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

渾天遭周秦之亂，師傳斷絕，而喪其文，唯渾儀尚在，臺是以不廢，故其詳可得言。至於纖微委屈，闕而不傳，周天裡數，無聞焉爾。而《洛書軌曜度》、《春秋考異郵》皆云：周天一百七萬一千里。至以日景驗之，違錯甚多，然其流行，布在眾書，通儒達士，未之考正。是以不敢背損舊術，獨據所見。故按其數，更課諸數，以究其意也。

The study of the spherical heavens suffered disruption during the Warring States period and the Qin dynasty, with teachers and disciples unable to pass on their knowledge, and the classics scattered and lost. Only the armillary sphere remained, so the observatory was not abandoned, and thus its general principles could still be explained. However, the subtle and intricate details were lost and missing; the exact number of li in a full circle of the heavens was unknown. The谶纬 texts, the "Luo Writing Illumination Measure" and the "Spring and Autumn Examination of Differences," both say that the heavens measure one million and seven ten-thousand li. When tested with the shadow of the sun, there are many errors and inconsistencies, yet this figure has been widely circulated and appears in many books. Even learned Confucian scholars have not examined or corrected it. Therefore, I dare not abandon or reduce the old calculations based solely on my own views; thus, I follow this number and recalculate the relevant data to investigate its original meaning.

古歷皆云周天三百六十五度四分度之一，皆分一百七萬一千里數為一度闊二千九百三十二里七十一步二尺七寸四分大強；斗下分為七百三十三里一十七步五尺一寸八分大弱。三光之行，不必有當，術家以算，追而求之；故諸家之歷，各有異同，

Ancient calendars all stated that the celestial circle is 365 and 1/4 degrees, and they divided the total of 1,071,000 li by degrees, resulting in a width of 2,932 li, 71 steps, 2 chi, 7 cun, and 4 fen approximately per degree. The remaining 1/4 degree, known in calendrical terms as "dou xia fen," equates to 733 li, 17 steps, 5 chi, 1 cun, and 8 fen approximately. The movement of the sun, moon, and five planets may not exactly match these figures, so calendrical experts can only use calculations to seek and determine them, hence different schools of calendrical calculation have similarities and differences.

漢靈帝之末，四分曆與天違錯，時會稽東部都尉太山劉洪，善於推候，乃考術史官自古至今曆法，原其進退之行，察其出入之驗，視其往來，度其終始。課斗下分不能四分之一，減以為五百八十九分之一百四十五，更造《乾象曆》，以追日、月、五星之行。比於諸家，最為精密，今史官所用，則其歷也。

In the late years of the Han Ling帝, the four-part calendar did not agree with celestial phenomena. At that time, Liu Hong, a person from Taishan who served as the eastern deputy commander of Kuaiji, was skilled in calculating and verifying astronomical phenomena. He examined various calendars used by historians since ancient times, traced the movement and retreat of the sun and moon, investigated their appearances and disappearances, observed their recurring patterns, and measured their beginning and end. He calculated that the "division below the斗" was less than a quarter of a degree, and adjusted it to 145/589. He then created the "Ganxiang Calendar" to recalculate the movements of the sun, moon, and five planets. Compared with other calendars, this one was the most accurate. The current calendar used by the officials is this one.

故所作渾象諸，分度節次及昏明中星，皆更以《乾象》法作之，擊天一百七萬一千里，以《乾象》法分之，得二千九百三十二里八十步三尺九寸五分弱，斗下分為七百二十一里二百五十九步四尺五寸地二強。《乾象》全度長古歷全度九步一尺二寸少弱，斗下分減古歷斗下分十一里五十八步六寸六分弱，其大數俱一百七萬一千里，斗下分減，則全度純數使其然也。

So when I made the celestial globe, I revised all the divisions and the stars at the meridian at dusk and dawn according to the calculations in the "Ganxiang Calendar." Taking the full circle of 1,071,000 li, I divided it according to the degrees in the "Ganxiang Calendar," resulting in one degree being approximately 2,932 li, 80 bu, 3 chi, 9 cun, and 5 fen, and the portion below the斗 (dou) being approximately 721 li, 259 bu, 4 chi, 5 cun, and 2 fen (the original text says 'four chi five cun ground two strong,' which seems to have some errors). One full degree in the "Ganxiang Calendar" is slightly longer than one degree in the old calendar by about 9 bu, 1 chi, and 2 cun. The portion below the斗 in the "Ganxiang Calendar" is slightly shorter than that in the old calendar by about 11 li, 58 bu, 6 chi, and 6 cun. The total remains 1,071,000 li, and the reduction in the portion below the斗 is precisely due to the increased actual length of the full degree.

又陸績云：‘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。’此蓋天黃赤道之徑數也。渾天蓋天黃赤道周天度同，故績取以言耳。此言周三徑一也。

Lu Ji also said: "The circumference of the sky is one million and seven ten thousand li, and the diameter in the east-west and north-south directions is fifty-five thousand seven thousand li, and the diameter vertically is the same." This actually refers to the diameter figures derived by the Gaitian school from the ecliptic and equatorial circles. The Huntuan and Gaitian schools both used the same circumference measure for the celestial circle, so Lu Ji was able to cite it. This calculation is based on the old ratio of "the circumference is three times the diameter."

古歷術日率圓周三，中徑一，臣更考之徑一不翅周三率；周百四十二，而徑四十五，以徑率乘一百七萬一千里，以周率約之，得徑三十二萬九千四百一里一百二十二步二尺二寸一分七十一分分之十，東西南北及立徑皆同，半之得十六萬九千七百里二百一十步一尺六寸百四十二分分之八十一，地上去天之數也。

Ancient methods of calculation stated that the circumference of a circle is three times its diameter; I have re-examined this and found that when the diameter is one, the circumference is more than three. Taking the circumference rate as 142 and the diameter rate as 45, multiply the diameter rate by 1,070,000 li, then divide by the circumference rate to obtain a diameter of 329,401 li, 122 bu, 2 chi, 2 cun, 1 fen, and 10/71. The east-west, north-south, and up-down diameters are all the same. Taking half of this, we get 169,700 li, 210 bu, 1 chi, 6 cun, and 81/142, which is the distance from the earth to the heavens.

夫周徑固前定物，為蓋天者，尚不考驗，而乃論天地之外，日月所不照，陰陽所不至，日精所不及，儀術所不測，皆為之說虛誕無證，是亦鄒子瀛海之類也。

The circumference and diameter are things that can be measured right before one's eyes. Those who advocate the theory that the heavens are a dome do not even bother to test these, yet they talk extensively about matters beyond the heavens—places that sunlight cannot reach, where yin and yang cannot extend, where sunlight cannot penetrate, and where instruments and calculations cannot measure. They fabricate explanations for all these places, which are baseless and without evidence. This is no different from the empty talk of Zou Yan of the Warring States period, who claimed there was a great sea beyond the nine provinces.

臣謹更以晷景考周天裡數，按《周禮大司徒》之職云：‘土圭之法，測土深，正日景，以求地中。日南則景短多暑；日北則景長多寒；日東則景夕多風；日西則景朝多陰。日到之景，盡有五寸，謂之地中。’鄭玄云：‘土圭之長，尺有五寸，以夏至立八尺表，其景適與圭等，謂之地中，今潁川陽城地為然。’鄭玄云：‘凡日景於地，千里而差一寸。景尺有五寸者，南戴日下萬五千里也。’

I now carefully use the shadow of the sun to measure the distance around the heavens. The Book of Rites, the Great Master of the People, says: "Use the method of the soil圭 to measure the distance of the land, correct the shadow of the sun, and find the center of the earth. In the southern direction, the shadow is short and there is more heat; in the northern direction, the shadow is long and there is more cold; in the eastern direction, the shadow is like that of the evening and there is more wind; in the western direction, the shadow is like that of the morning and there is more dampness. On the summer solstice, when the shadow is exactly one foot five inches, it is called the center of the earth." Zheng Xuan explains: "The soil圭 is one foot five inches long. On the summer solstice, if a pole eight feet high is erected, its shadow will be exactly the same length as the soil圭, and this is called the center of the earth. Now, the place in Yangcheng County, Yingshui Commandery is exactly like this." Zheng Xuan also says: "Whenever the shadow of the sun is cast on the ground, for every thousand li of distance, the shadow differs by one inch. A shadow of one foot five inches indicates that the place is fifteen thousand li south of the location directly under the sun."

誠以八尺之表，而有尺五寸景，是立八十而旁十五也；南萬五千里而當日下，則日當去其下地八萬里矣。從日斜射陽城，則天徑之半也，天體圓如彈丸，地處天之半，而陽城為中，則日春、秋、冬、夏、明、晝、夜，去陽城皆無盈縮矣，故知從日斜射陽城為天徑之半也。

If a gnomon eight chi high casts a shadow of one and a half chi, the ratio is vertical eighty to horizontal fifteen. If one travels south fifteen thousand li to reach the location directly beneath the sun, then the distance from the sun to the ground below should be eight ten-thousand li. The line from the sun slanting to Yangcheng is the radius of the celestial sphere. The celestial sphere is round like a ball, and the earth is at its center, with Yangcheng being the center of the earth. Therefore, the sun, regardless of the seasons, day or night, maintains the same distance from Yangcheng. Thus, the line from the sun slanting to Yangcheng is precisely the radius of the celestial sphere.

以勾股法言之，旁萬五千里勾也。立八萬里，則股也。從日斜射陽城，則弦也。以勾股求弦法入之，得八萬一千三百九十四里三十步五尺三寸六分，天徑之半，而地上去天之數也。倍之，得十六萬二千七百八十八里六十一步四尺七寸二分，天徑之數也。以周率乘之，徑率約之，得五十一萬三千六百八十七里六十八步一尺八寸二分，周天之數也。

Using the Pythagorean theorem: the horizontal distance of fifteen thousand five hundred li is the leg (勾), the vertical distance of eight ten-thousand li is the other leg (股), and the line from the Sun to Yangcheng is the hypotenuse (弦). Applying the Pythagorean method to calculate the hypotenuse, we get eighty-one thousand three hundred ninety-four li, thirty steps, five feet, three inches, and six parts, which is the radius of the celestial sphere, i.e., the distance from the earth to the heavens. Doubling this gives sixteen thousand two hundred seventy-eight li, sixty-one steps, four feet, seven inches, and two parts, which is the diameter of the celestial sphere. Multiplying this by the circumference rate and dividing by the diameter rate, we obtain fifty-one thousand three hundred sixty-seven li, sixty-eight steps, one foot, eight inches, and two parts, which is the length of the celestial circle.

減《甄曜度》《考異郵》五十五萬七千三百一十二里有奇，一度凡一千四百六里百二十四步六寸四分十萬七千五百六十五分分之萬九千四十九，減舊度千里五百二十五里二百五十六步三尺三寸二十一萬五千一百三十分分之十六萬七百三十分。

This number is more than fifty-five thousand seven hundred and thirty-two li less than what is stated in the "Luo Writing Verification and Measurement" and the "Spring and Autumn Examination of Miscellaneous Mail." According to this calculation, one degree of the celestial sphere corresponds to one thousand four hundred and six li, one hundred and twenty-four bu, six cun, four fen, and one ten-thousandth of nine thousand and forty-nine, which is one thousand five hundred and twenty-five li, two hundred and fifty-six bu, three chi, three cun, and sixteen thousand and seven hundred and thirty over twenty-one thousand five hundred and thirty parts less than the previous account. (The original text's phrase "a thousand five hundred and twenty-five li" seems to have an error.)

夫末世之儒，多妄穿鑿，補增《河洛》，竊作讖緯，其言浮虛難悉，據用六官之職，周公所制勾股之術，目前定數，晷景之度，事有明驗，以求推之，近為詳矣。

In the later ages, most Confucian scholars tended to indulge in fanciful interpretations and fabricated additions to the Yellow River Chart and the Luo Writing, privately creating omens and prophecies. These statements are superficial and baseless, difficult to fully trust. Now, based on the duties of the six officials as recorded in the Zhou Li, using the Pythagorean mathematics established by the Duke of Zhou, and relying on the definite numbers and measurements obtained from the shadows of the sun, everything can be verified clearly. By using these to make deductions, the account is comparatively more concrete and reliable.

黃赤道相與交錯，其間相去二十四度。以兩遊儀準之，二道俱三百六十五度有奇，是以知天體圓如彈丸。而陸績造渾象，其形如鳥卵；然而黃道應長於赤道矣。績云：‘東西南北徑三十五萬七千里。’然則績亦以天形正圓也。而渾象為鳥卵，則為自相違背。

The Yellow Path and the Equator intersect with each other, and the maximum distance between them is twenty-four degrees. Using two armillary spheres for calibration, both paths measure over three hundred and sixty-five degrees, thus indicating that the celestial sphere is round like a ball. However, Lu Ji made a celestial globe in the shape of an egg, implying that the Yellow Path should be longer than the Equator. Lu Ji himself also stated that the diameter of the sky in the four cardinal directions is thirty-five thousand seven hundred li, showing that he actually believed the sky to be a perfect sphere. Yet the celestial globe he made was egg-shaped, which is contradictory.

月行二十七日有奇而周天，其行半出黃道外，半入黃道內；在內謂之陰道，在外謂之陽道，其行陰陽道極遠者不過六度。黃道無常，諸家各異，各依其歷。節氣所行宿度，《尚書》、《月令》、《太初》、《三統》、《四分》、《乾象》各不同，昏明亦異，日行蹉跌，不遵常之所為也。

The moon takes about twenty-seven days to orbit the heavens once, with half its path outside the Yellow Path and half inside. The part inside is called the Yin Path, and the part outside is called the Yang Path. The maximum distance the moon travels in and out of the Yin and Yang Paths does not exceed six degrees. The fixed method of the Yellow Path varies among different schools, each based on their own calendar system. The positions of the sun during the solar terms, as recorded in the Book of Documents, the Rites of Zhou: Monthly Orders, and the Tai Chu Calendar, the San Tong Calendar, the Si Fen Calendar, and the Qian Xiang Calendar, differ from one another. The stars observed at sunset and at dawn also vary, because the movement of the sun is irregular and does not strictly follow a fixed pattern.

夫三光之行，雖有盈縮，天地之體，常然不變，故諸家之歷，皆不著渾象，唯黃道當各隨其歷而錯之。而今臣所施黃道，《乾象》法也，審校先代諸儒，於春秋二分多差，而冬夏二至恆在；二分限中，蕃按渾象之法，地當在天中，其勢不便故，反觀其形。地為外匡，於已解者，無異在內。詭狀殊體，而合於理，可謂奇巧。

The movements of the sun, moon, and stars, though they have periods of fullness and contraction, the form of heaven and earth remains constant. Therefore, various calendars do not need to create their own armillary spheres, but the Yellow Path must be placed according to each calendar. The Yellow Path I have placed here follows the method of the Ganyang Calendar. Careful examination of the previous scholars' conclusions shows that their errors were mainly in the spring equinox and autumn equinox, while the winter solstice and summer solstice remained accurate. The errors were all within the range of the two equinoxes. Wang Fan considered the method of the armillary sphere: the earth should be placed at the center of the celestial sphere, but that would make the structure inconvenient to observe. Therefore, he reversed the perspective, making the earth appear as an outer frame. For those who understand the principle, this is no different from placing the earth within the heavens. Though the shape is strange and the structure is unique, it conforms to the principles, and can be considered ingenious.

古舊渾象，以二分為一度，周七尺三寸半分。漢張衡更制，以四分為一度，凡周一丈四尺六寸一分。臣以古制局小，以布星辰，相去稠概，不得了察；而張衡所作，又復傷大，難可轉移。前表間以三分為一度事，許令臣新作周一丈九寸五分四分分之三，長古法三尺六寸五分四分分之一，減張衡亦三尺六寸五分四分分之三，渾象法黃道、赤道各廣一度有半，故今所作渾象，黃、赤道各廣四分半、相去七寸二分。渾儀中筒為璇璣，外規為玉衡。（吳太史令陳卓所作《渾天論》與王蕃大同。）

In ancient times, there was a device called the浑象, with each degree measured as two-thirds of a foot, making the entire circumference seven feet three inches and a half. In the Han dynasty, Zhang Heng redesigned it, measuring each degree as four-thirds of a foot, resulting in a total circumference of thirteen feet six inches and one part. I think the ancient design was too small, and after placing the stars, they were too close together to be clearly seen. Zhang Heng's design, on the other hand, was too large to be moved or rotated. Previously, I proposed using three-thirds of a foot as a degree, and was granted permission to create a new one. The total circumference of this new device is one zhang nine inches and five parts plus three parts of four, which is three feet six inches and one part plus three parts of four longer than the ancient method, and three feet six inches and three parts of four shorter than Zhang Heng's. According to the standard method of the浑象, the Yellow Path and the Red Path each have a width of one and a half degrees, so in this newly made浑象, the Yellow Path and the Red Path each have a width of one and a quarter, and the distance between the two paths is seven inches and two parts. The cylindrical part in the center of the浑仪 is called "Xuan Ji," and the outer circular ring is called "Yu Heng." (Original note: The "Hun Tian Lun" written by Chen Zhuo, the Astronomer of the State of Wu, is largely consistent with Wang Fen's description.)

姜岌《渾天論答難》曰：“渾天之說，天體包裹，地在其中。七曜躔麗，道有常率。天體旁倚，故日道南高而北下；運轉之樞，南下而北高。二樞為轂，日道為輪，周廻運移，終則復始。北樞謂之北極，出地上三十六度，故天北際七十二度，常見而不伏。南樞謂之南極，入地下亦三十六度，故天南際七十二度常伏而不見。

In the "Hol Tian Lun Da Nan" (Treatise on Answering Difficulties on the Theory of the Sphere), the astronomer and calendarist Jiang Ji of the Sixteen Kingdoms period said: The theory of the sphere holds that the celestial sphere surrounds the outside, and the earth is located at the center of the sky. The sun, moon, five planets, and seven luminaries move along the celestial sphere, each following a constant path. The celestial sphere is inclined, so the path of the sun is higher in the south and lower in the north. However, the axis of rotation of the celestial sphere is lower in the south and higher in the north. The northern and southern pivots are like the hub of a wheel, while the path of the sun is like the rim of the wheel, continuously rotating and moving. The northern pivot is called "Beiji" (North Pole), rising 36 degrees above the ground, so the area of the sky covering 72 degrees to the north is always visible and never hidden. The southern pivot is called "Nanjing" (South Pole), sinking 36 degrees below the ground, so the area of the sky covering 72 degrees to the south is always hidden and never visible.

或云：火，陽也，故外照。金、水，陰也，故內景。日為陽精，故為外照；月為陰之宗，應內景。而月復能外照，何也？對曰：月光者，日曜之所生，是故外景如日照也；是故瑩金、澄水得日之照，照物亦有景。

Someone asked: Fire belongs to Yang, so it radiates outward; metal and water belong to Yin, so they only contain images internally. The Sun is the essence of Yang, so it radiates outward; the Moon is the master of Yin, so it should only contain images internally. Yet the Moon can also radiate outward. Why is this? Answer: The light of the Moon is produced by the Sun's illumination, so the light it emits outward is like the Sun's illumination. Just as bright metal and clear water surfaces, when exposed to sunlight, can reflect light onto other objects and create images.

又曰：月無虧盈，虧盈由人也；日月之體，形如圓丸，各徑千里，月體向日，常有光也。月之初生，日曜其西，人處其東，不見其光，故名曰魄。魄一日之後，微東而南，故明生焉。八日正在南方，半之故見其弦也。望則人處日月之間，故見其圓也。

Jiang Ji also said: The moon itself does not have phases of being full or crescent; the appearance of fullness or crescent is caused by the position of the observer. The forms of the sun and moon are both like spheres, each with a diameter of one thousand li. The side of the moon facing the sun is always illuminated. When the moon is newly born, the sun shines on its western side, but people stand on its eastern side and cannot see the light on that side, so it is called "Pò". After one day, the moon moves slightly eastward and slightly southward, and the bright part becomes visible. By the eighth day, the moon is in the south, and the bright part is exactly half, so it appears as a crescent moon. On the full moon day, people are positioned between the sun and moon, so the moon appears full.

假使月初生時，移人在日月之間，東向以視，則月光圓若望也。夏至之日，日入戌，月初生時，則西北近日有光。及出於寅，未閱三日，以視月，則東北近日處光滿也。研之於心，驗之於目，月體向日，有光而形圓矣。

Suppose a person is moved to between the sun and moon at the beginning of the month, and looks east, the moonlight seen is still full, just like on a full moon day. On the summer solstice, the sun is at the戌 position, and at the beginning of the month, the side of the moon facing northwest near the sun is illuminated. When the sun rises at the寅 position and it has not yet been three days, looking again at the moon, one sees that the side facing northeast near the sun is already full. By thinking carefully and verifying with the eyes, one can know that the side of the moon facing the sun is always illuminated, and its shape is round.

難者又云：日曜星月，明乃生焉。然則月望之日，夜半之時，日在地下，月在地上，其間隔地，日光何由得照月，暗虛安得常日沖？對曰：日之曜天，不以幽而不至；不以明而不及。赫烈照於四極之中，明光曜煥乎宇宙之內，循天而曜星月，猶火之循突而升，及其光曜，無不周矣。惟沖不照，名曰暗虛，蓋日及天體，猶滿面賁鼓矣。

The person who raised the objection further said: The moon depends on the sun's light to produce brightness. But on the night of the full moon, the sun is beneath the earth, the moon is above the earth, and they are separated by the earth in between. How can the sunlight reach the moon? How can the dark void always face the sun? Answer: The sun shines upon the sky, not because a place is dark that it cannot reach, nor because a place is bright that it cannot reach. It brightly illuminates all four directions, its radiance thoroughly fills the universe. It follows the celestial bodies to illuminate the stars and the moon, just as fire rises along a chimney. Its light reaches everywhere. Only the spot directly opposite the sun is not illuminated, called the "dark void." This is because the sunlight covers the entire celestial sphere, just as a drumhead is stretched tightly.

日之光炎，在地之上，礙地不得直照而散，故薄虧，而照則近；在地之下，聚而直照，故滿盈，而照而遠。以斯言之，則日光應曜星月，有何礙哉？《易傳》：‘日夜食則星亡。’無日以曜之故也。

The light of the Sun, when it is above the ground, is obstructed by the earth and cannot shine straight ahead, but spreads outwards, appearing thin and dim, and its reach is short; when it is below the ground, it gathers and shines straight, appearing full and bright, and its reach is far. According to this, sunlight should naturally illuminate the stars and the moon, so what could prevent it? The Commentary on the Changes says, "When an eclipse occurs during the day, the stars disappear." This is precisely because there is no Sun to illuminate them.

難雲地上不得直照而散，故薄虧而照近。檢先望一日，日未入地而月已出，相去三十餘萬里，日在地上，散而直照，不應及月，而使月全明者，何也？對曰：薄虧而照則近，是言礙地光難周耳。水流溼，火就燥，類相從也。月星者，類也。日光散照，雖不及月，譬之燃燭，一燭在上，一燭在下，滅下燭使煙相當，則上滅之談，循煙而下，燃下燭矣。此類相從也。

Some people have raised an objection: if sunlight above the earth does not shine directly but spreads out and is weak and insufficient, then on the day before the full moon, when the sun has not yet set and the moon has already risen, the two are more than thirty thousand miles apart. At this time, the sun is above the earth, its light spreading out rather than shining directly, and should not reach the moon. Yet the moon is fully bright. Why is this? The answer is: the statement that sunlight is weak and insufficient and only reaches nearby places merely means that the earth obstructs the light, making it difficult for the light to reach everywhere. Water flows to the wet places, and fire burns in the dry places; similar things naturally follow each other. The moon and stars are things similar to the sun's light. Although the sun's light, which spreads out, cannot reach the moon, we can use a candle as an example: if one candle is placed above and another below, and the lower one is blown out, allowing the smoke to face the upper one, then the flame of the upper candle will follow the smoke downward and re-light the lower one. This is the principle of similar things following each other.

難者又曰：日夜食則眾星亡。按月體不大於地，今日在地下，月在地上，地體大尚不能掩日使不照月，月體小於地，安能掩日使不照曜星也。對曰：上元之初，日月如疊璧，五星如連珠，故曰重光。重光者，日在上，月次之，星居下。地在宿內，故掩日月光，星月存而曜亡也。月在星宿之外，故掩日，日光不得照星也。”

The person who raised the objection also said: When an eclipse occurs, making day like night, all the stars disappear. But according to the size of the moon, it is not bigger than the earth. Now the sun is below the earth and the moon is on the earth. Even the larger earth cannot block the sun's light from reaching the moon. Since the moon is smaller than the earth, how can it block the sun's light from reaching the stars? Answer: At the beginning of the calendar cycle, the sun and moon were like stacked jade disks, and the five planets were like connected pearls, so it was called "repeated brightness." The term "repeated brightness" refers to the sun being at the top, the moon next, and the stars at the bottom. The earth is inside the star mansions, so when it blocks the light of the sun and moon, the stars and moon are still there but their light is not visible. The moon is outside the star mansions, so when it blocks the sun's light, the sun's light cannot reach the stars.

梁武帝《嘗儀》云：“月體不全光，星亦自有光，非受明於日。若是日曜月所以成光，去日遠則光全，去日近則光缺。五星行度亦去日有遠近，五星安得不盈虧？當知不然。太陽之精，自有光景，但異於太陽，不得渾赫。星月及日，體質皆圓，非如圓鏡，當如丸矣。”

Liang Wu Di in the "Chang Yi" said: The form of the moon is not entirely luminous on its own; stars also have their own light, and are not dependent on the light from the sun. If the moon's brightness comes from the sun's illumination, then when the moon is far from the sun, its light should be full, and when it is close to the sun, its light should be diminished; the five planets also vary in their distance from the sun, so why would they not also experience waxing and waning? Clearly, this is not the case. The essence of the sun has its own light and shadow, but the light of stars and the moon is different from that of the sun, and cannot be as brilliant and grand as the sun's. The bodies of stars, the moon, and the sun are all round, unlike a flat circular mirror, but rather like a ball.

祖暅曰：“姜岌此言非也，星猶月，稟日之光，然後乃見。若星在日裡，則應盈魄，今既不然，故知星在日表而常明也。按星體自有光曜，非由稟日始明，今星宿有時食，月在魄中，分明質見，則是星行亦在月裡，不專在表。又姜岌承二燭為喻，理亦迂廻，非實驗也。”

Zugeng said: Jiang Ji's words are incorrect. Stars and the moon are both seen because they receive sunlight. If a star were on the inner side of the sun, it should be like the moon's dark and bright phases, sometimes full and sometimes crescent. But since it is not like that, we can conclude that the stars are on the outer side of the sun and thus always bright. However, we should also note that stars themselves emit light, not only by receiving sunlight. Now, when a star is eclipsed by the moon, and the moon is in its dark half, the star's shape is still clearly visible, showing that the stars sometimes lie on the inner side of the moon, not exclusively on the outer side. Moreover, Jiang Ji's use of the analogy of two candles is too indirect and not based on actual observation.

晉侍中劉智《論天》曰：“凡含天地之氣而生者，人其最貴而有靈智者也。是以動作營為，皆應天地之象。古先聖王，觀靈曜，造算數，準辰極，制渾儀，原性理，考徵詳。贊其幽義，而作歷術焉。

In the Jin Dynasty, the official Liu Zhi wrote in his work "Discourse on Heaven" that all beings that receive the vital energy of heaven and earth, are most esteemed and endowed with spiritual wisdom among them. Therefore, every action and endeavor of humans must correspond with the phenomena of heaven and earth. The ancient sages and kings observed the brightness of the sun, moon, and stars, created arithmetic, took the north star as their reference, made the armillary sphere, explored the nature and principles of all things, examined the details of verification, and explained the profound meanings, thus giving rise to the study of calendars.

渾儀象天之圓體，以含地方，輪轉周匝，中有二端，其可見者，極星是也，謂之北極。在南者，在地下不見，故古人不名。陰陽對合為群生，父母精象在於五星，其於共成天地之功也，則日月為政，五星為緯，天以七紀，七曜是也。

The浑仪 is modeled after the round shape of heaven, containing the square earth within it. When it rotates through a full circle, it covers everything completely. In the middle, there are two endpoints: the one that is visible is the polar star, called the North Star; the one in the south is located underground and cannot be seen, so the ancients did not give it a name. Yin and yang interact and generate all kinds of beings. The images of the essence of parents are reflected in the five planets. In terms of their shared contribution to the creation of heaven and earth, the sun and moon are called 'zheng' (government), and the five planets are called 'wei' (measuring lines). Heaven uses the seven to form its framework, which are known as the seven luminaries.

北極不過為眾星之君，命政指授，以斗建時。斗有七星，與曜同精，而布節氣於下者也。晦朔分於東西，消息辨於南北，取以定四方，天地配合，方氣有常，天以七紀，方修其政。故方有七宿，二十八星是也。

The North Star is merely the sovereign among the stars, issuing commands and directing affairs, determining the seasons by the position of the handle of the Big Dipper. The Big Dipper consists of seven stars, just like the seven luminaries, formed by condensed vital energy, spreading out the seasonal changes below. The alternation of new and old moons is observed in the east and west, while the waxing and waning of yin and yang is discerned in the north and south, thus determining the four directions. Heaven and earth correspond with each other, and the energies of each direction follow their own regular patterns. Heaven uses seven as its standard, and each direction also establishes its own governance, so each direction has seven stars, making up the twenty-eight constellations in total.

天有常度，日月成象。眾星有宮分，方物有體類，在朝十八星是也。天有常度，日月成象。眾星有宮分，方物有體類，在朝象官，在野象物，在人象事，理自然也。眾星定位，七曜錯行，盈縮有期節，故曆數立焉。

Heaven has constant measurements, the sun and moon form distinct celestial phenomena; the myriad stars each have their own assigned positions, and the things of the four directions each have their own forms and categories. In the court, they correspond to the eighteen stars. (This passage is a repetition from the original text, and it is recorded as such.) Heaven has constant measurements, the sun and moon form distinct celestial phenomena; the myriad stars each have their own assigned positions, and the things of the four directions each have their own forms and categories: in the court they correspond to the officials, in the fields they correspond to all things, in human affairs they correspond to various matters, this is the natural principle. Each star has its fixed position, the seven luminaries move in intersecting paths, their movements and variations have fixed limits and sequences, thus the study of calendars and astronomical calculations was established.

日，太陽也；施溫萬物生，施光則陰以明，眾所稟為倡先者也。君，尊之象也。月，太陰也。稟照於陽，虧盈隨時，有所稟受，臣卑之道也。五星象五常，托四時，成五事。

The Sun is the essence of the Sun; it spreads warmth, enabling all things to grow, and it spreads light, making dark things visible. It is what all things receive and follow, taking the leading position, thus symbolizing the sovereign's dignity. The Moon is the essence of the Too Yin; it receives illumination from the Sun, constantly waning and waxing, receiving all things, which embodies the way of a minister's humility and obedience. The Five Stars symbolize the Five Constants of benevolence, righteousness, propriety, wisdom, and faithfulness, which are寄托ed in the four seasons, accomplishing the five affairs of appearance, speech, gaze, hearing, and thought.

舊說日譬猶火也，月譬猶水也，火則施光，水則含景。故月光生於背日，當日則光盈，近日則明滅。然則月為臣象可知矣。又曰：當其衝月食者，陰性毀損，不受光也。群臣不等強，日月不等明。陰在於上，不自抑損，陽必侵之。望在交度，其應必食。故《詩》云：‘披月而食，則維期常。’道勢然也。侵甚，則既虛之象也。

Old interpretations compared the sun to fire and the moon to water; fire emits light outwardly, while water contains images within. Therefore, moonlight appears on the side facing away from the sun, and when the moon is directly opposite the sun, its light is full, but when it is close to the sun, its brightness fades. From this perspective, the moon symbolizes ministers, which can be understood. Old interpretations also state: when the sun and moon are in opposition and an eclipse occurs, it is because the nature of yin is damaged and cannot receive brightness. The strength of ministers varies, and the brightness of the sun and moon is not equal; when yin is in a superior position without self-restraint, yang will inevitably attack it. The full moon occurs precisely at the intersection of the yellow and white paths, and accordingly, an eclipse must occur. Therefore, the Book of Poetry says: "When the moon experiences an eclipse, it is still considered a common occurrence." This is due to the nature of things and the situation. When the attack is severe, it becomes the image of a total lunar eclipse.

日，尊君象也。月，卑臣象也。晦朔之會，交則同道。同道則形相蔽。天道君為尊，臣由臣道，雖度相值，月不掩日，卑下尊也。不由臣道，月掩日體，卑凌尊也。是故太平之時，交而不食，尊卑道順。”

The Sun symbolizes the noble and exalted sovereign, while the Moon symbolizes the humble minister. On the day when the new moon begins, the Sun and Moon meet; if they are precisely at the conjunction point, they travel along the same path. When they are on the same path, their forms will obscure each other. According to the heavenly order, the sovereign is the highest: if a minister follows the proper conduct of a minister, even if their positions align exactly, the Moon will not cover the Sun, for the humble should not take the place of the exalted. But if the minister does not follow the proper conduct, the Moon will cover the Sun, indicating that the humble has overstepped the exalted. Therefore, when the world is at peace, the Sun and Moon may meet but do not cause an eclipse, as the principles of hierarchy are maintained.

或問云：“顓頊氏造渾儀，黃帝為蓋天，其遺說雲蓋天象笠，極在其中，日月遠近，以為晦明。渾儀以天裹地，地載於氣，天以廻轉，而日月出入，以為晦明。二說誰其得之？”

Someone asked: Zhuan Xu established the浑仪, and the Yellow Emperor created the盖天. The remaining teachings of the盖天 say that the sky is like a bamboo hat, with the pole at the center of the hat, and the sun and moon, as well as people, are at varying distances, thus causing the phenomena of dawn, dusk, daylight, and darkness. The theory of the浑仪 says that the sky surrounds the earth, the earth rests on the气, and the sky rotates continuously, with the sun and moon rising and setting among it, thus causing the phenomena of dawn, dusk, daylight, and darkness. Which of these two theories is correct?

劉智曰：“蓋天之論謬矣，以春秋二分，日出卯入酉，若天象車蓋，極在其中，日、月、星辰，廻還藏明，二分之時，當晝短夜長。今以漏刻數之，則晝夜分等。以日入較之，則出卯入酉，此蓋天之說不通之驗也。然此二器，皆古之所制，但傳說義者，失其用耳。

Liu Zhi answered: The theory of the dome-shaped sky is wrong. Because on the days of the vernal and autumnal equinoxes, the sun rises from the Mao position in the due east and sets in the You position in the due west. If the sky were like a canopy with its apex in the center, and the sun, moon, and stars merely revolved around the top of the canopy, disappearing when they moved far away, then on the days of the equinoxes, the day should be short and the night long. But according to the water clock, day and night are exactly equal. Moreover, comparing the positions of sunrise and sunset, they indeed rise from Mao and set in You. This is evidence that the theory of the dome-shaped sky is incorrect. However, both the dome-shaped sky model and the armillary sphere were inventions of ancient people, but later those who transmitted their meanings lost their original purposes.

昔者，聖王治曆明時，作圓蓋，以圖列宿，極在其中。廻之以觀天象，未可正昏、明，分晝，夜；故作渾儀也。象天體，亦以極為中，而朱規為赤遊周環，去極九十一度有奇。考日所行，冬夏去極遠近不同，故復畫為黃道；夏至去極近，冬至去極遠，二分之際，交於赤道；二道有表裡，以定宿度之進退，為術乃密。

In ancient times, the sage kings established calendars and clarified the seasons by first making a circular dome and drawing the twenty-eight constellations on it, with the pole star at the center. They rotated it to observe celestial phenomena; however, this was insufficient for accurately determining dawn and dusk or dividing day and night. Therefore, they constructed the armillary sphere. The armillary sphere was modeled after the celestial sphere, also with the pole at its center. It was painted red with a circular equatorial ring surrounding it, at a distance of more than ninety-one degrees from the pole. They examined the path of the sun, noting that its distance from the pole varied between summer and winter. Hence, they also drew the Yellow Path. At the summer solstice, the sun was closest to the pole, while at the winter solstice, it was farthest away. At the equinoxes, the Yellow Path crossed the equator. The Yellow and Red Paths differed in their positions, used to determine the movement of the constellations and to make calculations more precise.

至漢順帝時，南陽張衡，考定進退。靈帝時，太山劉洪，步月遲疾。自此之後，天驗愈詳。自司馬遷、劉向、劉歆、揚雄、賈逵、張衡、蔡邕、劉洪、鄭玄，此九君者，不但精於算步，皆博索沈綜，才思弘遠，而不合論渾蓋之用。明定日行四時之道，雖或精考，雅有所得，亦或出心裁，失其本旨。人之不同，處意各異；道之難盡，效於斯矣。”

During the time of Emperor Shun of Han, Zhang Heng from Nanyang determined the degrees of the Yellow Path's advance and retreat; during the time of Emperor Ling of Han, Liu Hong from Taishan calculated the slow and fast movement of the moon. Since then, the verification of celestial phenomena has become more detailed. From Sima Qian, Liu Xiang, Liu Xin, Yang Xiong, Jia Kui, Zhang Heng, Cai Yong, Liu Hong to Zheng Xuan, these nine scholars not only excelled in calculation and verification but also widely collected and deeply thought, with broad and far-reaching intellect. However, they did not reach agreement on the use of the two instruments, the浑天 (homogeneous heaven) and the盖天 (covering heaven). As for determining the path of the Star of brightness and prestige through the four seasons, although some carefully examined and indeed had insights, others relied on their own assumptions and lost the original meaning. People differ, their intentions vary; the difficulty of understanding principles is evident here.

或問曰：“古歷論月食，或雲陰損，則不受明。或雲暗虛，暗虛所在。值月則月食，值星則星亡。今子不從，何也？”劉智曰：“言暗虛者，以為當日之沖，地體之蔭，日光不至，謂之暗虛。凡光之所照，光體小於所蔽，則蔭大於本質。今日以千里之徑，而地體蔽之，則暗虛之蔭，將過半矣。星亡月毀，豈但交會之間而已哉。由此言之，陰不受明，近得之矣。”

Another person asked: In ancient times, astronomers explained lunar eclipses, some saying it was due to the damage of yin energy, so that light could not be accepted; others said it was due to the presence of "dark emptiness" — when dark emptiness met the moon, a lunar eclipse occurred, and when it met a star, the star would disappear. Now you do not adopt the latter explanation, why is that? Liu Zhi answered: Those who talk about dark emptiness believe that the place directly opposite the sun is the shadow cast by the earth, where sunlight cannot reach, and this is called dark emptiness. Whenever light is cast, the light source is smaller than the object it illuminates, so the shadow is larger than the object itself. Now the sun's diameter is thousands of li, and the earth blocks it, so the shadow of dark emptiness would be larger than half the sky. If that were the case, the disappearance of stars and the eclipse of the moon should not only occur when they meet. Therefore, the explanation of "yin not receiving light" is closer to the truth.

又問曰：“若如所論，必有大蔭，月在日沖，何由有明？”劉智曰：“夫陰含陽面明，不待陽光明照之也。陰陽相應，清者受光，寒者受溫，無門而通，雖遠相應。是故觸石而雲出者，水氣之通也。相向而相及，無遠不至，無隔能塞者，至清之質，承陽之光，以天之圓，面向相背，側正不同，光魄之理也。陰陽相承，彼隆此衰，是故日月有爭明，日微則晝見，若但以形光相照，無相引受之氣，則當陽隆乃陰明隆；陽衰則陰明衰。二者之異，無由生矣。”

He then asked: According to your explanation, there must be a great shadow, yet the moon is in opposition to the sun, so how can it still shine? Liu Zhi answered: Yin contains the side facing yang and thus produces brightness, it does not necessarily require direct sunlight. Yin and yang respond to each other: clear things receive light, cold things receive warmth, and communication can occur without a direct path, even when they are far apart. Therefore, when a mountain or stone is touched, clouds rise, because of the connection of water vapour. When things face each other, they can reach each other, even if they are far apart, nothing can block them. The moon is of the clearest substance, receiving the light of yang. Because celestial bodies are round, the moon's face can be facing, back, or sideways, which is the reason for the formation of brightness and the moon's phases. Yin and yang mutually support each other, so when one is strong the other is weak, hence the phenomenon of competition between sun and moon. When the sun's light is weak, stars and the moon can still be seen in the daytime. If there were only mutual illumination between form and light, without mutual attraction and the transmission of energy, then when yang is strong, the brightness of yin would also be strong, and when yang is weak, the brightness of yin would also be weak. But the difference between them now arises precisely because of this.

後魏太史令晁崇，修渾儀以觀星象。按其儀以永興四年歲次困敦創造，傳至後魏末，入齊往周，隨至於大唐。歷年久遠，儀蓋日以傾墜。太史者，歷正也。

After the Northern Wei dynasty, the official Zhao Chong, who was in charge of astronomical observations, constructed a浑仪 to observe celestial phenomena. This instrument was created in the fourth year of Yongxing, which was the year of Zi (the Rat), and it was passed down until the end of the Northern Wei dynasty. Later, it was transferred to the Northern Qi dynasty and then to the Northern Zhou dynasty, and eventually passed on to the Tang dynasty. Due to the long period of time, the top cover of the instrument gradually tilted and deteriorated. The term "太史" refers to the official responsible for the calendar and astronomical affairs.

自景雲三年奉敕重令修造，使銀青光祿大夫、檢校將作少監楊務廉；與銀青光禪大夫、行太史令瞿曇悉達；正議大夫、行太史令李仙宗；試太史令殷知易；荊州都督兼秘書監、兼右衛率薛玉；銀青光祿大夫、檢校秘書監吳師道；正議大夫、行秘書少監閻朝隱等，首末共營，各盡其思。至先天二年，歲次赤備若，成其銘曰：“周天三萬七千里，分寸無敗；成歲三百六十日，盈縮有期。敬之敬之，以授人時。”今史司見用測候。

In the third year of Jingyun, they were ordered by imperial decree to reconstruct it, and appointed officials such as Yang Wulián, a silver and green officer of the Office of the Imperial Grace, and acting deputy supervisor of the Office of Works; Qutan Xida, a silver and green officer of the Office of the Imperial Grace, and acting Astronomer-in-Chief; Li Xianzong, a properly appointed officer of the Court, and acting Astronomer-in-Chief; Yin Zhiyi, a trial Astronomer-in-Chief; Xue Yu, the governor of Jingzhou and concurrently the Director of the Imperial Library, and concurrently the commander of the Right Guards; Wu Shidao, a silver and green officer of the Office of the Imperial Grace, and acting Director of the Imperial Library; and Yan Chaoyin, a properly appointed officer of the Court, and acting Deputy Director of the Imperial Library, to work together from the beginning to the end, each contributing their wisdom and thoughts. By the second year of Xiantian, in the year of Chi Fen Rao (the year of the Ox), they completed this instrument. The engraved inscription says: "The celestial sphere is thirty-seven thousand li, with precision in every inch, hair, and fraction; one year consists of three hundred and sixty days, with its expansion and contraction following a fixed cycle. Reverently, reverently, it is used to announce the seasons to the people." Now the office of the historians is using it to observe and verify.

守太子率更令何承天《論渾象體》曰：“詳尋前說，因觀渾儀，研求其意。有悟天形正圓，而水居其半。地中高處卑，水周其下。言四方者，東曰暘谷，日之所出。西日濛汜，日之所入。莊子又云：‘北溟有魚，化而為鳥，將徒於南溟。’斯亦古之遺記，四方皆水證也。

Heavenly Steward Hou Taizi Shu Geng Ling He Cheng Tian said in his work "On the Form of the Celestial Sphere": By carefully examining previous statements and observing the armillary sphere, he investigated the underlying principles and thus realized that the shape of heaven is a perfect circle, with water occupying half of it. The earth is higher in the center and lower around the edges, with water surrounding it below. Regarding the four directions, the east is called "Yanggu," where the Sun rises; the west is called "Mengsi," where the Sun sets. The "Zhuangzi" also says: "In the Northern Sea there is a fish that transforms into a bird, intending to migrate to the Southern Sea." This is also an ancient record that can prove that all four directions are surrounded by water.

四方皆水，謂之四海。凡五行相生，水生於金；是故百川發源，皆自山出，由高趨下，歸注於海。日為陽精，光曜炎熾，一夜入水，所以焦竭。百川歸注，足以相補。故旱不為減，浸不為增。”

All four directions are water, which is called "four seas." According to the principle of mutual generation in the Five Phases, water is generated from metal; thus, all rivers originate from mountains, flowing from high to low, eventually converging into the sea. The Sun is the essence of yang, its light and heat intense. When it sets at night, it sinks into the water, thereby drying up and depleting the sea. However, the continuous flow of rivers replenishes what is lost, so during severe droughts the sea does not decrease, and during prolonged flooding the sea does not increase.

又云：“周天三百六十五度三百四分之七十五。天常西轉一日一夜過周一度。南北二極，相去一百一十六度三百四分度之六十五強，即天徑也。黃道斜帶赤道，春分交於奎七度，秋分交於軫十五度，冬至斗十四度半強，夏至井十六度半弱。從北極抉天而南五十五度強，則居天四維之中最高處也，即天頂也。其下即地中也。”自外與王蕃大同。

He Chengtian also said: The celestial circle is 365 and 75/304 degrees. The sky constantly rotates westward, and in one day and night it rotates an extra degree. The distance between the north and south poles is 116 and 65/304 degrees, slightly more, which is the diameter of the sky. The Yellow Path is inclined with respect to the equator, crossing the equator at seven degrees of the Dipper mansion during the vernal equinox, and at fifteen degrees of the Chariot mansion during the autumnal equinox. The winter solstice is at fourteen and a half degrees of the Dipper mansion, and the summer solstice is at sixteen and a half degrees of the Well mansion. Measuring southward from the north pole through the celestial dome by fifty-five degrees, that is the highest point of the celestial sphere's four dimensions, which is the celestial pole; directly below it is the center of the earth. Other accounts are largely similar to Wang Fan's.

徐爰《宋書》云：“元嘉十三年有詔太史令錢樂之，依舊說鑄渾天銅儀。徑六尺八分，周一丈八尺二寸六分。地在天內，立黃、赤二道、南、北二極，規佈列二十八宿、北斗、極星。五分為一度，置日、月、五星於黃道之上。置立漏刻，以水轉儀，昏明中星，與天相符。

Xu Yuan's "Song Shu" states: In the thirteenth year of Yuanjia, the court issued an edict ordering the Astronomer Royal Qian Yuezi to cast a bronze celestial sphere according to the old methods. Its diameter was six feet eight parts, and its circumference was one zhang eight feet two cun six parts. The earth was placed inside the heaven, with the Yellow Path and the Red Path, as well as the South Pole and North Pole, set up. The twenty-eight constellations, the Big Dipper, and the Pole Star were arranged in circles according to the rules. Each degree was five parts long, and the sun, moon, and five planets were placed on the Yellow Path. Additionally, water clocks were set up to drive the instrument's rotation, so that the stars at the southern mid-heaven at dusk and dawn matched the actual celestial phenomena.

到十七年，又被敕作小渾天。徑二尺二寸，周六尺六寸。二分為一度，二十八宿、中外宮星備具，以白色珠及青黃三色珠為三家星。日、月、五星悉居黃道。”

By the seventeenth year of Yuanjia, Qian Lezi again received an imperial order to construct a small armillary sphere. Its diameter was two feet two inches, and its circumference six feet six inches. Each degree was represented by two inches. The twenty-eight constellations, as well as the stars of the central and outer palaces, were all complete. White beads and beads of green, yellow, and other colors were used to represent the stars as defined by Shi Shen, Gan De, and Wu Xian. The sun, moon, and five planets were all placed on the Yellow Path.

梁奉朝請祖暅《渾天論》曰：“自古論天者多矣，而群氏紛紜，互相非毀。竊覽同異，稽之典經。仰觀辰極，傍矚四維。睹日、月之升降，察五星之見狀。校之以儀象，覆之以晷漏。則渾天之理，信而有徵。輒遺眾說附渾儀雲《考靈曜》，先儒求得天地相去十七萬八千五百里，以晷景驗之，失之於過多。既不顯求之術，而虛設其數，蓋誇誕之辭，非聖人之旨也。學者多因其說，而未之革，豈不知尋其理歟？抑未能求其數故也。

In the "Tian Lun" (Treatise on the Heavenly Sphere) of Zhu Geng, a descendant of Liang Dynasty's Feng Chaqing, he said: Since ancient times, many people have discussed the celestial phenomena, but their views differ greatly, and they have criticized and attacked each other. I have personally examined the differences and similarities among various schools, verified them with the classics, observed the North Star and the celestial pole, looked around to see the rising and setting of the sun and moon, examined the appearance and movements of the five planets, checked with instruments and celestial devices, and confirmed with gnomon shadows and water clocks. Thus, the principles of the浑天 theory are indeed credible and verifiable. Therefore, I set aside all other theories and focused solely on the浑仪 (celestial sphere instrument). The谶纬 (prophecies and interpretations) book "Shangshu Kaoliang" (Book of Documents: Examination of the Celestial Signs) and previous scholars calculated the distance between heaven and earth as seventeen thousand eight hundred and fifty li, but when tested with the sun's shadow, this number was found to be excessive. They refused to clearly explain the method used to arrive at this figure, and instead arbitrarily set this number, which is merely an exaggerated and false statement, not the original intention of the sages. Later scholars have mostly followed this claim without correction. Is it because they do not understand the principles behind it? Perhaps it is because they cannot find the correct method to calculate it.

王蕃所考校之前說，不啻減半。雖非揆格所知而求之以理，誠未能遙趣其實，蓋近密乎。輒因王蕃天高數以求冬至、春分日高，及南戴日下去地中數法；令表高八尺與冬至景長一丈三尺各自乘並而開方除之為法，天高乘表高為實，實如法；得四萬二千六百五十八里有奇，即冬至日高也。以天高乘冬至景長為實，實如法；得六萬九千三百二十里有奇，即冬至南戴日下去地中數也。

Wang Fan's calculation of the number is more than half less than previous theories. Although it is not entirely known through measurement and calculation, following the principles to seek the truth, even if it cannot approach the actual situation very closely, it is roughly more accurate. I then used Wang Fan's figure for the height of heaven to calculate the height of the sun at winter solstice and spring equinox, as well as the distance from the place directly under the sun to the center of the earth. The method is: square the height of the gnomon, eight chi, and the length of the shadow at winter solstice, one zhang and three chi, add them together, and take the square root as the divisor. Multiply the height of heaven by the height of the gnomon to get the dividend, then divide the dividend by the divisor, resulting in more than forty-two thousand six hundred and fifty-eight li, which is the height of the sun at winter solstice. Then multiply the height of heaven by the length of the shadow at winter solstice to get the dividend, and divide by the same divisor, resulting in more than sixty-nine thousand three hundred and twenty li, which is the distance from the place directly under the sun to the center of the earth.

求春秋分數法，令表高及春秋分景長五尺三寸九分各自乘，並而開方除之為法；因冬至日高實而以法除之，得六萬七千五百二里有奇，即春秋分日高也。以天高乘春秋分景長實，實如法而一，得四萬五千四百七十九里有奇，即春秋分南戴日下去地中數也。南戴日下，所謂丹穴也。

The method for calculating the distances at the vernal and autumnal equinoxes: square the height of the gnomon and the shadow lengths at the vernal and autumnal equinoxes, which are five feet three inches and nine parts respectively, add them together, and take the square root as the divisor. Use the same dividend as when calculating the sun's height at the winter solstice, and divide by this divisor to get 67,502 li and more, which is the height of the sun at the vernal and autumnal equinoxes. Then multiply the height of heaven by the shadow length at the vernal and autumnal equinoxes as the dividend, and divide by the divisor to get one, resulting in 45,479 li and more, which is the distance from the center of the earth to the place directly beneath the sun at the vernal and autumnal equinoxes. The place directly beneath the sun is what the ancient texts refer to as the 'Red Pit'.

推北極裡數法；夜於地中表南傳地遙望北辰細星之末，令與端參合，以人目去表數及表高各自乘，並而開方除之為法，天高乘表高數為實，實如法而一，即北極細星高地數也。天高乘人目去表為實，實如法；即去北辰極下之數也。北戴斗極為空桐，日去赤道表裡二十四度，遠寒近暑而中和。

The method for calculating the distance to the North Star: at night, stand a gnomon in the center of the land and look northward towards the end of the North Star, aligning your line of sight with the top of the gnomon. Square the distance from the person's eye to the gnomon and the height of the gnomon, add them together, and take the square root of the sum as the divisor. Multiply the height of the sky by the height of the gnomon to get the dividend, then divide by the divisor to obtain the height of the North Star above the ground. Next, multiply the height of the sky by the distance from the person's eye to the gnomon to get another dividend, then divide by the same divisor to find the distance to the point directly beneath the North Star. The area directly above the Big Dipper and the North Star is called "Kongtong." The Sun rises and sets 24 degrees north and south of the equator; the farther it is, the colder the climate, the closer it is, the hotter the climate, and when it is in the middle, the climate is temperate.

二分之日，去天頂三十六度，日去地中，冬、夏、春、秋、晨、昏、晝、夜，皆同度也。而有寒暑者，地氣上騰，天氣下降故遠日下而寒，近日下而寒，非有遠近也。猶火居上，雖遠而炎；在旁，雖近而微。視日，在旁而大，居上而小者，仰矚為難，平視為易也。由視有夷險，非遠近之效也。今懸殊於百仞之上，或置之於百仞之前，從而觀之，則大小殊矣。先儒弗斯取驗取給，用繁翰墨，夷途頓轡，雄辭折辯，不亦迂哉。

On the day of the two equinoxes, the Sun is thirty-six degrees away from the zenith. The distance of the Sun from the center of the earth remains the same whether it is winter, summer, spring, or autumn, whether it is morning, evening, day, or night. Yet there are still cold and heat, which is due to the rising of the earth's qi and the descending of the heavenly qi: places farther from the Sun's direct overhead position are colder, while those closer are hotter (the original text has 'cold' here, but it should be 'heat'). It is not because the Sun itself is farther or closer. For example, a fire above one's head, though far away, still feels hot; a fire beside one, though near, seems weak. Looking at the Sun, when it is near the horizon, it appears larger, but when it is overhead, it seems smaller. This is because looking upward is more difficult, while looking level is easier, and it is the ease or difficulty of viewing that causes this, not the actual distance. Now, if an object is hung high up a hundred feet or placed a hundred feet ahead, when you walk past it, its size appears different. Previous scholars did not verify this by observing the actual situation or seek a simple explanation, but instead spent much ink arguing over this by stopping their horses on flat ground and using eloquent words to debate. Is this not too narrow-minded?

今大寒在冬至後二氣者，寒積而未消也。大暑在夏至後二氣者，暑積而未歇也。寒暑均和乃在春秋分後二氣者，寒暑積而未平也。譬之火始下室，而未甚溫，弗事加薪，久而愈熾，既已遷之，猶有餘熱也。月行有中道，有陰道，有陽道。

Now, the Great Cold comes two solar terms after the Winter Solstice, which is when the cold has accumulated but has not yet dissipated; the Great Heat comes two solar terms after the Summer Solstice, which is when the heat has accumulated but has not yet ceased; when the cold and heat are balanced and harmonious, coming two solar terms after the Spring Equinox and Autumn Equinox, it is when the cold and heat have accumulated but have not yet been balanced. This is like a fire just lit in a room, which is not yet warm. If the fire is kept burning and more fuel is added, the longer the time, the hotter it becomes. When the fire is moved away, the room still retains the residual heat. The moon's movement has a middle path, a yin path, and a yang path.

如姜岌說《周髀》云：‘日徑千二百五十里。’蓋天乖謬，已詳前識，無足採焉。以渾象言之，失於過大矣。張衡日、月其徑，當周天七百三十六分之一，地廣二百四十二分之一。按此而論，天周分數圓周率也。廣分數圓徑率也。以八約之，得周率九十二，徑率二十九；其率傷於周多徑少，衡之疏也。

As for Jiang Ji's citation of the "Zhou Bi Suan Jing" stating that the diameter of the Sun is 1,250 li, the errors in the theory of a dome-shaped sky have already been clearly analyzed and are not worth adopting; in terms of the浑象 (celestial sphere model), this number is excessively large. Zhang Heng claimed that the diameter of the Sun and Moon should be one seven hundred thirty-sixth of the entire celestial circle and one two hundred forty-second of the Earth's breadth. According to this, the fraction of the celestial circle corresponds to the value of pi, and the fraction of the Earth's breadth corresponds to the ratio of the diameter to the circumference. Simplifying this by dividing by eight, we get a pi value of ninety-two and a diameter-to-circumference ratio of twenty-nine. The problem with this ratio lies in the fact that the circumference is too large and the diameter too small, which is a mistake in Zhang Heng's calculation.

衡以日月之徑居一度之半，又言八極之維徑二億三萬二千三百里。既非考定日月徑又不明其理，飾辭華說，不足窮= 者也。望日月法，立於地中，以人目屬徑寸之管而望日月，令日月大滿管孔；及定管長，以管徑乘天高，管長除之，即日月徑也。

Zhang Heng thought that the diameter of the sun and moon occupied half a degree, and he also said that the diameter of the eight extremes and four directions was two hundred and thirty-two thousand three hundred li. He neither determined the actual diameter of the sun and moon nor explained the reasoning clearly, but only adorned it with ornate language, which was insufficient to thoroughly investigate the principles. The method of measuring the sun and moon is: stand in the center of the earth, press your eye against a tube with an inner diameter of one inch to look at the sun and moon, making sure the sun and moon exactly fill the tube's hole; then measure the length of the tube, multiply the tube's diameter by the height of the sky, and divide by the tube's length, the result will be the diameter of the sun and moon.

月，陰精也。其形圓，其質稟日之光而見。其體日光不照則謂之魄。故月望之日，日月相望，人居其間，盡觀其質，故形圓也。二絃之日，日照其側，人觀其旁，故半魄半明也。晦逆之日，月照其表，人在其裡，故形不見。舊說日月半魄半明也。晦朔之日，月照其表，人在其裡，故形不見。舊說日月行九道，斯蓋未究月行之變，而假為其說。

The moon is the essence of darkness, with a round shape. Its body can only be seen when illuminated by the sun. The part that is not illuminated by the sun is called 「魄」. On the full moon day, the sun and moon are opposite each other, and people are in the middle, seeing the fully illuminated side, hence the shape appears round. On the days of the new and last quarters, the sun shines on the side of the moon, and people view it from the side, hence it appears half-illuminated. On the days of the new moon and the last moon, the illuminated side of the moon faces away from people, and people are on the dark side of the moon, hence the moon's shape is not visible. (The text from 「旧说日月半魄半明也」 onwards is repeated, and is recorded as is.) The old saying mentions that the sun and moon travel along nine paths, but this is a fabricated explanation without investigating the actual changes in the moon's movement.

謹按：著作郎張東對三道九行曰：‘按《漢天文志》：日有三道，月有九行者，謂日、月行黃道，為得其正，或行黃道外，或行黃道內，內、外及正行，因名三道。故曰：房有四表三道。日、月經房星南門即為旱，經房星北門即為水。日有三道者，但內外差及正行之道也。

Carefully examined: The official Zhang Dong, in response to a question about the 'three paths and nine lines', said: According to the 'Astronomical Records' of the 'Book of Han', the sun has three paths and the moon has nine lines. This refers to the sun and moon moving along the Yellow Path being considered proper. Sometimes they move outside the Yellow Path, sometimes inside. Adding the inner, outer, and the central path, it is called three paths. Therefore, the Room Mansion has four gates and three paths: if the sun and moon pass through the southern gate of the Room Mansion, it indicates drought; if they pass through the northern gate, it indicates flooding. The saying that the sun has three paths merely refers to the deviation inside and outside the Yellow Path plus the central path.

月有九行，謂青道二，出於黃道東；赤道二，出於黃道南；白道二，出在黃道西；黑道二，出於黃道東；赤道二，出於黃道南；白道二，出在黃道西；黑道二，出在黃道北。故鄭玄注《月令》，立春、春分，日行青道，月為之佐；立夏、夏至，日行赤道，月為之佐；立秋、秋分，日行白道，月為之佐；立冬、冬至，日行黑道，月為之佐。

The moon has nine paths. This refers to two paths of the blue path emerging to the east of the Yellow Path, two paths of the red path emerging to the south of the Yellow Path, two paths of the white path emerging to the west of the Yellow Path, and two paths of the black path emerging to the north of the Yellow Path (the original text from "the black path two, emerging to the east of the Yellow Path" downwards is repeated and contains errors, and is recorded as is). Therefore, Zheng Xuan commented in his annotation of the "Book of Rites · Monthly Orders" that during the beginning of spring and the vernal equinox, the sun travels along the blue path, and the moon assists it; during the beginning of summer and the summer solstice, the sun travels along the red path, and the moon assists it; during the beginning of autumn and the autumn equinox, the sun travels along the white path, and the moon assists it; during the beginning of winter and the winter solstice, the sun travels along the black path, and the moon assists it.

故月之九行，非為黃道外別有九道，交橫綿絡，貫於躔舍，但為日月，既為動物，不能不小有盈縮，或行黃道內外。又按：日、月以立春合宿在營室，即北方之星，色當言黑。春分合宿在奎，奎為西方之宿，色當言白。而鄭云：立春、春風，日行青道者，考以歲起於春，春在東方，東方色青，故起青號。

Therefore, the nine paths of the moon are not nine separate roads outside the Yellow Path that cross and connect through the constellations passed by the sun and moon; rather, it is because both the sun and moon are moving bodies, and they naturally have slight variations in their movement, sometimes passing within the Yellow Path and sometimes outside of it. Also, according to the text, the sun and moon meet at the Encampment mansion at the Beginning of Spring, and Encampment is a northern star, which by nature should be black. At the Vernal Equinox, they meet at the Legs mansion, which is a western mansion, and by nature should be white. However, Zheng Xuan said that at the Beginning of Spring and the Vernal Equinox, the sun travels along the blue path, taking into account that the year begins in spring, and spring is in the east, where the color is blue, so he used the term blue to name it.

南、西及北，三方準此方，故日、月共有三道，實無九行。但以因八節之差，順四時之色，曲為九名，故月行言九；直舉其實，故日道稱三；三道九行，其義一也。則夫日之三道，躔於二十八宿，月之九行，經於八節也。

The southern, western, and northern directions are also compared in this way, so the sun and moon actually have only three paths, not nine lines. It is only because they change with the eight solar terms that there is a difference, following the colors of the four seasons, and thus nine names are established indirectly. Therefore, it is said that the moon has nine paths. When speaking directly about its actual situation, it is said that the sun's path is three. The three paths and nine lines are essentially the same thing. Thus, the three paths of the sun are among the twenty-eight mansions, while the nine paths of the moon are distributed among the eight solar terms.

染武帝支：‘自故以來，談天者多矣，皆是不識天象，各隨意造；家執所說，人著異見，非直毫釐之差，蓋實千里之謬。戴盆而望，安能見天，譬猶宅蝸牛之角，而欲論天之廣狹；懷蚌螺之南寧，而欲測海之多少；此可謂不知量矣。

Liang Wu Di said (the original text's four characters 'Ran Wu Di Zhi' are a mistake for 'Liang Wu Di Yun'): Since ancient times, many have talked about the heavens, but they all did not understand the heavenly phenomena and merely fabricated their own interpretations; each family clung to their own views, and everyone established different opinions, which is not just a slight error, but actually a great mistake. Trying to look at the sky with a pot on one's head, how can one see the sky? It is like living in the shell of a snail and yet trying to discuss the breadth and narrowness of the sky, or holding a clam shell or a ladle and yet trying to measure the volume of the sea. This can be called an act of self-overestimation.