Chapter 44 Section 10 The Peak of Ancient Mathematics Development

A group of outstanding mathematicians emerged during this period.Among them, Qin Jiushao (1202-1261 A.D.), Li Ye (1192-1279 A.D.), Yang Hui (about the middle of the 13th century), and Zhu Shijie (about the end of the 13th century and the beginning of the 14th century) are the most famous, known as Song Yuan Mathematics Four people.Their outstanding achievements include the following aspects: Numerical Solution of Higher-order Equations In the 11th century, mathematician Jia Xian created a new method for understanding higher-order equations, the "Origin Diagram of Square Extraction Method". Using the values ​​of the triangles in the figure, the coefficients of each higher-order expansion can be obtained.Later, Zhu Shijie extended and applied it to the eighth power.In Europe, this method was not derived by the German Apinas until the 16th century, and the Frenchman Pascal also obtained this result in the 17th century, and was called "Pascal's Triangle" by European mathematicians.Until Qin Jiushao extended this method to the numerical solution of arbitrary high-order equations, more than 600 years earlier than the same results of Europeans.

Tianyuan technique and Quaternary technique The so-called Tianyuan technique is to solve the problem of unary high-order equations and equations. "Yuan" represents an unknown number, which is equivalent to x in modern mathematics.Quaternion is the extension of unary to quaternary, that is, a system of high-order equations of four unknowns.Its solution uses the elimination method, which is consistent with the solution in current algebra.Zhu Shijie made a significant contribution in this regard.It was not until the 18th century in Europe that someone discussed the elimination method of multivariate high-order equations.

High-order arithmetic series Yang Hui inherited and developed Shen Kuo's gap accumulation technique, and Guo Shoujing used this method to calculate the movement of the sun, moon and five stars in "Shoushi Calendar".At the same time, Zhu Shijie created a general formula for high-level recruiting, and the formula obtained by Newton later was completely consistent with this. This is the development of the ancient Chinese method of solving simultaneous first-order congruences.The first-order congruence problem was first seen in "Sun Tzu Suan Jing" (written in the 4th and 5th centuries), which is also the famous Sun Tzu problem: "Today there are things that I don't know how many of. Three, the number of seven and seven is two, ask the geometry of things", "the answer is, twenty-three".This question is quite interesting in guessing riddles, and its solution is also very ingenious. It has been passed down to later generations, and it has names such as "King Qin's Secret Order of Soldiers", "Cutting Pipe Technique", "Guigu Suan", "Han Xin's Order of Soldiers", etc. A program of events.The solution to this problem is to find the common solution of the first degree congruence.Qin Jiushao extended this solution to solve various mathematical problems. The data in it are not only simple data such as three, five, and seven, but also integers, fractions, and decimals. He also systematically proposed general calculation steps. It was not until 500 years later that the famous European mathematicians Euler and Gauss conducted in-depth research on such problems.

From the above-mentioned achievements in several aspects, we can see the level of mathematics in Song and Yuan Dynasties of our country and the position it occupies in the history of mathematics in the world.