# Zu Chongzhi and his mathematics

Image:Zu Chongzhi. Author: Gisling, via Wikimedia Commons.

Zu Chongzhi was a Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. He did a lot of famous mathematics during his life. His three most important contributions were studying The Nine Chapters on the Mathematical Art, calculating pi, and calculating the volume of sphere.

As we know,  The Nine Chapters on the Mathematical Art is the most famous book in the history of Chinese mathematics. In ancient China, most people could not understand “The Nine Chapters on the Mathematical Art”. Zu Chongzi read the book and then he used his comprehension to explain the formulas of the book. Zu Chongzhi, and his father wrote the “Zhui Shu”(缀术) together. The book made The Nine Chapters on the Mathematical Art easier to read. And the book also added some important formula by Zu. For example, the calculation of pi and the calculation of sphere volume. “Zhui Shu” also become math textbook at the Tang Dynasty Imperial Academy. Unfortunately, the book was lost in the Northern Song Dynasty.

Zu’s ratio, also called milü is named after Zu Chongzhi. Zu’s ratio was an early accurate approximation of pi. It was recorded in the “Book Of Sui” and “Zhui Shu”. (Book Of Sui is the official history of the Sui dynasty). According to the “Book Of Sui”, Zu Chongzhi discovered that pi is between 3.14159276 and 3.14159277. Today, we know the actual number is in accord with Zu’s ratio. But “Book Of Sui” did not record the method used to get the number. Most historians and mathematicians think Zu Chongzhi used Liu Hui’s π algorithm to get the number. Liu Hui’s algorithm means approximating circle with a 24,576 sided polygon. Japanese mathematician Yoshio Mikami pointed out, “22/7 was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however milü π = 355/113 could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe”. Hence Mikami strongly urged that the fraction 355/113 be named after Zu Chongzhi as Zu’s fraction.( Yoshio Mikami)

Image:Zu Chongzhi’s method (similar to Cavalieri’s principle) for calculating a sphere’s volume includes calculating the volume of a bicylinder. Author: Chen Bai, via WIkimedia Commons.

Zu Chongzhi’s other important contribution was calculation volume of the sphere. Together with his son Zu Geng, Zu Chongzhi used an ingenious method to determine the volume of the sphere.(Arthur Mazer). In The Nine Chapters on the Mathematical Art, the author used Steinmetz solid to get the volume of the sphere. The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid.

Image:Steinmetz solid. Author: Van helsing, via Wikimedia Commons.

But the book did not give the formula of how to get the volume of the sphere. Zu Chongzhi used “Zu Geng principle” (another name: Cavalieri’s principle) to show the volume of the sphere formula is (π*d³)/6. In order to commemorate the fact that Zu Chongzhi found the significant contribution of the principle with his son, people called the principle “Zu Geng principle”. “Zu Geng principle” is the same as “Cavalieri’s principle”, but “Zu Geng principle” is earlier than “Cavalieri’s principle”. “Cavalieri’s principle” means two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.(Kern and Bland 1948, p. 26).

Work cited:

Yoshio Mikami , (1947). Development of Mathematics in China and Japan. 2nd ed. : Chelsea Pub Co;.

Arthur Mazer , (2010). The Ellipse: A Historical and Mathematical Journey. 1st ed. : Wiley;

Kern, W. F. and Bland, J. R. “Cavalieri’s Theorem” and “Proof of Cavalieri’s Theorem.” §11 and 49 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 25-27 and 145-146, 1948.

http://en.wikipedia.org/wiki/Cavalieri%27s_principle

http://en.wikipedia.org/wiki/Zu_Chongzhi