昔在包牺氏始画八卦，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。
远古时代，包牺氏开始绘制了八卦，用以沟通神和明的德行，以及分类万物的情状，他创作了乘法口诀表，以配合六爻的变化。到了黄帝时期，他将八卦进一步神圣化并且延伸发展，于是建立了历法，协调音律，用以考察自然规律的本源，这样以后天地阴阳和四象的精妙气息就可以被有效地体现了。记载中说，是隶首创造了数字，但具体的细节没有听说过。根据周公所制定的礼仪制度，就有了数学中的“九章算术”，它是数学知识流传下来的成果，即《九章算术》。
In ancient times, Baoxi Shi first drew the eight trigrams to communicate with the virtues of deities and categorize the sentiments of all things. He devised the technique of multiplying nine by nine to match the changes in the six lines of the trigrams. When it came to the Yellow Emperor, he spiritualized and extended these concepts, establishing a calendar, harmonizing musical pitches, and using them to investigate the principles of the natural way. Only then could the subtle essence of the dual cosmic forces and the four symbolic images be effectively manifested. Historical records attribute the invention of numbers to Li Shou, but the details are not well known. According to the rituals established by Duke Zhou, there are nine mathematical methods, which culminated in "The Nine Chapters on Mathematical Procedures."
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。苍等因旧文亡遗残，各称删补。故校其目则与古或异， 而所论者多近语也。
过去，残暴的秦朝焚烧书籍，学术经典因此散失破损。从那时起之后，汉朝的北平侯张苍和大司农中丞耿寿昌都因为他们擅长算术而闻名。张苍等人根据遗留下来的残缺不全的旧文献，各自进行了删减和补充。因此，当校对这些书目时，会发现它们与古代的内容或许有所不同，而且其中讨论的内容大多是接近当时的用语。
In the past, the tyrannical Qin Dynasty burned books, and academic classics were scattered and damaged as a result. Since that time, Zhang Cang of Han Dynasty's Beiping Marquis and Geng Shouchang, the Grand Agricultural Commissioner's Middle Counselor, both became famous for their skill in mathematics. Based on the remnants of old texts that had been lost or partially destroyed, each of them made their own deletions and additions. Therefore, when comparing these works with the ancient ones, it is found that they may differ from the originals, and the discussions within are mostly phrased in terms close to the language of the time.
徽幼习《九章》，长再详览。观陰陽之割裂，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，故枝条虽分而同本榦知，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。
刘徽从小就学习《九章算术》，长大后再次详细阅读。他观察阴阳的割裂，探索算术的根源，在深入探究之余，领悟了其中的意义。因此，他敢于竭尽自己的愚钝，采集自己的见解，为之作注。事物按照类别相互推导，各有归属，所以虽然分支不同，但都源于同一主干，只需启发一端即可。他又以理论分析，用图解解体，希望简约而周全，通达而不繁琐，让读者思考过半。况且算术在六艺之中，古代用以选拔贤能，教导国子；虽然称为九数，但其能力可以穷尽细微之处，探测无方；至于以法则相传，也如同规矩度量可以共享，不仅仅是难为之事。当今好之者少，所以世上虽然多有通才达学之人，但未必能综合于此。
Liu Hui studied "Jiuzhang Suanshu" (Nine Chapters on the Mathematical Art) from a young age and revisited it in detail when he grew up. He observed the division of Yin and Yang, delved into the origins of arithmetic, and, during his exploration of complexities, he grasped its essence. Therefore, he dared to exhaust his own dullness, gathering his insights to annotate the text. Things are categorized and derived from one another, each with its place, so although branches differ, they all stem from the same trunk, needing only to be initiated from one end. Moreover, he analyzed theories with words and dissected problems using diagrams, hoping to be concise yet comprehensive, clear yet not cluttered, allowing readers to ponder more than half of it. Furthermore, arithmetic is one of the six arts; in ancient times, it was used to select and educate the talented and virtuous. Although referred to as the 'Nine Numbers', its capability can penetrate the minutest details and explore limitless aspects; as for passing down the rules, it's like sharing measurements and standards, not merely a difficult task. Nowadays, few appreciate it, so even though there are many learned and knowledgeable people in the world, they may not necessarily master this field.
《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。案：《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。
周官·大司徒》的职责是，在夏至那天中午立一个八尺高的标杆。它的影子长度是一尺五寸，这被称为地中。据说，南方距离日下一万五千里。这样的说法，是通过技术推算出来的。根据《九章算术》中的四表望远和因木望山的技术，都是通过端旁互见，没有超出如此遥远的范围。然而苍等人的技术还不足以完全涵盖所有的数字。刘徽寻找九数中有重差的名称，原来其旨意就是要施加在这里。
In "Zhou Guan · Da Situ", the duty is to erect an eight-foot tall marker on the day of the summer solstice. Its shadow measures one foot and five inches, which is called the center of the earth. It is said that the distance southward from the sun is 15,000 li. Such a statement is derived through technical calculation. According to the methods in "Jiuzhang Suanshu" for establishing four markers to measure distance and using trees to sight mountains, all are based on mutual visibility from adjacent points, without exceeding such a vast range. However, the techniques of Cang and others are still insufficient to fully encompass all numbers. Liu Hui sought among the nine calculations for a name with a double difference, originally intending to apply it here.
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。立两表于洛陽之城，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。以径寸之筒南望日，日满筒空，则定筒之长短以为股率，以筒径为句率，日去人之数为大股，大股之句即日径也。
凡是望极高、测绝深并同时知道其远的，必须用重差、勾股，那就一定要以重差为率，所以称为重差。在洛阳城立两个表，使高八尺，南北各尽平地。在同一时间测量它们的正中时刻。以影差为法，以表高乘表间为实，实如法而一。所得加上表高，就是日离地的高度。以南表的影乘表间为实，实如法而一，就是从南表至南戴日下的距离。以南戴日下和日离地为勾、股，求出弦，就是日离人的距离。以直径一寸的竹筒向南望日，日充满竹筒的空间，就确定竹筒的长短作为勾率，以竹筒的直径为股率，日离人的数字为大股，大股的勾就是太阳的直径。
In all cases of observing the utmost height, measuring the extreme depth, and simultaneously knowing the distance, it is necessary to use the method of double difference and the Pythagorean theorem. The rate must be based on the double difference, hence it is called so. Two markers are erected in the city of Luoyang, each of which is eight feet high, at the farthest reaches of the flat ground, north and south. At the same time, measure their exact midpoints. Using the difference in their shadows as the standard, multiply the height of the markers by the distance between them as the actual value. If the actual value is equal to the standard, then add the height of the marker, which gives the height of the sun above the ground. Multiply the shadow of the southern marker by the distance between the markers as the actual value; if this is equal to the standard, then this gives the distance from the southern marker to the sun at its highest point. Taking the sun at its highest point and the sun's distance from the ground as the two sides of a right triangle, find the hypotenuse, which gives the sun's distance from the observer. Looking at the sun through a bamboo tube one inch in diameter, the sun fills the empty space within the tube. Then determine the length of the tube as the rate of the right side and the diameter of the tube as the rate of the left side. The number of the sun's distance from the observer becomes the large right side, and the left side of this large right triangle is the diameter of the sun.
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。
虽然天圆地方的意象仍然可以测量，更何况是泰山的高度和江海的广阔呢。刘徽认为现在的史籍只是简要地列举了天地之间的事物，考察讨论它们的数目，记载在志中，以阐述世间技术的美好。于是他创作了《重差》，并加以注解，以探究古人的意图，将其附在勾股定理之下。对于高度较高的物体，需要使用两个标记；对于深度较大的物体，需要使用多个标尺；对于孤立的物体，需要进行三次观测；对于偏离且旁边有其他物体的情况，需要进行四次观测。通过类比推理，即使是幽深隐秘、诡异伏藏的事物，也无所遁形。博学多识的君子，可以详细地阅读这本书。
Although the image of a round sky and a square earth is still measurable, how much more so the height of Mount Tai and the vastness of rivers and seas? Liu Hui believes that current historical records merely outline the objects between heaven and earth, examining and discussing their numbers, recording them in annals to illustrate the beauty of worldly techniques. Therefore, he created "Chong Cha" (Double Difference) and provided annotations to explore the ancients' intentions, appending it to the Pythagorean theorem. For objects of great height, two markers are needed; for those of great depth, multiple rulers are required; for isolated objects, three observations are necessary; for those deviating and adjacent to other objects, four observations are needed. By drawing analogies, even the most hidden and bizarre things can be revealed. The erudite gentlemen can read this book in detail.