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Among the many craters on the far side of the moon, five are named after ancient Chinese astronomers: Zu Chongzhi, Guo Shoujing, Zhang Heng, Shi Shen and Wanhu crater. Among the five craters, Guo, Zhang, Shi and Wan were officially approved by the International Astronomical Union in 1970, and the unique Zuchongzhi crater located at 17.16°N and menstrual longitude was recognized by the International Astronomical Society in 1961.
Zu Chongzhi crater got its name so early and has something to do with the Soviet Union. The crater, called the crater, was photographed by the Soviet Luna 3 satellite launched in 1959, and according to Boris Yevseyevich Chertok, a Soviet space control system designer involved in the mission, the Soviet Academy of Sciences decided to name the crater after famous ancient scientists and cultural figures, and specifically requested that an American and a Chinese scientist be included. They chose the well-known Edison as the representative of the United States, but because they were not familiar with Chinese scientists, they could only consult the Chinese embassy in the Soviet Union, and finally the Chinese side provided Zu Chongzhi with the reason that Zu Chongzhi had made outstanding mathematical achievements in the 5th century AD—that is, the well-known pi π exact approximations, and approximation and densities.
If Chertock's statement is true, Zu Chongzhi's name as a lunar crater does not seem to be based on astronomical achievements, but his inclusion in the lunar crater seems to imply his contribution to astronomy. This anecdote is actually similar to the later generations of Zu Chongzhi's understanding, because his achievements in pi estimation are too conspicuous, so people often unconsciously overlook other facts, that is, he is not only an excellent mathematician, but also an astronomer who has made an indestructible contribution to the calendar.
A scholar from a lower- and middle-class family
Zu Chongzhi, also known as Wen Yuan, was a native of Fanyang (present-day Laishui County, Hebei) in the north, and he himself was a native of Jiangnan. The Fan Yang Zu clan in the north ran to the south to take root, naturally because of the great chaos in Yongjia at the end of the Western Jin Dynasty, "Luojing overturned, and the Zhongzhou warriors and women avoided the chaos of Jiangzuo" ("Jin Shu Wang Dao Biography").
Statue of Zu Chong
In the north, the Fan Yang Zu clan has people for generations to achieve 2,000-stone level officials, and nine consecutive generations have been honored with filial piety, but in the south, they have not been able to enter the ranks of the gate valve clan, and the most famous ancestor in the family is only a posthumous riding general, which is completely incomparable with the Wang and Xie families. The humility and embarrassment of Zu's position in the court can be seen from the experience of Zu Chongzhi's great-grandfather Zutaizhi.
Around the seventeenth year of the Eastern Jin Dynasty (392), Zu Taizhi had become Shangshu Zuocheng, but was publicly insulted by Wang Guobao, a son of the Taiyuan Wang clan. According to history, after Wang Guobao got drunk at a banquet, he cursed Zutaizhi with his sleeves rolled up, and threw a coiled musical instrument at him, and Zutaizhi "did not dare to speak". When Sima Yao, the reigning emperor of Jin Xiaowu, heard this, he was removed from his post as a treasure of the kingdom, and he was also dismissed from office because he was cowardly and "not a supervisor".
Emperor Xiaowu disliked the cowardice of Zutai, but the kingdom treasure family was prominent, his father was Wang Tanzhi, an auxiliary minister when Emperor Xiaowu was young, and his father-in-law was Xie An, the name of the "pretentious town" in the Battle of Shuishui, and he himself had a close relationship with Emperor Xiaowu and Sima Daozi, who controlled the imperial government, how dare he contradict him? It's just a matter of swallowing your anger. Later, the political situation was repeated, Huan Xuan ruled to eradicate the royal family and the forces of the great clans such as Wang, Xie, and Yu, and Zutaizhi participated in it, impeaching Zhongshu Lang Fan Tai, former Situ Zuo Changshi Wang Zhunzhi, and auxiliary general Sima Xunzhi for "being rude" in the name of Imperial History Zhongcheng, forcing the three to withdraw from office, which can be regarded as a bad breath for the children of the family.
In the second year of the Jin Dynasty and the first year of the Yongchu Dynasty of the Liu Song Dynasty (420), Liu Yu destroyed the Eastern Jin Dynasty and established the Liu Song Dynasty, and the ancestral family continued to serve in the imperial court like most people. Zu Chong's grandfather Zu Chang served as a general master craftsman (renamed as Great Craftsman Qing during Emperor Wu of Liang), with a rank of Zhong 2,000 shi, responsible for civil engineering at the imperial court; His father, Zu Shuozhi, was invited by the dynasty to place the post of idle official for the Southern Dynasty. There is only one sentence in the original biography of Zutai in the official history, and Zu Chang and Zu Shuozhi do not even have a biography, but what is surprising is that although his father and ancestor are unknown, Zu Chongzhi turned over and entered the world's field of vision during the reign of Liu Jun, the fourth emperor of the Liu and Song dynasties, Xiaowu Emperor Liu Jun.
Zu Chongzhi was born in the sixth year of Yuan Jia (429), one year older than Liu Jun, and the history books do not record that the two had anything to do with each other in the early days, but after Liu Jun took the throne, he immediately asked Zu Chongzhi to "Zhihua Lin Xue Province" and "gave the house a car suit" ("The Book of Southern Qi, The Legend of Zu Chong"). There is no record of "Hualin Province" in the Book of Song, but it is frequently mentioned that Hualin Garden, Emperor Wu and Emperor Xiaowu personally heard lawsuits here many times, and the young emperor set up a stall in Hualin Garden before he was killed, which shows that it is a garden frequented by Emperor Liu Song, and it is reasonable to speculate that "Zhihua Forestry Province" should be a close attendant of the emperor who works in Hualin Garden. Not only that, in the fifth year of the Ming Dynasty (461), Zu Chongzhi's first appearance ("Shi Brown") was appointed as the official of Liu Ziluan's assassination in Southern Xuzhou, and the official mansion to join the army is more worthy of the world's deep thought: Liu Ziluan is Liu Jun's favorite son, as long as his father sees something good, "he will not enter Ziluan's mansion", in addition, it is Liu Song's practice to subdue the local noble family with the prince leading the county, in order to assist these princes who are not familiar with the world, he will often be assigned to a shrewd and capable imperial henchman as a staff member. Liu Ziluan was only 5 years old in the fifth year of the Ming Dynasty, and it was more likely that he would be in a remote lead, and it is unknown whether Zu Chongzhi was handling official affairs in Southern Xuzhou or beside Liu Ziluan in the capital, but he could be among the close ministers entrusted by the emperor to his beloved son, which shows that his relationship with Liu Jun is far closer than others imagine.
Around this time, Zu Chongzhi, who was fully trusted by the emperor, was able to make a major astronomical reform involving many academic aspects—revising the traditional calendar and introducing the Ming Calendar.
The revolutionary "Great Ming Calendar"
The Chinese ancestors who created a splendid agricultural civilization had a strong interest in the calendar very early on out of production needs, and after a long period of observation and summary of the operation law of the sun and moon, they developed a relatively rare combination of lunisolar calendars, that is, from one full moon (or moon absence) to the next full moon (or moon absence) as the base month (synodic month), and the winter solstice of the current year to the winter solstice of the following year as the base year (return year). The advantage of this setting is that the moon is determined by synodic moon, and the time can be determined by looking up, which is convenient to determine; The year is set with a return year, and the seasons are roughly the same every year, which is convenient for production.
However, the synodic moon is actually the length of the moon's orbital cycle, and the return year is the length of time the earth orbits the sun, and the two are not divisible, and the month is 30 or 29 days according to the size of the month, and the December month is 354 days, but the return year has a total of 365.25 days, and there is a gap of about 11 days. In order to solve this problem, ancient astronomical calendarists used the leap method to make up for it, that is, to add an additional "leap month" every two or three years, which led to a new question: So how many years should a leap month be placed?
Zu Chongzhi crater photographed by the US Lunar Orbiter 5
The solution was proposed as early as the pre-Qin period, and in practice it was found that the length of 19 return years was about the same as 235 synodic months, so the gap was leveled in addition to the 228 months of the normal 19 years. Since the ancients called 19 years a "chapter year", 19 years and 7 leaps were also called "chapter years". The "four-point calendar", which became popular from the Han Dynasty, was formulated based on the "chapter year law".
Obviously, the "chapter year" is only an approximation, and the error will become larger and larger over time, and by the time of the Southern and Northern Dynasties, people have found that although the "chapter year" can close the days, the time of the month has deviated from the original season of the month, which is undoubtedly a great bad news for a country that needs to arrange agricultural production according to the monthly solar terms, and the need to revise the calendar has become urgent.
The first to challenge the "Zhangshi Law" was the Northern Liang scholar Zhao, who formulated the Xuanshi Calendar (also known as the Yuan Calendar) in the first year of the Xuan Dynasty of Northern Liang (412), proposing to use the method of setting 221 leaps in 600 years instead. The pioneer of the Southern Dynasty's revision of the calendar was He Chengtian, who was also a member of the Liu and Song dynasties with Zu Chongzhi, and the calendar he compiled was implemented in the 22nd year of Emperor Yuanjia (445) of the Song Dynasty, and is known as the Yuanjia Calendar. Since the "Yuan Jia Li" still uses the "chapter year method", so there is still an error with the reality, Zu Chongzhi, after repeated calculations, believes that the 221 leap month in Zhao 600 is too little, but the "chapter year method" is too dense, and there will be one more day every 200 years, so he proposed to change it to 391 and set 144 leap month.
A bronze statue of He Chengtian, now in the Museum of the Six Dynasties in Nanjing
According to Zu Chongzhi's algorithm, the actual annual year is 365.24281481 days, while the year measured by modern astronomy is 365.24219879 days, with an error of only one in 650,000, about 50 seconds, this accurate record was not refreshed until 608 by Sui Dynasty astronomer Zhang Xuan with 365.24203170 days of the "Great Business Calendar".
Why was Zu Chongzhi able to determine the year of return so precisely? The main reason was that he introduced the most advanced astronomical discovery at that time, the equatorial precession confirmed by Yu Xi, an astronomer of the Eastern Jin Dynasty. The so-called equatorial precession is a phenomenon of spring equinox displacement caused by the movement of the Earth's rotation axis. Modern people know that the earth revolves around the sun on one side of the solar system, and rotates along the north and south axis on the other, but there is a gravitational influence between the earth and the sun, moon and even several other planets, so the axis of rotation is not in a stable state, if we imagine a line through the center of the earth and perpendicular to the ecliptic plane, we will find that the two ends of the earth's axis of rotation are slowly rotating around the imaginary line, and the result is that from the ground, the position of the sun on the winter solstice will have a slight displacement with the winter solstice of the following year. That's about 50.2 seconds per year, and it shifts one degree backward every 71.66667 years or so.
Bronze statue of Zu Chong
Precise pi ahead of millennia
What exactly is the ratio of circumference to radius π? This is not only an inevitable problem that people will encounter when studying astronomy, but also a problem they will encounter as long as they carry out production and life. The earliest human record of pi is in the Egyptian Rhind Mathematical Papyrus around the 16th century BC, which calculates pi as 3.1605. At that time, the ancient Egyptians often used empirical formulas to determine π value, and the method was also very simple: they placed the millet on the circumference and diameter, and by calculating the proportion of millet, they could obtain an approximation of the π.
One of the earliest mathematical works in ancient China, the "Zhou Qiu Suanjing" written at the end of the Western Han Dynasty, mentions "circle diameter one two Wednesday", apparently setting the π value as 3, that is, what the ancients called "ancient rate", although it is only π a rough approximation, but with the level of mathematical development at that time, there is no way to calculate a better value, so the "ancient rate" is also used in the "Nine Chapters of Arithmetic" written in the early Eastern Han Dynasty.
Chinese more accurate π values were derived from the Xinmang period at the end of the Western Han Dynasty. In Xinmang's "founding of the state", Wang Mang ordered Liu Xin, a national teacher, to imitate Zhou Li to make a copper hu, called "Lu Jia Lianghu" (also known as "Xinmang Jialiang"), which openly integrated five kinds of measuring tools, "the upper part is Hu, the lower is Dou, the left ear is the rise, the right ear is the combination, and the lower part is the gong", and the inscription on the back explains the specific dimensions of the Hu: "One foot square and round outside, nine centimeters and five millimeters next to the pot, one hundred and sixty-two inches, deep ruler, one thousand six hundred and twenty inches and a half, holding ten buckets." Notable in this text is the square, 庣, and power. The square is a square with a side length of 1 foot in the middle of the inner bottom of the measuring Hu, but is not in contact with the bottom wall, and the square is the distance from the vertex of the square to the circumference of the bottom of the Hu, so the diameter of the bottom circle is equal to the diagonal of the square plus the distance of two cylinders, according to the Pythagorean theorem, it is easy to calculate the radius of 0.7166 feet, and it is known that the "power" of the circle area is "162 inches", then it can be inferred that the π value used by Liu Xin is about 3.1547. In 1956, another Xinmang "bronze pinch of the first year of the founding of the People's Republic of China" was found in the Sui tomb of Liujiaqu in Shaanxi County, Henan Province, which was also engraved with similar squares, 庣, and powers, and estimated π value of about 3.1679.
Xinmang Jialiang, also known as Lujialianghu, is now in the National Palace Museum in Taipei
The history books do not explain how Liu Xin arrived at the π value, possibly based on an improved empirical formula. After that, Zhang Heng and Cai Yong of the Eastern Han Dynasty also used empirical formulas to give approximate π values, Zhang Heng believed that it was equal to 3.1622 (10 square meters); Cai Yong believed that it was equal to 25/8, and it was not until the Wei and Jin dynasties that the mathematician Liu Hui gave the first geometric method for finding pi - circle cutting when making notes on the "Nine Chapters of Arithmetic".
The full text of "Circle Circumcision" has a total of 1800 words, the core is to connect positive multi-variable sides in the construction circle, and the approximate circle area can be obtained by calculating the area of the regular polygon, and then the π value is inverted through the circle area formula (the circle area is equal to π multiplied by the square of the radius). In the example, Liu Hui first constructs an inscribed regular hexagon in a circle with a radius of 1 foot (starting from any point on the circle, intersecting the circle with the radius as a step, and the 6 points obtained are connected to the inscribed regular hexagon), and then constructing a regular dodecagonal shape according to the inscribed regular hexagon (making a line along the center of the circle to the midpoint of each side of the regular hexagon, and when the six lines extend to the circumference, 6 points are obtained, connecting the vertices of the regular hexagon and the new 6 points, and the regular 12-sided shape is obtained). By analogy, as long as there are more regular polygon sides, the closer its area is to a circle and the more accurate the π value.
Statue of Liu Hui, from the commemorative stamp "Ancient Chinese Scientists Group IV"
Since Liu Hui constructs regular polygons, its area can be simply calculated geometrically by the Pythagorean theorem. Liu Hui gave the area recursion formula from regular n-sided to regular 2n gons in the second part of "Circle Circumcision", and when derived to the regular 96-sided shape, he obtained π commonly used approximation of 3.14. However, what is more impressive is his subsequent processing of the data, when cutting to the regular 192 side, he gets an area of 314 and 64/625 inches squared, through the "power difference" method, that is, the positive 192 side and the positive 96 side area subtraction data multiplied by 2 on the positive 96 side area, you can get: positive 192 side area< circle area< positive 96 side area + difference power × 2, through weighted average calculation, you can get a π value of 3.1416.
Archimedes circular circumcision schematic, he constructed both the circle inscribed and circumscribed regular polygons, and then calculated their perimeter to obtain a perimeter approximation, repeating this step to obtain a pi of about 3.1409<π<3.1429. Archimedes' algorithm needs to calculate both inscribed and circumscribed regular polygons, which is much larger than that of circular cutting, and because the idea of "power difference" is close to the "least squares" method in numerical analysis, the numerical accuracy will be higher
It was on the basis of Liu Hui and others that Zu Chongzhi and his son Zu Wei (also recorded as Zu Weizhi) pushed pi to a new peak, accurate to 7 decimal places. The historical information about his calculation of pi is only found in the Sui Shu Law Chronicles: "Zu Chongzhi's more open secret method, with a circle diameter of 100 million as one zhang, the circumference of the profit number is 3 zhang 1 foot 4 inches 1 minute 5 centimeters 9 milliseconds 7 flicker, the number is 3 zhang 1 foot 4 inches 1 minute 5 centimeters 9 milliseconds 6 flicker, and the positive number is between the two limits of Yingxiao." The density rate circle diameter is 113, the circle is three hundred and fifty-five, the ratio circle diameter is about 7, and Tuesday 12. ”
This passage uses ancient mathematical expressions, so it is difficult to understand. "With a circle diameter of 100 million as one measure" can be understood as many as 9 digits (100 million) calculated by him, and the calculation can be made with this digit to calculate the result until 7 decimal places "flicker"; He said that his pi satisfies the inequalities 3.1415926 (朒) <π< 3.1415927 ( profit ) , and he also gives two convenient approximations : the approximate rate (22/7, approximately equal to 3.14285714) and the density ratio ( 355/113 , approximately equal to 3.14159292 ) .
Considering the level of mathematics at that time, later generations thought that Zu Chongzhi was advancing along the path of Liu Hui's circular cutting, and calculated the positive 3072 side and supplemented by the "interpolation method" to calculate the π value with such high accuracy. Although he still used geometric methods, it was not until the 15th century that the Central Asian mathematician Al-Kashi broke his record to calculate to 14 decimal places, and more accurate calculations were not achieved until after the middle of the 18th century, when Western mathematicians mastered modern mathematical tools such as infinite series, integrals, and power series expansions.
Liu Hui circumcision calculation schematic diagram, when the regular hexagon (green part) is cut into a regular dodecagonal (green, red, blue part), the red part shown in d can be obtained by using the Pythagorean theorem, which is the "difference power" (the difference between the area of the regular dodecagoon minus the regular hexagon), and the "difference power" multiplied by 2 is the rectangular ABCD area. It can be seen that the area of the circle is larger than the area of the regular dodecagonal shape, but less than the area of the regular hexagon plus ABCD (Liu Hui inequality), from which the π value between a large number and a small number can be determined, and a more accurate π value can be obtained by repeating this step
In addition, it cannot be ignored that Zu Chongzhi gave a simple and very accurate approximation rate and density rate, the approximation rate is probably based on the approximation of the π given by Liu Hui 157/50, by solving the indefinite equation, the first set of solutions is 22/7, and the density rate is about Zu Chongzhi's originality, but later generations do not know how he solved this solution, and can only guess that it may be the use of He Chengtian's "day adjustment method" (interpolation method of numerical approximation), or the use of the continuous fraction method to find the best asymptotic fraction. Or 355/113 was also solved using indefinite equations, but either method was not recalculated by German mathematicians until 1573. In terms of yes, Zu Chongzhi's calculation of pi is thousands of years ahead of the world.
Academic achievements have a bad fate
In the sixth year of the Ming Dynasty (462), Zu Chongzhi presented the Ming Calendar, which he had spent many years working on, to Emperor Xiaowu of Song, but was resolutely opposed by Emperor Xiaowu's favored Dai Faxing, and the ministers were afraid of Dai Faxing's power and agreed with his opinion. Zu Chongzhi repeatedly refuted the difficulties, and it was not until the eighth year of the Ming Dynasty that he persuaded Emperor Xiaowu to change the Yuan and the calendar the following year. Unexpectedly, Emperor Xiaowu died that year, and the matter was shelved. In the Southern Qi dynasty Liu and Song dynasties, Zu Chongzhi's "Ming Calendar" was again supported by Xiao Changmao, the prince of Wenhui, but Xiao Changmao died before it was ready for implementation, and the matter was again shelved. It was not until the ninth year of Tianjian (510) after Liang Dynasty Qi that it was implemented under Zu Wei, by which time Zu Chongzhi had died for 10 years, and the dynasty in which he formulated the calendar had changed to the third.
Fortunately, the glorious academic achievements of Zu Chongzhi and his son were far longer than all fateful blows. After a thousand years, the name of Zu Chong has not only not been forgotten, but also went abroad and landed on the moon, and Dai Faxing and the like, who once gave power and suppressed Zu Chong's achievements, in addition to being occasionally mentioned as the villain in the story, who else will remember?
(Source: National Humanities History)