Tag Archives: Chinese mathematics

Zu Chongzhi and his mathematics

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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)

https://i2.wp.com/upload.wikimedia.org/wikipedia/commons/e/e9/Sphere_volume_derivation_using_bicylinder.jpg

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.

https://i0.wp.com/upload.wikimedia.org/wikipedia/commons/2/20/Bicylinder_Steinmetz_solid.gif

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

Early Chinese Mathematics

Math is something that is found all throughout history.  It was used for may different reasons, in many different cultures.  What I find interesting is how these different cultures learned some of the same ideas without even having knowledge of the others’ work. These works could be anything from counting systems to Pascal’s triangle.  It can also include how one culture passed its knowledge on to another. This makes you wonder how some ideas that were known in western civilization could also be found in Asia.  As I was looking into this I found some very interesting facts about mathematics in China. Some small examples of math found in China begin with something called oracle bone scripts: scripts carved into animal bones or turtle shells. These scripts contain some of the oldest records in China.  This, like the clay from babylonian times, had many different uses including math.  Chinese culture also had something called the six arts: Rites, Music, Archery, Charioteerring, Calligraphy, and Mathematics.  Men who excelled in these arts were known as perfect gentlemen.

In China, like in India, one can find the use of a base ten numeral system.  This is quite different from the Babylonians, which makes it seem like there must have been some conduit of knowledge between India and China.  In China, around 200 BCE, they used something called “rod numerals.”  Rod numeral counting is very similar to what we use today.  This counting system consisted of digits that ranged from one to nine, as well as 9 more digits to represent the first nine multiples of 10.  The numbers one through nine were represented by rods going vertically, while the numbers of the power of 10 were horizontal.  This means that every other digit was horizontal while its neighbor was vertical.  For example 215 would be represented like this ||—|||||.  If one wanted to use a zero you would have to use an empty space.  The empty space is also something that can been seen in the Babylonian counting system.  As with the Babylonians, a symbol was eventually used for zero.  Interestingly enough, before there was a symbol for zero, counting rods included negative numbers. A number being positive or negative depended on its color: black or red.  This idea of having negative numbers didn’t come about in another culture until around 620 CE in India.  It seems quite apparent that several ideas that originated in China could possibly have been passed on to a neighboring country. 

Rod numerals. Image: Gisling, via Wikimedia Commons.

Rod numerals. Image: Gisling, via Wikimedia Commons.

The use of counting rods as a counting system brought about another very interesting mathematical concept, the idea of a decimal system.  China first used decimal fractions in the 1st century BCE.  Fractions were used like they are today, with one number on top of another.  For example, today if you used a faction for one half, it would be written like this: 1/2.  Using rod numbers you can do the same thing like this: | / ||.  Not only could this be represented as a fraction but it could also be written as a decimal.  To do this one would simply write the number out and insert a special character to show where the whole number started.  For example, if you wanted to say 3.1213, you would write it as a whole number like this: |||—||—|||.  To show where the left side of the decimal starts, you would mark it with a symbol under the number to the left of the decimal point, in this case under the first 3.  To me the use of rod numbers is so similar to how we use our numbers today that even the arithmetic that was used can be done easily by someone in our culture.  Addition is done almost the same except they would work from left to right.  Multiplication and division were used as well.   The use of base ten as well as using rod numerals made complicated equations much easier to attain, such as the use of polynomials and even Pascals triangle.

The Yanghui Triangle. Image: Public domain, via Wikimedia Commons.

The triangle known as “Pascal’s” in the west, in a Chinese manuscript from 1303 CE. Image: Public domain, via Wikimedia Commons.

Centuries before Pascal, the Chinese knew about Pascal’s triangle.  Shen Kuo, a polymathic Chinese scientist was known to have used Pascal’s triangle in the 12th century CE.  It appears that knowledge of Pascal’s triangle begins even before this. The first finding of Pascal’s triangle was in ancient India around 200 BCE.  We can see that this idea was sprouting around and found evidence in different cultures, from Persia to China and to Europe.  This again makes one wonder how this knowledge base was passed around from one culture to another.  Lacking historic details, it is hard to see if this idea of Pascal’s triangles was thought up individually or if this concept was somehow passed from one culture to another.

It seems that in all cultures there is a need for counting, which in turn brings about the need for math.  The cultural implications can mean that you are a “perfect gentlemen” by having mathematical knowledge, or it could lead a greater knowledge that can be passed on to other cultures.  In China, we see that many ideas of numbers and mathematics were thought up on their own without having other culture’s ideas intervening.  We can also see that the knowledge that was passed on was able to thrive and turn into something even more intriguing.  It is apparent that we can always learn and teach others to help our knowledge grow.

Source:

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

http://en.wikipedia.org/wiki/Decimal#History

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

http://nrich.maths.org/5961

http://en.wikipedia.org/wiki/Pascal’s_triangle