During the Zhou Dynasty, nobles often had six skills, namely ceremony, music, archery, imperial, calligraphy, and counting, which Confucius later called "the six arts of gentlemen", so the early literati were proficient in riding horses and archery calculations. During the Southern and Northern Dynasties, the literati had not yet been distorted as they had been in the Song, Yuan, Ming, and Qing dynasties, and mathematics had not yet been marginalized, and it was one of the "xianxue" at that time.
As a great scientist of the Southern and Northern Dynasties, Zu Chongzhi can be said to be a household name, with amazing achievements in mathematics, astronomical calendars and mechanical manufacturing, especially in the exploration of pi, for the first time to calculate "pi" to the seventh decimal place, that is, between 3.1415926 and 3.1415927, until the 16th century, the Arab mathematician Al Cassy broke this record.
As the saying goes, the father of the tiger has no dog, but what is the achievement of the son of Zu Chong? Less well known is that Zu Chongzhi's son, Gèng, also had astonishing achievements in mathematics and astronomy, including a world-class achievement that was not studied by Europe until 1100 years later. It is regrettable that after the Europeans studied it, they named the research results after this person, but zu Hui was forgotten by the world.
During the Two Han Dynasties, wei, and Jin dynasties, the atmosphere of scientific research in China was still relatively strong, and the Han Dynasty established the library building Shiqu Pavilion, and during the Southern Dynasty and song dynasties, there was a royal court research institute Hualin Academy, as well as an institution that integrates the collection, research and teaching of books - Zongmingguan. Without the distortion of Confucianism, stimulated by a relatively relaxed academic atmosphere, many famous scientists appeared in this period, such as Zhang Heng of the Eastern Han Dynasty, Liu Hui of wei and Jin, Zu Chongzhi and Zu Hui of the Southern Dynasty. Among them, some of the achievements of Zu Chongzhi and Zu Hui are closely related to Liu Hui in the Wei and Jin Dynasties, and one of Liu Hui's achievements is in the calculation of "circle".
Unlike the West, for example, ancient Greece suddenly appeared superb geometry without urgent practical needs and rich accumulation, while the Chinese mathematical context is more obvious, gradually escalating from simple to complex. First of all, there are nine singular numbers on the oracle bone and numbers such as "ten, hundred, and thousand", indicating that the "decimal system" has appeared at that time. Decimal is a great invention in the history of mathematics, only nine numbers are needed to express all numbers, at that time ancient Greece had to have a text representation for each number, Rome only had 7 numbers, Maya required 19 digits for 20, Babylon needed 59 digits for 60, so the emergence of decimal system laid the foundation for ancient Chinese mathematics to lead the world. Secondly, in the archaeological excavation of Qin Jian there is a "nine-nine multiplication table", and the history books record that the two Han Wei and Jin calculations have been able to calculate four operations and opening squares. After the development of Chinese mathematics to this step, new research ideas were naturally generated for some difficult problems, such as the problem of pi and the volume of the ball.
The application of pi is very extensive, especially in astronomy and calendar, and all problems involving "circle" must be calculated using pi. How to correctly derive the value of Pi was an important issue in the history of world mathematics at that time. The Zhou Hip Arithmetic Classic and the Nine Chapters of Arithmetic propose the ancient rate of the diameter of one Wednesday, and the fixed pi rate is three, that is, the circumference of the circle is three times the length of the diameter. However, confined to the mathematical knowledge at that time, it was difficult to further study in depth, and by the time of the Two Han Dynasties, Wei and Jin Dynasties, with the further development of Chinese mathematics, the basis for in-depth research was already available at this time, such as the pi value calculated by Zhang Heng in the Eastern Han Dynasty was 3.162, and the pi value calculated by Wang Fan during the Three Kingdoms period was 3.155.
Liu Hui was a great mathematician of the Wei and Jin dynasties, the first in China to explicitly advocate the use of logical reasoning to argue mathematical propositions, author of the "Nine Chapters of Arithmetic Notes" and "Island Arithmetic Classic", he contributed a lot in mathematics, one of which is called "circle cutting" to calculate pi. The Nine Chapters of Arithmetic Notes records that "the fineness of the cut is small, the loss is small, and the cut is cut so that it cannot be cut, then it is combined with the circle and there is nothing to lose", which is the method of constantly multiplying the number of sides of the regular polygon in the circle to find pi, calculating the pi rate of 3.14 or 157/50. Circumcision is not only the creation of an accurate method of exploring pi and the calculation of pi, but also the introduction of limits and infinitesimal segmentations into mathematical proof for the first time in human history, which is of extraordinary significance.
During the Southern and Northern Dynasties, Zu Chongzhi further studied the problem of pi on the basis of Liu Hui, and accurately calculated it to between 3.1415926 and 3.1415927, reaching the limit value that mathematical knowledge could study at that time.
After obtaining a more accurate pi, Zu Chongzhi began to study the calculation of the volume of the ball, but Zu Chongzhi did not solve it, and eventually wrote his life's research into the book "Embellishment".
After Zu Chongzhi's death, Zu Huan continued to engage in mathematical and astronomical research while tinkering and editing the "Embellishment Technique". In the process, Zu Hui made two world-class discoveries.
First of all, in the process of studying the volume of the sphere, he pioneered that "if the marginal power potential is the same, the product cannot be different", that is, the two geometries sandwiched between two parallel planes are intercepted by planes parallel to these two parallel planes, and if the area of the two sections is always equal, then the volumes of the two geometries are equal, which is the famous ancestral axiom.
Secondly, Zu Chongzhi and Zu Hui found that the formula for calculating the volume of the ball in the "Nine Chapters of Arithmetic" was wrong, and Liu Hui's theory of "Mou He Fang Gai" was also incorrect, and finally Zu Hui proposed the "Opening Circle Technique" according to the "Axiom of Zu Hui" found by himself, cleverly obtaining the correct formula for the volume of the ball, and solving the problem of the volume of the ball that Liu Hui had not yet solved. Later, Zu Hui added it to the "Suffixing Technique", but unfortunately the book has been lost.
Of course, although the "Art of Affixing" has been lost, the book has been mentioned in ancient Korean and Japanese texts, and it has also been mentioned in other ancient texts, and some of the contents of the book, including the axiom of zu huang and the opening of the circle, such as the mention of the opening of the circle when Li Chunfeng's annotation of the Nine Chapters of Arithmetic in the Tang Dynasty.
In 1635, the Italian mathematician Cavalieri proposed the principle of equal product in the article "Continuous Indivisible Geometry", but there was no rigorous proof process, and the actual conclusion was the Zu Wei principle. Although the Zuhui Principle predates Cavalieri by more than 1100 years, it is now called the "Cavalieri Principle" abroad, and only some books in China call it the "Zuhui Principle".
In the late Ming Dynasty, many European missionaries came to China, they brought only some sporadic European knowledge, but translated and disseminated a large number of Chinese books to the West, so whether Cavalieri was independently researched or referred to the "ancestral principle" is really a topic worth pondering. Before the exchange between China and the West in the late Ming Dynasty, Europe did not have any surprising discoveries in the field of mathematics, but after that, amazing results appeared in an endless stream, and perhaps this phenomenon has explained everything.