Chapter 37 Chapter 8 Tianyuanshu and Quaternaryjutsu

In the fifth section, we have seen how Liu Hui abstracted quadratic equations from practical problems.Tang Wang Xiaotong listed more complex equations.For example, in the 15th question of "Ji Gu Suan Jing", it is known that the multiplied power of Pythagoras is ab=706 (1/50), and the strings are more than hooks ca=36 (9/10). Find hook a, strand b, and string c.His essay is about how to hook a: a+1/2(ca)a=1/2(ab)/(ca).The way of listing self-notes is as follows: (ab)=ab, 1/2(ab)/(ca)=1/2ab/(ca)=1/2a(ca)/(ca)=1/2a(c+a)=a［a+1/2(ca)］ =a+1/2(ca)a However, how to think of changing from 1/2(ab)/(ca) is a very skillful problem.However, from Liu Hui to Wang Xiaotong, the common method is to take the multiplication of two numbers as an area, and the multiplication of three numbers as a volume, and solve it through the principle of complementation of entry and exit.This method developed into the segment method in the Song Dynasty, which is actually a method of listing equations through equal-area transformations.Here, the unknown quantity is put into graphic transformation and quantitative operation like the known quantity.The problem is that there is no unified symbol for the unknown, which is cumbersome.Tianyuanshu uses a symbol "Tianyuanyi" to represent the unknown number, and through the operation of Tianyuan polynomial to cancel, the opening method is listed, and the opening method has a standardized procedure.

Due to the loss of historical materials, the early development of Tianyuanshu is still unclear.Zu Yi talked about the development of Tianyuanshu in "Siyuan Yujian · Postscript".He said: "Pingyang (now Linfen, Shanxi) Jiang Zhou wrote "Yigu", Bolu (now Hebei Lixian) Li Wenyi wrote "Zhaodan", Luquan (now Hebei Huolu County) Shidao wrote "Qianjing", Liu Ruxie of Pingshui (now Linfen) wrote "Ruji Shisuo", which was written by Yuanyu Xicao, a native of Jiang (now Xinjiang County, Shanxi), and later generations knew that there was Tianyuan." That is to say, Tianyuanshu passed from Jiang Zhou to Yuan Yu series of mathematicians' continuous efforts to perfect it.It is a pity that except for some titles of "Yigu" which are preserved in Li Ye's "Yigu Yanduan", the rest of these works have disappeared.There is no mention of Li Ye here, and whether Yuan Yu is a friend of Li Ye or Yuan Yu is a question, and the academic circles have not yet reached a conclusion.In any case, in Li Ye's time, Tianyuanshu was already the common wealth of northern Jinyuan mathematicians, and "Cheyuan Haijing" and "Yigu Yanduan" are only the earliest works handed down to use Tianyuanshu.In Li Ye's notes, he talked about the development of Tianyuanshu in more detail.When he was young, he saw a calculation scripture in Dongping County, which used people to represent constant terms, centered, and Xian, Ming, Xiao, Han, Lei, Layer, Gao, Shang, Tian to represent the 9, 8..., 1 power of unknown numbers, Above people, earth, down, low, minus, falling, passing away, spring, darkness, and ghost represent the powers of -1, -2...8, -9 of unknowns, and they are below people.Later, in the works of Tianyuanshu that he saw, the Chinese characters representing each power were canceled, and Tianyuan was used to represent the positive power. On the top, the earth element represented the negative power, and on the bottom, the power was expressed according to the position value. The higher the order, the farther away from the earth element below, and the lower the order of power.Peng Ze and Yancai in Taiyuan made major reforms. They exchanged the positions of Tian Yuan and Di Yuan, adopting the method that Tian Yuan is at the bottom and Earth Yuan is at the top.We know that in the formula of square root, the constant term is on the top, and the first, second... powers of the unknown are arranged in turn below. Yancai's reform makes the expression of Tianyuanshu consistent with the formula of square root, which is much simpler.Li Ye's "Measuring Yuanhai Jing" shows that Dongyuan has already made clear steps to establish Tianyuanyi.Li Ye's major contribution to Tianyuanshu is that he canceled the earth element that represents negative powers, and only used a word "yuan" to represent the first power of an unknown number, or "too" to represent a constant term, and other powers are given according to the value of the position , which further simplifies the expression and operation of Tianyuan technique.In "Che Yuan Hai Jing", he still took the positive power at the top and the negative power at the bottom, but in "Yi Gu Yan Duan", he reversed it, with the positive power at the bottom and the negative power at the top. Later mathematicians all adopted this method. way.

People combined Tianyuan technique and equation technique to create binary technique, ternary technique and quaternary technique, that is, the solution method of binary, ternary and quaternary simultaneous high-order equations.After narrating the history of Tianyuanshu, Zu Yi went on to write: "Li Dezai of Pingyang wrote "The Collection of Liangyi Qunying Heroes" with both earth elements, and Mr. Xing of Huoshan (now Huo County) praised his younger brother Liu Dajian Runfu for writing "Qiankun Kuobag ", at the end there is only the second question of the human element. My friend Mr. Zhu Hanqing of Yanshan has been acting for many years. He has explored the depths of the three talents, explored the hidden meaning of "Nine Chapters", and established four elements according to heaven, earth, people, and things." The works of Li Dezai and Liu Dajian have also been lost.However, here the earth element no longer represents a negative power, but another unknown quantity juxtaposed with the sky element.The ternary art takes heaven, earth, and man as unknowns, and the quaternary art takes heaven, earth, man, and things as unknowns.