Chapter 32 Section 2 Cubic

"Nine Chapters of Arithmetic" also proposed a complete procedure for finding the cube. The method of opening the cube is: put the product into reality.Borrow one to count the steps, surpass the second class.To discuss the income, the method is to multiply the borrowed one and divide it.Apart from oneself, the three are the law.Removal, fold down.The number obtained by multiplying by three is placed in the middle row.Re-borrow and count down.Step by step, the first place in the Chinese Super League and the second place in the Lower Super League.Re-set the proposal, take the first multiplication, and then multiplication, all of which are supplemented by the method of addition.Divide by law.In addition to already, double down, and follow the law.Removal, fold down as before.Those who can't be opened are also incapable of opening.If there are points in the product, the inner child of the common points is a fixed reality.To determine the truth is to open it, and when it is finished, open its mother to repay and get rid of it.If the mother cannot open it, then use the mother to multiply it again to determine the reality, and then open it.After that, let's be one like a mother.

This text is not difficult to understand if you understand the method of prescribing alchemy.Here is the calculation of the five elements: Yide (root), reality, law, middle line, and borrowing.Liu Hui used a three-dimensional model to give an intuitive geometric explanation to the open cube.To open a cube is to find the side lengths of a cube given its volume.Assuming that its root is n+1 digits, subtracting A from the cube 10a obtained by one digit of the root is to subtract the cube whose side length is 10a from the cube whose volume is A. A-10a is its remaining part, and it is made up of three kinds of seven solids, as shown in Figure 28 (1).The first type is a small cube on the corner, as shown in Figure 28(2), and its volume is 10x; the second type is three flat small cuboids, each with a volume of (10a)10x=10ax, as shown in Figure 28(3) ; The third type is three strip-shaped small cuboids, each of which has a volume of 10a(10x)=10ax, which Liu Hui calls the three lengths, as shown in Figure 28(4), where Lian means side.Therefore, requiring x is equivalent to finding the positive root of the subtraction root equation 10x+3·10ax+3·10ax=A-10a, which is actually an open-band sub-cube problem.

"Sui Shu·Lv Li Zhi" said that Zu Chongzhi "set up Kai Chai Mi and Kai Chai Li, and also used positive and negative parameters to refer to precision, which is the best of the family." Qian Baocong believes that Kai Chai Power is Kai Chang , the area of ​​a rectangle with a difference in width, and the difference in width is the volume of a cuboid with a difference in length, width, and height.Set width x, length x+k, and height x+l, then x(x+k)=A or x+kx=A is the band from the square, which is the open difference power.And x(x+k)(x+l)=A or x+(k+l)x+klx=A is the open difference.If one or both of k and l are negative numbers, the coefficients of x and x may be negative, which is why both positive and negative parameters are used.That is to say, Zu Chongzhi has been able to solve the problems of square and cube of negative coefficient bands.Unfortunately his work has been lost.At present, the earliest open band from the cubic problem appeared in Wang Xiaotong's "Ji Gu Suan Jing", he solved the problem of solving the form x+ax+bx=A and x+ax=A, where a, A>0, b ≥0.The latter is solved by taking the square root twice.Jia Xian's Licheng release lock and Li method is the same as today's method of opening.