## Oct 6, 2013

### The Pythagorean theorem in "Zhou Bi Suan Jing" (周髀算经)

The celebrated Phythagorean theorem gives the relation between the sides of a right triangle: $a^2 + b^2 = c^2$ where $a,b,c$ are the lengths of the sides with $c$ the one opposite to the right angle.

In China, the theorem is known as "Gou Gu Theorem" (勾股定理) or "Gou Gu Xian Theorem" (勾股弦定理). It first appears in the oldest Chinese mathematical book that survives today, the "Zhou Bi Suan Jing" (周髀算经). "Zhou" refers to the ancient dynasty Zhou (周); "Bi" means arm and according to the book, it refers to the gnomon of the sundial. The book is dedicated to astronomical observation and calculation. "Suan Jing" or "classic of arithmetic" were appended in later time to honor the achievement of the book in mathematics.

In Chapter I.1, In a dialog between "the Duke of Zhou" (周公) and "Gao of Shang" (商高), Gao said,
"故折矩, 以為句廣三, 股修四，徑隅五."
"So for the half of the rectangle,  if Gou is three, Gu is four, then Xian is Five."

Gou (勾/句) , Gu (股), Xian (弦) are the Chinese names of the three sides of a triangle. These texts essentially give a Pythagorean triplet (3, 4, 5), a triple of numbers that satisfies the Pythagorean equation $a^2 + b^2 = c^2$. Further explanation appears in Chaper I.2 in a dialog between "Rong Fang" (荣方) and "Chen Zi"  (陈子). Chen Zi in explaining a problem concerning of the measurement of the position of the sun, said,
"若求邪至日者, 以日下為句, 日高為股, 句股各自乘, 并而開方除之, 得邪至日."
"Suppose the distance to the sun is wanted. Let the distance to the projection of the sun on the ground be Gou, the distance of the sun to the ground be Gu. Square them respectively, add together and take the square root. Then the distance to the sun is obtained."

Now the Pythagorean theorem is given.

The author of Zhou Bi is unknown, nor the year was it written. But according to the texts, it is a collection of the astronomical and mathematical methods of the Shang and Zhou dynasty ( 1100 - 1000 BC). The Duke of Zhou is the brother of the creator of Zhou dynasty, which had been a state in the Shang dynasty. Gao of Shang was apparently a figure in the Shang dynasty, as suggested by his name. If this is true, the Pythagorean theorem was proposed as early as 1000BC, contrast of Pythagoras who lived between 570 BC - 495 BC. Other people argue that Zhou Bi was compiled by more than one author and settled around 200 BC.

#### references:

http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Chinese_overview.html

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