Chapter 38 The second section of Tianyuan technique
The basic idea of Tianyuan technique is: first, set the desired quantity as Tianyuanyi, then give two equivalent Tianyuan formulas according to the conditions of the problem, and subtract the two Tianyuan formulas to obtain an opening method, that is, an equation.This latter process is called subtraction.However, such as product release lock, it includes the complete process of listing equations and solving equations.Obviously, setting a certain quantity as Tianyuan-1 is equivalent to setting XX as an unknown x, and Tianyuan formula is equivalent to today's polynomial with unknowns.For example, according to the notation -332x+27556x in "Measuring Round Sea Mirror".Let us take Question 2 of Volume 7 of "Che Yuan Hai Jing" as an example to demonstrate Tianyuan technique.This question is:
If there is a round city, I don't know the circumference.Or ask B to go out of the south gate and walk straight for 135 steps and stand there, and A to go out of the east gate and go straight for 16 steps to see him.What is the diameter?Another way is to say: the two elements are multiplied together to get a number, and from it, it is the triple multiplication.Combine the two steps, multiply the number of the two lines, and double it.The combination of the two elements is multiplied by itself, and the subtraction of the two elements is multiplied by itself and subtracted from the upper digit.Second cheap.One corner.The radius of benefit plots.
As shown in Figure 32, let the southbound EA be k, and the eastbound FB be l, then it is necessary to solve the equation -x+4klx+2kl(k+l)+(kl)=0
Figure 32 Measuring circle
Li Ye's method of deriving this quartic equation with Tianyuan technique is:
Cao said: "Establish Tianyuanyi as half the diameter of the city, and set it aside."Shangjia Southbound
For the stock, the lower position plus the east walk
for tick.Pythagorean
It is a direct product, divided by Tianyuan 
for the string.from it, get
For the power of the string, send it to the left.It is to hook oneself, to get
, and from the stock, get
, the two phases are combined
, is the same number.Canceled with the left, we get
.Kaiyi accumulates three squares, and you get one hundred and twenty steps, which is half the city diameter.
The translation of this text with modern symbols is: Let x be the radius OC of the round city. x+k is stock OA, x+l is hook OB.Since AB·OC=OA·OB, namely AB·x=(x+k)(x+l), the string AB=(x+k)(x+l)/x=[x+(k+l)x +kl]/x=x+(k+l)+klx, AB=[x+(k+l)+klx]=x+2(k+l)x+k+l+4kl+2(k+l) klx+(kl)x and AB=OA+OB=(x+k)+(x+l)=2x+2(k+l)x+k+l, the two are subtracted, and both ends are multiplied by x, that is Get the above-mentioned opening method.
It can be seen that in Li Ye's time, he had fully mastered the addition, subtraction, multiplication of Tianyuan polynomials and the division of Tianyuan monomials, as well as the multiplication and division of exponents and the combination of similar terms.Because the expression of Tianyuan formula adopts the position value system, so the multiplication and division is the multiplication and division of Tianyuan power, as long as the word "yuan" or "tai" is moved up and down.
Figure 32 Measuring circle
For the stock, the lower position plus the east walkfor tick.PythagoreanIt is a direct product, divided by Tianyuan for the string.from it, getFor the power of the string, send it to the left.It is to hook oneself, to get , and from the stock, get , the two phases are combined , is the same number.Canceled with the left, we get .Kaiyi accumulates three squares, and you get one hundred and twenty steps, which is half the city diameter.
The translation of this text with modern symbols is: Let x be the radius OC of the round city. x+k is stock OA, x+l is hook OB.Since AB·OC=OA·OB, namely AB·x=(x+k)(x+l), the string AB=(x+k)(x+l)/x=[x+(k+l)x +kl]/x=x+(k+l)+klx, AB=[x+(k+l)+klx]=x+2(k+l)x+k+l+4kl+2(k+l) klx+(kl)x and AB=OA+OB=(x+k)+(x+l)=2x+2(k+l)x+k+l, the two are subtracted, and both ends are multiplied by x, that is Get the above-mentioned opening method.
It can be seen that in Li Ye's time, he had fully mastered the addition, subtraction, multiplication of Tianyuan polynomials and the division of Tianyuan monomials, as well as the multiplication and division of exponents and the combination of similar terms.Because the expression of Tianyuan formula adopts the position value system, so the multiplication and division is the multiplication and division of Tianyuan power, as long as the word "yuan" or "tai" is moved up and down.