Chapter 39 Section 3 Quaternary Art

Quaternion is the solution method of multivariate high-order equations, which actually includes two parts: quaternion notation and quaternion elimination. The expression method of quaternion is that the constant term is in the center, and the word "Tai" is written beside it. The power coefficient of Tianyuan is at the bottom, the earth element is at the left, the person element is at the right, and the matter element is at the top. It is determined by the relationship, it is not necessary to memorize the characters of heaven, earth, people, things, etc., the farther away from the word "tai", the higher the power, and the product of the powers of adjacent two elements is recorded at the intersection of each row and column, and the non-adjacent elements There is no corresponding position for the product of the power of , and it is stored in the gap, as shown in Figure 33.One equation is equivalent to one equation today, two equations list two equations, three equations list three equations, and four equations list four equations.This is a representation of separation coefficients, which is very convenient for listing high-degree equations and eliminating elements.Unfortunately, since the plane has only four directions of up, down, left, and right, at most four elements can be listed, and the equations higher than the four elements are helpless.

The core of the quaternary technique is the quaternary elimination method, that is, the quaternary and four forms are eliminated into three elements and three forms, and then converted into two elements and two forms, and finally transformed into one element and one form, that is, the high-level opening method.Zhu Shijie's "Four Elements Jade Mirror" lists the examples of Tianyuan, Binary, Sanyuan and Quaternary in "False Order and Fine Grass" at the beginning of the volume.I would like to explain the elimination method of the third question "Sancai Yunyuan" as follows.

Cao said: The first element of heaven is the hook, the first element of earth is the stock, and the first element of human is the string.Compatibility of the Three Talents to Find the Present Style Solution: Let x be hook a, y be strand b, z be string c, and list x+y+z-xy(zy)=0 or -xy+xyz-xyz=0 (present formula) from known conditions. same -x+x+xz+yz=0 (cloud style). x+yz=0 (ternary formula). Subtract the current formula from the cloud formula, divide by x, and substitute y=zx, y=-x+x+xz-z to get the previous formula: x+x-xz+xz-z+xz-2z-2= 0 Substitute y=-x+x+xz-z into the ternary formula to get the following formula: x-2x+2x-2xz+4xz-2z+xz-2z=0

And put Ren Yuan on Tian Yuan.Mutual implicit division and elimination, get (-z+3z+7)x+(z-3z-7z-6)=0 For the left formula, (-2z+5z+11z+13)x+(2z-5z-15z-13z-14)=0 is the right formula. Multiply the previous formula by the left row (-z+1), (-z+1)x+(2z-2)x+(-z-3z+2z+2)x+(2z-2z)=0 Multiply the previous formula by x to get (-z+1)x+(z+z+1)x+(-2z-z-2)x=0 The two are eliminated to get (zz-3)x+(-z-z+3z+4)x+(2z-2z)x=0 Multiply the previous formula (-z+z)x+(z+z+z)x+(-2z-z-2z)x=0 by z Subtract it and get -3x+(4z+4)x+(-z-4z)=0 [Multiply the left row (-z+1) of the previous formula, get (3z-3)x+(-4z+4)x+(z+3z-4z)=0

Multiply the previous formula by 3, get (-3x+3)x+(3z+3z+3)x+(-6z-3z-6)=0 The two are eliminated, and (-z+3x+7)x+(z-3z-7z-6)=0 is the left formula. Multiply the previous formula by the coefficient of the left formula x to get (z-4z-4z+7)x+(-z+2z+9z+10z+7)x+(2z-5z-15z-13z-14)=0 Multiply the left formula by the coefficient (-z+1) of the previous formula x and x, and get (z-4z-4z+7)x+(-z+4z+4z-z-6)x=0 and cancel the two to get the right formula : (-2z+5z+11z+13)x+(2z-5z-15z-13z-14)=0] Multiply the inner two rows to get (z-3z-7z-6)(-2z+5z+11z+13)=-2z+11z+10z-43z-146z-157z-78 Multiply the outer two lines to get (-z+3z+7)(2z-5z-15z-13z-14)=-2z+11z+14z-67z-130z-133z-98

The subtraction of the two should be 0 4z-24z+16z+24z-20=0 z-6z+4z+6z-5=0 z=5 The question is: "Nowadays there are chords compared with divided chords and sums and direct products. It is only said that hooked chords are compared with divided chords compared with sums and hooks. Ask the chord geometry?" That is, it is known that (a+b+c)÷(cb) =ab, (-a+b+c)÷(ca)=a, and the Pythagorean theorem a+b=c, find c.The solution is: (see pages 133-135) Because Zhu Shijie's text is too succinct, Luo Shilin's fine grass quoted in "Mutual Implication, Tongling, Division and Elimination" only provides an example to roughly explain the process of eliminating elements.As for whether it is in line with Zhu's original intention, it is not known.In fact, many scholars have different fine grasses.