Pythagoras Theorem

The only geometrical theorem which we can be certain that the ancient Chinese were acquainted is that the area of the square described on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares described on the sides; Pythagoras theorem.

-- Rouse Ball, 1960, p9

Could the origin of "Pythagoras Theorem" be in China? Swetz and Kao believe so (1977). Why? A search back in history finds a proof in an ancient Chinese mathematical text, 'Chou pei Suan Ching'. 'This book is the oldest Chinese mathematics text known. While the exact date of its origin is controversial, with estimates ranging as far back as 1100 B.C.'(Swetz and Kao, 1977, p 14). The diagram that describes the proof (Diagram 2) was so well known in China that it had a special name, 'hsuan-thu'. This diagram shows 'the square on the hypotenuse folded backwards ... and demonstrably containing three further identical triangles together with a square of the difference between the base and the altitude' (Needham, p95). This picture was described as

The diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known.

-- J.Needham, 1959, p96


Diagram 2. The 'hsuan-thu'.

Problems from Chiu chang suan shu also offering evidence of the knowledge of right angled triangles

The Chiu chang suan shu is the most famous Chinese mathematical text. Its context is a summary of the mathematical knowledge possessed in China up to the middle of the 3rd century. Alexander Wylie 'was the first to translate parts of the text into English and he was marvelled at his findings' (Swetz and Kao, 1977, p17).

A few examples are as follows :

Problem 1
Given: kou = 3 ch'ih, ku = 4 ch'ih. What is the length of hsien?
Answer
hsien = 5 ch'ih
Method
Add the square of kou and ku. The square root of the sum is equal to hsien ( Look at diagram 3).


Diagram 3. Pictorial representation of Method 1.
Problem 2
Given: hsien = 5 ch'ih, kou = 3 ch'ih. What is the length of ku?
Answer
ku = 4 ch'ih
Method
The length of ku is equal to the square root of the difference of the squares of hsien and kou OR square of ku equals (hsien + kou) x (hsien - kou) (Look at diagram 4).


Diagram 4. Pictorial representation of Method 2.
Problem 3
Given: ku = 4 ch'ih, hsien = 5 ch'ih. What is the length of kou?
Answer
kou = 3 ch'ih
Method
The length of kou is equal to the square root of the difference of the squares of hsien and ku OR square of kou equals (hsien + ku) x (hsien - ku) (Look at diagram 5).


Diagram 5. Pictorial representation of Method 3.
NOTE: In a right triangle the short adjacent to the right angle was called kou. The longer adjacent side to the right angle was called ku. Therefore the side opposite the right angle (now known as the hypotenuse) was called the hsien.

Therefore kou was shorter than ku, and ku was shorter than hsien.

The Chinese characters for kou and ku originally were used to designate 'leg' and 'thigh'. Hsien being a string strung between two points.

**This example is from Swetz and Kao pages 26 - 28**

Conclusion

The problems like those found in Chiu chang suan shu indicate that the Chinese had accumulated a wealth of experience in working with the right triangle from an early time. And also as the diagram of the 'hsuan thu' represents the oldest recorded proof of the "Pythagoras Theorem", like Swetz and Kao (1977) I pose this question, was Pythagoras Chinese?

No other ancient society could boast on a similar level of accomplishment.

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