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 inquire how Fu-Hsi anciently established the degrees of the celestial sphere? There are no steps by which one may ascent the heavens, and the earth is not measurable with a footrule. 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 (lit. T-square or carpenter's square).
3. The rectangle originates from the fact that 9 x 9 = 81 (i.e., the multiplication table or 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 two 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 (of area 24); then (when this is subtracted from the square plate of area 49) the remainder is of area 25. This (process) is called 'piling up the rectangles' (chi chu).
5. The methods used by Yu the Great in governing the world were derived from these numbers."
6. Chou Kung exclaimed, "Great indeed is the art of numbering. I should like to ask about the Tao of the use of the right-angled triangle" (lit. T-square).
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 reverse 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 (deduced) 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."