "I heard that doctors are good at counting. Could you please tell me that the former Bao Xi set up a calendar of the whole week? The husband and the heaven can't rise without steps, and the ground can't be measured. How about counting?"
Shanggao said, "The method of number comes from the square, the circle comes from the square, the square comes from the moment, and the moment comes from the ninety-nine eighty-one. Therefore, when we break the moment, we think that it is three broad, four strands are trimmed, and five corners are cut. Since it is square, the outer half of it is one moment, and the ring is connected together, we can get 345. The two moments are twenty and five in length, which is called the product moment. Therefore, the reason why Yu ruled the world was born of this number."
Duke Zhou's comments on ancient timesFuxi(Victim) The story of constructing the calendar of the week is inconceivable (the sky can't be stepped up, and the earth can't be measured), so please ask where the mathematics knowledge of Shang Gao comes from.So he took the proof of Pythagorean theorem as an example to explain the origin of mathematical knowledge."The method of number comes from the square, the circle comes from the square, the square comes from the moment, and the moment comes from the ninety-nine eighty-one." The explanation and development context - the method of number comes from the circle (circumference three) square (four sides), and the circle comes from the square (circular area=circumscribedsquareArea * Pi/4), the square is derived from the moment (the square is derived from the equal moment on both sides), and the moment is derived from the ninety-nine eighty-one (the length multiplied by the width area is calculated from the ninety-nine multiplication table).
"Therefore, the moment ① is broken, thinking that the three corners are wide, the four corners are narrow, and the five corners are narrow."Mapping——Select a moment with three ticks (Pi 3) and four strands (square), and the line connecting the two ends of the moment should be 5 (radius corner five).
Pythagorean theorem
"② If it is square, the outer half of it is a moment, and the ring is connected together, it will be three to four to five." This is the keyProof process——Draw a square with two sides of the moment (square, thigh), and draw another moment according to the outside of the chord of the moment（carpenter 's square, actually used asright triangle）, which is obtained by "one moment of the outer half"triangleCut the circle and copy it to form a big square. You can see that there are three squares with side length of three hooks, four strands and five strings.
"The two moments have a total length of ③ twenty and five, which is called the product moment." This is a checking calculation - the sum of the areas of the square and the thigh square, which is equal to the area of the chord square 25 - from the graphic point of view, the big square minus fourTriangle areaAfter that, it is the chord square, and then the big square minus the area of the upper right and lower left rectangles is the sum of the square strands.Since a triangle is half of the area of a rectangle, it can be deduced that the area of four triangles is equal to the area of the two rectangles on the upper right and the lower left, so square hook+strand square=chord square.
Note: ① Moment, also called curve ruler, is an L-shaped carpenter's tool, which is a right angle composed of two long and short pieces of wood.In ancient times, "moment" refers to L-shaped curved ruler, and "rectangle" is the rectangle derived from "moment".
② There is a dispute about the sentence "one square, one half of the other".the Qing dynastyComplete Library of Four Branches of BooksThe version is defined as "one half of the square",And itMost of the previous versions were "half the square".By Chen Liangzuo, Li GuoweiLi Jimin、Qu AnjingAccording to other scholars' research, it is more logical to "make a square, and make the outer half a moment".
③ The long refers to the area.In ancient times, different dimensions ofdimensionComparisons do not invent new terms, but are collectively referred to as "long".Zhao ShuangNote: "If two moments are equal, they should be multiplied by each other. If the two moments are equal, they should be equal to each other.
Due to the long history, the chord diagram of Zhou Gong was lost, and the handed down version only printedZhao Shuang String Diagram（PapermakingIt was invented in the Han Dynasty.Therefore, some scholars mistakenly thought that Shang Gao had not proved it (just said a paragraph of inexplicable words), and then Zhao Shuang gave a proof.In fact, it's not true. Extract what Zhao Shuang did when he annotated the Zhoubi Suanjing《Pythagorean circle square》- "The sentence shares are multiplied separately, and the combination of them is the chord, and the square root division is the chord. Case:Chordal graphYou can also multiply sentence shares into Zhu Shi 2, and multiply sentence shares into Zhu Shi 4, and multiply the difference between sentence shares into medium yellow solid, and add difference solid into string solid.
