What are Tianyuan's masterpieces in the Jin and Yuan Dynasties?

According to historical records, there were a number of works about celestial arts in Jin and Yuan Dynasties, especially the works of mathematicians Ye Li and Zhu Shijie, which clearly expounded celestial arts.

In her mathematical monograph "Measuring the Round Sea Mirror", Ye Li comprehensively discussed the steps, skills, algorithms and symbolic representations of establishing unknowns and equations through the Pythagorean inclusion problem, which made celestial art develop to a quite mature new stage.

Yi Gu Yan Duan is a concise and easy-to-learn introductory book written by Ye Li for Tianyuan beginners. He is the author of Jing Zhai Gu Hu Jin, Jing Zhai Anthology, Bi Shu Cong Jian, Pan Shuo and so on. The former has a collection of 12 volumes, while the last three volumes have been lost.

Zhu Shijie's "Arithmetic Enlightenment", which includes commonly used data, conversion between weights and measures and field area units, four calculation algorithms, calculation simplification, fraction, proportion, area, volume, surplus and deficiency, higher-order arithmetic progression summation, solution of numerical equations, solution of linear equations, celestial theory, etc., is a relatively comprehensive mathematical enlightenment book.

Zhu Shijie's masterpiece "Siyuan Encounter" records the establishment and solution of the higher-order equation he created, as well as his important achievements in higher-order arithmetic progression summation and higher-order interpolation.

In addition to Ye Li and Zhu Shijie, the General Theory of River Defense written by Yuan Semu Shan Si also has the application of natural science in water conservancy projects.

During the Song and Yuan Dynasties, astronomy and mathematics were more closely related. The establishment, development and application of seeking differences is a great achievement of world significance in the history of ancient mathematics and astronomy in China. When I was in North Song Zhenzong, there was a fire in the palace one year, and many buildings were burned down, so the restoration work needed a lot of earthwork. At that time, Shen Kuo's plan was adopted because the soil outside the city was too far away:

Borrow soil from a nearby street, dig the street into a huge ravine, and then pump Bianshui into a river, so that ships carrying goods can reach the palace gate along the river. Upon completion, the garbage will be filled into huge ditches and restored to the streets.

Shen Kuo's plan solved the problems of soil borrowing, material transportation and waste disposal in one fell swoop. In addition, Shen Kuo's ideas of "grain against the enemy", "superb combination of dragons" and "diverting water to fill embankments" are all examples of the application of operational research.

Shen Kuo was a great scientist in the Northern Song Dynasty. He is knowledgeable and has written astronomy, local chronicles, laws and calendars, music, medicine and divination. Shen Kuo paid attention to the application of mathematics and applied it to astronomy, calendar, engineering, military and other fields, and achieved many important results.

Shen Kuo's mathematical achievements mainly include the differential product technique, calculation, measurement, water transport countermeasures and so on. The "gap product" is a high-order arithmetic progression summation method, which paved the way for Yang Hui's "overlapping product" in the Southern Song Dynasty and Guo Shoujing and Zhu Shijie's "calling for differences" in the Yuan Dynasty.

Superposition, that is, superposition quadrature. Because many accumulation phenomena are high-order arithmetic progression, accumulation technology has become a special method to study high-order arithmetic progression summation in ancient China mathematics.

Shen Kuo said in Meng Qian Bi Tan: There are all kinds of methods to calculate geometric volume in arithmetic, such as right-angled prism, right-angled cylinder with right-angled triangles at two bottom surfaces, triangular pyramid and quadrangular pyramid, but there is no such algorithm as gap product.

The so-called gap products are stacked with gaps, such as chess pieces piled up in hotels and jars piled up. Although they are shaped like buckets, and all four measuring surfaces are inclined, due to the internal clearance, if they are calculated by the quadrangular pyramid method, the results are often smaller than the actual ones.

What Shen Kuo said made clear the relationship between gap product and volume. The same is quadrature, but there are gaps in the "gap product", just like playing chess, stacking cans layer by layer.

However, the volume formula of right-angled prism can not be applied to the gap products such as restaurant product altar. But it is not incomparable. After all, the stack with gaps is like a right-angle prism, and there should be some connections in the algorithm.

How did Shen Kuo get this correct formula? Meng Xi Bitan did not elaborate. At present, there are many kinds of guesses, some people think that it is obtained by induction after many stacking experiments of products with different lengths, widths and heights; Some people think that it may be obtained by cutting and repairing geometric figures with the method of "loss width and narrowing"

Shen Kuo's method of summing series by comparing series with volume provides a way of thinking for future generations to study the problem of summing series. First of all, Yang Hui, a mathematician at the end of the Southern Song Dynasty, made some achievements in this idea.

Yang Hui enriched and developed Shen Kuo's gap product results in Nine Chapters Detailed Explanation of Arithmetic Algorithms and General Variations of Algorithms, and put forward a new superposition formula.

The sequence discussed by Shen Kuo and Yang Hui is different from the general arithmetic progression. The difference between the two terms is not equal, but the difference of item by item or the difference of higher order is equal. This kind of high-order arithmetic progression's research is generally called "piling up" after Yang Hui.

Zhu Shijie, a mathematician in the Yuan Dynasty, put forward Shen Kuo and Yang Hui's work on the summation of higher-order arithmetic progression in "Siyuan Yujian".

Zhu Shijie made a further study on the superposition technique, and obtained a series of important summation formulas of high-order arithmetic sequence, which was another outstanding achievement of mathematics in Yuan Dynasty. He also studied more complex superposition formulas and their practical applications in various problems.

For arithmetic progression and geometric progression in general, China had a preliminary research achievement in ancient times. It is not easy to summarize these formulas, but it is quite difficult. Shen Kuo, Yang Hui and Zhu Shijie have made outstanding contributions to this.

Seeking Difference is also an important achievement in the field of ancient mathematics in China. It was once used by the great scientist Newton and had a far-reaching influence in the world.

The first interpolation method has been used in ancient astronomy in China, and the second interpolation method with equal spacing and unequal spacing was established in Sui and Tang Dynasties to calculate the visibility of the sun, moon and five stars. This work was first started by Liu Zhuo.

Liu Zhuo was a Confucian scholar and astronomer in Sui Dynasty. Many of his proteges have become celebrities. Among them, Kong and Ge Wenda of Hengshui County were his favorite students, and later became masters of Confucian classics in the early Tang Dynasty.

Yang Di ascended the throne, and Liu Zhuo was appointed doctor of imperial academy. There were many fallacies in the calendar at that time, so he worked hard to make the emperor's calendar, and for the first time, he considered the unevenness of the apparent motion of the sun and created the "equal interval quadratic interpolation formula" to calculate the running speed.

Huang is much more accurate than the previous calendar in calculating the daily profit and loss, the profit and loss of the ecliptic and the moon, the number of solar and lunar eclipses, and their places and times.

Because the apparent motion of the sun is not a quadratic function about time, even the quadratic interpolation formula with unequal intervals can not accurately calculate the speed of the sun and the moon. Therefore, Liu Zhuo's interpolation method needs further study.

During the Song and Yuan Dynasties, astronomy and mathematics were more closely related. Many important mathematical methods, such as numerical solution of higher-order equations and summation of higher-order arithmetic progression, have been absorbed by astronomy and become important tools for making new calendars. The chronograph calendar in Yuan Dynasty is a typical example.

Chronological calendar is an advanced calendar work written by Guo Shoujing, an astronomer and mathematician in Yuan Dynasty. One of its advanced achievements is the application of tricks.

Guo Shoujing created an algorithm equivalent to the spherical triangle formula, which was used to calculate the ecliptic coordinates and equatorial coordinates of celestial bodies and their mutual conversion, abolished the decimal calculation in calendars compiled in previous dynasties and adopted decimal system, greatly simplifying the operation process.

Many academic leaders emerged in the field of ancient mathematics in China, who made classical mathematics shine. If no one had studied mathematics in history, there would be no such books as Zhou Kuai Shu Jing and Nine Chapters Arithmetic handed down. Without mathematicians, Zhou Wang opened a mine and Qin Shihuang built a mausoleum.