Chapter 19 The volume of the fourth circle

The circle here mainly refers to the relevant solids such as cylinders, cones, truncated cones, and spheres. "Nine Chapters" called the cylinder a round wall [baodaobaodao], or a round wall, and called the round platform a round pavilion. "Nine Chapters" proposes: the volume formula of a cylinder is V=(1/12)hL, the volume formula of a cone is L+L+L)h, where L is the next week, L and L are the upper and lower weeks, and h is the height.Apparently, these formulas all correspond to the circle area formula S=(1/12)L, so they are inaccurate.Liu Hui used pi 157/50 to correct its coefficient 1/12 to 25/314.

The ball is called a vertical circle in "Nine Chapters", and Liu Hui calls it a ball. "Nine Chapters" does not give a clear formula for the volume of a sphere, but Shao Guangzhang's formula for calculating the diameter d from the volume V of a sphere is d=√(16V/9), which means that V=(9/16)d.Liu Hui pointed out that this formula is incorrect.He designed the Mouhe square cover, and Zu Dingzhi completely solved this problem. Quchi is a three-dimensional plane whose section is a ring, and its shape is shown in Figure 13. "Nine Chapters" classifies it as a child, but the upper long b in the formula is replaced by 1/2(L+L), and the lower long b is replaced by 1/2(l+l). The L and L are The middle and outer circumference of the upper bottom of Quchi, l, l are the middle and outer circumference of the lower bottom, so the formula should be V=(1/6)［(2a+a)(L+L)/2+(2a+ a) (l+l)/2] h.