The outstanding mathematician and astronomer of ancient my country Zu Chongzhi
Few people know that, in addition to mathematics, Zu Chongzhi has studied in many fields such as astronomy and machinery, and is even proficient in music theory and phonology.
indemnity fine cut, the loss less Mi ......
- LiuHui (Wei Jin)
Strong36strong Written | Cai Tianxin (School of Mathematical Sciences, Zhejiang University)
Strong36strong editor in charge | Li Juan, Han Kun
01 Zu Chongzhi and Pi
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The ancestor's family was a family of officials and eunuchs. Zu Chongzhi's great-grandfather was in the Eastern Jin Dynasty from the official to the minister, Guanglu doctor, equivalent to the prime minister and national policy adviser. Both his grandfather and father served as officials in the Southern Dynasties. His grandfather was a great master, in charge of palaces, ancestral temples, mausoleums, etc. 's civil construction; his father was invited by the court, this is the idle official. In ancient times, the meeting in the spring was called the court, and the meeting in the autumn was called the invitation. Most of the successive members of this family have studied the astronomical calendar.
A stamp commemorating Zu Chongzhi
In contrast, Zu Chongzhi has held a lower official position, but he has achieved more outstanding achievements in the fields of astronomy and mathematics, and even mechanical manufacturing than his predecessors . During the Liu Song period, he served as a historical post in Southern Xuzhou, which was an official post of urging documents and prosecuting illegal acts. It must be pointed out that the south Xuzhou is not the south of Xuzhou, but Zhenjiang, which is more than 300 kilometers southeast of Xuzhou. He also served as the magistrate of Lou County (now Kunshan, Suzhou) ,He also acted as a veteran of court etiquette and mission. In the Xiaoqi dynasty, from Zu Chong's official to Changshui's lieutenant, this was the highest official position he held in his life. (fourth grade) .
From his youth, Zu Chongzhi had a keen interest in mathematics and astronomy. He once stated in his books that since childhood, " special skills, 37span Ancient and modern" . Zu Chongzhi collected and studied various documents from the time of the ancient from to the time of his life. At the same time, he personally carried out precise measurements and careful calculations, and did not tie himself to the stale ideas of the ancients. It can be said that Zu Chongzhi has critically accepted the academic heritage of his predecessors and has the courage to put forward his own new insights. This is an excellent quality shared by all outstanding scientists throughout the ages.
In the field of mathematics, Zu Chongzhi inherited Liu Hui from the Wei and Jin dynasties, who was two hundred years earlier than him. The latter invented the "circle circle technique" for calculating the pi and the method for calculating the volume of the ball. From the circle area calculation formula
, it is easy to know that only the area of the circle is required, and then divided by the square of the radius, it is the pi. As for how to find the area of a circle, Liu Hui wrote in the notes of "Nine Chapters Arithmetic":
Cut it and cut it again, so that it cannot be cut, then it merges with the circle without losing anything.
Liu Hui starts to calculate the area from the circle inscribed with the regular hexagon, and doubles the number of sides in turn to find the inscribed regular dodecagon, regular twiddle, regular forty-eight, etc Area. As the number of sides increases, the area of the inscribed regular polygon gets closer and closer to the area of the circle, and the accuracy of the circle area and pi ratio becomes higher.
In ancient times, some peoples, including China and Babylon, used 3 as the pi. In this regard, the calculations of the ancient Egyptians were more accurate, and they got a pi ratio of 3.1. Liu Hui used his circular cutting technique to find the pi to be 3.14, which is consistent with the pi calculated by the ancient Greek mathematician Archimedes , which is six centuries earlier than Liu Hui. Some people think that Liu Hui calculated 3.1416, but there is not enough evidence. Calculation
Zu and pi
Zu out pi range:
3.1415926 <<>
i.e. decimal place 7 bits. In addition, he also got the pi of the score called
, which is only accurate to 6 decimal places, but it is equally amazing.It was not until after 962 that Zu Chongzhi's pi was improved by the Persian mathematician Cassie under Arab rule. Cassie used the half-angle formula of the cosine function to simplify the calculation and was accurate to 17 decimal places. The German Otto obtained the secret rate more than a thousand years later than that of Zu Chong.
Hong Kong, China commemorates Zu Chongzhi’s contribution to Pi
How did Zu Chongzhi’s achievements be achieved? No historical materials have been handed down, because Zu Chongzhi's works have all been lost, and the "Span37span Sui Shu " which records the value of Pi has no specific explanation. Since there was only one method of Liu Hui's circle-cutting technique at that time, we can only guess that Zu Chongzhi used the same method. In that case, he needs to calculate the area of the square 24576 polygon inscribed in the circle, and the density rate (some Japanese scholars suggest it is called the ancestral rate) may be obtained with the help of the predecessor astronomer He Chengtian invented the "sun adjustment" law".
02 The volume of the sphere and the Great Ming calendar
Compared to the area of the circle, the sphere (the ancients called the vertical circle) is more skillful in calculating the volume of 37span. In the classic Chinese mathematics "Nine Chapter Arithmetic", the volume of the sphere is calculated according to the following proportional formula:
Obviously, the three data of square area, circle area and cylinder volume It is relatively easy to find.
Ancient Chinese mathematics book "Nine Chapters Arithmetic"
Zu Chongzhi knew that this formula was wrong, he wrote in "Refutation", "to 37 round the old mistakes said that but Fugai...this is considered a fault... the minister used his free time to correct all the errors." It can be seen that the ball volume is also Zu Chongzhi’s work, but four hundred years later, the Tang Dynasty scholar However, Li Chunfeng attributed this work to Zu Chongzhi's son Zu Wei in his annotations for "Nine Chapters Arithmetic", and cited it as "Zu Wei Kai Li Yuan Shu". In any case, we can regard this as the common research result of the grandfather and his son.
Speaking of which, Zu Chong's father and son's achievement was also completed on the basis of Liu Hui's work. Liu Hui first discovered the error of the above-mentioned proportions and proposed a new concept of "Mou He Fang Gai", which is the common part of two perpendicularly intersecting cylinders of the same diameter. In the above proportional formula, Liu Hui replaces the volume of the cylinder with the volume of the Muhe square cap, that is,
This idea and method are correct. Unfortunately, Liu Hui I cannot find the volume of Mou Hefang Gai. This step was completed by the ancestors and his sons. They divided the part other than the square cover of Mouhe into three pieces, called "Wai Sanqi". After proper adjustment and collage, they found that the part occupies one-third of the volume of the cube. In that way, Mou Hefang Gai's volume naturally occupies two-thirds. Using the above proportional equation again, the sphere volume formula can be obtained as:
It’s not hard to imagine,When Zu Chong's father and son got this formula, how happy and excited they were. However, what they didn't know was that this formula existed as early as the 3rd century BC, and it also came from the ancient Greek mathematician Archimedes, who has the elegant title of "the god of mathematics."
In addition to mathematics, Zu Chongzhi also has achievements in astronomy. After actual observation, he found that He Chengtian's Yuan Jia calendar, which was adopted by the Liu Song dynasty at that time, had many errors. For example, during the winter solstice, the sun is three degrees away from the actual measurement, the winter solstice and the summer solstice are one day away, and the five-star (golden wood, water, fire and soil) has a difference of 40 days. So, Zu Chongzhi started to compile a new calendar-the "Da Ming Li", which was the most accurate calendar of that era.
In 462 AD, 33-year-old Zu Chong went to the table of Emperor Xiaowu Liu Jun, and asked him to discuss the new calendar. Unexpectedly, he was opposed by the emperor's favored minister 戴法兴. The forces that the hundreds of officials in the DPRK and the Central Government were afraid of Dai were mostly attached to. Zu Chongzhi boldly and boldly debated, and wrote the famous "Refutation" and presented it to the emperor. There are two famous sayings in the article, "I would like to hear the evidence, and verify the truth" "Floating remarks and derogatory words, not to be afraid of stealing." This debate reflects the struggle between the two forces of progress and conservation, and science and anti-science. This is also something that every progress of science will often encounter.
Due to various obstacles, until half a century later, after the Liu Song and Xiao Qi dynasties, they were finally in the Liang Dynasty for nine years (510 years) , due to the determination of the ancestors. request,After verification of actual astronomical phenomena, the Ming calendar was officially issued under the name of "Jiazi Yuanli". At that time, Zu Chongzhi had passed away 10 years. "Jiazi Yuanli" introduced the concept of "precession", changing every 19 years of leap 7 years in the old calendar to 144 years of leap every 391 years, making the error of one year only 50 seconds. It was not until the Song Dynasty that a better calendar appeared.
03 Compass and Thousand Miles Ship
In addition to his work in mathematics and astronomy, Zu Chongzhi has also made all kinds of ingenious machinery, including compasses and thousands of miles, and he is also good at music. Known as the Pythagorean or Archimedesian scientific figures. The name of the compass has been around for a long time in our country, but its mechanism structure has not been circulated. It is said that Ma Jun, the great inventor of the Three Kingdoms period, once made a compass, but it had long been lost in Jin Dynasty.
Speaking of Ma Jun, he is a native of Xingping, Shaanxi. He is not good at words and stutters, a bit like Tartaglia, the Italian who gave a general solution to the cubic equation in the 16th century. In addition to the compass, he also ordered a variety of puppets, known as "water turns a hundred operas". The weaving machine was also transformed, which increased the work efficiency by four or five times. Ma Jun also improved the keel waterwheel used for agricultural irrigation, and the military mechanical continuous crossbow invented by Zhuge Liang, which is a device that can shoot arrows continuously.
It is said that in the late Eastern Jin Dynasty, the founding emperor of Liu Song in the Southern Dynasties Liu Yu invaded Chang'an, the capital of the post-Qin period during the Sixteen Kingdoms period, and many of the artifacts obtained contained compasses, but " number is not refined, although it is said that it is a guide, it is often not reviewed, and the steps of the song are still required." During the reign of Emperor Shun, the last emperor of the Southern Dynasties, Xiao Daocheng , who served as the auxiliary government, "makes Chong Zhi follow the ancient method. Chong Zhi transforms the bronze machine,The circle turns endlessly and the division enters into one, which has not happened since Ma Jun. "_Span38span
In addition to the compass, Zu Chongzhi also "made a device with a wooden ox and a horse from Zhuge Liang." This device is "not due to feng shui, but it does not take much labor." But because of the lack of image data, we can't Imagine what kind of machinery this is. However, Zu Chongzhi “built a boat for thousands of miles and tried it on the Xinting River, traveling eight hundred miles a day.” Obviously this is a kind of fast boat, but I don’t know where the Xinting River is? Is it on the Yangtze River? What about the section? He also "built a water mill in Le Youyuan, Emperor Wu (Emperor Qi Wu, reigned 483-494) personally came to see it."
is not limited to Zu Chongzhi In the natural sciences, he is also proficient in music theory and has a lot of research on rhythm. There are historical records that "Bose was unique at that time, and there is no one who can do it." (Note: The above quotes are from "Southern Qi Shu" or " "The Biography of Zu Chong" in Southern History.) In addition, Zu Chongzhi also authored "Yi Yi", "Lao Zi Yi", "Zhuang Zi Yi", "Interpretation of Analects" and other philosophical works. Unfortunately, like his mathematics books, they have all been lost. His literary works include "Su Yi Ji", and you can still see excerpts of this article in ancient books such as "Tai Ping Yu Lan" in the Song Dynasty.
Han dynasty bone counting chips, pictures are from the Han dynasty
wikipedia
In the age of Zu Chongzhi’s life, the abacus had not yet been invented. People used a calculation tool called counting chips. It was a few inches long square or flat sticks, including bamboo, wood, iron, and jade. And other materials. The more digits of the calculation number, the larger the area needed to place it. When you plan to calculate the results once, you will need to write down the results, so that you can’t get intuitive graphics and calculations. Therefore, as long as there are errors, you can only start from scratch. Start.Zu Chongzhi strives for perfection and repeatedly calculates to obtain the precise value of Pi.
As for Zu Xuan, the date of his birth and death is unknown. It is only known that he was the Crown Prince, which was an official position in charge of Jinbo Caibo in the Southern Dynasty. Due to the influence of his family, especially his father, Zu Xuan had a keen interest in mathematics since he was a child. Zu Chongzhi's "Da Ming Li" was completed on the basis of Zu Xuan's three suggestions. Zu Chongzhi's representative work, the "Zhushu" that was once included in the mathematics textbooks of the Tang Dynasty and circulated in Korea and Japan has been verified by scholars, and some entries are made by Zu Xuan. As for the calculation formula of the ball volume, it should be the most representative discovery of Zu Xuan's life: (if he found it) .
Looking at the two major mathematical achievements of the father and son of Zu Chong, because Archimedes has already given the formula for calculating the volume of the ball, the work on pi is more praised by the world. But that is like the record of a sports game. It was who appeared to be broken by the people.
Since there were infinite series representations and computers, the artificial competition of Pi has lost its meaning. Personally, I think that the two achievements of the Southern Song mathematician Qin Jiushao- Chinese surplus theorem and Qin Jiushao's algorithm are more meaningful and more important. But the conclusions and stories of Pi are undoubtedly easier to understand by ordinary people, and they are more in line with the heroic imagination of Chinese people.
About the author
Prof. Tianxin Cai , span37 _span38strongspan, Zhejiang UniversityPoet and essayist, his recent works include "Legend of Mathematics", "Book of Numbers", "Temptation of Rio: Memories of Latin America", "Pinch her a few times", etc.
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