1
志第十二曆一
Treatise 12: Calendars, Part One
2
太古聖人，體二氣之權輿，賾三才之物象，乃創紀以窮其數，畫卦以通其變。 而紀有大衍之法，卦有推策之文，由是曆法生焉。 殷人用九疇、五紀之書，《周禮》載馮相、保章之職，所以辨三辰之躔次，察九野之吉凶。 歷代疇人，迭相傳授，蓋推步之成法，協用之舊章。 暨秦氏焚書，遺文殘缺，漢興作者，師法多門。 雖同徵鐘律之文，共演蓍龜之說，而建元或異，積蔀相懸，旁取證於《春秋》，強乩疑於《系》、《象》，靡不揚眉抵掌，謂甘、石未稱日官； 運策播精，言裨、梓不知天道。 及至清台眎祲，黃道考祥，言縮則盈，少中多否，否則矯云差算，中則自負知時。 章、亥不生，憑何質證？
In remote antiquity the sage rulers took up the genesis of yin and yang and fathomed the forms of Heaven, Earth, and humanity; they devised chronology to exhaust number and drew the hexagrams to trace change. Chronology held the method of the Great Expansion, and the hexagrams held the texts for casting and counting—out of this calendrical science was born. The Yin drew on the books of the Nine Categories and the Five Regulators; the Rites of Zhou sets down the posts of the court astronomers, by which one traced the courses of sun, moon, and stars and read fortune and misfortune across the nine regions. Down the generations calendrical experts passed their craft from hand to hand—the settled rules of computation and the long-standing regulations they applied together. After the Qin burned the classics, what survived was broken; when the Han arose, each new author followed a different school. They might cite the same texts on pitch-pipes and bells and rehearse the same lore of milfoil and tortoise, yet their era-origins diverged and their cycle constants clashed; some dragged in the Spring and Autumn Annals as proof, others forced the Appended Texts and Images into doubt—every party arched a brow and clapped a palm, insisting that Gan De and Shi Shen were unworthy to be called officers of the sun; Those who cast stalks and sifted the subtle said that Gongshu Ban and Lu Ban did not understand the Way of Heaven. When they turned to the observatory to read omens and tested fortune along the ecliptic, what they said shrank when fullness came; few predictions hit and many failed—on a miss they blamed faulty arithmetic, on a hit they boasted that they knew the seasons. Zhang Shou and Guo Hehai were not living in our day—what proof can settle the matter?
3
高齊天保中，六月日當蝕朔，文宣先期問候官蝕何時，張孟賓言蝕申，鄭元偉、董峻言蝕辰，宋景業言蝕巳。 是日蝕于申酉之間，言皆不中時。 景業造《天保曆》則疏密可知矣。 昔鄧平、洛下閎造漢《太初曆》，非之者十七家。 後劉洪、蔡伯喈、何承天、祖沖之，皆數術之精粹者，至於宣考曆書之際，猶為橫議所排。 斯道寂寥，知音蓋寡。 所以張胄玄佩印而沸騰，劉孝孫輿棺而慟哭，俾諸後學，益用為疑。 以臣折衷，無如舊法。
Under the Northern Qi, in the Tianbao era, a solar eclipse was due at new moon in the sixth month. Emperor Wenxuan asked the astronomy clerk in advance when the eclipse would occur. Zhang Mengbin said at shen, Zheng Yuanwei and Dong Jun said at chen, and Song Jingye said at si. That day the eclipse fell between the shen and you hours—not one prediction matched the moment. From the Tianbao Calendar that Jingye devised, one can see how coarse or fine it was. Long ago Deng Ping and Luoxia Hong devised the Han Taichu Calendar, and seventeen schools rose against it. Later Liu Hong, Cai Yong, He Chengtian, and Zu Chongzhi were all masters of mathematical astronomy; yet when their calendars came up for imperial review, partisan clamor still shut them out. The art has grown desolate, and those who truly understand it are few. Hence Zhang Zaiyuan wore his seal while the court seethed, and Liu Xiaosun brought a coffin and wept in protest—so that later students only grew more uncertain. In this subject's judgment, nothing surpasses the established methods.
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高祖受隋禪，傅仁均首陳七事，言戊寅歲時正得上元之首，宜定新曆，以符禪代，由是造《戊寅曆》。 祖孝孫、李淳風立理駁之，仁均條答甚詳，故法行於貞觀之世。 高宗時，太史奏舊曆加時浸差，宜有改定，乃詔李淳風造《麟德曆》。 初，隋末劉焯造《皇極曆》，其道不行，淳風約之為法，時稱精密。 天后時，瞿曇羅造《光宅曆》。 中宗時，南宮說造《景龍曆》，皆舊法之所棄者，復取用之。 徒云革易，寧造深微，尋亦不行。 開元中，僧一行精諸家曆法，言《麟德曆》行用既久，晷緯漸差。 宰相張說言之，玄宗召見，令造新曆。 遂與星官梁令瓚先造《黃道遊儀圖》，考校七曜行度，准《周易》大衍之數，別成一法，行用垂五十年。 肅宗時，韓潁造《至德曆》。 代宗時，郭獻之造《五紀曆》。 德宗時，徐承嗣造《正元曆》。 憲宗時，徐昂造《觀象曆》。 其法今存，而元紀蔀章之數，或異前經； 而察斂啟閉之期，何殊舊法。 至論征驗，罕及研精。 綿代流行，示存經法耳。
When Gaozu took the throne from the Sui, Fu Renjun was first to submit seven proposals, arguing that the wuyin year fell at the head of the upper origin and a new calendar should be set to match the transfer of mandate—thus the Wuyin Calendar was created. Zu Xiaosun and Li Chunfeng argued against him on principle; Renjun rebutted each point at length, and his system was adopted in the Zhenguan reign. Under Gaozong the Grand Astrologer reported that the old calendar's time corrections were drifting and ought to be revised; the emperor ordered Li Chunfeng to create the Linde Calendar. Earlier, at the end of the Sui, Liu Zhuo had devised the Huangji Calendar, but his method had not been adopted; Chunfeng distilled it into a system that contemporaries hailed as precise. Under the Empress Wu, Gautama Siddhartha devised the Guangzhai Calendar. Under Zhongzong, Nangong Shuo devised the Jinglong Calendar—methods the older tradition had already cast aside, yet they were revived. They spoke only of change and could not produce anything truly deep; in the end none of these was kept in use. In the Kaiyuan era the monk Yixing mastered every school's calendrical methods and reported that the Linde Calendar had been in use so long that gnomon readings and celestial coordinates were drifting. Chief Minister Zhang Yue relayed this; Xuanzong summoned Yixing and ordered a new calendar. He and the court astronomer Liang Lingzan first built the armillary chart of the ecliptic, verified the courses of the seven luminaries, took the Great Expansion numbers from the Book of Changes as their standard, and fashioned a new method that remained in use for nearly fifty years. Under Suzong, Han Ying devised the Zhide Calendar. Under Daizong, Guo Xianzhi devised the Wuji Calendar. Under Dezong, Xu Chengsi devised the Zhengyuan Calendar. Under Xianzong, Xu Ang devised the Guanxiang Calendar. Their methods still survive, yet the constants for era-cycles, obscurations, and intercalation sometimes diverge from earlier canons; yet in fixing the seasons of solstice, equinox, and the quarter-points, how do they differ from the old methods? When it comes to empirical proof, few achieve real precision. Handed down through the ages, they merely show that the canonical methods still endure.
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前史取傅仁均、李淳風、南宮說、一行四家歷經，為《曆志》四卷。 近代精數者，皆以淳風、一行之法，曆千古而無差，後人更之，要立異耳，無逾其精密也。 《景龍曆》不經行用，世以為非，今略而不載。 但取《戊寅》、《麟德》、《大衍》三曆法，以備此志，示於疇官爾。
The earlier history drew the calendrical treatises of Fu Renjun, Li Chunfeng, Nangong Shuo, and Yixing from four masters and compiled them into four fascicles of the Treatise on Calendars. Recent masters of calculation all hold that the methods of Chunfeng and Yixing stand for ages without measurable error; later reformers sought chiefly to distinguish themselves and could not surpass their precision. The Jinglong Calendar was never adopted in practice; the world judged it unsound, and it is passed over here. Only the methods of the Wuyin, Linde, and Dayan calendars are included in this treatise, set before the calendrical officers.
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戊寅歷經
The Wuyin Calendar Classic
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已上闕文日。 自入立秋，初日加四千八十分，後日減七十六分，置初日所加之分，計後日減之數以減之。 訖，餘以行分法約之，為日數。 及加平見日及分，滿行分法，又去之，從日一，為定見日及分。 後皆放此。 畢于秋分。 自入寒露，日減一百二十七分，減若不足，即一日加行分法，反減之，為定見日及分。 後皆放此。 畢于立冬。 自入小雪，畢于大雪，均減八日。 初見去日十四度。
The text above is lost as of this record. From the Start of Autumn, add 4,080 parts on the first day and subtract 76 parts on each later day; take the first day's addition and subtract the running total of later reductions. When this is done, reduce the remainder by the motion-part divisor to obtain the day count. Add these to the mean appearance day and parts; whenever the total fills the motion-part divisor, remove one divisor and carry one day—this yields the fixed appearance day and parts. All later cases follow this rule. This continues through the Autumn Equinox. From Cold Dew, subtract 127 parts per day; if the subtraction cannot be completed, add one motion-part divisor for one day and subtract in reverse to obtain the fixed appearance day and parts. All later cases follow this rule. This continues through the Start of Winter. From Minor Snow through Major Snow, reduce evenly by eight days. At first appearance it stands fourteen degrees from the sun.
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熒惑
Mars
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平見：入冬至，初日減一萬六千三百五十四分，後日減五百四十五分，畢於小寒。 自入大寒，日加四百二十六分，畢於啟蟄。 自入雨水，畢于穀雨，均加二十九日。 入立夏，初日加一萬九千三百九十二分，後日減二百一十三分，畢於大暑。 自入立秋，依平。 自入處暑，日減一百八十四分，畢于立冬。 自入小雪，畢于大雪。 均減二十五日。 初見去日十七度。
Mean appearance: from the Winter Solstice, subtract 16,354 parts on the first day and 545 parts on each later day, through Minor Cold. From Major Cold, add 426 parts per day through the Start of Insects. From Rain Water through Grain Rain, add evenly twenty-nine days. From the Start of Summer, add 19,392 parts on the first day and subtract 213 parts on each later day, through Major Heat. From the Start of Autumn, follow the mean rate. From End of Heat, subtract 184 parts per day through the Start of Winter. From Minor Snow through Major Snow. Reduce evenly by twenty-five days. At first appearance it stands seventeen degrees from the sun.
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鎮星
Saturn
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平見：入冬至，初日減四千八百一十四分，後日加七十九分，畢於氣盡。 自入小寒，畢于大寒。 均減九日。 入立春，均減八日。 入啟蟄，均減七日。 入雨水，均減六日。 入春分，均減五日。 入清明，均減四日。 入穀雨，畢芒種，均減三日。 入夏至，畢十日內，均減二日。 十日外，入小暑，畢五日內，均減一日。 五日外，畢於氣盡，依平。 自入大暑，日加一百八十一分，畢于立秋。 自入處暑，均加九日。 自入白露，初日加六千二分，後日減一百三十三分，畢於寒露。 自入霜降，日減七十九分，畢于大雪。 初見去日十七度。
Mean appearance: from the Winter Solstice, subtract 4,814 parts on the first day and add 79 parts on each later day, until the qi period ends. From Minor Cold through Major Cold. Reduce evenly by nine days. At the Start of Spring, reduce evenly by eight days. At the Start of Insects, reduce evenly by seven days. At Rain Water, reduce evenly by six days. At the Spring Equinox, reduce evenly by five days. At Clear Brightness, reduce evenly by four days. From Grain Rain through Grain in Ear, reduce evenly by three days. From the Summer Solstice, within the first ten days reduce evenly by two days. After the tenth day, from Minor Heat, within the first five days reduce evenly by one day. After the fifth day, through the end of the qi period, follow the mean rate. From Major Heat, add 181 parts per day through the Start of Autumn. From End of Heat, add evenly nine days. From White Dew, add 6,002 parts on the first day and subtract 133 parts on each later day, through Cold Dew. From Frost Descent, subtract 79 parts per day through Major Snow. At first appearance it stands seventeen degrees from the sun.
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太白
Venus
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晨平見：入冬至，依平。 自入小寒，日加六十六分，畢于大寒。 自入立春，畢于立夏，均加三日。 自入小滿，初日加一千九百六十四分，後日減六十六分，畢於芒種。 自入夏至，依平。 自入小暑，減六十分，畢於大暑。 自入立秋，畢于立冬，均減三日。 自入小雪，初日減一千九百六十四分，後日減六十六分，畢大寒。
Morning mean appearance: from the Winter Solstice, follow the mean rate. From Minor Cold, add 66 parts per day through Major Cold. From the Start of Spring through the Start of Summer, add evenly three days. From Minor Fullness, add 1,964 parts on the first day and subtract 66 parts on each later day, through Grain in Ear. From the Summer Solstice, follow the mean rate. From Minor Heat, subtract 60 parts through Major Heat. From the Start of Autumn through the Start of Winter, reduce evenly three days. From Minor Snow, subtract 1,964 parts on the first day and 66 parts on each later day, through Major Cold.
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夕平見：入冬至，日減一百分，畢于立春。 自入啟蟄，畢于春分，均減九日。 自入清明，初日減五千九百八十六分，後日減一百分，畢于小滿。 自入芒種，依平。 自入夏至，日加一百分，畢于立秋。 自入處暑，畢于秋分，均加九日。 自入寒露，初日加五千九百八十六分，後日減一百分，畢于小雪。 自入大雪，依平。 初見去日十一度。
Evening mean appearance: from the Winter Solstice, subtract 100 parts per day through the Start of Spring. From the Start of Insects through the Spring Equinox, reduce evenly nine days. From Clear Brightness, subtract 5,986 parts on the first day and 100 parts on each later day, through Minor Fullness. From Grain in Ear, follow the mean rate. From the Summer Solstice, add 100 parts per day through the Start of Autumn. From End of Heat through the Autumn Equinox, add evenly nine days. From Cold Dew, add 5,986 parts on the first day and subtract 100 parts on each later day, through Minor Snow. From Major Snow, follow the mean rate. At first appearance it stands eleven degrees from the sun.
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辰星
Mercury
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晨平見：入冬至，均減四日。 自入小寒，畢于大寒，依平。 自入立春，畢啟蟄，減三日。 其在啟蟄氣內，去日一十八度外、四十度內，晨無木、土、金一星已上者，不見也。 自入雨水，畢于立夏，應見不見。 其在立夏氣內，去日度如前，晨有木、火、土、金一星已上者，亦見之。 自入小滿，畢於寒露，依平。 自入霜降，畢于立冬，加一日。 自入小雪，畢于大雪十二日，依平。 若在大雪十三日，即減一日。 在十四日，減二日。 在十五日，減三日。 在十六日，減四日。
Morning mean appearance: from the Winter Solstice, reduce evenly four days. From Minor Cold through Major Cold, follow the mean rate. From the Start of Spring through Awakening of Insects, subtract three days. If within Awakening of Insects, more than eighteen degrees but less than forty degrees from the sun, and no Jupiter, Saturn, or Venus is visible in the morning, it is not seen. From Rain Water through the Start of Summer, it should appear but does not. If within Start of Summer, at the same solar distance as described above, and Jupiter, Mars, Saturn, or Venus is visible in the morning, it is also seen. From Minor Fullness through Cold Dew, follow the mean rate. From Frost's Descent through the Start of Winter, add one day. From Minor Snow through the twelfth day before Major Snow, follow the mean rate. If it falls on the thirteenth day before Major Snow, subtract one day. On the fourteenth day before Major Snow, subtract two days. On the fifteenth day before Major Snow, subtract three days. On the sixteenth day before Major Snow, subtract four days.
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夕平見：入冬至，畢于清明，依平。 自入穀雨，畢於芒種，減二日。 自入夏至，畢於大暑，依平。 自入立秋，畢于霜降，應見不見。 其在立秋及霜降二氣之內，夕有星去日如前晨者，亦見。 自入立冬，畢于大雪，依平。 初見去日十七度。
Evening mean appearance: from the Winter Solstice through Clear Brightness, follow the mean rate. From Grain Rain through Grain in Ear, subtract two days. From the Summer Solstice through Major Heat, follow the mean rate. From the Start of Autumn through Frost's Descent, it should appear but does not. If within Start of Autumn or Frost's Descent, at the same evening solar distance as the morning rule above, it is also seen. From the Start of Winter through Major Snow, follow the mean rate. At first appearance it stands seventeen degrees from the sun.
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行五星法
Method for Computing the Five Planets' Motion
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各置星定見之前夜半日所在宿度算及分，各以定見去朔日算及一分加之。 小分滿法十四分，從行分一。 行分滿法六百七十六分，從度一。 又以星初見去日度數，晨減夕加之。 命度以次，即星初見所在度及分。 自此已後，皆棄此小分也。
For each planet, set the lodge degree count and parts of the sun's position at midnight before its fixed appearance; to each add the fixed appearance's offset from new moon in days and one part. When minor parts fill the divisor of fourteen parts, carry one to the motion-parts. When motion-parts fill the divisor of 676 parts, carry one degree. Take the star's first-appearance distance from the sun; subtract for morning appearances and add for evening appearances. Count off through the degrees in order to obtain the star's first-appearance degree and parts. From this point onward, discard these minor parts.
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求次日術
Procedure for the Next Day
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各加一日所行度及分。 其火、金之行而有小分者，各以日率為母。 小分滿其母，去從行分一。 行分滿法，去從度一。 其行有益疾遲者，副置一日行分。 各以其分疾益遲損，乃加之。 留者因前，退則減之，伏不注度。 順行出斗去其分，行入斗先加分。 訖，皆以二十六副行分為度分。
Add each day's motion in degrees and parts. For Mars and Venus, when minor parts are involved, use the day-rate as denominator. When minor parts fill their denominator, cast out the excess and carry one to the motion-parts. When motion-parts fill the divisor, cast out the excess and carry one degree. When motion varies between faster and slower phases, set aside one day's motion-parts. Apply the daily differential for acceleration or deceleration, then add the result. During stationary motion, continue from the prior value; during retrograde motion, subtract accordingly; during invisibility, do not record longitude. In direct motion exiting the Dipper asterism, subtract the fractional parts; in motion entering the Dipper, first add the fractional parts. When complete, divide all motion-parts by 26 to convert to degree-parts.
22
歲星
Jupiter
23
初見：順，日行一百七十六分五十秒，日益遲一分。 一百一十四日行十九度二百九分。 而留，二十八日。 乃退，日九十七分。 八十四日退十二度五十分。 又留，二十六日五百九十六，小分七四分。 即以初定見日分而加之，若滿行分法，即去之，從月去之，從一日。 乃順，初日行六十分，日益疾一分。 一百十四日行十九度四百三十七分而伏。
At first appearance it moves directly, 176 parts 50 seconds per day, slowing by 1 part each day. In 114 days it travels 19 degrees 209 parts. Then it stations for 28 days. Then it retrogrades 97 parts per day. In 84 days it retreats 12 degrees 50 parts. It stations again for 26 days 596 parts, 74 minor parts. Add the parts from the first fixed-appearance day; whenever the total fills the motion-part divisor, cast out the excess and carry one day. Then it moves directly again: 60 parts on the first day, speeding by 1 part each day. In 114 days it travels 19 degrees 437 parts and becomes invisible.
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熒惑
Mars
25
初見：入冬至，初率二百四十一日行一百六十三度。 已後二日損日及度各一。 盡一百二十八日，率一百七十七日行九十九度。 畢一百六十一日皆同。 已後三日損日及度各一。 盡一百八十二日，率一百七十日行九十二度。 畢一百八十八日皆同。 已後三日益日及度各一。 盡二百二十七日，率一百八十三日行一百五度。 已後二日益日及度各一。 盡二百四十九日，率一百九十四日行一百一十六度。 已後一日益日及度各一。 盡三百一十日，率二百五十五日行一百七十七度。 畢三百三十七日皆同。 已後二日損。 盡三百六十五日，復二百四十一日行一百六十三度。
At first appearance upon entering the Winter Solstice, the initial rate is 241 days for 163 degrees of travel. Thereafter decrease the day-rate and degree-rate by 1 every 2 days. Through 128 days the rate becomes 177 days for 99 degrees. This rate holds through day 161. Then decrease the day-rate and degree-rate by 1 every 3 days. Through 182 days the rate becomes 170 days for 92 degrees. This rate holds through day 188. Then increase the day-rate and degree-rate by 1 every 3 days. Through 227 days the rate becomes 183 days for 105 degrees. Then increase the day-rate and degree-rate by 1 every 2 days. Through 249 days the rate becomes 194 days for 116 degrees. Then increase the day-rate and degree-rate by 1 each day. Through 310 days the rate becomes 255 days for 177 degrees. This rate holds through day 337. Thereafter decrease by 1 every 2 days. Through 365 days it returns to the initial rate of 241 days for 163 degrees.
26
初見：入小寒已後，三日去日率一，畢於啟蟄。 自入雨水，畢于立夏，均去日率二十。 自入小滿，初去日率二十。 以次三日去十九，日日去十八。 以次三日去一日，畢於小暑，即依平，為定日之率。 若入處暑，畢于秋分，皆去度率六，各依冬至後日數而損益之，又依所入之氣以減之，名為前疾。 日數及度數之率，若初行。 入大寒，畢於大暑，皆差行，日益遲一分。 其餘皆平行。 若入白露，畢于秋分，初日行半度，四十日行二十度。 即去日率四十，度率二十，別為半度之行，訖，然後求平行之分以續之。 平行分者，置定行度率，以分法乘之，以定日率除之，所得即平行一日之分，不盡為小分。 求差行者，置日率之數，減一。 訖，又半之，加平行一日之分，為初日行分。 各盡其日度而遲。 初日行三百二十六分，日益遲一分半，六十日行二十五度五分。 其前疾去度六者，此遲初日加六十七分、小分三十六。 小分滿六十，去之，從行分一，即六十日行三十一度，分同。 而留，十二日。 前去日分日於二留，奇後從後留。 乃退，日一百九十二分，六十日退十七度二十八分。 又留，十二日六百二十六分、小分三十分。 亦如初定見之分，滿去如前。 又順，後遲。 初日行二百三十八分，日益疾一分半，六十日行二十五度三十五分。 此遲在立秋至秋分者，加一日，行六十七、小分三十六。 滿去如前，即六十日行三十一度。 分同也。 而後疾。 入冬至，初率二百一十四日行一百三十六度。 已後一日損日及度各一。 盡三十七日，率一百七十七日行九十九度。 已後二日損日及度各一。 盡五十七日，率一百六十七日行八十九度。 畢七十九日皆同。 已後三日益日及度各一。 盡一百三十日，率一百八十四日行一百六十度。 已後二日益日及度各一。 盡一百四十四日，率一百九十一日行一百一十三度。 已後一日益日及度各一。 盡一百九十日，率二百三十七日行一百五十九度。 已後一日益日及度各一。 盡二百一十日，率二百六十七日行一百八十九度。 畢二百五十九日皆同。 已後二日損日及度各一。 盡三百六十五日，復率二百一十四行一百三十六度。 後遲加六度者，此後疾去度率六，為定度。 各依冬至後日數而損益之，為後疾日及度之率。 若入立夏，於夏至，日行半度，盡六十日，行三十度。 若入小暑，於大暑，盡四十日，行二十度。 皆去日及度之率，別為半度之行，訖，然後求平行之分以續之。 各盡其日度而伏。
At first appearance after Minor Cold, remove 1 from the day-rate every 3 days through Awakening of Insects. From Rain Water through the Start of Summer, uniformly remove 20 from the day-rate. From Minor Fullness, initially remove 20 from the day-rate. Next remove 19 every 3 days, then 18 each day. Next remove 1 day every 3 days through Minor Heat, then follow the mean rate—this is the fixed day-rate. If the motion begins between End of Heat and the Autumn Equinox, subtract 6 from the degree-rate throughout; adjust according to the number of days past the Winter Solstice and subtract according to the solar term entered—this is called initial rapid motion. The day-rate and degree-rate follow the same rules as the initial motion phase. If the motion falls between Major Cold and Major Heat, use variable motion, slowing by 1 part each day. In all other seasons, use uniform motion. If entering between White Dew and the Autumn Equinox, start at half a degree per day, covering 20 degrees in 40 days. Set aside 40 days of the day-rate and 20 degrees of the degree-rate as a separate half-degree segment; when that segment is complete, calculate the uniform-motion parts and continue from there. To find uniform-motion parts, multiply the fixed degree-rate by the part divisor and divide by the fixed day-rate; the quotient is one day's uniform motion, and the remainder gives the minor parts. For differential motion, take the day-rate and subtract 1. Halve the result, add the one-day uniform-motion parts—that yields the initial day's motion-parts. Continue until the allotted days and degrees are used up, then enter slow motion. Initial daily motion is 326 parts, slowing by 1.5 parts each day; in 60 days it covers 25 degrees and 5 parts. If the initial rapid phase had 6 degrees subtracted from the degree-rate, on the first day of this slow phase add 67 parts and 36 minor parts. When minor parts fill 60, cast out the excess and carry one to the motion-parts; in 60 days it covers 31 degrees with the same fractional value. Then it stands still for 12 days. If days were subtracted during initial rapid motion, split those days between the two stationary periods, with any odd days assigned to the later one. Then it retrogrades at 192 parts per day, covering 17 degrees 28 parts in 60 days. It stands still again for 12 days 626 parts, 30 minor parts. Apply the parts from the first fixed appearance as before; when full, cast out the excess as above. Then it resumes direct motion in the later slow phase. In the later slow phase, initial daily motion is 238 parts, accelerating by 1.5 parts each day; in 60 days it covers 25 degrees 35 parts. If this slow phase falls between the Start of Autumn and the Autumn Equinox, add one day, with motion of 67 parts and 36 minor parts. When full, cast out the excess as before; in 60 days it covers 31 degrees. The fractional values match. Then comes the later rapid phase. Entering the Winter Solstice, the initial rate is 136 degrees in 214 days. Thereafter decrease the day-rate and degree-rate by 1 each day. Through 37 days the rate becomes 99 degrees in 177 days. Then decrease the day-rate and degree-rate by 1 every 2 days. Through 57 days the rate becomes 89 degrees in 167 days. This rate holds through day 79. Then increase the day-rate and degree-rate by 1 every 3 days. Through 130 days the rate becomes 160 degrees in 184 days. Then increase the day-rate and degree-rate by 1 every 2 days. Through 144 days the rate becomes 113 degrees in 191 days. Then increase the day-rate and degree-rate by 1 each day. Through 190 days the rate becomes 159 degrees in 237 days. Then increase the day-rate and degree-rate by 1 each day. Through 210 days the rate becomes 189 degrees in 267 days. This rate holds through day 259. Then decrease the day-rate and degree-rate by 1 every 2 days. Through 365 days the rate returns to 136 degrees in 214 days. If the later slow phase added 6 degrees, subtract 6 from the degree-rate in this later rapid phase—that is the fixed degree-total. Adjust each by the day-count after the Winter Solstice to obtain the day- and degree-rates for the later rapid phase. If the phase falls between the Start of Summer and the Summer Solstice, use half a degree per day, covering 30 degrees in 60 days. If it falls between Minor Heat and Major Heat, it covers 20 degrees in 40 days. Set aside the corresponding day- and degree-rates as a separate half-degree segment; when that segment is complete, calculate the uniform-motion parts and continue from there. Each phase runs through its allotted days and degrees, then the planet disappears.
27
鎮星
Saturn
28
初見：順，日行六十分，八十三日行七度二百四十八分。 而留，三十八日。 乃退，日四十一分，一百日退六度四十四分。 又留，三十七日六十一分小分四。 亦以初定見日分加之。 滿去如前。 乃順，日行六十分，八十三日行七度二百四十八分而伏。
At first appearance it moves directly at 60 parts per day, covering 7 degrees 248 parts in 83 days. Then it stands still for 38 days. Then it retrogrades at 41 parts per day, covering 6 degrees 44 parts in 100 days. It stands still again for 37 days 61 parts, 4 minor parts. Also add the parts from the first fixed appearance day. When full, cast out the excess as before. Then direct motion at 60 parts per day, 7 degrees 248 parts in 83 days, then it disappears.
29
太白
Venus
30
晨初見：乃退，日一度半，十日退十五度。 而留，九日。 乃順遲，差行。 先遲，日益疾八分，四十日行三十度。 若此遲入大雪已後，畢于小滿，即依此為定而求行分。 自入芒種，十日減一度為定度，畢於夏至。 自入小暑，畢於霜降，均減三度。 自入立冬，初日減三度，後十日減一度，畢于霜降，小雪，皆為定度。 求一日行分者，以行分法乘定度，以四十餘之，為平分，不盡為小分。 又以四乘三十九，以減平分，為初日行分。 平行，日一度，十五日行十五度。 若此平行入小寒後，十日益日及度各一，畢於啟蟄。 自入雨水之氣，皆二十一日行二十一度。 自入春分後，十日減一，畢于立夏，即十五日。 自入處暑，畢於寒露，即無此平行。 自入霜降，即四日益一，畢于大雪，後十五日行十五度。 疾，百七十日行二百四度。 前順遲減度者，計所減之數，以益此度為定度。 求一日行度及分者，以百七十日減度數，餘行以分法乘，以百七十餘之，所得為之日平行度分。 晨伏東方。
At morning first appearance it retrogrades at 1.5 degrees per day, covering 15 degrees in 10 days. Then it stands still for 9 days. Then it resumes direct slow motion with variable rate. Beginning slow, it accelerates by 8 parts each day, covering 30 degrees in 40 days. If this slow phase falls between Great Snow and Lesser Full Grain, use this as the fixed rate to calculate the motion-parts. From Grain in Ear, subtract 1 degree every 10 days as the fixed total, through the Summer Solstice. From Minor Heat through Frost Descent, subtract an even 3 degrees. From the Start of Winter, subtract 3 degrees on the first day and 1 degree every 10 days thereafter, through Frost Descent and Minor Snow—these are the fixed degrees. To find one day's motion-parts, multiply the fixed degrees by the motion-parts factor and divide by 40 for the equal parts; the remainder gives the minor parts. Multiply 4 by 39, subtract from the equal parts—that yields the initial day's motion-parts. Uniform motion: 1 degree per day, 15 degrees in 15 days. If this uniform-motion phase begins after Minor Cold, increase both the day-count and degree-count by 1 every 10 days, through Awakening of Insects. From Rain Water onward, the rate is 21 degrees in 21 days throughout. After the Spring Equinox, decrease by 1 every 10 days through the Start of Summer, ending at 15 days. From End of Heat through Cold Dew, this uniform-motion phase does not apply. From Frost Descent, increase by 1 every 4 days through Great Snow; then the rate is 15 degrees in 15 days. Rapid motion: 204 degrees in 170 days. If the earlier direct slow phase subtracted degrees, add that amount to this phase's degree-total as the fixed value. To find one day's motion in degrees and parts, subtract the degree-total from 170 days, multiply the remaining travel by the part divisor, and divide by 170—the quotient gives the daily uniform-motion degrees and parts. It disappears in the morning in the east (at conjunction).
31
夕初見：順疾，百七十日行二百。 畢于立夏，依此順疾。 入冬至已後，畢于立夏，依此率為定。 自入小滿，六日加一度。 自入大暑初，畢於芒種，自入夏至，畢於小暑，均五度。 自入大暑，初加五度，後三日減一度，畢於氣盡。 自入立秋，畢于大雪，還依本率。 從白露畢春分，皆差行。 先疾，日益遲一分半。 自入清明，畢於處暑，並平行，同晨疾。 求差行者，半一百六十九，乃以一分半乘之，以加平行分，為初日行度分也。 平行，日一度，十五日行十五度。 此平行入冬至後，十日減日及度各一，畢于立春。 自入啟蟄，畢於芒種，皆均九日行九度。 自入夏至後，五日益一，畢於小暑。 自入大暑，畢於氣盡，皆十五日行十五度。 自入立秋後，六日一，畢于小雪。 自入大雪，畢於氣盡，皆十五日行十五度者也。 順遲，差行。 先疾，日益遲八分，四十日行三十度。 前加度者，此依數減之，求一日行分，如晨遲准減者為加之。 又留，九日。 乃退，日半度，十日退五度，而夕伏西方。
At evening first appearance, direct rapid motion: 200 degrees in 170 days. Through the Start of Summer, follow this direct rapid rate. From the Winter Solstice through the Start of Summer, use this as the fixed rate. From Lesser Full Grain, add 1 degree every 6 days. From the beginning of Major Heat through Grain in Ear, and from the Summer Solstice through Minor Heat, adjust by an even 5 degrees. From Major Heat, add 5 degrees initially, then subtract 1 degree every 3 days until the qi period ends. From the Start of Autumn through Great Snow, revert to the base rate. From White Dew through the Spring Equinox, all phases use variable motion. Beginning fast, it slows by 1.5 parts each day. From Clear Brightness through End of Heat, use uniform motion matching the morning rapid phase. For variable motion, halve 169, multiply by 1.5 parts, add to the uniform-motion parts—that yields the initial day's motion in degrees and parts. Uniform motion: 1 degree per day, 15 degrees in 15 days. If this uniform-motion phase begins after the Winter Solstice, decrease both the day-count and degree-count by 1 every 10 days, through the Start of Spring. From Awakening of Insects through Grain in Ear, the rate is 9 degrees in 9 days throughout. After the Summer Solstice, increase by 1 every 5 days, through Minor Heat. From Major Heat through the end of the qi period, the rate is 15 degrees in 15 days throughout. After the Start of Autumn, adjust by 1 every 6 days, through Minor Snow. From Great Snow through the end of the qi period, the rate is 15 degrees in 15 days throughout. Direct slow motion with variable rate. Beginning fast, it slows by 8 parts each day, covering 30 degrees in 40 days. If degrees were added earlier, subtract that amount here; to find one day's motion-parts, apply the morning slow procedure in reverse—where that called for subtraction, add instead. It stands still again for 9 days. Then it retrogrades at half a degree per day, covering 5 degrees in 10 days, and disappears in the evening in the west.
32
辰星：晨初見，留，六日。 順遲，日行一百六十九分，四日行一度。 若初見入大寒，畢於啟蟄之內，即不須此遲行。 平行，日一度，十日行十度。 此平行若入大寒已後，二日去日及度各一，畢於二十日，日及度俱盡，即無此平行。 疾，日行一度六百九十分，十日行十九度六分。 前無遲行者，此疾日減二百三分，十日行十七度四分。 晨伏東方。
Mercury: at morning first appearance it stands still for 6 days. Direct slow motion at 169 parts per day, covering 1 degree in 4 days. If first appearance falls between Major Cold and Awakening of Insects, omit this slow-motion phase. Uniform motion: 1 degree per day, 10 degrees in 10 days. If this uniform-motion phase begins after Major Cold, decrease both the day-count and degree-count by 1 every 2 days; at 20 days both are used up—there is no uniform-motion phase. Rapid motion at 1 degree 690 parts per day, covering 19 degrees 6 parts in 10 days. If there was no slow phase earlier, reduce this rapid motion by 203 parts per day, covering 17 degrees 4 parts in 10 days. It disappears in the morning in the east (at conjunction).
33
夕初見：順疾，日行一度六百九分，十日行十九度六分。 此疾者，入小暑畢於處暑之內，日減二百三分，十日行十六度四分。 平行，日一度，十日行十度。 此平行若入大暑已後，於二日去日及度各一。 畢於二十日，日及度俱盡，即無此平行。 遲，日行一百六十九分，四日行一度。 若疾減二百三分者，即不須此遲行。 又留，六日九分。 夕伏西方。
At evening first appearance, direct rapid motion at 1 degree 609 parts per day, covering 19 degrees 6 parts in 10 days. If this rapid phase falls between Minor Heat and End of Heat, reduce daily motion by 203 parts, covering 16 degrees 4 parts in 10 days. Uniform motion: 1 degree per day, 10 degrees in 10 days. If this uniform-motion phase begins after Major Heat, decrease both the day-count and degree-count by 1 every 2 days. At 20 days both day- and degree-rates are used up—there is no uniform-motion phase. Slow motion at 169 parts per day, covering 1 degree in 4 days. If the rapid phase was reduced by 203 parts, omit this slow-motion phase. It stands still again for 6 days 9 parts. It disappears in the evening in the west (at conjunction).
34
推交會
Method for Computing Conjunctions and Crossings
35
交會法：一千二百七十四萬一千二百五分。
Conjunction divisor: 12,741,205 parts.
36
交分法：六百三十七萬六百二十九分。
Node-parts factor: 6,370,629 parts.
37
朔差：一百八萬五千四百九十二分。
New-moon difference: 1,085,492 parts.
38
望分：六百九十一萬三千三百五十分。
Full-moon parts: 6,913,350 parts.
39
交限：五萬八十二萬七千八百五十八分。
Conjunction limit: 5,827,858 parts.
40
望差：五十四萬二千七百四十七一分。
Full-moon difference: 542,747 parts.
41
外限：六百七十六萬七百八十二九分。
Outer limit: 6,760,782 parts.
42
中限：一千二百三十五萬一千二十五八分。
Middle limit: 12,351,025 parts.
43
內限：一千二百一十九萬八千四百五十八七分。
Inner limit: 12,198,458 parts.
44
交時法：二萬九千一十八。
Node-time factor: 29,018.
45
推交分術
Method for Computing Node Parts
46
置入上元已來積月，以交會法去之。 餘，以朔差乘之。 滿交會法，又去之。 仁均本術，武德年加交差七百七十五萬五千一百六十四分。 餘為所求年天正朔入平交分。 求望平交分術，以望分加之，滿去如前，為平分。 次月平分術，其朔望，入冬至氣內，依平為定。 若入小寒已後，日加氣差一千六百五十分，畢于立春。 自入啟蟄，畢于清明，均加七萬六千一百分。 後日減一千六百五十分，畢于小滿。 置初日所加之分，計後日減之數以減之，餘以加平交分。 自入芒種，畢於夏至，依平為定。 加之，滿交會法，即去。 餘為定交分。 其朔入災交，若入小寒，畢于雨水，及立夏，畢于小滿，值盈二時已下，皆半氣差而加之。 二時已上，皆不加。 其朔入時交分，如望差分已下，外限已上，有星伏，木土去見十日外，火去見四十日外，金星伏去見二十二日外。 有一星者，不加氣差。 其朔望，入小暑已後，日減氣差一千二百分，畢於處暑。 自入白露，畢於霜降，均減九萬五千八百二十分。 自入立冬，初日減六萬三千三百分，後日減二千一百一十分，畢于小雪。 置初日所減之分，計後日減之數以減之，餘以減平交分也。 自入大雪，亦依平為定。 減若不足者，加交會法，乃減之。 餘為定交分。 其朔入交分，如交限內限已上，交分中限已下，有星伏如前者，不減氣差。
Take the accumulated months since the upper origin and divide by the conjunction divisor. Multiply the remainder by the new-moon difference. When the product fills the conjunction divisor, discard full multiples again. In Renjun's original procedure, the Wude-era revision adds 7,755,164 parts to the node difference. The remainder is the mean node parts for the new moon of the year's Heavenly First Month. To find the mean node parts at full moon, add the full-moon parts; discard full multiples as before, and the result is the mean parts. For subsequent months, when new moon or full moon falls within the Winter Solstice solar term, take the mean value as fixed. From entry into Slight Cold, add 1,650 parts of solar-term difference per day until Start of Spring. From Awakening of Insects through Clear Brightness, add 76,100 parts in total. On each subsequent day, subtract 1,650 parts until Grain Fills. Take the amount added on the first day, subtract the cumulative reductions for later days, and add the remainder to the mean node parts. From Grain in Ear through the Summer Solstice, again take the mean as fixed. Add the result; when it fills the conjunction divisor, discard the multiple. The remainder is the fixed node parts. When the new moon falls in an eclipse-prone conjunction—between Slight Cold and Rain Water, or between Beginning of Summer and Grain Fills—and the excess amounts to less than two hours, add half the solar-term difference in each case. If two hours or more, add nothing. When the new moon's timed node parts fall below the full-moon difference and above the outer limit, and planets are invisible—Jupiter and Saturn more than ten days from appearance, Mars more than forty days from appearance, or Venus in morning or evening concealment more than twenty-two days from appearance— if any one of these planets is present, do not add the solar-term difference. For new and full moons after Slight Heat, subtract 1,200 parts of solar-term difference per day until End of Heat. From White Dew through Frost's Descent, subtract 95,820 parts in total. From Beginning of Winter, subtract 63,300 parts on the first day and 2,110 parts on each subsequent day until Slight Snow. Take the amount subtracted on the first day, subtract the cumulative reductions for later days, and use the remainder to reduce the mean node parts. From Great Snow onward, again take the mean as fixed. If the remainder is insufficient for subtraction, add one conjunction divisor before subtracting. The remainder is the fixed node parts. When the new moon's node parts lie above the inner conjunction limit and below the middle node limit, and planets are invisible as described above, do not subtract the solar-term difference.
47
推道在內外及先後去交術，其定交分不滿交分法者，為在外道。 滿去之，餘為在內道。 其餘如望差已下，即是去先交分。 以時法約之得一，為去先交時數。 交限已上，即以減交分法。 餘為去後交分，亦以時法約之，為時數。 望則月蝕也。 其朔在內道者，朔則日蝕。 或雖在內道去交而遠，在外道去交而近，亦為蝕也。
Method for determining inner versus outer path and distance before or after conjunction: if the fixed node parts amount to less than the node-parts factor, the moon is on the outer path. Discard one full node-parts factor; the remainder places the moon on the inner path. If the remainder is less than the full-moon difference, it is the distance before conjunction. Divide by the hour factor; each unit is the number of hours before conjunction. If above the conjunction limit, subtract the node-parts factor instead. The remainder is the distance after conjunction; divide again by the hour factor to obtain the number of hours. At full moon there is a lunar eclipse. If the new moon is on the inner path, there is a solar eclipse at new moon. Even if the moon is on the inner path but far from conjunction, or on the outer path but near it, an eclipse may still occur.
48
推月蝕加時術
Method for Computing the Hour of Lunar Eclipse
49
置有蝕之望定小餘。 若入曆一日，即減二百八十。 入十五日，即加之。 若入十四日，即加五百五十。 入二十八日，即減之。 自入諸日，值盈皆加二百八十，值縮皆減之，為定餘。 乃以十二乘之，以時法六千五百三除之，所得為半辰之數。 命以子半起算外，即所在辰。 初命子半以一算，自後皆以二算為一辰。 不盡為時餘。 若時餘在辰半之前者，乃倍之。 如法無所得，為辰初。 又以三因之，如法得一，名為強。 若得強，若得二強，即名少弱。 若倍之，如法得一，為少。 凡四分一為少，二為半，三為太。 不盡者，又三之，如法得一，名為強。 若得二強者，即名為半弱。 若時餘在辰半之後者亦倍之。 如法無所得，為正在辰半。 以三因之，如法得二，名為強，即名半強。 若得二強，即名太弱。 若倍之，如法得一，為態。 不盡者，又三之，如法得一，為強，即名太強； 若得者，又二強者，為辰末。 亦可前辰名之。 月在沖上蝕，日出後入前各一時半外，不注蝕。
Take the fixed minor remainder of the eclipsed full moon. If the moon is on the first day of the sequence, subtract 280. On the fifteenth day, add it back. If on the fourteenth day, add 550. On the twenty-eighth day, subtract it. For each day entered thereafter, add 280 when the moon is in excess and subtract when it is in contraction; the result is the fixed remainder. Multiply by twelve and divide by the hour factor, 6,503; the quotient is the number of half double-hours. Count from midnight, and the double-hour reached outside the count is the position. Midnight counts as one; each subsequent double-hour counts as two. The remainder is the fractional part within the double-hour. If the fractional remainder falls before the middle of the double-hour, double it. If division by the factor yields nothing, the time is the start of the double-hour. Multiply by three; each unit obtained by division is called a strong. One strong, or two strongs, yields the designation minor-weak. If doubled, each unit obtained is minor. In quarter divisions, one is minor, two is half, and three is major. For whatever remains, multiply by three again; each unit obtained is a strong. Two strongs yield the designation half-weak. If the fractional remainder falls after the middle of the double-hour, double it as well. If division yields nothing, the time is exactly the middle of the double-hour. Multiply by three; two units obtained are one strong, yielding the designation half-strong. Two strongs yield the designation major-weak. If doubled, each unit obtained is major. For whatever remains, multiply by three again; one unit obtained is a strong, yielding the designation major-strong; If two strongs are obtained, the time is the end of the double-hour. It may also be named from the preceding double-hour. When the moon eclipses at opposition, if the eclipse falls more than one and a half hours after sunrise or before sunset, do not record it.
50
推日蝕加時術
Method for Computing the Hour of Solar Eclipse
51
置有蝕之朔定小餘。 若入曆一日，即減三百。 入十五日，即加之。 若入十四日，即加五百五十。 入二十八日，即減之以為定。 自後不入四時加減之限。 春三月，內道，去交四時已上，入曆，值盈加二百八十，值縮反減之。 夏三月，內道，值盈加二百八十，值縮反減之。 秋三月，內道，去交十一時已下，值盈加二百八十，值縮不加； 十一時已上，值盈加五百五十，值縮不加一百八十。 冬三月，內道，去交五時已下，值盈加二百八十，縮不加。 皆為定餘。 乃以十二乘之，以時法除之，所得半辰之數，命以子半起算外，即所在辰。 命辰如前法。 不盡為時餘，別置為副。 若入仲辰半前，即以副減法，餘為差率。 若在半後，即退其半辰，還以法加餘，即以副為差率。 若入季辰半前，即以法加副，而為差率。 若在半後，即其半辰，還以法加餘，乃倍法以加副，而為差率。 若入孟辰半前，即三因其法，而以副減之，餘為差率。 若半後，即退其半辰，還以法加餘，又以法加副，乃三因其法而以副減之，為差率。 又置去交時數，三已下加三，六已下加二，九已下加一，九已上依數，十二以上從十二，以乘差率。 若在季辰半後，孟辰半前，去交六時以上者，皆從其六，以乘差率。 六時已下，自依數，不須加。 如十四得一，為時差。 子至卯半，午至酉半，以時餘加之； 卯至午半，酉至子半，以減時餘。 加之若滿時法者，乃去之，加於辰，即進之於前也。 減之若不足者，減半辰，加時法，乃減之，即退之於後也。 餘為定時餘。 乃如月蝕法，子午卯酉為仲，辰戌醜未為季，寅申已亥為孟。 日出前後各一時半外，不注日蝕。
Take the fixed minor remainder of the eclipsed new moon. If the moon is on the first day of the sequence, subtract 300. On the fifteenth day, add it back. If on the fourteenth day, add 550. On the twenty-eighth day, subtract it to obtain the fixed value. Thereafter the seasonal addition-and-subtraction limits no longer apply. In spring, on the inner path, when the moon is more than four hours from conjunction and has entered the sequence, add 280 if in excess and subtract if in contraction. In summer, on the inner path, add 280 if in excess and subtract if in contraction. In autumn, on the inner path, when the moon is within eleven hours of conjunction, add 280 if in excess and do not add if in contraction; if more than eleven hours from conjunction, add 550 if in excess and subtract 180 if in contraction. In winter, on the inner path, when within five hours of conjunction, add 280 if in excess and do not add if in contraction. These yield the fixed remainder. Multiply by twelve and divide by the hour factor to obtain the number of half double-hours; count from midnight, and the double-hour reached outside the count is the position. Determine the double-hour by the method above. The remainder is the fractional part within the double-hour; set it aside as an auxiliary value. If the time falls before the middle of a cardinal double-hour, subtract the auxiliary from the factor; the remainder is the difference rate. If after the middle, retreat one half double-hour, add the remainder back to the factor, and use the auxiliary as the difference rate. If before the middle of a seasonal double-hour, add the auxiliary to the factor to obtain the difference rate. If after the middle, take one half double-hour, add the remainder back to the factor, double the factor and add the auxiliary, and the result is the difference rate. If before the middle of an angular double-hour, triple the factor and subtract the auxiliary; the remainder is the difference rate. If after the middle, retreat one half double-hour, add the remainder to the factor and the factor to the auxiliary, then triple the factor and subtract the auxiliary to obtain the difference rate. Also take the hours from conjunction: if three or fewer, add three; if six or fewer, add two; if nine or fewer, add one; if more than nine, use the number itself; if twelve or more, use twelve; multiply the difference rate by this value. If the time falls after the middle of a seasonal double-hour and before the middle of an angular double-hour, and the moon is six or more hours from conjunction, use six to multiply the difference rate. If six hours or fewer, use the number itself without addition. Divide by fourteen; the quotient is the time difference. From zi through the middle of mao, and from wu through the middle of you, add the hour remainder; from mao through the middle of wu, and from you through the middle of zi, subtract the hour remainder. If addition fills the hour factor, discard the multiple and carry one to the double-hour, advancing to the preceding one. If subtraction is insufficient, retreat one half double-hour, add the hour factor, then subtract, moving back to the following one. The remainder is the fixed fractional part within the double-hour. Then, as with the lunar-eclipse method, name the fractional part: zi, wu, mao, and you are cardinal double-hours; chen, xu, chou, and wei are seasonal; yin, shen, si, and hai are angular. If the eclipse falls more than one and a half hours before or after sunrise, do not record it.
52
推內道日不蝕術
Method for When the Inner Path Does Not Eclipse
53
夏五月朔，加時在南方三辰，先交十三時外，六月朔，後交十三時外者，不蝕。 啟蟄畢清明，先交十三時外，值縮，加時在未巳西者，亦不蝕。 入處暑，畢寒露，後交十三時，值盈，加時在己巳東者，亦不蝕。
On the fifth-month new moon of summer, if the computed hour falls in the three southern double-hours and the moon is more than thirteen hours before conjunction; or on the sixth-month new moon, if it is more than thirteen hours after conjunction—no eclipse occurs. From Awakening of Insects through Clear Brightness, if the moon is more than thirteen hours before conjunction, is in contraction, and the computed hour falls in wei, si, or the western double-hours, there is also no eclipse. From End of Heat through Cold Dew, if the moon is thirteen hours after conjunction, is in excess, and the computed hour falls in si, ji, or the eastern double-hours, there is also no eclipse.
54
推外道日蝕術
Method for Solar Eclipse on the Outer Path
55
不問交之先後，但去交一時內者，皆蝕也。 若先交二時內者，值盈二時外者，亦蝕。 若後交二時內，值縮二時外者，亦蝕。 其夏去交二時在南方三辰者，亦蝕。 若去分至十二時內，去交六時內者，亦蝕。 若去交春分三日內，後交二時內者，亦蝕。 秋分三日內，先交二時內者，亦蝕。 諸去交三時內，星伏如前者，亦蝕。
Regardless of whether conjunction comes before or after, if the moon is within one hour of conjunction, there is always an eclipse. If before conjunction within two hours and the moon is in excess more than two hours from conjunction, there is also an eclipse. If after conjunction within two hours and the moon is in contraction more than two hours from conjunction, there is also an eclipse. In summer, if the moon is two hours from conjunction and the computed hour falls in the three southern double-hours, there is also an eclipse. If the eclipse falls within twelve hours of an equinox or solstice and within six hours of conjunction, there is also an eclipse. If within three days of the Spring Equinox and within two hours after conjunction, there is also an eclipse. Within three days of the Autumn Equinox, if the moon is within two hours before conjunction, there is also an eclipse. Whenever the moon is within three hours of conjunction and planets are invisible as described above, there is also an eclipse.
56
推月蝕分術
Method for Computing Lunar Eclipse Magnitude
57
置去交分。 其在冬，先後交皆去不蝕分二時之數。 若在於春，先交去半時，後交去二時。 夏即依定。 若在於秋，先交去二時，後交去半時。 若不足去者，蝕既，乃以三萬六千一百八十三為法除之，所得為不蝕分。 不盡者，半法已上為半強，已下為半弱，而以減十五，餘為蝕之大分。
Take the parts of departure from conjunction. In winter, subtract two hours' worth of non-eclipse parts whether conjunction comes before or after. In spring, subtract half an hour before conjunction and two hours after conjunction. In summer, use the fixed value unchanged. In autumn, subtract two hours before conjunction and half an hour after conjunction. If the subtraction cannot be completed, the eclipse is total; otherwise divide by 36,183, and the quotient is the non-eclipse parts. For the remainder, count above half the divisor as half-strong and below as half-weak; subtract from fifteen, and the remainder is the eclipse magnitude in great parts.
58
推月蝕所起術
Method for Determining Where the Lunar Eclipse Begins
59
若在外道，初起東北，蝕甚西北。 若在內道，初起東南，蝕甚西南。 十三分已上，正東起。 推皆據正南而言。
On the outer path, obscuration first appears in the northeast and reaches its maximum in the northwest. On the inner path, obscuration first appears in the southeast and reaches its maximum in the southwest. If the magnitude is thirteen parts or more, obscuration begins due east. All these directions are reckoned from due south.
60
推日蝕分術
Method for Computing Solar Eclipse Magnitude
61
置去交分。 若入冬至已後，畢于立春，皆均減十二萬八百，餘為不蝕分。 不足減者，反以交分減之，餘為不蝕分。 亦減望差為定法。 其後交值縮者，直以望差為定法，不須減之。 自入啟蟄，初日減二十二萬八百分，後日減一千八百一十分，置初日所減之分，計後日減之數以減之，餘以減交分。 畢於芒種。 自入夏至，日減二千四百分，畢于白露。 自入秋分，畢于大雪，皆均減二十二萬八百分。 但不足減者，皆如前，反以交分減之，訖，皆為不蝕。 若入冬至，畢於小寒，不蝕分依定。 若入大寒，畢于立夏，後去交五時外，皆去不蝕分一時。 時差值減者，先交減之，後交加之。 不足減者，蝕既。 時差值加者，先交加之，後交減之。 不足減者，蝕既。 乃為定分，以十五乘之，以定法除之，所得為不蝕分。 不盡者，半法已上為半強，已下為半弱，而以減十五，餘為蝕之大分也。
Take the parts of departure from conjunction. From after the Winter Solstice through the Start of Spring, subtract 120,800 in total; the remainder is the non-eclipse parts. If the remainder is insufficient, subtract from the node parts instead; what remains is the non-eclipse parts. Also subtract the full-moon difference to obtain the fixed factor. When conjunction comes after and the moon is in contraction, take the full-moon difference itself as the fixed factor without subtracting it. From Awakening of Insects, subtract 228,000 parts on the first day and 1,810 parts on each later day; take the first day's subtraction, subtract the running total of later reductions, and use the remainder to reduce the node parts. Continue until Grain in Ear. From the Summer Solstice, subtract 2,400 parts per day until White Dew. From the Autumn Equinox through Great Snow, subtract 228,000 parts in total. Whenever subtraction is insufficient, reverse the operation against the node parts as before; when this is done, the result is the non-eclipse portion throughout. From the Winter Solstice through Slight Cold, the non-eclipse parts follow the fixed value unchanged. From Great Cold through Beginning of Summer, if the moon is more than five hours after conjunction, subtract one hour's worth of non-eclipse parts in every case. When the time difference requires subtraction, subtract from the before-conjunction figure and add to the after-conjunction figure. If subtraction cannot be completed, the eclipse is total. When the time difference requires addition, add to the before-conjunction figure and subtract from the after-conjunction figure. If subtraction cannot be completed, the eclipse is total. Take this as the fixed parts, multiply by fifteen, and divide by the fixed factor; the quotient is the non-eclipse parts. For the remainder, count above half the divisor as half-strong and below as half-weak; subtract from fifteen, and the remainder is the eclipse magnitude in great parts.
62
推日蝕所起術
Method for Determining Where the Solar Eclipse Begins
63
若在外道，初起西南，蝕甚東南。 若在內道，初起西北，蝕甚東北。 十三度已上，正西起。 亦據正南而言之。
On the outer path, obscuration first appears in the southwest and reaches its maximum in the southeast. On the inner path, obscuration first appears in the northwest and reaches its maximum in the northeast. If the obscuration is thirteen degrees or more, it begins due west. These directions, too, are reckoned from due south.
64
求日出入所在術
Method for Finding Sunrise and Sunset Times
65
以所入氣辰刻及分，與後氣辰刻及分相減。 餘乘入氣日算，以十五除之。 所得以加減所入氣為定日出人。 從冬至至夏至，日出減之，日入加之。 從夏至至冬至，日出加之，日入減之。 入餘為定刻及分。
Subtract the entered solar term's sunrise or sunset double-hour quarters and parts from those of the next solar term. Multiply the remainder by the number of days elapsed within the solar term and divide by fifteen. Add or subtract this result to the entered solar term's value to obtain the fixed times of sunrise and sunset. From the Winter Solstice to the Summer Solstice, subtract for sunrise and add for sunset. From the Summer Solstice to the Winter Solstice, add for sunrise and subtract for sunset. The remainder within the double-hour is the fixed quarter and fractional parts.
66
五月二日校曆人前曆博士臣南宮子明
On the second day of the fifth month: calendar verifier, former Calendar Doctor Nangong Ziming.
67
校曆人前曆博士臣薛弘疑
Calendar verifier, former Calendar Doctor Xue Hongyi.
68
校曆人算曆博士臣王孝通
Calendar verifier, Computational Calendar Doctor Wang Xiaotong.
69
監校曆大理卿清河縣公崔善為
Superintendent of calendar verification, Chief Minister of the Court of Judicial Review and Duke of Qinghe County, Cui Shanwei.
70
夜漏半
Half the night clepsydra.
71
右依經，加於漏刻日出沒二十四氣下。
As prescribed in the classic, enter these values below the clepsydra quarters for sunrise and sunset under each of the twenty-four solar terms.
72
推月蝕加時術
Method for Computing the Hour of Lunar Eclipse
73
右加有蝕之望，以百刻乘定小餘，日法而一，以課所近氣不滿夜半者，命日以甲子算上注曆。
For each eclipsed full moon, multiply the fixed minor remainder by a hundred quarters and divide by the day factor; check whether the nearest solar term falls short of midnight, assign the day, and enter it on the calendar from the jiazi count.
74
推月蝕虧初復滿先造每箭更籌用刻
Method for Lunar Eclipse First Contact and Full Recovery: Pre-establishing the Quarter-Use for Each Clepsydra Arrow and Watch Tally
75
倍月蝕日所入氣夜漏半，二十五而一，為籌刻分，亦注於曆下。
Double the half-night clepsydra for the solar term on the day of the lunar eclipse and divide by twenty-five; the result is the tally quarter-parts, entered below on the calendar.
76
月蝕分用刻率置月蝕分
Lunar Eclipse Magnitude Quarter-Rate: set the lunar eclipse parts.
77
推日月蝕加時定刻術
Method for Fixed Quarters of Solar and Lunar Eclipse Hour-Addition
78
置日月蝕加時定餘。 在辰半後者，加時法于時餘，以二十五乘之，三萬九千一十八而一刻，命刻算外，即所入辰刻。
Take the fixed remainder of the solar or lunar eclipse hour-addition. If the time falls after the middle of the double-hour, add the hour-addition factor to the hour remainder, multiply by twenty-five, and take one quarter for every 39,018; count off the quarters, and the result is the entered double-hour quarter.
79
求虧初復滿術
Method for First Contact and Full Recovery
80
置蝕分，用刻率副之，以乘所入曆損益率，四千五十七而一。 值盈反其損益，值縮依其損諡，副為蝕定用刻數，乃六乘之，十而一，以減蝕加時辰刻，為虧初。 丈四乘餘之用刻數，十而一，以加蝕加時辰刻，為復滿。
Set the eclipse parts with the quarter-rate as auxiliary, multiply by the entered calendar increase-decrease rate, and divide by 4,057. If in excess, reverse the increase and decrease; if in contraction, apply the increase and decrease as given; take the auxiliary as the fixed eclipse quarter-count, multiply by six, divide by ten, and subtract from the eclipse hour-addition double-hour quarter to obtain first contact. Again multiply the remainder's fixed quarter-count by four, divide by ten, and add to the eclipse hour-addition double-hour quarter to obtain full recovery.
81
求所蝕夜初甚末更籌刻術
Method for First, Maximum, and Last Watch Tallies of the Eclipse Night
82
因其日日所入辰殘刻及分，依次加辰刻及分，至蝕初辰刻及分，減二刻十二分，從其更用刻及分除之，不滿更，即初蝕更籌。 依所求得至甚刻加之，命即甚。 依求得甚後刻數加之，命即末更籌刻及分。 日出前復滿，日入後初虧，皆不注蝕。
From that day's entered double-hour with its remaining quarters and parts, add double-hour quarters and parts in sequence until first contact; subtract two quarters and twelve parts; divide by the watch's quarter-use and parts, and if the result is less than one full watch, that is the first-eclipse watch tally. Add the quarters to maximum as obtained by the method and assign the time; that is maximum obscuration. Add the post-maximum quarters as obtained and assign the time; that is the last watch tally's quarters and parts. If full recovery occurs before sunrise or first contact after sunset, do not record the eclipse.