The Kaiyuan Treatise on Astrology (Tang Dynasty, compiled by Qutan Xida under imperial commission)

唐瞿曇悉達撰。《唐書·藝文志》載一百十卷。《玉海》引《唐志》亦同。又注云：《國史志》四卷，《崇文目》三卷。此本一百二十卷，與諸書所載不符，當屬後人分卷之異。自一卷“天占”至一百十卷“星圖”，均占天象。自一百十一卷“八穀占”至一百二十卷“龍魚蟲蛇占”，均占物異。或一百十卷以前為悉達原書，故與《唐志》及《玉海》卷數相符。其後十卷，後人以雜占增附之歟？

This book was compiled by Qutan Xida of the Tang dynasty. The "New Book of Tang · Art and Literature Catalogue" records it as 110 volumes. Wang Yinglin of the Song dynasty, in his "Jade Sea," also cites the same number of volumes from the "Tang Catalogue." The "Jade Sea" further notes that the "National History Catalogue" records it as four volumes and the "Chongwen General Catalogue" as three. However, the current edition is 120 volumes, which does not match the records of these previous bibliographies. This discrepancy is likely due to later editors reorganizing the volumes. From volume one "Heavenly Divination" to volume 110 "Star Chart," the content deals with celestial phenomena. From volume 111 "Divination of the Eight Grains" to volume 120 "Divination of Dragons, Fish, Insects, and Snakes," the content focuses on abnormal changes in creatures. Perhaps the first 110 volumes are the original work of Qutan Xida, matching the records of the "Tang Catalogue" and the "Jade Sea." The subsequent ten volumes were likely added by later scholars as supplementary miscellaneous divinations.

卷首標銜，悉達曾官太史監事。考《玉海》開元六年詔瞿曇悉達譯《九執曆》，則悉達之為太史監，當在開元初。卷首又標奉敕撰，而奉敕與成書年月皆無可考，惟其中載歷代曆法止於唐《麟德曆》，且雲李淳風見行《麟德曆》。考唐一行以開元九年奉詔創《大衍曆》，以開元十六年頒之，其時《麟德曆》遂不行，此書仍雲見行《麟德曆》，知其成於開元十六年以前矣。

The official title at the beginning of the volume indicates that Qutan Xida once held the position of Taishi Jian. According to the Zuyi Hai, in the sixth year of Kaiyuan, the court issued an edict ordering Qutan Xida to translate the Indian text Jiuzhizhi Li. This suggests that he was appointed Taishi Jian during the early years of Kaiyuan. The volume also states that it was "compiled by imperial order," but the time when the imperial edict was issued and the date when the book was completed are unknown. However, the book records historical calendars up to the Tang dynasty's Linde Li, and mentions that the Linde Li established by Li Chunfeng was the calendar in use at that time. According to the Tang dynasty monk Yixing, who was ordered by the court in the ninth year of Kaiyuan to create the Dayan Li, which was promulgated nationwide in the sixteenth year of Kaiyuan, the Linde Li was then discontinued. Since this book still refers to the Linde Li as the current calendar, it must have been written before the sixteenth year of Kaiyuan.

所言占驗之法，大抵術家之異學，本不足存。惟其中卷一百四、一百五全載麟德、九執二歷。《九執曆》不載於《唐志》，他書亦不過標撮大旨。此書所載，全法具著，為近世推步家所不及窺。又《玉海》載《九執曆》以開元二年二月朔為歷首。今考此書，明雲今起明慶二年丁巳歲二月一日以為歷首，亦足以訂《玉海》所傳之誤。

The method of divination and verification discussed in the book generally belongs to the peripheral studies of the school of numerology and calculation, and is not worth preserving. What is valuable, however, is that volumes 104 and 105 of the book completely include two astronomical calendars, the "Lingde Calendar" and the "Jiu zhi Calendar." The "Jiu zhi Calendar" is not recorded in the "Tang Shu · Yiwen Zhi," and other books that mention it only briefly describe it; this book, however, records the entire set of calculation methods, which are unknown to modern astronomers who calculate celestial phenomena. Additionally, the "Yuhai" states that the "Jiu zhi Calendar" takes the second day of the second lunar month of the second year of Kaiyuan as its starting point. Upon checking this book, it clearly states, "Now take the first day of the second lunar month of the Dingsi year of Mingqing as the beginning of the calendar," which is sufficient to correct the error in the "Yuhai."

至《麟德曆》雖載《唐志》，而以此書較之，多有異同。若推入蝕限術、月食所在辰術、日月蝕分術諸類，《唐志》俱未之載。又此書載章歲、章月、半總、章閏、閏分歷、周月法、弦法、氣法、曆法諸名，與《新唐書》所載全不合。其相合者，惟辰率、總法等目。蓋悉達所據當為《麟德曆》，見行本《唐志》遠出其後，不無傳聞異詞。又是可訂史傳之訛，有裨於考證不少矣。

As for the "Linde Calendar," although it is recorded in the "Tang Shu · Yiwen Zhi," when compared with this text, there are many discrepancies. For example, calculations regarding the method of determining when the sun and moon enter the eclipse limit, the method of determining the hour when a lunar eclipse occurs, and the method of calculating the extent of solar and lunar eclipses are not mentioned in the "Tang Shu." This text also lists terms such as "Zhang Sui," "Zhang Yue," "Ban Zong," "Zhang Run," "Run Fen Li," "Zhou Yue Fa," "Xian Fa," "Qi Fa," and "Li Fa," which are entirely different from those recorded in the "Xin Tang Shu." Only a few items such as "Chen Lv" and "Zong Fa" correspond. It is likely that Qu Tan Xida based his work on the original "Linde Calendar" in use at that time, while the "Tang Shu" was compiled much later, inevitably leading to inaccuracies and inconsistencies due to oral transmission. This can also help correct errors in historical texts and greatly assist in the study of historical verification.

又徵引古籍，極為浩博。如《隋志》所稱緯書八十一篇，此書尚存其七八，尤為罕見。然則其術可廢，其書則有可採也。卷首有萬曆丁巳張熙識語，謂是書歷唐迄明，約數百年，始得之挹元道人。鉤沉起滯，非偶然已。

This book cites a vast number of ancient texts. For example, the "Jingji Zhi" section of the "Sui Shu" mentions eighty-one books of prophecy and omens, and this book preserves more than seven or eight tenths of them, which is especially rare. In this sense, the divination techniques it discusses may be discarded, yet the book itself has much value. At the beginning of the book there is a preface written by Zhang Xi in the Dingsi year of the Wanli era of the Ming dynasty, stating that the book has been passed down from the Tang dynasty to the Ming dynasty, spanning several hundred years, and he obtained it from Yi Yuan Dao Ren. To rediscover ancient texts that had been buried for a long time does not seem to be a mere coincidence.

按後漢河間相張衡《靈憲》曰：“昔在先王，將步天路，用定靈軌，尋考本元。先準之於渾體，是為正儀、立度，而皇極有由建也，樞運有由稽也。乃建乃稽，斯經天常。聖人無心，因茲以生心。故《靈憲》作興。

According to Zhang Heng, who once served as the governor of Hejian during the Eastern Han dynasty, in his astronomical work "Ling Xian," he said: In ancient times, the former kings wished to calculate the paths of the sun, moon, and stars, to determine the rules of the movement of celestial bodies, and to seek the origin of heaven and earth. They always first used the "hun ti" (a spherical model of the heavens) as a standard, using it to calibrate instruments and define measurements. Thus, the "huang ji" (the great central principle, the fundamental way of the sovereign's rule) could be established, and the movement of the celestial axis could be examined. Being able to establish standards and examine movements, they thus grasped the constant laws of heaven. The sages had no deliberate intent, but through these celestial phenomena, they developed the mind for governing the world. Therefore, the book "Ling Xian" was written at the right time.

曰：太素之前，幽清玄靜，寂漠冥默，不可為象。厥中惟虛，厥外惟無，如是者永久焉，期謂溟涬，蓋乃道之根也。道根既建，自無生有。太素始萌，萌而未兆，並氣同色，混沌不分。故道志之言云：‘有物混成，先天地生。’其氣體固未可得而形，其遲速固未可得而紀也。如是者又永久焉，斯謂龐鴻，蓋乃道之干也。

Zhang Heng said: Before the "Tai Su" stage (the initial stage of primordial qi), the universe was profoundly empty, dark and quiet, silent and without light, and could not be described by any form. Inside it was only "Xu" (void), and outside it was only "Wu" (non-being). This state lasted for an extremely long time, which was called "Ming Xing." This was the root of the "Dao." Once the root of the Dao was established, "You" (being) emerged from "Wu" (non-being). The Tai Su began to stir, yet its signs were still not visible. Qi and color were blended together, forming a chaotic and undivided state. Therefore, the classic on the Dao, the "Laozi," says: "There was something formed in a harmonious and complete way, existing before heaven and earth." At that time, the form of primordial qi was not yet distinguishable, and its speed of movement could not be recorded. This state lasted for an extremely long time, which was called "Pang Hong." This was the body of the Dao.

道干既育，有物成體。於是元氣剖判，剛柔始分，清濁異位。天成於外，地定於內。天體於陽，故圓以動；地體於陰，故平以靜。動以行施，靜以合化，堙鬱構精，時育庶類，斯謂天元，蓋乃道之實也。

When the body of Dao has matured, all things have taken form. Then the original qi splits apart, and the strong and the gentle begin to separate, the clear and the turbid each occupying their proper place: Heaven forms outside and generates, while Earth settles inside and establishes its position. Heaven takes yang as its substance, thus having a round shape and ceaseless movement; Earth takes yin as its substance, thus having a flat shape and stillness. Heaven uses movement to spread out the transformation of qi, while Earth uses stillness to assist in generation. The two qi accumulate and interact, combining to form the essence, and at the proper time they give birth to all kinds of creatures. This is called "Heavenly Origin," which is precisely the fruit borne by "Dao."

在天成象，在地成形。天有九位，地有九域。天有三辰，地有三形。有象可效，有形可度。情性萬殊，旁通感薄，自然相生，莫之能紀。於是人之精者作聖，實始紀綱而經緯之。

Heaven forms various celestial phenomena, and the earth forms various physical forms. Heaven has nine directions, and the earth has nine regions; Heaven has the sun, moon, and stars as the three celestial bodies, and the earth has mountains, rivers, and plains as the three types of terrain. Celestial phenomena can be depicted, and earthly forms can be measured. The nature of all things is diverse, they communicate with each other, influence each other closely, and naturally generate and form each other. No one can record them all. Thus, among people, the most intelligent became sages, who first established order and divided the structure.

八極地維，徑二億三萬二千三百里，南北則短減千里，東西則廣增千里。自地至天半丁八極，則地之深亦如之。通而度之，則是渾也。將覆其數，用重差鉤股，懸天之景，薄地之儀，皆移千里而差一寸得之；過此而往，未之或知也。未之或知者，宇宙之謂也。宇之表無極，宙之端無窮。

The diameter of the distance between the eight directions at the ends of the earth is twenty-three million two thousand three hundred li; the distance from north to south is shorter by one thousand li, while the distance from east to west is longer by one thousand li. The height from the ground to the sky is equal to half the diameter of the eight extremes; the depth below the earth is also this same number. When all directions above and below are taken into account, the whole forms a round sphere. To verify these figures, one must use the methods of "repeated differences" and "Pythagorean theorem": measure the length of the shadow cast by the Sun at a height, and compare it with a vertical gnomon placed on the ground, using the proportion that "for every thousand li of distance, the shadow differs by one inch." Beyond this range, there is no way to know. What is referred to as "no way to know" is the "cosmos": "Yu" refers to space, which has no boundary outside; "Zhou" refers to the past and future, which has no end at either end.

天有兩儀，以舞道中。其可觀樞星，是謂之北極。在南者不著，故聖人弗之名焉。其世之遂九分而減二。陽道左廻，故天運左行。有驗於物，則人氣左嬴，形右繚也。

Heaven has two central pivots, north and south, rotating in the center of the celestial path. The visible pivot star is called the "North Pole"; the one in the south is hidden and therefore not given a name by the sages. The phrase "its world is divided into nine parts and reduced by two" seems to have some errors in the original text, but its general meaning is that the degrees of the celestial circle should be calculated by dividing into nine parts and subtracting two. The path of yang energy is to rotate to the left, so the movement of celestial bodies also proceeds to the left. This can be verified in all things: the qi of the human body is more full on the left side, while the form is coiled to the right.

天以陽廻，地以陰淳。是故天致其動，稟氣舒光。地致其靜，承施候明。天以順動，不失其中，則四時順至；寒暑不減，致生有節，故品物用生。地以靈靜，作合承天，清化致養，四時而後育，故品物用成。

Heaven relies on yang energy to rotate, while Earth relies on yin energy to become dense and solid. Therefore, Heaven reaches the extreme of motion, receiving the original energy and spreading out its brightness; Earth reaches the extreme of stillness, receiving the gifts from Heaven and waiting for illumination. Heaven moves in accordance with the seasons without losing its center and uprightness, so the four seasons come in order; cold and heat do not diminish, and growth and nourishment are measured, thus allowing all things to flourish. Earth, with its emptiness, clarity, and stillness, matches and supports Heaven, using the harmonious transformation of clear energy to nurture all things. After the four seasons have passed and all things have matured, thus all things are completed.

凡至大者莫如天，至厚者莫若地，至質者曰地而已。至多莫若水，水精為漢，漢周於天而無列焉，思次質也。地有山嶽，以宣其氣，精鍾為星。星也者，體生於地，精成於天，列居錯峙，各有由屬。

The greatest thing in the world is heaven, the thickest is earth, and the most substantial is also earth. Water is the most abundant, and the essence of water rises to form the "Heavenly River" (Milky Way); the Heavenly River surrounds the entire sky but is not included among the star mansions, because its substance is of a lower quality. On earth there are mountains, which release the earth's energy. The essence of the earth's energy condenses into stars. Stars take shape on earth and their refined energy coalesces in heaven. They are scattered and arranged in opposition, each with its own master.

紫宮為皇極之居，太微為五帝之延。明堂之房，大角有席，天市有座。蒼龍連踡於左，白虎猛據於右，朱雀奮翼於前，靈龜圈首於後，黃神軒轅於中。六擾既畜，而狼蚖魚鱉，罔有不具。在野象物，在朝象官，在人象事，於是備矣。

The Purple Micro Palace is the dwelling place of the Heavenly Emperor. The Taiwei Enclosure is the court where the Five Imperial Emperors receive their ministers. In the heavens, there is a 'Bright Hall' building, with a seat set aside for the 'Dajiao' star, and a place designated for the 'Tianshi' Enclosure. To the east, the Azure Dragon coils and winds on the left, while to the west, the White Tiger stands majestically on the right. To the south, the Vermilion Bird soars ahead, and to the north, the spiritual turtle looks back. In the center is the divine form of the Yellow Emperor, Xuanyuan. The six domesticated animals, horse, ox, sheep, pig, dog, and chicken, are all present. Creatures such as wolf, lizard, fish, and turtle are also found. The celestial phenomena correspond to various creatures in the wild, various official positions in the court, and various affairs on earth. Here, everything is complete.

懸象著明，莫大乎日月。其徑當天周七百三十六分之一，地廣二百四十三分之一。日者，陽精之宗，積而成鳥，象烏，有三趾，陽之類，其數奇。月者，陰精之宗，積而成獸，象兔，陰之類，其數偶。

The celestial phenomena suspended in the sky, whose light is clearly visible, none are greater than the sun and the moon. The diameter of the sun and moon is equivalent to one seven hundred thirty-sixth part of the celestial sphere, and one two hundred forty-third part of the earth's breadth. The sun is the master of yang essence; accumulated yang essence transforms into the form of a bird, symbolized by a crow, which has three feet, belonging to the category of yang, hence taking an odd number. The moon is the master of yin essence; accumulated yin essence transforms into the form of an animal, symbolized by a jade rabbit, belonging to the category of yin, hence taking an even number.

其後有馮焉者，羿請無死之藥於西王母，姮娥以之奔月。將往，枚筮之於有黃，有黃占之曰：‘吉，翩翩歸妹，獨將西行，逢天晦芒，毋驚毋恐，後且大昌。’姮娥遂託身於月，是為蟾蜍。

Later there was also a legend involving the moon: Hou Yi obtained the elixir of immortality from Xi Wang Mu, and Heng娥 (who is also known as Chang'e) took it and flew to the moon palace. Before she left, she asked a shaman named You Huang to use yarrow stalks for divination. You Huang's interpretation was: "It is auspicious. A graceful woman flying away, traveling alone to the west, encountering dark skies with no light on the way, do not be alarmed or frightened, for afterwards there will be great prosperity." Heng娥 then entrusted her body to the moon and transformed into a toad.

夫日譬猶火，月譬猶水，火則外光，水則含景。故月光生於日之所照，魄生於日之所蔽，當日則光盈，就日則光盡也。眾星被耀，因水轉光。當日之沖，光常不合者，蔽於地也。是謂暗虛。在星星微，月遇則食。

The sun is like fire, and the moon is like water; fire emits light outward, while water holds images within. Therefore, the brightness of the moon arises from the part illuminated by the sun, and the dark part of the moon comes from where the sun is obscured. When the moon is directly opposite the sun, its light is full; when it is close to the sun, its light fades. All the stars receive the illumination of the sun and reflect light like water. The area directly facing the sun often does not receive light because it is blocked by the earth, which is called "dark void." When the star mansions enter the dark void, their light becomes dim; when the moon enters the dark void, an eclipse occurs.

日之薄地，暗其明也。繇暗視明，明無所屈，是以望之若大。方於中天，天地同明。繇明視暗，暗還自奪，故望之若小。火當夜而揚光，在晝則不明也。月之於夜，與日同而差微。星則不然，強弱之差也。

When the Sun is near the horizon, the surroundings are dark while the Sun itself is bright. Looking at a bright object from darkness, the brightness is not suppressed, so it appears larger. When the Sun rises to the middle of the sky, the brightness above and below is the same. Looking at a dim object from brightness, the dimness is overshadowed by the brightness, so it appears smaller. Fire also behaves similarly; it shines brightly at night but is not visible in daylight. The Moon behaves in the same way as the Sun, but the difference is smaller. Stars are different; their size and brightness are only differences in their own light.

眾星列布，其以神著，有五列焉，是謂三十五星。一居中央，謂之北斗。動變定占，實司王命。四佈於方，為二十八宿。日月運行，歷示吉凶，五緯更次，用告禍福，則天心於是見矣。

The stars are arranged in a scattered manner, appearing remarkable due to their divine nature, forming a total of five rows, known as the "thirty-five stars." Among them, one row is located in the center, called the Big Dipper; the movements and changes of the Big Dipper determine the divination, actually governing the fate of the emperor. The other four rows are distributed in the four cardinal directions—east, south, west, and north—known as the Twenty-Eight Mansions. The sun and moon move through these mansions in sequence, displaying auspicious or inauspicious signs; the five elements—gold, wood, water, fire, and earth—successively pass through each mansion, predicting misfortune or fortune. Thus, the will of heaven is revealed.

中外之官，常明者百有二十四，可名者三百二十，為星二千五百，而海人之占未存焉。微星之數，蓋萬一千五百二十。庶物蠢動，咸得繫命。不然何以總而理諸。

The Central Palace and the Outer Palace contain a total of one hundred and twenty-four stars that are always bright, with three hundred and twenty of them having names, making a total of two thousand five hundred stars. The southern celestial phenomena used by sailors are not included in this count. The number of faint and dim stars is approximately eleven thousand five hundred and twenty. All things under heaven are constantly moving, and each has its destiny recorded among the stars; otherwise, how could they be governed and managed?

夫三光同形，有似珠玉，神守精存，麗其職而宣其明。及其衰，神歇精斁，於是乎有隕星。然則奔星之所墮，至則石矣。

The sun, moon, and stars are three luminous bodies with the same form and quality, resembling pearls and jade. They are guarded by deities and retain their vital energy, each shining brightly in their respective positions. When they decline, the deities cease their protection and their vital energy is destroyed, leading to falling stars. Therefore, where a meteor falls, it becomes a stone upon reaching the earth.

文耀麗乎天，其動者七，日、月、五星是也。周旋右廻。天道者，貴順也。近天則遲，遠天則速，行則屈，屈則留廻，留廻則逆，逆則遲，迫於天也。行遲者觀於東，觀於東屬陽；行速者觀於西，觀於西屬陰，日與月以配合也。攝提、熒惑、地候見晨，附於日也。太白、辰星見昏，附於月也。二陰三陽、參天兩地，故男女取焉。

The brilliant light shines in the sky, with seven moving bodies: the sun, moon, and the five planets of metal, wood, water, fire, and earth. They revolve around the celestial sphere in a clockwise direction, as the heavenly way values smooth movement. Those closer to the celestial sphere move more slowly, while those farther away move faster. In their movement, there are bends, and bends lead to halts and retrograde motion, which in turn cause slower movement. All these are due to the compulsion of the celestial movements. The slow-moving ones are observed in the east, and observing in the east is yang; the fast-moving ones are observed in the west, and observing in the west is yin. This corresponds exactly with the sun and moon. The star of the year (Jupiter), the malefic of violence (Mars), and the earth monitor (Saturn) appear in the morning, following the sun; the white star (Venus) and the morning star (Mercury) appear in the evening, following the moon. Two are yin and three are yang, conforming to the numerical principle of 'parting heaven and dividing earth,' thus the distinction between male and female is modeled after this.

方星巡鎮，必因常度，苟或盈縮，不逾於次。故有列司作使，曰老子四星，周伯、王逢、絮芮各一，錯乎五緯之間，其見無期，其行無度，實妖經星之所，然後吉凶宣周，共詳可盡。”

The four cardinal stars patrol and guard their respective directions, necessarily following fixed degrees. Even if they sometimes advance or lag, they will not exceed their original positions. In addition, there are stars that serve as officials or envoys, called the "Four Old Fathers" stars. There are also individual stars named "Zhou Bo," "Wang Feng," and "Xu Rui," which are scattered among the five visible planets. These stars do not appear at fixed times nor follow regular paths, and they are indeed anomalous stars that disturb the region of the fixed stars. With their presence, auspicious and inauspicious omens are fully manifested, and the details of celestial phenomena can be thoroughly understood.

張衡《渾儀註》曰：“渾天如雞子，天體圓如彈丸，地如雞子中黃，孤居於內。天大而地小，天表裡有水。天之包地，猶殼之裹黃。天地各乘氣而立，載水而浮。

Zhang Heng's "Commentary on the浑仪" says: The shape of the浑天 is like an egg. The celestial sphere is round, like a ball, and the earth is like the yolk in the egg, alone in the center. The sky is vast and the earth is small. Both the outer and inner surfaces of the sky have water. The sky surrounds the earth, just as the eggshell surrounds the yolk. Both heaven and earth rest on qi and stand firm, floating on water.

周天三百六十五度四分度之一；又中分之則一百八十二度八分之五覆地上，一百八十二度八分之五繞地下。故二十八宿，半見半隱，其兩端謂之南北極。

The celestial sphere has a total of 365 and 1/4 degrees. When divided equally, 182 and 5/8 degrees are above the earth's surface, and 182 and 5/8 degrees are below the earth's surface. Therefore, the twenty-eight constellations are always half visible and half hidden. The two endpoints around which the celestial sphere rotates are called the South Pole and the North Pole.

北極，乃天之中也。在正北，出地上三十六度，然則北極上規，徑七十二度，常見不隱。南極，天之中也。在南，入地三十六度，南極下規七十二度，常伏不見。兩極相去一百八十二度半強。

The North Pole is the central point of the sky, located due north, and is thirty-six degrees above the horizon. Drawing a circle around the North Pole as the center, the diameter of the "upper circle" is seventy-two degrees, and the stars within this circle are always visible and never set. The South Pole is also the central point of the sky, located due south, and is thirty-six degrees below the horizon. Drawing a circle around the South Pole as the center, the diameter of the "lower circle" is also seventy-two degrees, and the stars within this circle are always hidden and never visible. The distance between the North and South Poles is a little more than one hundred and eighty-two degrees.

天轉如車轂之運也，周旋無端，其形渾渾，故曰渾天也。赤道橫帶，天之腹；去南北二極，各九十一度十九分度之五。（橫帶者，東西圍天之中要也。然則北極小規去赤道五十五度半，南極小規亦去赤道五十五度半，並出地、入地之數，是故各九十一度半強也。）

The rotation of Heaven is like the hub of a wheel turning continuously, cycling without beginning or end, forming a single, undivided body, hence it is called "Holistic Heaven." The equator is like a belt wrapped around the middle of Heaven, at a distance of ninety-one and five degrees and nineteen parts from the north and south poles. (Original note: This belt runs east-west around the center of the celestial sphere. Thus, the small circle near the north pole is fifty-five and a half degrees from the equator, and the same applies to the small circle near the south pole. Adding the thirty-six degrees by which the north pole is above the horizon and the south pole is below, each side totals ninety-one and a half degrees or so.)

黃道斜帶，其腹出赤道表裡各二十四度。（日之所行也，日與五星行黃道，無虧盈。月行九道：春行東方青道二，夏行南方赤道二，秋行西方白道二，冬行北方黑道二，四季還行黃道，故月行有虧盈。東西南北隨八節也。

The Yellow Path is slanted around the celestial abdomen, extending 24 degrees outside and inside the equator. (Original book note: The Yellow Path is the path that the Sun travels along. The Sun and the five planets all move along the Yellow Path without deviating, experiencing neither excess nor deficiency. The Moon, however, travels along nine paths: in spring it follows two paths of the eastern green path, in summer two paths of the southern red path, in autumn two paths of the western white path, and in winter two paths of the northern black path. At the end of each season, it returns to the Yellow Path, hence the Moon's waxing and waning. The four groups of eastern, southern, western, and northern paths change according to the eight solar terms: the Beginning of Spring, the Vernal Equinox, the Summer Solstice, and the Autumnal Equinox.)

日最短，經黃道南，在赤道外二十四度，是其表也。日最長，經黃道北，去赤道內二十四度，是其裡，故夏至去極六十七度而強，冬至去極百一十五度亦強。

At the time when day is shortest, the Sun passes through the southernmost part of the Yellow Path, twenty-four degrees south of the equator, which is called "biao"; at the time when day is longest, the Sun passes through the northernmost part of the Yellow Path, twenty-four degrees north of the equator, which is called "li". Therefore, on the summer solstice, the Sun is more than sixty-seven degrees south of the North Pole, and on the winter solstice, the Sun is more than one hundred fifteen degrees south of the North Pole.

日行而至斗二十一度，則去極一百一十五度少強，是故日最短，夜最長，景極長，日出辰、入申，晝行地上一百四十六度強，夜行地下二百一十九度少強。夏至日在井二十五度，去極六十七度少強。是故日最長，夜最短，景極短，日出寅，日入戌，晝行地上二百一十九度少強，夜行地下一百四十六度強。）

When the Sun reaches the 21st degree of the Southern Dipper mansion, it is about 115 degrees away from the North Pole, so at this time the day is shortest, the night is longest, and the shadow is longest. The Sun rises from the Chen position (southeast偏east) and sets at the Shen position (southwest偏west). During the day, it travels above the ground for about 146 degrees, and during the night, it travels below the ground for about 219 degrees. On the summer solstice, the Sun is at the 25th degree of the Well mansion, about 67 degrees away from the North Pole, so at this time the day is longest, the night is shortest, and the shadow is shortest. The Sun rises from the Yin position (northeast偏east) and sets at the Xu position (northwest偏west). During the day, it travels above the ground for about 219 degrees, and during the night, it travels below the ground for about 146 degrees.

然則黃道斜截赤道者，即春、秋分之去極也。（斜截赤道者，東西交也。然則春分日在奎十四度少強，西交於奎也。秋分日在角五度弱，東交於角也。在黃赤二道之交中，去極俱九十一度少強，故景居二至長短之中，奎十四、角五，出卯入酉，日晝行地上，夜行地下，俱一百八十二度半強，故晝夜同也。）

This suggests that the two points where the Yellow Path intersects the equator are precisely where the Sun is located at the time of the vernal and autumnal equinoxes, in terms of its distance from the North Pole. (Original book note: When the Yellow Path intersects the equator obliquely, it means the Yellow and ecliptic paths intersect at the east and west sides. At the vernal equinox, the Sun is slightly more than 14 degrees in the constellation Qiu Xu, which is where it intersects the west side of Qiu Xu; at the autumnal equinox, the Sun is slightly less than 5 degrees in the constellation Jiao Xu, which is where it intersects the east side of Jiao Xu. The Sun is located at the intersection points of the Yellow and ecliptic paths, and its distance from the North Pole is slightly more than 91 degrees, so the length of the shadow is exactly between the extremes of the winter and summer solstices. At these two points, 14 degrees in Qiu Xu and 5 degrees in Jiao Xu, the Sun rises exactly in the east at the卯 position and sets exactly in the west at the酉 position. During the day, it is above the ground, and during the night, it is below the ground, moving slightly more than 182 and a half degrees, so day and night are equal.)

張衡《渾儀圖注》曰：“今此春分去極九十一度少強，秋分去極九十一度少強者，就夏曆景去極之法以為率也。是以作小渾，盡赤道、黃道，乃各調賦三百六十五度四分之一，從冬至所在始起，令之相當值也。

Zhang Heng's "Commentary on the Diagram of the Armillary Sphere" says: Now the spring equinox is said to be ninety-one degrees more than the north pole, and the autumn equinox is also ninety-one degrees more than the north pole. This ratio is determined by using the length of shadows according to the summer calendar. Therefore, a small armillary sphere is made, with both the celestial equator and the ecliptic fully drawn. Then, the 365 and 1/4 degrees are evenly divided on both the celestial equator and the ecliptic. Starting from the position of the winter solstice, the degrees on the ecliptic and celestial equator are made to correspond with each other.

取北極及衡，各針穿之為軸，取薄竹篾穿其兩端，令兩穿中間與渾半等以貫之，令察之與渾相切摩，乃從減半起，以為八十二度八分之五，盡衡減之半焉。

At the north and south poles, two needles are used to pierce through as the axis. Then a thin bamboo strip is taken, with holes drilled at both ends so that the distance between the two holes is exactly half the circumference of the armillary sphere. The bamboo strip is then fitted over the two axes, allowing it to slide along the surface of the armillary sphere. Starting from the midpoint, it is set to measure eighty-two and five-eighths degrees (the original text seems to be missing the word 'one hundred', so it should be one hundred eighty-two and five-eighths degrees), and this measurement is continued until the midpoint at the end of the southern axis.

又中分其篾，拗去其半，令其半之際，正直與兩端減半相直，令篾半之際，從冬至起一度一移之，視篾之半際多少，黃赤道幾也。其所多少，則進退之數也。從北極數之，則去極之度也。

Now take the bamboo strip and split it exactly in the middle, then remove half of it so that the broken half aligns straight with the reduced ends. Let the midpoint of the bamboo strip start from the winter solstice and move degree by degree. By observing the degree indicated by the midpoint, one can determine the difference between the Yellow Path and the Ecliptic at that point. The amount of difference indicates the number of degrees by which the Yellow Path advances or retreats. Counting from the North Pole, this also shows the degree of distance from the pole at that point.

各分赤道、黃道為二十四氣，一氣相去十五度十六分之七，每一氣者，黃道進退一度焉。所以然者，黃道直時去南北極近，其處地少而橫行，與赤道且等，故以篾度之，於赤道多也。

The equator and the Yellow Path are each divided into twenty-four segments called solar terms. The distance between each solar term is seventeen and seven sixteenths degrees. As each solar term passes, the Yellow Path advances or retreats by one degree. This is because when the Yellow Path is near the north and south poles, the portion it covers is small and almost horizontal, nearly parallel to the equator. Therefore, when measured with bamboo strips, the degrees measured on the equator appear greater.

設一氣令十六日皆常率，四日差少半也。令一氣十五日，不能半耳。故使中道三日之中差少半，三氣一節，故四十六日而差今三度也。至乎差三之時而五日同率者一，其實一節之間不能四十六日也。今殘日居其策，故五日同率也。其率雖同，先之皆強，後之皆弱，不可勝記耳。

If each solar term is considered as sixteen days, and calculated at the same rate, then four days would differ by less than half a degree; if each solar term is considered as fifteen days, the difference would still be less than half a degree. Therefore, within the middle three days, the difference would be less than half a degree. When three solar terms are combined into one, forty-six days would result in a difference of three degrees. When the difference reaches three degrees, there will be a situation where five consecutive days share the same rate. In reality, one solar term does not reach forty-six days, but the remaining days are included in the calculation, leading to the phenomenon of five days sharing the same rate. Although these rates are the same, the earlier ones are slightly stronger, and the later ones are slightly weaker. The minor discrepancies are numerous and cannot all be recorded.

至於三而復有進退者，黃道稍斜於橫行，不得度故也。春分、秋分所以退者，黃道始起更斜矣，於橫行不得度故也。亦每氣一度焉。故三氣一節亦差三度也。至三氣之後，稍遠而直，故橫行得度而稍進也。

After the difference reaches three degrees, there is both advancement and retreat, because the Yellow Path gradually tilts away from the horizontal direction, and the measured amount is insufficient by the proper degree. At the Spring Equinox and Autumn Equinox, the retreat occurs because the Yellow Path begins to tilt more sharply from here, so measuring horizontally results in a deficiency of the degree, and each solar term differs by one degree. Therefore, three solar terms together result in a difference of three degrees. After three solar terms, the Yellow Path gradually moves away from the pole and becomes more level, so measuring horizontally allows for the proper degree to be reached, and thus it gradually advances again.

立春、立秋橫行稍退矣，而度猶雲進者，以其所退減其所進，猶有盈餘未盡故也。立夏、立冬橫行稍進矣，而度猶雲退者，以其所進增其所退，猶有不足未畢故也。以斯言之，日行非有進退也，而以赤道量度黃道使之然也。本二十八宿相去度數，以赤道為強耳，故黃道亦有進退也。

At the time of the beginning of spring and autumn, the degrees along the ecliptic are gradually retreating, yet they are still said to be advancing. This is because the retreat is being used to offset the previous advance, and there is still a surplus that has not been fully accounted for. Similarly, at the beginning of summer and winter, the degrees along the ecliptic are gradually advancing, yet they are still said to be retreating. This is because the advance is being used to compensate for the previous retreat, and there is still a deficit that has not been fully made up. According to this reasoning, the movement of the sun itself does not have any real advance or retreat. It is only when measuring the ecliptic with the equatorial scale that the appearance of advance and retreat becomes evident. Originally, the degrees between the twenty-eight constellations were determined based on the equator. Since the equator measures more strongly, the ecliptic also appears to have advance and retreat as a result.

冬至在斗二十一度少半，最遠時也，而此歷斗二十度，俱一百一十五度強矣，冬至宜與之同率焉；夏至在井二十一度半強，最近時也。而此歷井二十三度，俱六十七度強矣，夏至宜與之同率焉。”

At the winter solstice, the sun is at 21.5 degrees of the Dipper mansion, which is the time when it is farthest from the north pole; however, this calendar fixes the sun at 20 degrees of the Dipper mansion, and the distance from the pole is over 115 degrees in both cases. Therefore, the winter solstice should be calculated according to this proportion. At the summer solstice, the sun is at 21.5 degrees more of the Well mansion, which is the time when it is closest to the north pole; however, this calendar fixes the sun at 23 degrees of the Well mansion, and the distance from the pole is over 67 degrees in both cases. Therefore, the summer solstice should also be calculated according to this proportion.

漢靈帝時議郎蔡伯喈於朔方上書曰：“論天體者有三家：《宣夜》之學，絕無師法。《周髀》術數具存，考驗天狀，多所違失。唯渾天近得其情。今史官所用候臺銅儀，則其法也。立八尺圓體而具天地之形，以正黃道，占察發斂，以行日月，以步五緯，精微深妙，百世不易之道也。”

During the reign of Emperor Ling of Han, the official Cai Yong (also known as Bojie), who held the position of Yilang, submitted a memorial while in exile in Shuofang. He stated that there were three schools of thought regarding the celestial sphere. The Xuanye school had no established tradition or method to rely upon. The Zhou Bi school, although possessing numerical methods for calculation, often failed when tested against actual celestial phenomena. Only the Hun Tian theory was closer to the truth. The bronze armillary sphere currently used by the historians at the observatory is based on the Hun Tian theory. It features a circular frame eight feet in height, incorporating the forms of both heaven and earth. It is used to correct the Yellow Path, observe the north-south movement of the Sun, measure the degrees of the Sun and Moon, and calculate the positions of the five planets. Its subtle and profound principles are beyond change for generations.

蔡氏《月令章句》曰：“天者純陽精剛，轉運無窮。其體深而包地，地上者一百八十二度八分之五，地下亦如之。其上中北偏出地三十六度，謂之北極，極星是也。史官以玉衡，長八尺，孔徑一寸，從下端望之，此星常見於孔端，無有移動，是以知其為天中也。

Cai Yong's "Chapter on the Monthly Ordinances" says: Heaven is a refined and strong body formed by the pure yang energy, and it rotates endlessly. Its form is deep and vast, encompassing the earth: the part above the ground measures one hundred eighty-two and five-eighths degrees, and the part below the ground is the same. The center of heaven is slightly to the north of the central position, thirty-six degrees above the ground, which is called the "North Pole." The North Star is located there. The historians used the Yu Heng to measure it. The Yu Heng is eight feet long and has a one-inch diameter. Looking through the lower end, this star always appears at the other end of the hole without any movement, so they concluded that it is the central axis of heaven.

其下中南偏，入地亦三十六度，謂之南極，從上端望之，當孔下端是也。此兩中者，天之輻軸所在，轉運所由也。

Heaven's center lies slightly south of the middle, and it sinks into the earth at a depth of thirty-six degrees, which is called the "South Pole." Looking from the top of the Yu Heng, the place directly opposite the bottom of the hole is where it is located. These two central points, north and south, are the hub and axle of the great celestial wheel, and the entire rotation of heaven is generated through them.

天左旋出地上而西，入地下而東，其繞北極徑七十二度，常見不伏，圖內赤小規是也。繞南極徑七十二度，常伏不見，圖外赤大規是也。據天地之中，而察東西，則天半見半不見，圖中赤規截婁、角者是也。”

The sky rotates to the left, moving westward when it appears above the ground and eastward when it disappears beneath the ground. The circle around the North Pole, with a diameter of seventy-two degrees, contains stars that are always visible and never hidden, which is represented by the red small circle in the star chart. The circle around the South Pole, also with a diameter of seventy-two degrees, contains stars that are always hidden and never visible, which is represented by the red large circle at the outer edge of the star chart. Standing at the center of the heavens and observing east and west, the sky is half visible and half invisible, which is represented by the red circle crossing theBond mansion andHorn mansion in the chart.

後漢末吳人陸公紀《渾天》曰：“先生之道，存乎治曆明時，本之驗著，在於天儀。夫法象莫若渾天，渾天之設久矣。昔在顓頊，使南正重司天。而帝嚳亦序三辰，堯命羲和，欽若昊天，曆象日、月、星辰。舜之受禪，在璇璣玉衡，以齊七政。以是數者言之，曩時已立渾天之象明矣。

In the late Eastern Han dynasty, the Wu native Lu Ji (style name Gongji) wrote in his book "Holysphere": The way of the sage ancestors is realized in establishing the calendar and clarifying the seasons; however, its fundamental verification and manifestation lie in the instruments used to observe the heavens. To model the image of heaven, nothing is more appropriate than the holysphere. The establishment of the holysphere has a long history. In the time of Zhuanxu, the southern official Chong was entrusted with matters of the heavens; Emperor Ku arranged the order of the three celestial bodies, the sun, moon, and stars; Yao ordered Xi Shi and He Shi to respectfully follow heaven's course, observing and calculating the movement of the sun, moon, and stars; after Shun accepted the mandate, he used the "Xuanji Yuheng" instrument to correct the paths of the sun, moon, five planets, and the seven major celestial bodies. From these instances, it is clear that the holysphere model was already established long ago.

周公序次六十四卦，兩者相承，反覆成象，以法天行，周而復始，晝夜之義，故《晉卦象》曰：“晝日三接。”《明夷象》曰：“初登于天，後入于地。”仲尼說之，曰：“明出地上，晉進而麗乎大明，是以晝日三接，明入地中。”明夷夜也，先晝後夜，先晉後明夷，故曰初登于天，照四國也。後入于地，失則也。日月麗乎天，隨天轉運，出入乎地，以成晝夜也。渾天之義，蓋與此同。

The Duke of Zhou arranged the order of the sixty-four trigrams, with adjacent trigrams connected in sequence, forming the trigram images whether viewed normally or reversed. This was done to follow the movement of Heaven, which cycles continuously, and it contains the principle of day and night. Therefore, the image text of the "Progress" trigram says, "Three times visiting the Son of Heaven during the day," and the image text of the "Darkness over Light" trigram says, "At first rising up to the sky, then sinking into the earth." Confucius explained, "Light comes from the earth, which is the Progress trigram—advancing and attaching to the great brightness, hence three times visiting during the day; light sinking into the earth is the Darkness over Light trigram." Darkness over Light refers to night. Since day comes first and night follows, the Progress trigram is arranged first, followed by the Darkness over Light trigram. Therefore, "at first rising up to the sky" refers to shining over all the surrounding states, while "later sinking into the earth" means losing the established order. The sun and moon attach to the sky, rotating with Heaven, appearing and disappearing on the earth, thus forming day and night. The principles of the theory of the spherical heavens are roughly similar to this.

仲尼歿，大道乖，諸子穿鑿妄作，乃有蓋天之說。其以虛偽，較然可知，又曰：“渾天以日出地上則晝，故《易》曰：‘明出地上，晝日三接。’又曰：晉，晝也。日入於地則夜，夜則明傷。故《易》曰：‘明夷，傷也。”又曰：“‘初登于天，照四國也。後入于地，失則也。’《尚書》：‘寅賓出日，寅餞納日。’以此言之，知出入於地，審矣。若日不出於地，則何緣得有晝、夜、明、暗乎。

After Confucius' death, the great principles became distorted, and the various schools of thought fabricated and invented new theories, leading to the theory of "Covered Heaven." Its falsehood is clearly evident. Lu Ji said: The theory of the spherical heaven holds that when the Sun rises above the earth, it is day, so the Zhou Changes say, "Light comes from above, and in the daytime, there are three times of appearance," and also say, "Jin means daylight." When the Sun sinks below the earth, it is night, and the light is diminished, so the Zhou Changes say, "Mingyi means injury." He also said, "At first, when the Sun rises into the sky, it illuminates all the four directions; later, when it sinks into the earth, it loses its regular course." The Book of Documents, Yao's Classic, says, "With reverence, welcome the sunrise, and with reverence, send off the sunset." Based on this, we can know that the Sun indeed rises and sets on the earth, which is clearly evident. If the Sun does not rise from beneath the earth, then on what basis can there be distinctions between day and night, and between light and darkness?

天半覆地上，半周地下，繞地而運。故二十八宿半見半隱，如天半右覆地上，半周地下，二十八宿何故再見更隱乎。由此言之，天乃裹地而運，信矣。此是昏明之大術也。

Heaven occupies half above the earth and half around below the earth, circling the earth, so the twenty-eight constellations are always half visible and half hidden. If we follow the theory of the Covering Heaven, which says that Heaven is slanting, half covering the earth and half going around beneath it, how can the twenty-eight constellations appear and disappear repeatedly? Therefore, the idea that Heaven revolves around the earth is indeed credible. This is the principle behind the alternation of day and night.

天之形狀，圓周渾然，運於無窮，故曰渾。《易》曰：‘乾為天，為圜。’又曰：‘先甲三日，後甲三日，終則有始。’天行也，此之謂矣。天大地小，天統地；半覆地，上半周地下，譬如卵白，白繞黃也。”

The shape of Heaven is round and complete, continuously rotating without end, so it is called "hun." The "Zhou Yi · Shuo Gua" says: "Qian represents Heaven and roundness." The "Gu Gua" also says: "Three days before the Jia day and three days after the Jia day, it concludes and begins again." This precisely describes the movement of Heaven, explaining this principle. Heaven is vast, while Earth is small; Heaven governs Earth. Half of Heaven covers the Earth, and the other half surrounds the Earth beneath, just like the egg white surrounding the yolk.

揚子雲《太玄經》曰：“天穹隆而周乎下，地旁薄而向乎上，故知天裹地下。周天三百六十五度四分度之一，一百八十二度八分度之五覆地上，一百八十二度八分度之五周地下，故二十八宿半見半隱，日月出入以成昏明也。

In the "Classic of Supreme Mystery," Yang Xiong, who was known as Zi Yun, said: Heaven is lofty and arches downward, while Earth is vast and flat, extending upward to receive it. Therefore, it is known that Heaven encompasses the area beneath Earth as well. The full circle of the heavens is 365 and 1/4 degrees. Of this, 182 and 5/8 degrees cover the land, and 182 and 5/8 degrees encircle the land beneath. Thus, half of the twenty-eight mansions are visible and half are hidden. The sun and moon rise and set among them, thereby creating the periods of dawn, daylight, dusk, and night.

北極星北方偏出於地三十六度，南極中偏入於地亦三十六度，南極、北極天軸所在，轉運所由，譬車之有輪，所以自行也。眾星皆移無常，惟北極守中不易，是以知其為天中也。天傾，故極在中北。仲尼曰：‘譬如北辰，居其所而眾星共之。’”

The North Star is located in the north, deviating 36 degrees from the ground, while the South Star is in the center, deviating 36 degrees southward into the ground. The North and South Stars are the locations of the celestial axis, from which the rotation of the sky originates. Just as a cart has a wheel axle that allows it to move on its own, the sky rotates due to this axis. All the stars are in motion and their positions are ever-changing, but the North Star remains fixed in the central position, never moving. Therefore, it is known to be the center of the sky. The sky is tilted, so the celestial pole is located slightly north of the central position. Confucius said, "Just as the North Star remains in its place, while all the other stars revolve around it."

又《太玄經》曰：“天圓地方，極樞中央，動以歷靜，時乘十二，以建七政。’是以《堯典》曰：‘在璇璣玉衡，以齊七政。’此之謂也。繞南極七十度，常在地下不出地。周天一百七萬一千里，東西南北徑三十五萬七千里，立徑亦然。”自外諸說，度次交會，與蔡氏、張衡同，故略雲。

The Classic of Supreme Mystery also says: Heaven is round and Earth is square, with the pole at the center. It moves through stillness, rotating in sequence according to the twelve earthly branches, thus establishing the paths of the sun, moon, five planets, and the seven luminaries. Therefore, the Classic of Yao in the Book of Documents says, "Using the璇玑 and玉衡 to regulate the seven luminaries," which refers to this very concept. The area around seventy degrees south of the south celestial pole remains beneath the earth and does not appear above the surface. The circumference of the celestial sphere is one million and seventy thousand li, with the diameter from east to west and from north to south being thirty-five thousand seven hundred li. The diameter from above to below is the same. There are also other schools of thought discussing the order of degrees and the intersections of the ecliptic and the equator, which are all consistent with what Cai Yong and Zhang Heng have said, so they are omitted here.

吳將廬江王蕃《渾天象說》曰：‘幽平之後，周室遂卑，天子不能頒朔，魯曆不正，百有餘年；以建申之月為建亥，而怪蟄蟲不伏也。曆紀廢壞，道術浸亂，渾天之義，傳之者寡，末世之儒，或不聞見，各以私意，為天作說；故有《周髀》、《宣夜》之論。《宣夜》絕無師法。《周髀》見行於世，考驗天伏，多所違失。

In the book "Hun Tian Xiang Shuo" (Treatise on the浑 Heaven Model), Wang Fen, a general of the Wu state during the Three Kingdoms period and a native of Lujiang, said: After King You of Zhou and King Ping of Zhou, the royal house of Zhou gradually declined, and the Son of Heaven could no longer issue the calendar and the beginning of the month to the feudal lords. The calendar of Lu state became disorderly and incorrect, lasting for more than a hundred years. They even took the month of Jiashen (the seventh month of the Xia calendar) as the month of Jiahai (the tenth month of the Xia calendar), and were puzzled why insects did not retreat underground to hide. The principles of the calendar were abandoned, and the study of astronomy gradually became chaotic. The meaning of the theory of the浑 heaven was passed down by few people. Some late scholars had never even heard of it, and thus created their own theories to explain the heavens, leading to the emergence of two schools of thought: the "Zhou Bi" and the "Xuanye" theories. The Xuanye theory had no teacher's transmission or established rules. The theory of Zhou Bi was popular among people, but when tested against the appearances and disappearances of celestial phenomena, it often showed discrepancies.

依劉洪《乾象曆》之法而論渾天曰：‘前儒舊說，天地之體，狀如鳥卵，天包地外，猶殼之裹黃也。周旋無端，其形渾渾然，故曰渾天也。周天三百六十五度五百八十九分度之百四十五，東西南北，辰轉周規，半覆地上，半在地下，故二十八宿半見半隱；以赤儀準之，其見者常百八十二度有奇，是以知其半覆地上，半隱地下，其二端謂之南極、北極，天之中也。

Wang Fan discussed the theory of the浑天 using the calculation method from Liu Hong's "Ganxiang Li". He said: previous scholars of the Confucian tradition believed that the shape of heaven and earth was like that of a bird's egg, with heaven surrounding the earth, just as the eggshell surrounds the yolk. It rotates continuously without beginning or end, forming a complete sphere, hence it is called the浑天. The entire circle of the sky is 365 and 145/589 degrees. The directions east, west, south, and north move along the twelve earthly branches in a circular path, with half of it covering the land and the other half hidden beneath the ground. Therefore, the twenty-eight constellations are half visible and half hidden. Using the equatorial armillary sphere for verification, the visible portion is usually about 182 degrees more. From this it can be known that half of the sky covers the land and the other half is hidden beneath it. Its two endpoints are called the South Pole and the North Pole, which are the central axis of the sky.

北極在正北，出地三十六度。南極在正南，入地亦三十六度。兩極相去一百八十二度半強。眾星皆移，而北極不徙，猶車輪之有釭軸也。繞北級徑七十二度，常見不隱，謂之上規。繞南極七十二度，常隱不見，謂之下規是也。上規去南極，下規去北極，皆一百四十度半強。以二規於渾儀。

The North Pole is located in the due north, thirty-six degrees above the horizon; the South Pole is in the due south, thirty-six degrees below the horizon. The distance between the two poles is one hundred eighty-two and a half degrees. All the stars are moving, but the North Pole remains fixed, like the axle of a cart wheel. A circle around the North Pole with a diameter of seventy-two degrees contains stars that are always visible and never set, called the "upper circle." A circle around the South Pole with a diameter of seventy-two degrees contains stars that are always hidden and never appear, called the "lower circle." The upper circle is one hundred forty-two and a half degrees away from the South Pole, and the lower circle is the same distance from the North Pole. These two circles must both be marked on the armillary sphere.

為中規赤道帶天之紘，去兩極各九十一度少強。黃道，日所行也。半在赤道外，半在赤道內，與赤道東交於角五弱，西交於奎十四少強。其出赤道外極遠者，去赤道二十四度，斗二十一度是也。其入赤道內極遠者，入赤道二十四度，井二十五度是也。

The central circle is the equator, like a belt called "Capping" around the middle of the sky, located about 91 degrees from each of the north and south poles. The Yellow Path is the path of the Star of brightness and prestige, half of it lying outside the equator and half inside. It intersects the equator slightly before the fifth degree of the Horn mansion in the east and slightly after the fourteenth degree of the Legs mansion in the west. The farthest point of the Yellow Path outside the equator is 24 degrees away from the equator, which corresponds to the twenty-first degree of the Dipper mansion. Similarly, the farthest point of the Yellow Path inside the equator is also 24 degrees within the equator, which corresponds to the twenty-fifth degree of the Well mansion.

日南至斗二十一度，去極百一十五度少強，是日最南，去極最遠，故景最長。黃道斗二十一度，出辰、入申，故日亦出辰、入申，晝行地上百四十六度強，故日短；夜行地下二百一十九度少弱，故夜長。

On the day of the winter solstice, the sun moves southward to the 21st degree of the Dipper mansion, which is 115 degrees plus a little away from the north pole. This is the day when the sun is furthest south and farthest from the north pole, resulting in the longest shadow. At the 21st degree of the Dipper mansion on the Yellow Path, the sun rises at the Dragon position and sets at the Monkey position, so the sun also rises at the Dragon position and sets at the Monkey position. During the day, the sun travels 146 degrees plus a little above the ground, so the daylight is short; at night, the sun travels 219 degrees minus a little below the ground, so the night is long.

自南至之後，日去極稍近，故景稍短；日晝行地上度稍多，故日稍長；夜行地下稍少，故夜稍短；日行度稍北；故日出入稍北；以至於夏至日在井二十五度，去極六十七度少強，是日最近北，去極最近，故景最短。

After the winter solstice, the sun gradually approaches the north pole, so the shadow becomes shorter gradually; the degrees the sun travels above the ground gradually increase, so the day becomes longer gradually; the degrees the sun travels below the ground gradually decrease, so the night becomes shorter gradually; the sun's path gradually leans north, so the direction of sunrise and sunset also gradually leans north. Until the summer solstice, the sun reaches 25 degrees in the Well mansion, slightly more than 67 degrees from the north pole, on this day the sun is closest to the north, so the shadow is the shortest.

黃道井二十五度出寅、入戌，故日亦出寅、入戌，日晝行地上二百一十九度少弱，故日長；夜行地下百四十六度強，故夜短；自夏至之後，日去極稍遠，故景稍長；日晝行地上度稍少，故日稍短；夜行地下度稍多，故夜稍長；日所在度稍南，故日出入稍南；以至於南至，而復初焉。斗二十一度，井二十五度，南北相應四十八度。

The Yellow Path passes through the Well mansion covering 25 degrees. This begins rising at the Yin position and sets at the Xu position, so the Star of brightness and prestige also rises at Yin and sets at Xu. During the day, it travels 219 degrees slightly less above the ground, hence the day is long; at night, it travels 146 degrees more below the ground, hence the night is short. After the summer solstice, the Star of brightness and prestige gradually moves farther from the North Pole, so the shadow becomes longer; the degrees it travels above the ground gradually decrease, so the day gradually shortens; the degrees it travels below the ground gradually increase, so the night gradually lengthens; the position of the Star of brightness and prestige gradually shifts southward, so the direction of sunrise and sunset also gradually shifts southward. Until the winter solstice, it returns to its original state. The Well mansion covers 21 degrees and the Dipper mansion covers 25 degrees, corresponding north and south, differing by 48 degrees.

春分日在奎十四少強，秋分日在角五少弱；此黃赤二道之中交也。去極俱九十一度少強，南北處斗二十一、井二十五之中，故景居二至長短之中，奎十四、角五，出卯、入酉，故日亦出卯、入酉，日晝行地上，夜行地下，俱百八十二度半強，故日見之漏五十刻，不見之漏五十刻，謂之晝夜同。

At the time of the vernal equinox, the sun is located at the 14th degree of the Qiu mansion with a little more, and at the autumnal equinox, it is at the 5th degree of the Jiao mansion with a little less. These two points are exactly the midpoint where the ecliptic and the celestial equator intersect. Both points are about 91 degrees away from the north celestial pole, and their north-south positions are precisely in the middle between the 21st degree of the Dou mansion and the 25th degree of the Jing mansion. Therefore, the length of the shadow is between the extremes of the winter solstice and the summer solstice. The points at the 14th degree of the Qiu mansion and the 5th degree of the Jiao mansion rise at the Mao position and set at the You position, so the sun also rises at the due east (Mao) position and sets at the due west (You) position. During the day, it travels above the earth, and at night, it travels below the earth, covering a little more than 182 and a half degrees. Therefore, the visible hours of the sundial are fifty parts, and the invisible hours are also fifty parts, which is called equal day and night.

夫天之晝夜，以日出入為分；人之晝夜，以昏明為限。日未出二刻半而明，日入後二刻半而昏。故損夜五刻，以增晝刻。是以春秋分之漏，晝五十五刻。

The heavenly division of day and night is based on the rising and setting of the sun; the human division is based on the brightness of the sky and darkness. Before the sun rises, two and a half measures of time have already passed with the sky beginning to brighten, and after the sun sets, two and a half measures of time pass before the sky becomes dark. Therefore, five measures should be subtracted from the night time and added to the daytime. Thus, during the vernal equinox and autumnal equinox, according to the water clock, the daytime is fifty-five measures.