Zu Geng and Zu Geng’s axiom

Zu Geng was a famous Chinese mathematician who lived in the Southern and northern dynasties (420-589 AD). He was interested in math and science since childhood, by the effect of his father, famous Chinese mathematician Zu Chongzhi. In Chinese mathematical history, Zu Geng was seen as the successor of his father, Zu Chongzhi. He was famous when he was young. It is said that when he was thinking about his problems he could not hear loud thunder. Even there was a time that when he was thinking while walking, he bumped a minister. His most famous contributions about math are finishing the famous Chinese mathematic book Zhui shu (缀术), with his father and putting forward “Zu Geng axiom”. In addition, with “Zu Geng axiom” he successfully computed out the volume of sphere and many other geometrical objects.
Now let us focus on “Zu Geng axiom”. The statement of “Zu Geng axiom” is if two objects can be so oriented that there exists a plane such that any plane parallel to it intersects equal areas in both objects, then the volumes of the two objects are equal. In Europe, it is called Cavalieri’s principle, but the discovery of Cavalieri’s principle is 1100 years later than the discovery of “Zu Geng axiom”. Also Cavalieri did not give a serious proof of his theory. In modern analytic geometry and measure theory, “Zu Geng axiom” is a particular case of Fubini’s theorem, which gives conditions under which it is possible to compute double integral using iterated integrals:

Fubini's theorem.

Fubini’s theorem.

About the motivation of “Zu Geng principle”, Zu Geng was motivated by discovering a mistake made by “Nine Chapters on Mathematical Art” about computing the volume of inscribed sphere of cube. In “Nine Chapters on Mathematical Art”, the author first made two perpendicular identical intersecting inscribed cylinders in a cube, like figure 1,

Figure 1. Image: courseware.eduwest.com.

Their overlap is called “Mou he fang gai (牟合方盖) (in Europe, it is called Steinmetz solid), which is like figure 2.

Figure 2.

With the help of “Mou he fang gai”, the author gets the volume of the inscribed sphere is 3/4 of the volume of one of the cylinders. Then by computing the volume of the cylinder, we can get the volume of the sphere. But in another problem of “Nine Chapters on Mathematical Art”, the author shows the volume of the inscribed sphere is 3/4 of the volume of “Mou he fang gai”. Note the volume of volume of cylinder is larger than the volume of “Mou he fang gai”. Therefore, apparently, both solutions from “Nine Chapters on Mathematical Art” are wrong. Another Chinese mathematician, Liu Hui, claimed we should compute the volume of “Mou he fang gai” first, then we can get the volume of the sphere. But he could not get the solution. Zu Geng inherited the ideas from Liu Hui and developed his “Mou he fang gai” theory to compute the volume. Finally, he put forward “Zu Geng’s axiom” so that he can get the volume of sphere by another way, with the help of “Zu Geng’s axiom”. First, we consider the half sphere with radius r, which is inscribed in a cylinder with radius r and height r. By “Zu Geng’s axiom” the volume of the half sphere equals to the volume of the cylinder minus the volume of its inscribed cone (considering the parallel cross section, the area of the section of half sphere is π(r2-d2), while the area of the section of the cone is πd2, thus we can apply “Zu Geng’s axiom” here.) Therefore, we can get the volume of half sphere i.e πr3(1-1/3)=2/3πr3. Thus the volume of a sphere with radius r is 4/3πr3.
Although the computation of the volume of sphere had been discovered by Archimedes, Zu Geng got the volume and his “Zu Geng’s axiom” independently. Also the applications of “Zu Geng’s axiom” are more various. It is a distinguished discovery and creation in the development of history, which makes Zu Geng go down in history.

Citation:
http://en.wikipedia.org/wiki/Steinmetz_solid
http://baike.baidu.com/view/284989.htm
http://courseware.eduwest.com/sharecourse/courseware/0350/content/zhonghe/tc3.html (figure 1)
http://www.0737weal.com/d/whddfhv/ (figure 2)

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