周髀算經·卷上（選） The Gnomon of the Zhou: Shang Gao and Chen Zi Anonymous 佚名 (Western Han, compiled c. 1st century BCE) Qin–Han · compiled c. 1st century BCE Cold Library · Open Stacks · S-006004 · https://coldlibrary.com/s/006004 Rights: Classical Chinese text of the Han dynasty, about two thousand years old. Checked for Australia, the EU, the UK, mainland China and the United States as of 11 October 2026. Guidance, not legal advice. Source: Wikisource, https://zh.wikisource.org/wiki/周髀算經 (retrieved 2026-10-11) English translation: Claude (an AI), 2026 · CC0 1.0 Generated 2026-10-14 from https://coldlibrary.com/en/open-stacks/zhoubi-suanjing-shang-gao ---- 周髀算經 卷上之一 昔者周公問于商高曰：竊聞乎大夫善數也。 請問古者包犧立周天歷度，夫天不可階而升；地不可得尺寸而度，請問數安從出？ 商高曰：數之法，出于圓方，圓出于方，方出于矩，矩出于九九八十一。 故折矩，以為句廣三，股修四，徑隅五。 既方之，外半其一矩，環而共盤。得成三四五。 兩矩共長二十有五。是謂積矩。 故禹之所以治天下者。此數之所生也。 周公曰：大哉言數，請問用矩之道？ 商高曰：平矩以正繩，偃矩以望高，覆矩以測深，臥矩以知遠，環矩以為圓，合矩以為方。 方屬地，圓屬天，天圓地方。方數為典，以方出圓，笠以寫天。 天青黑，地黃赤，天數之為笠也。青黑為表，丹黃為裏，以象天地之位。 是故，知地者智，知天者聖。智出于句，句出于矩。 夫矩之于數，其裁制萬物，惟所為耳。 周公曰：善哉。 周髀算經 卷上之二 昔者，榮方問于陳子。 曰：今者竊聞夫子之道，知日之高大，光之所照，一日所行。 遠近之數，人所望見，四極之窮，列星之宿，天地之廣袤。 夫子之道，皆能知之，其信有之乎？ 陳子曰：然。 榮方曰：方雖不省，願夫子幸而說之。今若方者，可教此道耶？ 陳子曰。然，此皆算術之所及。 子之于算，足以知此矣，若誠累思之。 于是榮方歸而思之，數日不能得，復見陳子曰：方、思之不能得，敢請問之。 陳子曰：思之未熟。此亦望遠起高之術，而子不能得，則子之於數，未能通類。 是智有所不及，而神有所窮。 夫道術、言約而用博者，智類之明。問一類而以萬事達者，謂之知道。 今子所學，算數之術，是用智矣，而尚有所難，是子之智類單。 夫道術所以難通者。既學矣，患其不博；既博矣，患其不習；既習矣，患其不能知。 故同術相學，同事相觀，此列士之愚智，賢不肖之所分。 是故能類以合類，此賢者業精習智之質也。 夫學同業而不能入神者，此不肖無智，而業不能精習，是故算不能精習。 吾豈以道隱子哉，固復熟思之。 榮方復歸思之，數日不能得，復見陳子曰： 方思之以精熟矣，智有所不及，而神有所窮，知不能得，願終請說之。 陳子曰：復坐，吾語汝。 ---- 周髀算经 卷上之一 昔者周公问于商高曰：窃闻乎大夫善数也。 请问古者包牺立周天历度，夫天不可阶而升；地不可得尺寸而度，请问数安从出？ 商高曰：数之法，出于圆方，圆出于方，方出于矩，矩出于九九八十一。 故折矩，以为句广三，股修四，径隅五。 既方之，外半其一矩，环而共盘。得成三四五。 两矩共长二十有五。是谓积矩。 故禹之所以治天下者。此数之所生也。 周公曰：大哉言数，请问用矩之道？ 商高曰：平矩以正绳，偃矩以望高，覆矩以测深，卧矩以知远，环矩以为圆，合矩以为方。 方属地，圆属天，天圆地方。方数为典，以方出圆，笠以写天。 天青黑，地黄赤，天数之为笠也。青黑为表，丹黄为里，以象天地之位。 是故，知地者智，知天者圣。智出于句，句出于矩。 夫矩之于数，其裁制万物，惟所为耳。 周公曰：善哉。 周髀算经 卷上之二 昔者，荣方问于陈子。 曰：今者窃闻夫子之道，知日之高大，光之所照，一日所行。 远近之数，人所望见，四极之穷，列星之宿，天地之广袤。 夫子之道，皆能知之，其信有之乎？ 陈子曰：然。 荣方曰：方虽不省，愿夫子幸而说之。今若方者，可教此道耶？ 陈子曰。然，此皆算术之所及。 子之于算，足以知此矣，若诚累思之。 于是荣方归而思之，数日不能得，复见陈子曰：方、思之不能得，敢请问之。 陈子曰：思之未熟。此亦望远起高之术，而子不能得，则子之于数，未能通类。 是智有所不及，而神有所穷。 夫道术、言约而用博者，智类之明。问一类而以万事达者，谓之知道。 今子所学，算数之术，是用智矣，而尚有所难，是子之智类单。 夫道术所以难通者。既学矣，患其不博；既博矣，患其不习；既习矣，患其不能知。 故同术相学，同事相观，此列士之愚智，贤不肖之所分。 是故能类以合类，此贤者业精习智之质也。 夫学同业而不能入神者，此不肖无智，而业不能精习，是故算不能精习。 吾岂以道隐子哉，固复熟思之。 荣方复归思之，数日不能得，复见陈子曰： 方思之以精熟矣，智有所不及，而神有所穷，知不能得，愿终请说之。 陈子曰：复坐，吾语汝。 ---- English translation by Claude (an AI) 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.