Explaining the Proof of Pythagoras' Theorem in the Chinese Chou Pei Suan Ching.
From Joseph Needham's Science and Civilisation in China, Volume 3, P. 22, we read...
1. Of old, Chou Kung addressed Shang Kao, saying, "I have heard that the Grand Prefect(Shang Kao) is versed in the art of numbering. May I venture to enquire how Fu-Hsi anciently established the degrees of the celestial sphere? There are no steps by which one may ascend the heavens, and the earth is not measurable with a foot-rule. I should like to ask you what was the origin of these numbers?
2. Shang Kao replied, 'The art of numbering proceeds from the circle and the square. The circle is derived from the square and the square from the rectangle."
3. The rectangle originates from 9 × 9= 81 (i.e. the multiplication table or the properties of numbers as such).
4. Thus, let us cut a rectangle(diagonally), and make the width 3 units wide, and the length 4 units long.
The diagonal between the corners will then be 5 units long. Now after drawing a square on this diagonal,
circumscribe it by half-rectangles like that which has been left outside, so as to form a (square) plate.
Thus the four outer half-rectangles of width 3, length 4, and diagonal 5, together make two rectangles;
then (when this is subtracted from the square plate of area 49) the remainder is of area 25. This is called "pilling up the rectangles".
JC COMMENT The area of the prior single red square formed on the original hypotenuse remains, yet is now split into two smaller red squares, one formed from a shorter side and the other formed from a longer side.
Here we have a = 3 (short leg of triangle) and b = 4 (long leg of triangle) and c = 5 (hypotenuse).
The big red square on the hypotenuse c, (left above), can be rearranged into two smaller squares on a and b, (right above), to show why, in a right angle triangle, a² + b² = c².
5. The methods used by Yu the Great in governing the world were derived from these numbers [Yu was the patron saint of hydraulic engineers and all those concerned with water control, irrigation and conservancy.]
6. Chou Kung exclaimed, "Great indeed is the art of numbering, I would like to ask about the Tao of the use of the right-angled triangle
7. Shang Kao replied, " The plane right-angled triangle(laid on the ground) serves to lay out(works) straight and square(by the aid of) cords. The recumbent right-angled triangle serves to observe heights. The reversed right-angled triangle serves to fathom depths. The flat right-angled triangle is used for ascertaining distances.
8. By the revolution of a right-angled triangle(compasses) a circle may be formed. By uniting right-angled triangles squares and oblongs are formed.
9. The square pertains to earth, the circle belongs to heaven, heaven being round and the earth square. The numbers of the square being the standard, the(dimensions of the) circle are from those of the square.
10. Heaven is like a conical sun-hat. Heaven's colors are blue and black, earth's colors are yellow and red. A circular plate is employed to represent heaven, formed according to the celestial numbers; above, like an outer garment, it is blue and black, beneath, like an inner one, it is red and yellow. Thus is represented the figure of heaven and earth.
11. He who understands the earth is a wise man, and he who understands the heavens is a sage. Knowledge is derived from the straight line. The straight line is derived from the right angle. And the combination of the right angle with numbers is what guides and rules the ten thousand things.
12. Chou Kung exclaimed, "Excellent indeed!"
[NOTE, the Chinese diagram above simplifies as Proof 1 in the diagram below. This also reveals a proof of Proposition 4 from Book II of Euclid's Elements. (If a straight line is cut at random, then the square on the whole equals the sum of the squares on the segments plus twice the rectangle contained by the segments.)
Science and civilisation in China. / Volume 3, Mathematics and the sciences of the heavens and the earth, Joseph Needham and Ling Wang, Pp. 22-23, Cambridge University Press, Cambridge, 1959.