Chapter 21 The second section unraveling the Pythagorean

The Pythagorean Chapter of "Nine Chapters" puts forward a number of known factors such as the sum and difference of the two sides of the Pythagorean shape, and the example problem of finding the length of the side.Zhao Shuang, Liu Hui, and Jia Xian successively made further developments and proposed general formulas and proofs. The topic of Indian lotus, which is popular at home and abroad, is actually a rewriting of the topic "Introduction to the Shore" in "Nine Chapters".The title is: There is a pool with a square of 1 zhang, a reed [jiajia, a newborn reed] grows in the center, 1 chi above the water surface, and leads the reed to the shore, just at the same level as the shore.How much is the depth of the water and the length of the Jia?As shown in Figure 15, Liu Hui pointed out that half of the side length of the pool is the hook a, the water depth is the strand b, the length of the reed is the chord c, and the height of the reed above the water surface is the difference cb of the chord, which is the known difference between the hook and the chord. Questions about strands and strings:

b=[a-(cb)]/[2(cb)], c=(cb)+b. In the 1989 Chinese college entrance examination volume, there is an ancient Chinese translation question, which is taken from the Pythagorean Chapter of "Nine Chapters" "Bamboo High and Breaking the Ground" Question: Today there is a bamboo that is 1 zhang high, and it was broken. , ask how much is the remaining high?As shown in Figure 16, Liu Hui pointed out that the distance from the ground to the bamboo root is the hook a, the remaining height is the strand b, and the broken part is the string c, then the height of the bamboo is the strand sum c+b, which is the known hook and strand Chord and the problem of seeking shares: b=[(c+b)-a]/[2(c+b)].

These two types of questions answer each other, and Liu Hui proved it with the principle of complementarity and complementarity. There is a door, the height is 6 feet 8 inches more than the width, and the distance between the two corners is 1 zhang.What is the height and width of this portal?Liu Hui believes that if the width of the household is the hook a, the height is the strand b, and the distance between the two corners is the chord c, then this is a problem of finding the hook and strand given the difference ba between the chord c and the hook. The interpretation of "Nine Chapters" was rewritten by Liu Hui as

Liu Hui’s method of proving the complement of in and out is: take Pythagorean and b+a as side lengths to make a square, which is called Dafang, with an area of ​​(a+b); make a Chinese square inside it, and its vertices are on each side a and b of Dafang. At the subpoint of , its side length is naturally c, and its area is c; inside the Chinese square, four Pythagorean shapes with side lengths of a, b, and c are made, each with an area of ​​1/2ab, called Zhu Mi.The Chinese side removes the four Pythagorean shapes, leaving a square with ba as the side length, called the yellow square, with an area of ​​(ba), as shown in Figure 17, Dafang has eight Zhu powers, one Huang power, and the Chinese side has four Zhu powers , a yellow power, therefore, the Chinese side minus half of the yellow power is equal to half of Dafang: 1/2(b+a)=c-1/2(ba), 1/4(b+a)=1/2［ c-1/2(ba)], so

There was a gate whose height and width were unknown. Someone was holding a bamboo pole. He did not know its length. He went out horizontally and it was 4 feet long. He went out vertically and it was 2 feet long.What is the height, width and slope of the door?Liu Hui regards the height, width, and inclination of the portal as the hook, strand, and string respectively. This question is a problem of finding the hook, strand, and string given the hook difference ca and the string difference cb. The formula given in "Nine Chapters" is:

In order to prove these formulas, Liu Hui first pointed out the mutual position of the hook power and the joint power in the string power, or the moment is on the surface, or the square is in the inside: if the joint power is a square, then the hook power is located on the surface of the square as the hook moment , as shown in Figure 18(1); and vice versa, as shown in Figure 18(2).Liu Hui rotates one of the figures by 180° and overlaps with the other, which becomes the situation in Figure 18(3).The sum of the areas of the hook moment cb and the strand moment ca should be the chord power c.In the figure, the two coincide at the two corners in two rectangles whose width is cb and length is ca, the area of ​​which is 2(ca)(cb).However, there is a small yellow square whose side length is a+bc in the string power, which is not filled with hook moments and strand moments.Obviously, the area (a+bc) of the little yellow square should be equal to 2(ca)(cb), and the square root is given by

a=(a+bc)+(cb), b=(a+bc)+(ca), c=(a+bc)+(cb)+(ca) The above three formulas are proved. Jia Xian of the Northern Song Dynasty abstracted the four types of methods for solving the Pythagorean shape in "Nine Chapters" into general formulas.Yang Hui further summed up the 13 relationships of a, b, c, c±a, c±b, b±a, a+b±c, c±(ba) and the number of segments that become ba, cb, a+bc, It is called "Thirteen Famous Figures of Pythagorean Changes".These 13 kinds of relations include all possible relations of hook, strand, chord and their sum and difference in Pythagorean, which play an outline role in Pythagorean theory.