V-ball =4πr3÷3

The calculation of ball volume is a very important problem in the history of mathematics, especially in ancient times. In a sense, how to solve this problem marks the mathematical level of a country or a nation. We Chinese nation can be proud of our outstanding achievements in this field.

As early as BC 1 century, the calculation of China's sphere volume was completed by actual measurement. As a result, the calculation formula of sphere volume was obtained:, where V- sphere volume and D- sphere diameter, why? It's simple. Make a cube with a cubic inch of gold and a ball with a diameter of 1 inch, and weigh it with a scale to get an approximate formula for calculating the volume of the ball. The above formula has been kept until the publication of Nine Chapters of Arithmetic. It can be said that this is the first stage of ball volume calculation in China: actual measurement.

In the 3rd century A.D., Liu Hui raised an objection to this formula when commenting on Nine Chapters of Arithmetic. To illustrate Liu Hui's point of view, we first introduce the following models, as shown in figure 1.

V1-the inscribed cylinder of a cube with a side length of d, and the intersection of two inscribed cylinders of v2-v1and v3-v1. V-a ball with a diameter equal to D, and V3 was specially introduced by Liu Hui and named as "the square cover for seeking peace", that is, two identical square umbrellas integrated from top to bottom. The inaccuracy of Liu Hui's analysis is caused by the following reasoning:

But he immediately suggested that V2: V = 4: π was wrong, because V3:V = 4: π(V3 (any cross section of V3 and V is 4:π). Liu Hui's assertion is very correct. In fact, he pointed out an effective method to calculate the volume of the ball, that is, trying to find out the volume of the "square cover". Unfortunately, at that time, Liu Hui had not found a way to find the volume of the "square cover". He said, "Let's look at the solid volume inside the cube and outside the lid. It's getting thinner and thinner from top to bottom, and the numbers are not clear enough. Because it is mixed with Fiona Fang, its cross section is extremely irregular, and there is actually no standard model to compare with it. I'm afraid it's against the principle to make a judgment without respecting the graphic characteristics. Do you dare to leave no doubt? Let the people on the street talk about it. " This shows that Liu Hui is not superstitious about the spirit of seeking truth from facts of his predecessors. This is the second stage of China sphere volume calculation: improvement.

Mouhe Fanggai

By the 6th century, China's sphere volume calculation had entered the third stage of strict deduction. Zuqu, the son of the famous mathematician Zu Chongzhi, filled it out again, and that part of the volume was "outside the cover, within the cube" that Liu Hui didn't know how to deal with. When the filled cube is cut horizontally at the height z, it can be cut into squares, which are composed of F 1, F2, F3 and F4. Among them (from Pythagorean theorem), there are. Therefore, Zu put forward the famous conclusion that "if the potential is the same, the products cannot be different", which was later called "Zu principle". As shown in Figure 3, because F2+F3+F4=F*=Z2. And B* is an inverted cube horse, which is the volume of B. Obviously, B 1 is the volume of B. Using Liu Hui's conclusion V3: V = 4: π, we can get the formula for calculating the volume of the ball, where d is the diameter of the ball.

At this point, we can say that Liu Hui's method is really wonderful, while Zu's deduction is perfect. In the west, Archimedes prepared 33 propositions by exhaustive method in the first volume of On Spheres and Columns in the 3rd century BC, and drew a conclusion from 34 propositions. By the 7th century BC, cavalli had used the so-called "irreducibility principle" which was the same as the "ancestor principle" and reached a new conclusion, but only adopted the form he adopted, which is also the method adopted by the current middle school textbooks. Students can compare the characteristics of these methods by themselves.