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The technique of opening a circle, an ancient Chinese algorithm, refers to the method of finding the diameter of a sphere when the volume of the sphere is known.It was first recorded in the "Nine Chapters of Arithmetic" in the few chapters.
Chinese name
Openning roundness
Definition
The method of finding the diameter of a sphere when the volume of the sphere is known
The earliest written records can be found in《Chapter Nine Arithmetic》Shao Guangzhang: "The method of opening a circle is to set the number of square feet, multiply it by sixteen, and divide it by nine to one, so that the diameter of the vertical circle is obtained." This method is equivalent to giving the formula for calculating the volume of the sphere
(where d is the diameter of the ball).The origin of this formula is not given in Chapter Nine Arithmetic.
Liu Hui's Interpretation
Announce
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stayLiu HuiWhen commenting on the Nine Chapters of Arithmetic, two explanations are given:
1. It is actually measured from "the gold square weighs 16 liang and the gold pill diameter weighs 9 liang".
2. The ratio of the volume of a cube to that of its inscribed cylinder is 4/3 based on the section principle, and then it is assumed that the ratio of the volume of a cylinder to that of its inscribed sphere is also 4/3, and then it is obtained from the continuous ratio.
Liu Hui further investigated his second interpretation, pointing out that if π=3, the volume ratio of the cube to its inscribed cylinder is 4/3, but the volume ratio of the cylinder to its inscribed sphere is not 4/3, but "Muhe square cover" - the common part of two orthogonal equal diameter cylinders - the volume ratio of its inscribed sphere is 4/3.Thus, it is pointed out that this formula is wrong, and it is clarified that the correct way to obtain the calculation formula of spherical volume is to find out the volume of the square cover.Zu Xuan investigated the relationship between the cube and its content, the Mouhe square cover, and applied the section principle named after him to get the ratio of the volume of the Mouhe square cover and its external cube as 2/3.Thus, the correct volume calculation formula of the sphere is
