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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 5 craters, 4 of them, Guo, Zhang, Shi and Wan, were officially approved by the International Astronomical Union in 1970 and are uniquely located at the latitude of 1716°N, menstruation 145The 16°E Ancestral Crater was recognized by the International Astronomical Society in 1961 nine years earlier.
When Zu Chongzhi is mentioned, perhaps the first thing that comes to mind is his outstanding mathematical achievement in the 5th century A.D., which is known as the exact approximation of pi and the approximation and density ratio.
But it is also overlooked that he was not only a brilliant mathematician, but also an astronomer who made an indestructible contribution to the calendar.
A scholar from a lower-middle-class family
Zu Chongzhi, the word is far away, the ancestor is a native of Fanyang (now Laishui County, Hebei) in the north, and he himself is a native of Jiangnan. The Fan Yang Zu clan in the north ran to the south to take root, naturally because of the Yongjia chaos at the end of the Western Jin Dynasty.
Statue of Zu Chong. Zu Chongzhi was an outstanding mathematician and astronomer in the Song and Qi dynasties of the Southern Dynasties. He is best known for his work in calculating an exact approximation of pi and giving an approximation and density, the result of which is thousands of years ahead of the world. In addition to this, he was also a prominent astronomer and compiled the Great Ming Calendar
In the second year of the reign of Yuanxi of the Jin Dynasty and the first year of Yongchu 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 Chongzhi's grandfather, Zu Chang, served as a master craftsman (renamed Dajiang Qing when Emperor Wu of Liang was represented), with a rank of 2,000 stones, and was responsible for the civil engineering of the imperial court;His father, Zu Shuozhi, was invited by the court to place the position of an idle official for the Southern Dynasty. There is only one sentence in the official history of Zu Taizhi's biography, Zu Chang and Zu Shuozhi are not even 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 Liu Song.
Zu Chongzhi was born in the sixth year of Yuanjia (429), one year older than Liu Jun, and there is no record in the history books that the two had anything to do in the early days, but after Liu Jun ascended the throne, he immediately let Zu Chongzhi"Zhihua Forestry Province", and"Give the house a car suit"("The Book of Nanqi: The Biography of Zu Chong"). Not in the Book of Song"Hualin Province"But frequently mention Hualin Garden, Emperor Wu, Emperor Xiaowu heard the lawsuit here many times, the young emperor was killed before the stall in Hualin Garden, it can be seen that Emperor Liu Song often went to a garden, speculation"Zhihua Forestry Province"It should be reasonable for the emperor's close attendants to work in Hualin Garden. Not only that, in the fifth year of the Ming Dynasty (461), Zu Chongzhi made his first appearance ("Release brown"He was appointed as the assassin of Southern Xuzhou Liu Ziluan's engagement, and the public government 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"Don't go into Ziluan's house"In addition, it was the practice of Liu Song to subdue the local wealthy clans with the prince's leadership of the county, and in order to assist these princes who were ignorant of the world, he would often be assigned 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 is more likely that he will be led remotely, although it is unknown whether Zu Chongzhi is handling official business in Southern Xuzhou or by Liu Ziluan's side in the capital, but he can be among the close ministers entrusted by the emperor to his 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 that involved a lot of extra-academic implications—revising the traditional calendar and introducing the "Great Ming Calendar".
The U.S. Lunar Orbiter 5 photographed the Zu Chongzhi crater, which is located at the latitude of 17 months16°N, menstruation 14516° E, diameter 283 km, depth 198 km. The crater was first photographed by the Soviet Luna 3 satellite, which was launched in 1959, and the USSR Academy of Sciences named it Zu Chongzhi Crater after consulting the Chinese side.
The revolutionary "Da 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 laws of the sun and moon, they developed a relatively rare set of lunisolar calendars combined with the calendar, that is, the moon is determined by the full moon (or the absence of the moon) from one full moon to the next full moon (or the absence of the moon), and the winter solstice of the current year to the winter solstice of the next year is the basis for the year (return year). The advantage of this determination is that the synodic moon is used to determine the month, and the time can be determined by looking up, which is convenient to determine;The year is determined by the return year, and the season is roughly the same every year, which is convenient for production.
However, the synodic month is actually the length of the moon's cycle around the earth, and the return year is the length of the earth's orbit around the sun, and the two are not divisible, and the month is 30 or 29 days according to the size of the month, which is 354 days in December, but the return year is 36525 days, there is a difference of about 11 days between the two. In order to solve this problem, the ancient astronomical calendarists used the intercalation method to complete it, that is, to add one more every two or three years"Leap month"This leads to a new question: how many years should a leap month be placed?
The solution was proposed as early as the pre-Qin period, and it was found in practice that the length of 19 return years was about the same as 235 synodic months, so that the difference was equalized by adding 7 leap months to the 228 months of the normal 19 years. Since the ancients called 19 years one"Chapter year", 19 years 7 leap is also called"Chapter age method"。It has been popular since the Han Dynasty"Quadrangular calendar", which is based on"Chapter age method"formulated.
Obviously,"Chapter age method"It is just an approximation, and the error will become larger and larger over time, and people have already discovered it by the time of the Northern and Southern Dynasties"Chapter age method"Although the days could be closed, the time of the month deviated from the original season of the month, which was undoubtedly great news for a country that needed to arrange agricultural production according to the solar terms of the month, and the need to revise the calendar became urgent.
Finally, after repeated calculations, Zu Chongzhi believed that the annual reality was 36524281481 days, while modern astronomy measured a year at 365On 24219879, the error was only 1/650,000, or about 50 seconds. Therefore, it was proposed to change it to 144 leap months in 391.
The Book of Southern Qi and the Biography of Zu Chong, Zu Chongzhi was born in the sixth year of Emperor Yuanjia of Liu Song and Wen of the Southern Dynasty (429) and died in the second year of Yongyuan of Southern Qi (500), so it was included in the biography of the Book of Southern Qi. However, the "Great Ming Calendar" compiled by his life's work was not implemented in the Song and Qi dynasties for various reasons, and it was not implemented until the ninth year of the Tianjian of Emperor Wu of Liang (510), 10 years after his death.
Why was Zu Chongzhi able to determine the return year so precisely?The main reason is that he introduced the most advanced astronomical discovery at that time - the equatorial precession confirmed by the Eastern Jin Dynasty astronomer Yu Xi.
The so-called equatorial precession is a phenomenon of displacement of the vernal equinox caused by the movement of the earth's axis of rotation.
Bronze statue of Zu Chong. Zu Chongzhi was not only an outstanding mathematician, but also an astronomer who made immortal contributions to the calendar. He introduced the precession of the equator in the "Great Ming Calendar", accurately calculated the monthly value of the node (the time it takes for the moon to pass through the intersection of the ecliptic and the white path twice), calculated the orbital period of Jupiter, and determined the time required for Mercury and Venus to orbit for one week.
Having an accurate calendar and using precession to determine the length of each year is one thing, and the other is naturally determining the time of the winter solstice. The ancient Chinese calendar has always taken the winter solstice point as the return year, and determining the specific winter solstice point time has also become the top priority of the calendar. For a long time, the ancients used a rough method of determining the winter solstice, that is, the day with the longest noon shadow in a year was set as the winter solstice, and the error could be measured in days. Since the Western Han Dynasty, astronomers realized that the accuracy needed to be improved from the sky to the specific moment, and began to try to find the specific winter solstice time, and by He Chengtian, the accuracy had been improved to about 50 moments by improving the measurement methods. However, it is on the basis of these measurement methods that Zu Chongzhi has improved the accuracy of the winter solstice point by a large margin through an extremely ingenious mathematical processing method.
Zu Chongzhi mathematically converted hard-to-measure time into geometric calculations, thus greatly reducing the error. According to his method, people did not need to make continuous observations at all, but only around the time of the winter solstice, and considering that it was only the fifth century AD, his mathematical thinking was really impressive. In fact, in Zu Chongzhi's "Great Ming Calendar", there are exquisite calculations everywhere, and he not only accurately calculated the monthly value of the node (the time it takes for the moon to pass through the ecliptic and the white path twice) to 27On 21,223 days, the difference with modern observations is only one millionth, which makes the estimation of lunar eclipses more accurate - the "Great Ming Calendar" can accurately calculate the four solar eclipses from the thirteenth year of Yuanjia (436) to the third year of the Ming Dynasty (459). In addition, Zu Chongzhi also calculated that Jupiter's orbital period is 11858 (Modern Determination 11.)In 862), it was determined that the time required for Mercury and Venus to orbit for one week was similar to that of modern astronomical observations, and if it were not for other reasons, it would have been enough to say that he was a great mathematician and astronomer based on a copy of the Great Ming Calendar.
But there is no need to regret it, because the mathematical talent in the obscuring calendar is none other than his other great achievement in mathematics - the calculation of the exact pi, the approximation (22 7) and the density (355 113).
Millennially accurate pi
What is the ratio of the circumference to the radius of a circle?This is not only a problem that people will inevitably encounter when studying astronomy, but also a problem that they will encounter as long as they carry out production and life. The earliest record of pi was written in the Egyptian Mathematical Papyrus in the 16th century BC, which calculated that pi was 31605。At that time, the ancient Egyptians used empirical formulas to determine the value, and the method was simple: they placed the millet on the circumference and diameter, and the approximate value could be obtained by calculating the proportion of the millet.
One of the earliest mathematical works in ancient China, it was mentioned in the "Zhou Ji Sutra" at the end of the Western Han Dynasty"Circle diameter one, two, Wednesday", apparently setting the value at 3, which was called by the ancients"Ancient rate"Although it is only a very rough approximation, with the level of mathematical development at that time, there was no way to calculate a better value, so it was also used in the "Nine Chapters of Arithmetic", which was written in the early Eastern Han Dynasty"Ancient rate"。
Portrait of Liu Hui from the commemorative stamp "Ancient Chinese Scientists Group IV". Liu Hui (c. 250—?).As a mathematician of the Wei and Jin dynasties, his "Nine Chapters of Arithmetic Notes" and "Island Arithmetic" are the most valuable mathematical heritage of our country. In the "Nine Chapters of Arithmetic Notes", he specifically introduced the specific process of finding pi by classical geometric methods - circumcision.
The Chinese came up with more accurate values in the late Western Han Dynasty.
Xinmang Jialiang, also known as Lijia Lianghu, is 25 high6 cm, now in the National Palace Museum, Taipei. The bronze measuring tool integrates five kinds of measuring tools into one, "the upper part is the Hu, the lower part is the bucket, the left ear is the ascending, the right ear is the close, and the lower part is the Gong", and the inscription on the back indicates the specific size of the Hu, from which it can be deduced that the value used by the makers at that time was about 31547
Since then, Zhang Heng and Cai Yong of the Eastern Han Dynasty have also used empirical formulas to give approximate values, and Zhang Heng believes that it is equal to 31622 (10 squares);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 - circumcision when he made notes to the "Nine Chapters of Arithmetic".
Schematic diagram of Archimedes' circumcision, in which he constructs both inscribed and inscribed regular polygons of a circle, and then calculates their perimeters to obtain an approximation of the perimeter, repeating this step to obtain a pi of about 31409
It was on the basis of Liu Hui and others that Zu Chongzhi and his son Zu Xuan (also recorded as Zu Xuanzhi) pushed pi to a new peak, accurate to 7 decimal places.
Although he still used geometric methods, it was not until the 15th century that the Central Asian mathematician Al-Kashi broke his record and calculated to 14 decimal places, and more accurate calculations were not realized until the middle of the 18th century, when Western mathematicians mastered modern mathematical tools such as infinite series, integrals, and power series expansions.
In addition, it should not be ignored that Zu Chongzhi gave a simple and very accurate approximation rate and density ratio, the approximate rate is probably based on the approximation value of 157 50 given by Liu Hui, 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 no longer know how he found this solution, can only guess that He Chengtian may have been used"The method of adjusting the sun"(numerical approximation of interpolation), or the method of continuous fractions was used to find the best asymptotic fraction, but either way, it was not until 1573 that German mathematicians recalculated the method. In fact, Zu Chongzhi's calculation of pi is thousands of years ahead of the world.
Schematic diagram of the principle of Zu Xuan. The two piles of coins on the left and right meet the condition that the cross-sectional area of the three-dimensional contour of the same height is equal, so the volume is equal, that is, the so-called "power potential is the same, then the product cannot be different". This principle is visually illustrated in the diagram, but strict mathematical proof requires the application of definite integrals in higher mathematics.
Academic achievements are ill-fated
Whether it is the "Great Ming Calendar" or pi, Zu Chongzhi's achievements can be described as shocking to the past and the present, and the most fundamental reason for his achievements is undoubtedly superhuman mathematical thinking. However, to the regret of later generations, the "Fixation Technique" that recorded the mathematical thoughts of him and Zu Wei was lost in the Tang Dynasty, so that later generations had no way to understand the calculation method of Zu's father and son, and could only appreciate the style of the two from the fragments relayed by others. For example, Zu Wei once mentioned it in "Fixation"."If the power potential is the same, the product cannot be different", which means that two three-dimensional dimensions of the same height, if the cross-sectional area at the same height is equal, then the volume is equal. In other words, two solids between two parallel planes are truncated by either plane parallel to these two planes, and if the areas of the two sections are equal, the volume of the two three-dimensional planes is equal. It was not until the 17th century that the principle of Zu was discovered by the Western scholar C**Alieri, and in China, it was skillfully used by the Zu father and son to find the volume of the square lid and then calculate the volume of the sphere.
In fact, the proof of Zu Xuan's principle required the use of definite integrals, which required a rather abstract three-dimensional geometry ability to understand at that time, which can also be imagined as the difficulty of the "Fixation". During the reign of Emperor Gaozong of the Tang Dynasty, this book was included in the "Ten Classics of Arithmetic" and was one of the mathematics textbooks of Guozijian, but the study period of the "Fixation" was four years, which was the longest of the "Ten Classics of Arithmetic". The abstract and difficult to understand of "Fixation" even led to a public case, and the author of "The Ancient Arithmetic", mathematician Wang Xiaotong, was a doctor of arithmetic and Taishi Cheng during the Tang Gaozu period, and openly criticized "Fixation"."It doesn't make sense"As a result, the compilation team of Taizong and the ministers secretly ridiculed in the "Sui Book and Legal Calendar":"Scholars can't investigate its profundity, and they ignore it"。
Fortunately, Zu Chongzhi and his son's glorious academic achievements are far longer than all fateful blows. After a thousand years, the name of Zu Chongzhi has not only not been forgotten, but also went abroad and landed on the moon.
References:
Jin Kaicheng and Guo Rui "Mathematics Taidou Zu Chongzhi".
Wei Xiaoni, "An Exploration of Pi in History".
*丨National Humanities and History (text by Li Sida) Editor丨Gan Xiaobo