九章算術注序九章算术注序

昔在庖犧氏始畫八卦，以通神明之德，以類萬物之情，作九九之術，以合六爻之變。暨於黃帝神而化之，引而伸之，於是建曆紀，恊律呂，用稽道原，然後兩儀四象精微之氣可得而效焉。記稱「隸首作數」，其詳未之聞也。按周公制禮而有九數，九數之流，則《九章》是矣。往者暴秦焚書，經術散壞。自時厥後，漢北平侯張蒼、大司農中丞耿壽昌皆以善筭命世。蒼等因舊文之遺殘，各稱刪補。故校其目則與古或異，而所論者多近語也。

昔在庖牺氏始画八卦，以通神明之德，以类万物之情，作九九之术，以合六爻之变。暨于黄帝神而化之，引而伸之，于是建历纪，协律吕，用稽道原，然后两仪四象精微之气可得而效焉。记称「隶首作数」，其详未之闻也。按周公制礼而有九数，九数之流，则《九章》是矣。往者暴秦焚书，经术散坏。自时厥后，汉北平侯张苍、大司农中丞耿寿昌皆以善算命世。苍等因旧文之遗残，各称删补。故校其目则与古或异，而所论者多近语也。

Long ago, Fu Xi first drew the eight trigrams, to communicate with the virtue of the spirits and to classify the nature of the ten thousand things, and he made the art of the nine-nines table to match the changes of the six lines. Then the Yellow Emperor transformed it with his genius, drew it out and extended it. So the calendar was established and the pitch pipes were harmonised, and the source of the Way was investigated; only then could the subtle breaths of the two modes and the four images be put to use. The records say that Lishou invented numbers, but we have not heard the details. According to the rites established by the Duke of Zhou, there were the nine arts of calculation, and the Nine Chapters are what came down from them. In the past the tyrannical Qin burned the books, and the learning of the classics was scattered and ruined. Since that time Zhang Cang, Marquis of Beiping, and Geng Shouchang, Assistant to the Grand Minister of Agriculture, both of the Han, became famous in their age for their skill in calculation. Working from what remained of the old texts, Zhang Cang and the others each made deletions and additions. So when the chapter headings are compared with antiquity, some differ, and much of what is discussed is in recent language.

徽幼習《九章》，長再詳覽。觀陰陽之割裂，總筭術之根源，探賾之暇，遂悟其意。是以敢竭頑魯，采其所見，為之作注。事類相推，各有攸歸，故枝條雖分而同本幹知，發其一端而已。又所析理以辭，解體用圖，庶亦約而能周，通而不黷，覽之者思過半矣。且筭在六藝，古者以賓興賢能，教習國子。雖曰九數，其能窮纖入微，探測無方。至於以法相傳，亦猶規矩度量可得而共，非特難為也。當今好之者寡，故世雖多通才達學，而未必能綜於此耳。

徽幼习《九章》，长再详览。观阴阳之割裂，总算术之根源，探赜之暇，遂悟其意。是以敢竭顽鲁，采其所见，为之作注。事类相推，各有攸归，故枝条虽分而同本干知，发其一端而已。又所析理以辞，解体用图，庶亦约而能周，通而不黩，览之者思过半矣。且算在六艺，古者以宾兴贤能，教习国子。虽曰九数，其能穷纤入微，探测无方。至于以法相传，亦犹规矩度量可得而共，非特难为也。当今好之者寡，故世虽多通才达学，而未必能综于此耳。

As a boy I studied the Nine Chapters, and when I grew up I went through them again in detail. I observed how yin and yang divide, and brought together the roots of the arts of calculation, and in the time I spent searching out what was hidden, I came at last to understand their meaning. So I have dared to give my poor and limited best, gathering what I have seen, to write a commentary on them. Things of the same kind can be inferred from one another, and each has its proper place. So although the branches divide, they share the same trunk; it is enough to bring out one end of it. I have also analysed the principles in words and explained the shapes with diagrams, in the hope of being brief yet complete, thorough yet not tiresome, so that readers will grasp more than half of it. Moreover, calculation is one of the six arts. In ancient times it was used to recommend the worthy and able, and to teach the sons of the state. Though it is called the nine arts of calculation, it can reach into the finest and most minute things, and probe without limit. As for handing down its methods, it is like the compass and square and the measures of length and capacity: they can be shared by everyone, and are not particularly difficult. Today few people are fond of it, and so, although the world has many people of wide talent and broad learning, they have not necessarily mastered this.

《周官‧大司徒》職，夏至日中立八尺之表，其景尺有五寸，謂之地中。說云，南戴日下萬五千里。夫云爾者，以術推之。按《九章》立四表望遠及因木望山之術，皆端旁互見，無有超邈若斯之類。然則蒼等為術猶未足以博盡羣數也。徽尋九數有重差之名，原其指趣乃所以施於此也。凡望極高、測絕深而兼知其遠者必用重差、句股，則必以重差為率，故曰重差也。立兩表於洛陽之城，令高八尺。南北各盡平地，同日度其正中之時，以景差為法，表高乘表間為實，實如法而一。所得加表高，即日去地也。以南表之景乘表間為實，實如法而一，即為從南表至南戴日下也。以南戴日下及日去地為句、股，為之求弦，即日去人也。以徑寸之筩南望日，日滿筩空，則定筩之長短以為股率，以筩徑為句率，日去人之數為大股，大股之句即日徑也。雖夫圓穹之象猶曰可度，又況泰山之高與江海之廣哉。徽以為今之史籍且略舉天地之物，考論厥數，載之於志，以闡世術之美。輒造《重差》，并為注解，以究古人之意，綴於《句股》之下。度高者重表，測深者累矩，孤離者三望，離而又旁求者四望。觸類而長之，則雖幽遐詭伏，靡所不入。博物君子，詳而覽焉。

《周官‧大司徒》职，夏至日中立八尺之表，其景尺有五寸，谓之地中。说云，南戴日下万五千里。夫云尔者，以术推之。按《九章》立四表望远及因木望山之术，皆端旁互见，无有超邈若斯之类。然则苍等为术犹未足以博尽群数也。徽寻九数有重差之名，原其指趣乃所以施于此也。凡望极高、测绝深而兼知其远者必用重差、句股，则必以重差为率，故曰重差也。立两表于洛阳之城，令高八尺。南北各尽平地，同日度其正中之时，以景差为法，表高乘表间为实，实如法而一。所得加表高，即日去地也。以南表之景乘表间为实，实如法而一，即为从南表至南戴日下也。以南戴日下及日去地为句、股，为之求弦，即日去人也。以径寸之筩南望日，日满筩空，则定筩之长短以为股率，以筩径为句率，日去人之数为大股，大股之句即日径也。虽夫圆穹之象犹曰可度，又况泰山之高与江海之广哉。徽以为今之史籍且略举天地之物，考论厥数，载之于志，以阐世术之美。辄造《重差》，并为注解，以究古人之意，缀于《句股》之下。度高者重表，测深者累矩，孤离者三望，离而又旁求者四望。触类而长之，则虽幽遐诡伏，靡所不入。博物君子，详而览焉。

In the office of the Grand Minister of Education in the Rites of Zhou, an eight-foot gnomon is set up at noon on the summer solstice, and where its shadow is a foot and a half long, that is called the centre of the earth. The commentary says that it is fifteen thousand li from the point directly beneath the sun to the south. This was worked out by calculation. But in the Nine Chapters, the methods of sighting distances with four posts and of sighting a mountain from a tree all show the end and the side together; there is nothing that reaches so far as this. So the methods of Zhang Cang and the others were not yet enough to cover all kinds of number. I found that among the nine arts there was one named double difference, and when I traced its purpose, I saw that it was meant for exactly this. Whenever one sights something very high, or measures something very deep, and wants to know its distance as well, one must use the double difference together with the base and upright, taking the double difference as the ratio; that is why it is called the double difference. Set up two gnomons eight feet high in the city of Luoyang, one to the south and one to the north, on level ground. On the same day, measure their shadows at noon. Take the difference of the shadows as the divisor, multiply the height of the gnomon by the distance between the gnomons as the dividend, and divide. Add the height of the gnomon to the result, and you have the height of the sun above the earth. Multiply the shadow of the southern gnomon by the distance between the gnomons as the dividend and divide by the same divisor, and you have the distance from the southern gnomon to the point directly beneath the sun. Take the distance to the point beneath the sun and the height of the sun as base and upright, and find the hypotenuse: this is the distance of the sun from the observer. Look south at the sun through a tube one inch across, so that the sun just fills the opening; then take the length of the tube as the upright ratio and its diameter as the base ratio, and the distance of the sun from the observer as the great upright; the base that goes with the great upright is the diameter of the sun. If even the image of the round vault of heaven can be measured, how much more the height of Mount Tai and the breadth of the rivers and seas. I believe that the histories of our day ought at least to give a brief account of the things of heaven and earth, examining and discussing their numbers and recording them in their treatises, to show the beauty of the arts of the age. So I have made the Double Difference, with a commentary, to search out the intentions of the ancients, and attached it after the chapter on base and upright. To measure height, use repeated gnomons; to measure depth, use stacked set-squares; for what stands isolated, sight it three times; for what is isolated and must also be found from the side, sight it four times. Extend this to things of the same kind, and there is nothing, however hidden, remote, strange or concealed, that it cannot reach. May the learned gentlemen read it closely.