周髀算經·卷上（選）周髀算经·卷上（选）

Zhoubi Suanjing, Book One, Part One#

Long ago the Duke of Zhou asked Shang Gao: I have humbly heard, sir, that you are skilled in number.

May I ask how Baoxi in ancient times set out the degrees of the circuit of heaven? Heaven cannot be climbed by steps, and the earth cannot be measured with foot and inch. Where, then, did the numbers come from?

Shang Gao said: The method of number comes from the circle and the square. The circle comes from the square, the square comes from the carpenter's square, and the carpenter's square comes from nine nines are eighty-one.

So bend a carpenter's square, making the base three across and the upright four long; then the diagonal between the corners is five.

Square it on the outside, and halve the outer square to give one carpenter's square; set these round in a ring on a single board, and three, four and five are formed.

The two carpenter's squares together measure twenty-five. This is called piling up the squares.

So the means by which Yu set the world in order were born of these numbers.

The Duke of Zhou said: Great indeed is your account of number. May I ask the way of using the carpenter's square?

Shang Gao said: Lay the square level to true a cord; tilt it up to sight a height; turn it over to sound a depth; lay it down to know a distance; turn it round to make a circle; put squares together to make a square.

The square belongs to the earth, the circle belongs to heaven: heaven is round and the earth is square. The numbers of the square are the standard; from the square the circle is drawn out; the bamboo hat is used to picture heaven.

Heaven is blue-black and the earth is yellow-red. The numbers of heaven make the shape of a hat: blue-black on the outside, cinnabar and yellow on the inside, to show the places of heaven and earth.

Therefore one who knows the earth is wise, and one who knows heaven is a sage. Wisdom comes from the base of the triangle, and the base comes from the carpenter's square.

The carpenter's square, applied to number, can shape and order the ten thousand things; it does whatever it is set to do.

The Duke of Zhou said: Excellent.

Zhoubi Suanjing, Book One, Part Two#

Long ago Rong Fang asked Chen Zi.

He said: I have heard that by your way, master, you know the height and size of the sun, how far its light reaches, how far it travels in a day,

the numbers of near and far, what a person can see, the limits of the four extremes, the lodges of the stars, and the breadth and length of heaven and earth.

Is it true that your way can know all of these?

Chen Zi said: It is.

Rong Fang said: Though I am slow to understand, I hope you will be kind enough to explain it. Can someone like me be taught this way?

Chen Zi said: Yes. All of this lies within the reach of calculation.

Your skill in calculation is enough to know it, if you truly think it over and over.

So Rong Fang went home and thought about it. For several days he could not get it. He came to Chen Zi again and said: I have thought about it and cannot get it. May I venture to ask you?

Chen Zi said: You have not thought it through. This is only the art of sighting distances and working out heights, and if you cannot get it, then your grasp of number does not yet reach from one kind of thing to another.

That is, your understanding has places it does not reach, and your spirit has places where it runs out.

A method that is brief in words and wide in use comes from seeing things by their kinds. To ask about one kind and so reach through ten thousand affairs: that is called knowing the way.

What you are learning now, the art of calculation, is a use of the understanding; yet you still find difficulty in it, because your understanding of kinds is thin.

The reason a method is hard to master is this: once you have studied, the trouble is that you are not broad; once you are broad, the trouble is that you have not practised; once you have practised, the trouble is that you do not understand.

So those who learn the same method learn from one another, and those in the same work watch one another; this is where scholars part into the foolish and the wise, the worthy and the unworthy.

Therefore to be able to join one kind to another is the very substance of a worthy person's refined work and practised understanding.

Those who study the same work and cannot enter into its spirit are unworthy and lack understanding, and their work cannot be refined by practice; so their calculation cannot be refined by practice.

Do you think I am hiding the way from you? Go and think it over again, thoroughly.

Rong Fang went home again and thought about it. For several days he could not get it. He came to Chen Zi again and said:

I have thought it through as finely and fully as I can. My understanding has places it does not reach, and my spirit has run out. I know I cannot get it, and I beg you at last to explain it.

Chen Zi said: Sit down again, and I will tell you.