1
曆法尚矣。 自堯命羲、和，曆象日月星辰，以閏月定四時成歲，其事略見於《書》。 而夏、商、周以三統改正朔，為曆固已不同，而其法不傳。 至漢造曆，始以八十一分為統母，其數起于黃鐘之龠，蓋其法一本於律矣。 其後劉歆又以《春秋》、《易象》推合其數，蓋傅會之說也。 至唐一行始專用大衍之策，則曆術又本于《易》矣。 蓋曆起於數，數者，自然之用也。 其用無窮而無所不通，以之于律、于《易》，皆可以合也。 然其要在於候天地之氣，以知四時寒暑，而仰察天日月星之行運，以相參合而已。 然四時寒暑無形而運于下，天日月星有象而見於上，二者常動而不息。 一有一無，出入升降，或遲或疾，不相為謀。 其久而不能無差忒者，勢使之然也。 故為曆者，其始未嘗不精密，而其後多疏而不合，亦理之然也。 不合，則屢變其法以求之。 自堯、舜、三代以來，曆未嘗同也。
The art of calendrics has been esteemed since antiquity. When Yao charged Xi and He to observe the heavens and set the calendar by intercalary months, fixing the seasons and the year, the gist appears in the Book of Documents. Xia, Shang, and Zhou each adjusted the new-year reckoning by the Three Conformities system, so their calendars already diverged, but none of those methods survive. When the Han first devised a calendar, they took eighty-one as the mother of conformity, deriving the figure from the Yellow Bell tube—calendrics rooted in musical pitch. Liu Xin later forced the numbers to agree with the Spring and Autumn and the Book of Changes—a contrived harmonization. In Tang, the monk Yixing relied chiefly on the Great Expansion methods, so calendrical science again took the Changes as its foundation. Calendars begin in number, and number is the working tool of the natural order. Number's applications are boundless; whether joined to musical pitch or to the Changes, everything can be fitted together. The heart of the matter is to read seasonal qi on earth and compare it with the observed courses of sun, moon, and stars in the sky. Seasonal warmth and cold move invisibly on earth, while the luminaries show visible patterns above; both realms move ceaselessly. They alternate presence and absence, rise and fall, speed up and slow down, without any mutual agreement. Given enough time, error is inevitable—that is the nature of the case. New calendars start precise, then drift out of alignment; that pattern is itself predictable. When predictions fail, reformers revise the rules again and again. Since Yao, Shun, and the Three Dynasties, no two reigns have kept the same calendar.
2
唐終始二百九十餘年，而曆八改。 初曰《戊寅元曆》，曰《麟德甲子元曆》，曰《開元大衍曆》，曰《寶應五紀曆》，曰《建中正元曆》，曰《元和觀象曆》，曰《長慶宣明曆》，曰《景福崇玄曆》而止矣。
Across Tang's roughly two hundred ninety years, the official calendar was replaced eight times. In order: the Wuyin Origin, Lindé Jiazi Origin, Kaiyuan Dayan, Baoying Five Reckonings, Jianzhong Zhengyuan, Yuanhe Guanxiang, Changqing Xuanming, and finally the Jingfu Chongxuan calendars.
3
高祖受禪，將治新曆，東都道士傅仁均善推步之學，太史令庾儉、丞傅弈薦之。 詔仁均與儉等參議，合受命歲名為《戊寅元曆》。 乃列其大要，所可考驗者有七，曰：「唐以戊寅歲甲子日登極，曆元戊寅，日起甲子，如漢《太初》，一也。 冬至五十餘年輒差一度，日短星昴，合於《堯典》，二也。 周幽王六年十月辛卯朔，入蝕限，合於《詩》，三也。 魯僖公五年壬子冬至，合《春秋命曆序》，四也。 月有三大、三小，則日蝕常在朔，月蝕常在望，五也。 命辰起子半，命度起虛六，符陰陽之始，六也。 立遲疾定朔，則月行晦不東見，朔不西朓，七也。」 高祖詔司曆起二年用之，擢仁均員外散騎侍郎。
After Gaozu took the throne he planned a new calendar; the Daoist Fu Renjun of the eastern capital, expert in astronomical computation, was recommended by Yu Jian and Fu Yi. Renjun worked with Yu Jian and others and named the system the Wuyin Origin Calendar after Tang's inaugural year. Renjun set out seven verifiable claims: Tang enthroned on jiazi day in a wuyin year, with wuyin as calendar origin and days counted from jiazi—as in Han Taichu (point one). Winter solstice shifts one degree every fifty-some years; short days with Mao culminating matches the Yao canon (point two). Zhou King You's sixth year, tenth month xinmao new moon within eclipse limits, matches the Odes (point three). Lu Duke Xi's fifth year renzi winter solstice agrees with the Spring and Autumn calendar sequence (point four). Three long and three short months place solar eclipses at new moon and lunar eclipses at full moon (point five). Hours reckoned from mid-zi, degrees from Xu 6, aligning yin-yang origins (point six). Slow-fast tables for true new moon keep the moon invisible in the east at month-end and not a western sliver at new moon (point seven). Gaozu ordered adoption from year two and made Renjun Extraordinary Gentleman of the Scattered Cavalry.
4
三年正月望及二月、八月朔，當蝕，比不效。 六年，詔吏部郎中祖孝孫考其得失。 孝孫使算曆博士王孝通以《甲辰曆》法詰之曰：「'日短星昴，以正仲冬。 '七宿畢見，舉中宿言耳。 舉中宿，則餘星可知。 仁均專守昴中，執文害意，不亦謬乎？ 又《月令》仲冬'昏東壁中'，明昴中非為常准。 若堯時星昴昏中，差至東壁，然則堯前七千餘歲，冬至昏翼中，日應在東井。 井極北，去人最近，故暑； 斗極南，去人最遠，故寒。 寒暑易位，必不然矣。 又平朔、定朔，舊有二家。 三大、三小，為定朔望； 一大、一小，為平朔望。 日月行有遲速，相及謂之合會。 晦、朔無定，由時消息。 若定大小皆在朔者，合會雖定，而蔀、元、紀首三端並失。 若上合履端之始，下得歸餘於終，合會有時，則《甲辰元曆》為通術矣。」 仁均對曰：「宋祖沖之立歲差，隋張胄玄等因而修之。 雖差數不同，各明其意。 孝通未曉，乃熱南斗為冬至常星。 夫日躔宿度，如垂阝傳之過，宿度既差，黃道隨而變矣。 《書》云：'季秋月朔，辰弗集于房。 '孔氏云：'集，合也。 不合則日蝕可知。 '又云：'先時者殺無赦，不及時者殺無赦。 '既有先後之差，是知定朔矣。 《詩》云：'十月之交，朔月辛卯。 '又《春秋傳》曰：'不書朔，官失之也。 '自後曆差，莫能詳正。 故秦、漢以來，多非朔蝕。 宋御史中丞何承天微欲見意，不能詳究，乃為散騎侍郎皮延宗等所抑。 孝通之語，乃延宗舊說。 治曆之本，必推上元，日月如合璧，五星如連珠，夜半甲子朔旦冬至。 自此七曜散行，不復餘分普盡，總會如初。 唯朔分、氣分，有可盡之理，因其可盡，即有三端。 此乃紀其日數之元爾。 或以為即夜半甲子朔冬至者，非也。 冬至自有常數，朔名由於月起，月行遲疾匪常，三端安得即合。 故必須日月相合與至同日者，乃為合朔冬至耳。」 孝孫以為然，但略去尤疏闊者。
In year three, predicted eclipses at the first-month full moon and at second- and eighth-month new moons failed to appear. In year six, Zu Xiaosun of the Ministry of Personnel was ordered to review the calendar's errors. Xiaosun had Wang Xiaotong challenge him with the Jiachen Calendar: "'Days short, stars at Mao' fixes mid-winter. When all seven lodges are visible, the text names only the one at culmination. Name the culminating lodge and the rest follow. Renjun insists only on Mao at culmination, slave to the letter—is that not wrong? The Monthly Ordinances place Wall at culmination in mid-winter, so Mao is not the fixed norm. If Yao had Mao culminating at winter solstice, extrapolating back seven thousand years puts Wings culminating and the sun at Well. Well lies far north and nearest the observer, hence summer heat; Dipper lies far south and farthest away, hence winter cold. Swapping summer and winter seasons—that cannot be right. Mean versus true new moon had long been two rival methods. Three long and three short months belong to true conjunction reckoning; one long and one short to mean conjunction reckoning. Sun and moon move unevenly; their meeting is conjunction. Last day and new moon shift with seasonal irregularities. Forcing every long/short month to new moon fixes conjunction but breaks obscuration, era, and cycle-head alignment. If cycle-heads align at the year start and remainders at year end, the Jiachen Origin Calendar is the sound general method. Renjun answered: Zu Chongzhi of Liu-Song founded precession; Zhang Zexuan of Sui and others refined it. Their constants differed, but each school knew its own aim. Xiaotong missed the point and still treated Dipper as the fixed winter-solstice star. The sun's motion through lodges shifts like a gnomon shadow; lodge drift moves the ecliptic too. The Documents record: 'Autumn's last month, new moon—the chronogram not assembled at Fang.' Kong Yingda: 'Gathered means in conjunction.' If not united, a solar eclipse is implied. It also says: 'Too early—execute without mercy; too late—execute without mercy.' That timing error proves true new moon reckoning. The Odes: 'At the tenth month's turn, new moon on xinmao.' The Zuo Commentary: 'Failure to record new moon was the clerks' fault.' Later calendars drifted and no one could fix them fully. Hence Qin and Han records often place eclipses off new moon. He Chengtian of Song glimpsed the idea but could not finish; Pi Yanzong and others blocked him. Xiaotong merely repeated Yanzong's old argument. Calendar-making must reckon a high origin when sun and moon conjoin and the five planets align at midnight jiazi on winter-solstice new moon. Thereafter the seven wanderers diverge until remainders cycle and all align again. Only new-moon and qi fractions can fully cycle; that exhaustibility yields the three origins. That is simply the epoch for counting days. Some equate the three origins with midnight jiazi winter-solstice new moon—that is wrong. Winter solstice is regular; new moon follows the moon; lunar speed varies—the three origins cannot coincide by definition. True epoch new-moon winter solstice requires sun and moon to conjoin on the solstice day itself. Xiaosun accepted this, trimming only the worst flaws.
5
九年，復詔大理卿崔善為與孝通等較定，善為所改凡數十條。 初，仁均以武德元年為曆始，而氣、朔、遲疾、交會及五星皆有加減。 至是復用上元積算。 其周天度，即古赤道也。
In year nine, Cui Shanyi and Xiaotong revised dozens of rules. Renjun had begun from Wude year one with adjustments to qi, new moons, lunar speed, nodes, and planets. Now they reverted to high-origin accumulated counts. Its celestial circuit degrees follow the old equator.
6
貞觀初，直太史李淳風又上疏論十有八事，復詔善為課二家得失，其七條改從淳風。 十四年，太宗將親祀南郊，以十一月癸亥朔，甲子冬至。 而淳風新術，以甲子合朔冬至，乃上言：「古曆分日，起於子半。 十一月當甲子合朔冬至，故太史令傅仁均以減餘稍多，子初為朔，遂差三刻。」 司曆南宮子明、太史令薛頤等言：「子初及半，日月未離。 淳風之法，較春秋已來晷度薄蝕，事皆符合。」 國子祭酒孔穎達等及尚書八座參議，請從淳風。 又以平朔推之，則二曆皆以朔日冬至，於事彌合。 且平朔行之自古，故《春秋傳》或失之前，謂晦日也。 雖癸亥日月相及，明日甲子，為朔可也。 從之。 十八年，淳風又上言：「仁均曆有三大、三小，雲日月之蝕，必在朔望。 十九年九月後，四朔頻大。」 詔集諸解曆者詳之，不能定。 庚子，詔用仁均平朔，訖麟德元年。
Early Zhenguan, Li Chunfeng petitioned on eighteen points; seven revisions followed his method after Shanyi compared both schools. In year fourteen Taizong would sacrifice at the southern altar when guihai new moon and jiazi winter solstice fell in the eleventh month. Chunfeng's new astronomy placed jiazi new moon on winter solstice and argued days should begin at mid-zi. Renjun's excess subtraction made new moon fall at zi start, missing the solstice by three quarters. Ziming and Xue Yi replied that at zi start the luminaries had not yet separated. Chunfeng's tables matched gnomon and eclipse records back to Spring and Autumn. Kong Yingda and the Secretariat ministers voted for Chunfeng. By mean new moon both systems still placed winter solstice on new-moon day—better for the rite. Mean new moon was ancient practice; Zuo sometimes records eclipse a day early (on month-end). Even if conjunction fell on guihai, jiazi the next day could count as new moon. The court agreed. In year eighteen Chunfeng noted Renjun's three-long/three-short rule claiming eclipses only at new and full moon. After the ninth month of year nineteen, four consecutive long new moons appeared. Experts convened but could not settle the dispute. In a gengzi year the court reverted to Renjun's mean new moon through Lindé year one.
7
仁均曆法祖述胄玄，稍以劉孝孫舊議參之，其大最疏於淳風。 然更相出入，其有所中，淳風亦不能逾之。 今所記者，善為所較也。
Renjun followed Zhang Zexuan with bits of Xiaosun; overall it was coarser than Chunfeng. Yet each method won on some points neither could beat. What follows is Shanyi's comparative revision.
8
戊寅歷
The Wuyin Calendar
9
《戊寅曆》上元戊寅歲至武德九年丙戌，積十六萬四千三百四十八算外。
From wuyin high origin to Wude year 9 (bingxu): 164,348 accumulated counts beyond the epoch.
10
章歲六百七十六。 亦名行分法。 章閏二百四十九。 章月八千三百六十一。
Rule-years (zhang sui): 676. Also termed the motion-parts method. Rule intercalations (zhang run): 249. Rule months (zhang yue): 8,361.
11
月法三十八萬四千七十五。 日法萬三千六。 時法六千五百三度法、氣法九千四百六十四氣時法千一百八十三。
Month-factor (yue fa): 384,075. Day-factor (ri fa): 13,006. Hour-factor 6,503; degree-factor; qi-factor 9,464; qi-hour-factor 1,183.
12
歲分三百四十五萬六千六百七十五。 歲餘二千三百一十五。 周分三百四十五萬六千八百四十五半。 斗分一千四百八十五半。 沒分七萬六千八百一十五。 沒法千一百三。
Year-parts (sui fen): 3,456,675. Year remainder: 2,315. Circuit divisor: 3,456,845.5. Dipper parts: 1,485.5. Hidden qi parts: 76,815. Hidden qi factor: 1,103.
13
曆日二十七，曆餘萬六千六十四。 曆周七十九萬八千二百。 曆法二萬八千九百六十八。 餘數四萬九千六百三十五。
Sequence days: 27; sequence remainder: 10,064. Sequence cycle: 798,200. Sequence factor: 28,968. Remainder number: 49,635.
14
章月乘年，如章歲得一，為積月。 以月法乘積月，如日法得一，為朔積日； 餘為小餘。
Multiply the rule-months by the number of years and divide by the rule-years to obtain the accumulated months. Multiply the accumulated months by the month-factor and divide by the day-factor to obtain the accumulated new-moon days; the remainder is the minor remainder.
15
日滿六十，去之； 餘為大餘。 命甲子算外，得天正平朔。 加大餘二十九、小餘六千九百一，得次朔。 加平朔大餘七、小餘四千九百七十六、小分四之三，為上弦。 又加，得望。 又加，得下弦。 餘數乘年，如氣法得一，為氣積日。 命日如前，得冬至。 加大餘十五、小餘二千六十八、小分八之一，得次氣日。 加四季之節大餘十二、小餘千六百五十四、小分四，得土王。 凡節氣小餘，三之，以氣時法而一，命子半算外，各其加時。 置冬至小餘，八之，減沒分，餘滿沒法為日。 加冬至去朔日算，依月大小去之，日不滿月算，得沒日。 餘分盡為減。 加日六十九、餘七百八，得次沒。
When the day count reaches sixty, cast out full sexagenary cycles; the remainder is the major remainder. Count off from jiazi beyond the tally to obtain the mean new moon of the celestial first month. Add 29 to the major remainder and 6,901 to the minor remainder to obtain the next new moon. Add 7 to the major remainder, 4,976 to the minor remainder, and 3/4 minor parts to the mean new moon to obtain the first quarter moon. Add again to obtain the full moon. Add again to obtain the last quarter moon. Multiply the remainder number by the number of years and divide by the qi-factor to obtain the accumulated qi days. Count off the day as before to obtain winter solstice. Add 15 to the major remainder, 2,068 to the minor remainder, and one-eighth minor parts to obtain the next qi day. Add 12 to the major remainder, 1,654 to the minor remainder, and 4 minor parts at each of the four seasonal nodes to obtain the Earth sovereign period. For every solar term, triple the minor remainder, divide by the qi-time factor, and count from midnight beyond the tally to obtain the time of occurrence. Take the winter solstice minor remainder, multiply it by 8, subtract the hidden qi parts, and convert any remainder that fills the hidden qi factor into days. Add the winter solstice count of days from new moon, cast out full months according to their lengths, and when the days no longer fill a month count, obtain the hidden qi day. When the fractional remainder is exhausted, treat it as a decrement. Add 69 days and 708 remainder to obtain the next hidden qi.
16
以平朔、弦、望入氣日算乘損益率，如十五得一，以損益盈縮數，為定盈縮分。 凡不盡半法已上亦從一。 以曆法乘朔積日，滿曆周去之； 餘如曆法得一，為日。 命日算外，得天正平朔夜半入曆日及餘。 次日加一，累而裁之。 若以萬四千四百八十四乘平朔小餘，如六千五百三而一，不盡，為小分，以加夜半入曆日。 加之滿曆日及餘，去之，得平朔加時所入，加曆日七、餘萬一千八十四、小分三千九百九十五，命如前，得上弦。 又加，得望、下弦及後朔。
Multiply the day-count into qi for the mean new moon, quarter moons, and full moon by the excess-deficit rate, divide by 15, and apply the result to increase or decrease the elongation number to obtain the fixed elongation parts. Whenever the remainder is half the divisor or greater, round up by one. Multiply the accumulated new-moon days by the sequence factor and cast out full sequence cycles; divide the remainder by the sequence factor to obtain the day. Count off the day beyond the tally to obtain, for the mean new moon of the celestial first month at midnight, the day and remainder entering the sequence. For each subsequent day add one, accumulating and trimming as needed. Multiply the mean new moon minor remainder by 14,484, divide by 6,503, take the undivided remainder as minor parts, and add it to the midnight entry into the sequence day. When the addition fills the sequence days and remainder, cast it out to obtain the entry at the mean new moon time of occurrence; then add sequence day 7, remainder 10,084, and minor parts 3,995, and count as before to obtain the first quarter moon. Add again to obtain the full moon, the last quarter moon, and the following new moon.
17
曆行分與次日相減，為行差，後多為進，後少為退。 減去行分六百七十六，為差法。 各置平朔、弦、望加時入曆日餘，乘所入日損益率，以損益其下積分，差法除，為定盈縮積分。 置平朔、弦、望小餘，各以入氣積分盈加、縮減之，以入曆積分盈減、縮加之，滿若不足、進退日法，皆為定大小餘，命日甲子算外。 以歲分乘年為積分，滿周分去之； 餘如度法得一，為度。 命以虛六，經斗去分，得冬至日度及分。 以冬至去朔日算及分減之，得天正平朔前夜半日度及分。 以小分法十四約度分為行分。 凡小分滿法成行分，行分滿法成度。 若注曆，又以二十六約行分。 月星准此。 斗分百七十七，小分七半。 累加一度，得次日。 以行分法乘朔、望定小餘，以九百二十九除為度分，又以十四約為行分。 以加夜半度，為朔、望加時日度。 定朔加時，日月同度。 望則因加日度百八十二、行分四百二十六、小分十太。 以夜半入曆日餘乘行差，滿曆法得一，以進加、退減曆行分，為行定分。 以朔定小餘乘之，滿日法得一，為行分。 以減加時月度，為朔、望夜半月度。 求次日，加月行定分，累之。
Subtract the next day's sequence motion parts from the current day's to obtain the motion difference: if the later value is greater, the motion advances; if lesser, it retreats. Subtract 676 motion parts to obtain the difference divisor. For each mean new moon, quarter moon, and full moon, take the time-of-occurrence entry day and remainder in the sequence, multiply by that day's excess-deficit rate, apply the increase or decrease to the accumulated parts below, and divide by the difference divisor to obtain the fixed elongation accumulated parts. Take the mean new moon, quarter moon, and full moon minor remainders; add excess or subtract deficit using the entry-qi accumulated parts, and subtract excess or add deficit using the entry-sequence accumulated parts; on overflow or underflow, carry or borrow using the day-factor to obtain the fixed major and minor remainders, then count off the day from jiazi beyond the tally. Multiply the number of years by the year-parts to obtain accumulated parts, then cast out full circuit divisors; divide the remainder by the degree factor to obtain degrees. Count from xu 6, passing through the Dipper and removing the dipper parts, to obtain the winter solstice solar longitude in degrees and parts. Subtract the winter solstice count of days and parts from new moon to obtain the solar longitude at midnight before the mean new moon of the celestial first month. Reduce degree parts to motion parts using the minor-parts factor of 14. Whenever minor parts fill their factor they become motion parts; whenever motion parts fill their factor they become degrees. When annotating the calendar, further reduce motion parts by 26. The moon and stars follow this same rule. Dipper parts: 177; minor parts: 7.5. Accumulate by adding one degree each day to obtain the next day's value. Multiply the true new moon and full moon minor remainders by the motion-parts factor, divide by 929 to obtain degree parts, then reduce by 14 to obtain motion parts. Add the result to the midnight solar longitude to obtain the solar longitude at the new moon and full moon times of occurrence. At the true new moon time of occurrence, the sun and moon share the same longitude. For the full moon, add 182 degrees of solar longitude, 426 motion parts, and 10.5 minor parts. Multiply the midnight entry-into-sequence day remainder by the motion difference, divide by the sequence factor when full, and add on advance or subtract on retreat from the sequence motion parts to obtain the fixed motion parts. Multiply by the true new moon minor remainder and divide by the day-factor when full to obtain motion parts. Subtract the result from the time-of-occurrence lunar longitude to obtain the midnight lunar longitude at new moon and full moon. To find the next day, add the fixed lunar motion parts and accumulate.
18
率三百七十七萬五千二十三。
Sidereal rate: 3,775,023.
19
終日三百九十八，行分五百九十六，小分七。
Total days: 398; motion parts: 596; minor parts: 7.
20
平見，入冬至初日，減行分五千四百一十一。 自後日損所減百二十分。 立春初，日增所加六十分。 春分，均加四日。 清明畢穀雨，均加五日。 立夏畢大暑，均加六日。 立秋初日，加四千八十分。 乃日損所加六十七分。 入寒露，日增所減百一十七分。 入小雪，畢大雪，均減八日。
At mean appearance, on the first day after entering winter solstice, subtract 5,411 motion parts. Thereafter decrease the subtracted amount by 120 parts each day. At the start of Establishment of Spring, increase the added amount by 60 parts each day. At the spring equinox, uniformly add 4 days. From Clear Brightness through Grain Rain, uniformly add 5 days. From Establishment of Summer through Major Heat, uniformly add 6 days. On the first day of Establishment of Autumn, add 4,080 parts. Then decrease the added amount by 67 parts each day. Upon entering Cold Dew, increase the subtracted amount by 117 parts each day. From entering Minor Snow through Major Snow, uniformly subtract 8 days.
21
初見，順，日行百七十一分，日益遲一分，百一十四日行十九度二百九分。 而留，二十六日。 乃退，日九十七分，八十四日退十二度三十六分。 又留，二十五日五百九十六分，小分七。 凡五星留日有分者，以初定見日分加之。 若滿行分法，去之，又增一日。 乃順，初日行六十分，日益疾一分，百一十四日行十九度四百三十七分。 而伏。
At first appearance it moves directly, 171 parts per day, slowing by 1 part each day; in 114 days it travels 19 degrees 209 parts. Then it stations for 26 days. Then it retrogrades 97 parts per day; in 84 days it retreats 12 degrees 36 parts. Again it stations for 25 days 596 parts, minor parts 7. For all five planets, when stationary days carry fractional parts, add the parts from the first fixed appearance day. If they fill the motion-parts factor, cast out the excess and add 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. Then it becomes invisible.
22
率七百三十八萬一千二百二十三。
Sidereal rate: 7,381,223.
23
終日七百七十九，行分六百二十六，小分三。
Total days: 779; motion parts: 626; minor parts: 3.
24
平見，入冬至初日，減萬六千三百五十四分。 乃日損所減五百四十五分。 入大寒，日增所加四百二十六分。 入雨水後，均加二十九日。 立夏初日，加萬九千三百九十二分。 乃日損所加二百一十三分。 入立秋初，依平。 入處暑，日增所減百八十四分。 入小雪後，均減二十五日。
At mean appearance, on the first day after entering winter solstice, subtract 16,354 parts. Then decrease the subtracted amount by 545 parts each day. Upon entering Major Cold, increase the added amount by 426 parts each day. After entering Rain Water, uniformly add 29 days. On the first day of Establishment of Summer, add 19,392 parts. Then decrease the added amount by 213 parts each day. Upon entering the first day of Establishment of Autumn, follow the mean rate. Upon entering End of Heat, increase the subtracted amount by 184 parts each day. After entering Minor Snow, uniformly subtract 25 days.
25
初見，入冬至，初率二百四十一日行百六十三度。 自後二日損日度各一，自百二十八日，率百七十七日行九十九度，畢百六十一日。 又三日損一，盡百八十二日，率百七十日行九十二度，畢百八十八日。 乃三日益一，盡二百二十七日，率百八十三日行百五度。 又二日益一，盡二百四十九日，率百九十四日行百一十六度。 又每日益一，盡二百一十日，率二百五十五日行百七十七度，畢三百三十七日。 乃二日損一，盡大雪，復初見。 入小寒後，三日去日率一。 入雨水，畢立夏，均去日率二十。 自後三日減所去一日，畢小暑，依平，為定日率。 若入處暑，畢秋分，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，為前疾日度率。 若初行入大寒，畢大暑，皆差行，日益遲一分； 其餘皆平行。 若入白露，畢秋分，初遲，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，然後求平行分，續之。 以行分法乘度定率，如日定率而一，為平行分。 不盡，為小分。 求差行者，減日率一，又半之，加平行分，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，行三十一度五分。 此遲初日加六十七分、小分六十分之三十六。
At first appearance upon entering 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; from day 128 the rate is 177 days for 99 degrees, completing 161 days. Then decrease by 1 every 3 days; through 182 days the rate is 170 days for 92 degrees, completing 188 days. Then increase by 1 every 3 days; through 227 days the rate is 183 days for 105 degrees. Then increase by 1 every 2 days; through 249 days the rate is 194 days for 116 degrees. Then increase by 1 each day; through 210 days the rate is 255 days for 177 degrees, completing 337 days. Then decrease by 1 every 2 days; through Major Snow it returns to first appearance. After entering Minor Cold, remove 1 from the day-rate every 3 days. From entering Rain Water through Establishment of Summer, uniformly remove 20 from the day-rate. Thereafter decrease the removed amount by 1 day every 3 days; through Minor Heat follow the mean rate—this is the fixed day-rate. If the motion begins between End Heat and Autumn Equinox, subtract 6 from the degree-rate throughout. Adjust each according to the number of days past the winter solstice and subtract according to the solar term entered, to obtain the day-rate and degree-rate for the initial rapid phase. If the initial motion falls between Great Cold and Great Heat, use variable motion, slowing by 1 part each day; In all other seasons, use uniform motion. If entering between White Dew and Autumn Equinox, start with slow motion 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. Multiply the fixed degree-rate by the motion-parts factor and divide by the fixed day-rate to get the uniform-motion parts. The remainder gives the minor parts. For differential motion, subtract 1 from the day-rate, halve the result, add to the 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, the distance covered is 31 degrees and 5 parts. On the first day of this slow phase add 67 parts and 36/60 minor parts.
26
而留，十三日。 前疾去日者，分日於二留，奇從後留。 乃退，日百九十二分，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。
Then it stands still for 13 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, 3 minor parts.
27
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，加六度，行三十一度三十五分。 此遲初日加行分六十七、小分六十分之三十六。 而後疾。 入冬至，初率二百一十四日行百三十六度。 乃每日損一，盡三十七日，率百七十七日行九十九度。 又二日損一，盡五十七日，率百六十七日行八十九度，畢七十九日。 又三日益一，盡百三十日，率百八十四日行百六度。 又二日益一，盡百四十四日，率百九十一日行百一十三度。 又每日益一，盡百九十日，率二百三十七日行百五十九度。 又每日益二，盡二百日，率二百五十七日行百七十九度。 又每日益一，盡二百一十日，率二百六十七日行百八十九度，畢二百五十九日。 乃二日損一，畢大雪，復初。 後遲加六度者，此後疾去度率六，為定。 各依冬至後日數而損益之，為後疾日度率。 若入立夏，畢夏至，日行半度，盡六十日，行三十度。 若入小暑，畢大暑，盡四十日，行二十度皆去日度率，別為半度之。 行訖，然後求平行分，續之。 各盡其日度而伏。
Then it resumes direct motion. 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 Start of Autumn and Autumn Equinox, add 6 degrees—for a total of 31 degrees 35 parts. On the first day of this slow phase, add 67 motion-parts and 36/60 minor parts. Then comes the later rapid phase. Entering winter solstice, the initial rate is 136 degrees in 214 days. Then decrease by 1 each day for 37 days, yielding a rate of 99 degrees in 177 days. Next, decrease by 1 every 2 days for 57 days, giving 89 degrees in 167 days—a segment of 79 days total. Then increase by 1 every 3 days for 130 days, yielding 106 degrees in 184 days. Next, increase by 1 every 2 days for 144 days, giving 113 degrees in 191 days. Then increase by 1 each day for 190 days, yielding 159 degrees in 237 days. Next, increase by 2 each day for 200 days, giving 179 degrees in 257 days. Then increase by 1 each day for 210 days, yielding 189 degrees in 267 days—a segment totaling 259 days. Then decrease by 1 every 2 days until Great Snow, returning to the initial rate. If the later slow phase added 6 degrees, subtract 6 from the degree-rate in this later rapid phase—that is the fixed amount. Adjust each according to days past the winter solstice to obtain the later rapid day-rate and degree-rate. If entering between Start of Summer and summer solstice, move half a degree per day for 60 days, covering 30 degrees. If entering between Lesser Heat and Greater Heat, for 40 days covering 20 degrees—set these aside from the day- and degree-rates as a separate half-degree segment. When that segment is complete, calculate the uniform-motion parts and continue from there. Continue until its allotted days and degrees are used up, then it disappears (conjunction).
28
率三百五十七萬八千二百四十六。
Circulation rate: 3,578,246.
29
終日三百七十八，行分六十一。
Total days: 378; motion-parts: 61.
30
平見，入冬至初日，減四千八百一十四分。 乃日增所減七十九分。 入小寒，均減九日。 乃每氣損所減一日。 入夏至初日，均減二日。 自後十日損所減一日。 小暑五日外，依平。 入大暑，日增所加百八十一分。 入處暑，均加九日。 入白露初日，加六千二分。 乃日損所加百三十三分。 入霜降，日增所減七十九分。
For mean appearance, on the first day of winter solstice subtract 4,814 parts. Then each day add back 79 parts of what was subtracted. Entering Lesser Cold, subtract an even 9 days. Then reduce the subtraction by 1 day per solar term. On the first day of summer solstice, subtract an even 2 days. Thereafter, reduce the subtraction by 1 day every 10 days. After the fifth day of Lesser Heat, use the mean rate. Entering Greater Heat, each day add 181 parts to the increment. Entering End Heat, add an even 9 days. On the first day of White Dew, add 6,002 parts. Then each day reduce the increment by 133 parts. Entering Frost's Descent, each day add back 79 parts of what was subtracted.
31
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，百日退六度四十四分。 又留，三十七日六十一分。 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, in direct motion at 60 parts per day, covering 7 degrees 248 parts in 83 days. Then stationary for 38 days. Then retrograde at 41 parts per day, covering 6 degrees 44 parts in 100 days. Stands still again for 37 days 61 parts. Then direct motion at 60 parts per day, 7 degrees 248 parts in 83 days, then disappears.
32
率五百五十二萬六千二百。
Circulation rate: 5,526,200.
33
終日五百八十三，行分六百二十，小分八。
Total days: 583; motion-parts: 620; minor parts: 8.
34
晨見伏三百二十七日，行分六百二十，小分八。
Morning visibility and invisibility: 327 days; motion-parts: 620; minor parts: 8.
35
夕見伏二百五十六日。
Evening visibility and invisibility: 256 days.
36
晨平見，入冬至，依平。 入小寒，日增所加六十六分。 入立春，畢立夏，均加三日。 小滿初日，加千九百六十四分。 乃日損所加六十分。 入夏至，依平。 入小暑，日增所減六十分。 入立秋，畢立冬，均減三日。 小雪初日，減千九百六十四分。 乃日損所減六十六分。
For morning mean appearance at winter solstice, use the mean rate. Entering Lesser Cold, each day add 66 parts to the increment. Entering Start of Spring through Start of Summer, add an even 3 days. On the first day of Lesser Full Grain, add 1,964 parts. Then each day reduce the increment by 60 parts. Entering summer solstice, use the mean rate. Entering Lesser Heat, each day add back 60 parts of what was subtracted. Entering Start of Autumn through Start of Winter, subtract an even 3 days. On the first day of Minor Snow, subtract 1,964 parts. Then each day reduce the subtraction by 66 parts.
37
初見，乃退，日半度，十日退五度。 而留，九日。 乃順，遲，差行，日益疾八分，四十日行三十度。 入大雪畢小滿者，依此。 入芒種，十日減一度。 入小暑，畢霜降，均減三度。 入立冬，十日損所減一度，畢小雪。 皆為定度。 以行分法乘定度，四十除，為平行分。 又以四乘三十九，以減平行，為初日行分。 平行，日一度，十五日行十五度。 入小寒，十日益日度各一。 入雨水後，皆二十一日行二十一度。 入春分後，十日減一。 畢立夏，依平。 入小滿後，六日減一。 畢立秋，日度皆盡，無平行。 入霜降後，四日加一。 畢大雪，依平。 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。 而晨伏。
At first appearance, it retrogrades at half a degree per day, covering 5 degrees in 10 days. Then stationary for 9 days. Then direct slow motion with variable rate, accelerating by 8 parts each day, covering 30 degrees in 40 days. If entering between Great Snow and Lesser Full Grain, follow this procedure. Entering Grain in Ear, subtract 1 degree every 10 days. Entering Lesser Heat through Frost's Descent, subtract an even 3 degrees. Entering Start of Winter, reduce the subtraction by 1 degree every 10 days until Minor Snow. These are the fixed degrees. Multiply the fixed degrees by the motion-parts factor and divide by 40 to get the uniform-motion parts. Multiply 4 by 39, subtract from the uniform-motion parts—that yields the initial day's motion-parts. Uniform motion: 1 degree per day, 15 degrees in 15 days. Entering Lesser Cold, increase both the day-count and degree-count by 1 every 10 days. After Entering Rain Water, the rate is 21 degrees in 21 days throughout. After the spring equinox, decrease by 1 every 10 days. Through Start of Summer, then use the mean rate. After Lesser Full Grain, decrease by 1 every 6 days. Through Start of Autumn the day- and degree-rates are used up—there is no uniform-motion phase. After Frost's Descent, increase by 1 every 4 days. Through Great Snow, then use the mean rate. Rapid motion: 204 degrees in 170 days. If the earlier direct slow phase subtracted degrees, add that amount to this phase's degrees as the fixed total. Then it disappears in the morning (conjunction).
38
夕平見，入冬至，日增所減百分。 入啟蟄，畢春分，均減九日。 清明初日，減五千九百八十六分。 乃日損所減百分。 入芒種，依平。 入夏至，日增所加百分。 入處暑，畢秋分，均加九日。 寒露初日，加五千九百八十六分。 乃日損所減百分。 入大雪，依平。
For evening mean appearance at winter solstice, each day add back 100 parts of what was subtracted. Entering Awakening of Insects through spring equinox, subtract an even 9 days. On the first day of Clear and Bright, subtract 5,986 parts. Then each day reduce the subtraction by 100 parts. Entering Grain in Ear, use the mean rate. Entering summer solstice, each day add 100 parts to the increment. Entering End Heat through Autumn Equinox, add an even 9 days. On the first day of Cold Dew, add 5,986 parts. Then each day reduce the subtraction by 100 parts. Entering Great Snow, use the mean rate.
39
初見，順疾，百七十日行二百四度。 入冬至畢立夏者，依此。 入小滿，六日加一度。 入夏至，畢小暑，均加五度。 入大暑，三日減一度。 入立秋，畢大雪，依平。 從白露畢春分，皆差行，日益疾一分半。 以一分半乘百六十九而半之，以加平行，為初日行分。 入清明，畢于處暑，畢平行。 乃平行，日一度，十五日行十五度。 入冬至後，十日減日度各一。 入啟蟄，畢芒種，皆九日行九度。 入夏至後，五日益一。 入大暑，依平。 入立秋後，六日加一。 畢秋分，二十五日行二十五度。 入寒露，六日減一。 入大雪，依平。 順遲，日益遲八分，四十日行三十度。 前加度者，此依數減之。 又留，九日。 乃退，日半度，十日退五度。 而夕伏。
At first appearance, direct rapid motion: 204 degrees in 170 days. If entering between winter solstice and Start of Summer, follow this procedure. Entering Lesser Full Grain, add 1 degree every 6 days. Entering summer solstice through Lesser Heat, add an even 5 degrees. Entering Greater Heat, subtract 1 degree every 3 days. Entering Start of Autumn through Great Snow, use the mean rate. From White Dew through spring equinox, all variable motion, accelerating by 1.5 parts each day. Multiply 1.5 parts by 169 and halve the result, add to uniform motion—that yields the initial day's motion-parts. Entering Clear and Bright through End Heat, the uniform-motion phase ends. Then uniform motion: 1 degree per day, 15 degrees in 15 days. After winter solstice, decrease both day-count and degree-count by 1 every 10 days. Entering Awakening of Insects through Grain in Ear, the rate is 9 degrees in 9 days throughout. After summer solstice, increase by 1 every 5 days. Entering Greater Heat, use the mean rate. After Start of Autumn, increase by 1 every 6 days. Through Autumn Equinox, the rate is 25 degrees in 25 days. Entering Cold Dew, decrease by 1 every 6 days. Entering Great Snow, use the mean rate. Direct slow motion, slowing by 8 parts each day, covering 30 degrees in 40 days. If degrees were added earlier, subtract that amount here. Stands still again for 9 days. Then retrograde at half a degree per day, covering 5 degrees in 10 days. Then it disappears in the evening (conjunction).
40
率百九萬六千六百八十三
Circulation rate: 1,096,683
41
終日百一十五，行分五百九十四，小分七。
Total days: 115; motion-parts: 594; minor parts: 7.
42
晨見伏六十三日，行分五百九十四，小分七。
Morning visibility and invisibility: 63 days; motion-parts: 594; minor parts: 7.
43
夕見伏五十二日。
Evening visibility and invisibility: 52 days.
44
晨平見，入冬至，均減四日。 入小寒，依平。 入立春後，均減三日。 入雨水，畢立夏，應見不見。 其在啟蟄、立夏氣內，去日十八度外、三十六度內，晨有木、火、土、金一星者，亦見。 入小滿，依平。 入霜降，畢立冬，均加一日。 入小雪，至大雪十二日，依平。 若在大雪十三日後，日增所減一日。
For morning mean appearance at winter solstice, subtract an even 4 days. Entering Lesser Cold, use the mean rate. After Start of Spring, subtract an even 3 days. Entering Rain Water through Start of Summer, it should appear but does not. If within Awakening of Insects or Start of Summer, the sun's distance is between 18 and 36 degrees and Jupiter, Mars, Saturn, or Venus is visible in the morning, Mercury also appears. Entering Lesser Full Grain, use the mean rate. Entering Frost's Descent through Start of Winter, add an even 1 day. Entering Minor Snow through the twelfth day of Great Snow, use the mean rate. If after the thirteenth day of Great Snow, each day add back 1 day of what was subtracted.
45
初見，留，六日。 順遲，日行百六十九分。 入大寒，畢啟蟄，無此遲行。 乃平行，日一度，十日行十度。 入大寒後，二日去日度各一，畢於二十日，日度俱盡，無此平行。 疾，日行一度六百九分，十日行十九度六分。 前無遲行者，此疾日減二百三分，十日行十六度四分。 而晨伏。
At first appearance, stationary for 6 days. Direct slow motion at 169 parts per day. Entering Great Cold through Awakening of Insects, omit this slow-motion phase. Then uniform motion: 1 degree per day, 10 degrees in 10 days. After Great Cold, decrease both 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 609 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 16 degrees 4 parts in 10 days. Then it disappears in the morning (conjunction).
46
夕平見，入冬至後，依平。 入穀雨，畢芒種，均減二日。 入夏至，依平。 入立秋，畢霜降，應見不見。 其在立秋、霜降氣內，夕有星去日如前者，亦見。 入立冬，畢大雪，依平。
For evening mean appearance after winter solstice, use the mean rate. Entering Grain Rain through Grain in Ear, subtract an even 2 days. Entering summer solstice, use the mean rate. Entering Start of Autumn through Frost's Descent, it should appear but does not. If within Start of Autumn or Frost's Descent, another planet is at the same solar distance as described above, Mercury also appears in the evening. Entering Start of Winter through Great Snow, use the mean rate.
47
初見，順疾，日行一度六百九分，十日行十九度六分。 若入小暑，畢處暑，日減二百三分。 乃平行，日一度，十日行十度。 入大暑後，二日去日及度各一，畢於二十日，日度俱盡，無此平行。 遲，日行百六十九分。 若疾減二百三分者，即不須此遲行。 又留，六日七分。 而夕伏。
At first appearance, direct rapid motion at 1 degree 609 parts per day, covering 19 degrees 6 parts in 10 days. If entering Lesser Heat through End Heat, reduce daily motion by 203 parts. Then uniform motion: 1 degree per day, 10 degrees in 10 days. After Greater Heat, decrease both day-count and degree-count by 1 every 2 days; at 20 days both are used up—there is no uniform-motion phase. Slow motion at 169 parts per day. If the rapid phase already had 203 parts subtracted, omit this slow-motion phase. Stands still again for 6 days 7 parts. Then it disappears in the evening (conjunction).
48
各以星率去歲積分，餘反以減其率，餘如度法得一為日，得冬至後晨平見日及分。 以冬至去朔日算及分加之，起天正，依月大小計之，命日算外，得所在日月。 金、水各以晨見伏日及分加之，得夕平見。 各以其星初日所加減之分，計後日損益之數以損益之。 訖，乃以加減平見為定見。 其加減分皆滿行分法為日。 以定見去朔日及分加其朔前夜半日度，又以星初見去日度，歲星十四，太白十一，熒惑、鎮星、辰星皆十七，晨減、夕加之，得初見宿度。 求次日，各加一日所行度及分。 熒惑、太白有小分者，各以日率為母。 其行有益疾遲者，副置一日行分，各以其差疾益、遲損，乃加之。 留者因前，退則依減，伏不注度。 順行出斗，去其分； 退行入斗，先加分。 訖，皆以二十六約行分，為度分。
For each planet, subtract the year's accumulated parts from its circulation rate; take the remainder, subtract it from the rate; divide the remainder by the degree-factor to get days—this yields the morning mean appearance date and parts after winter solstice. Add the winter-solstice-to-new-moon day count and parts, starting from the year's first month and counting by month lengths, to find the month and day. For Venus and Mercury, add the morning visibility-and-invisibility days and parts to get the evening mean appearance. For each planet, take the first day's correction parts and apply the daily increment or decrement through subsequent days. When complete, apply the corrections to the mean appearance to get the fixed appearance. When correction parts fill the motion-parts factor, convert to whole days. Add the fixed appearance's offset from new moon to the midnight longitude before new moon; subtract (morning) or add (evening) the planet's solar distance—Jupiter 14°, Venus 11°, Mars, Saturn, and Mercury 17° each—to get the first-appearance lodge and degree. For the next day, add that day's motion in degrees and parts. For Mars and Venus, when minor parts are involved, use the day-rate as denominator. When motion varies between faster and slower phases, set aside one day's motion-parts and apply the daily differential—adding for acceleration, subtracting for deceleration. During stationary, continue from the prior value; during retrograde, subtract accordingly; during invisibility, do not record longitude. In direct motion exiting the Dipper asterism, subtract the fractional parts; In retrograde motion entering the Dipper, first add the fractional parts. When complete, divide all motion-parts by 26 to convert to degree-parts.
49
交會法千二百七十四萬一千二百五八分。 交分法六百三十七萬六百二九分。
Conjunction divisor: 12,741,258 parts. Node-parts factor: 6,370,629 parts.
50
朔差百八萬五千四百九十四二分。 望分六百九十一萬三千三百五十。 交限五百八十二萬七千八百五十五八分。 望差五十四萬二千七百四十七一分。
New-moon difference: 1,085,494 parts. Full-moon parts: 6,913,350. Conjunction limit: 5,827,855 parts. Full-moon difference: 542,747 parts.
51
外限六百七十六萬七百八十二九分。 中限千二百三十五萬一千二十五八分。 內限千二百一十九萬一千四百五十八七分。
Outer limit: 6,760,782 parts. Middle limit: 12,351,025 parts. Inner limit: 12,191,458 parts.
52
以朔差乘積月，滿交會法去之； 餘得天正月朔入平交分。 求望，以望分加之。 求次月，以朔差加之。 其朔望，入大雪，畢冬至，依平。 入小寒，日加氣差千六百五十分。 入啟蟄，畢清明，均加七萬六千一百分。 自後日損所加千六百五十分。 入芒種，畢夏至，依平。 加之滿法，去之。 若朔交入小寒畢雨水，及立夏畢小滿，值盈二時已下，皆半氣差加之。 二時已上則否。 如望差已下、外限已上有星伏，木、土去見十日外，火去見四十日外，金晨伏去見二十二日外，有一星者，不加氣差。 入小暑後，日增所減千二百分。 入白露，畢霜降，均減九萬五千八百二十五分。 立冬初日，減六萬三千三百分，自後日損所減二千一百一十分。 減若不足，加法，乃減之，餘為定交分。 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。 不滿交分法者，為在外道； 滿去之，餘為在內道。 如望差已下，為去先交分，交限已上，以減交分，餘為去後交分。 皆三日法約，為時數。 望則月蝕，朔在內道則日蝕。 雖在外道，去交近，亦蝕。 在內道，去交遠，亦不蝕。
Multiply accumulated months by the new-moon difference and discard full multiples of the conjunction divisor; the remainder is the mean node parts at the first month's new moon. For full moon, add the full-moon parts. For the next month, add the new-moon difference. For new and full moons entering Great Snow through winter solstice, use the mean node parts. Entering Lesser Cold, add 1,650 parts of qi-difference each day. Entering Awakening of Insects through Clear and Bright, add an even 76,100 parts. Thereafter, reduce the increment by 1,650 parts each day. Entering Grain in Ear through summer solstice, use the mean rate. When the sum fills the divisor, discard the full amount. If the new-moon node falls between Lesser Cold and Rain Water, or Start of Summer and Lesser Full Grain, and the moon is waxing through the second double-hour or earlier, add half the qi-difference. At the second double-hour or later, do not. If below the full-moon difference and above the outer limit a planet is invisible—Jupiter or Saturn more than 10 days from appearance, Mars more than 40 days, Venus morning invisibility more than 22 days—and one such planet is present, do not add the qi-difference. After Lesser Heat, each day add back 1,200 parts of what was subtracted. Entering White Dew through Frost's Descent, subtract an even 95,825 parts. On the first day of Start of Winter, subtract 63,300 parts; thereafter reduce the subtraction by 2,110 parts each day. If subtraction would go negative, add the divisor and then subtract—the remainder is the fixed node parts. If at new moon the node parts are above the inner limit of the conjunction limit and below the middle limit, and a planet is invisible as described above, do not subtract. If less than the node-parts factor, the moon is on the outer path; If full, discard the factor; the remainder means the inner path. If below the full-moon difference, this is the distance from the prior node; at the conjunction limit and above, subtract from the node parts—the remainder is the distance from the subsequent node. Divide all by 3 and then by the time-factor to get the double-hour count. At full moon there is a lunar eclipse; at new moon on the inner path there is a solar eclipse. Even on the outer path, if close to the node, there is still an eclipse. On the inner path, if far from the node, there is no eclipse.
53
置蝕望定小餘。 入曆一日，減二百八十； 若十五日，即加之； 十四日，加五百五十； 若二十八日，即減之； 餘日皆盈加、縮減二百八十：為月蝕定餘。 十二乘之，時法而一，命子半算外； 不盡，得月蝕加時。 約定小餘如夜漏半已下者，退日算上。
Set aside the fixed minor remainder for the eclipse full moon. On sequence day 1, subtract 280; On day 15, add instead; On day 14, add 550; On day 28, subtract instead; On all other days, add 280 when waxing and subtract 280 when waning—this is the fixed remainder for lunar eclipse. Multiply by 12, divide by the time-factor, and count from midnight of zi; The remainder gives the lunar eclipse time. If the fixed minor remainder is below half the night's clepsydra outflow, move back one day on the day count.
54
置蝕朔定小餘。 入曆一日，即減二百八十； 若十五日，即加之； 十四日，加五百五十； 若二十八日，即減之； 為定。 後不入四時加減之限。 其內道，春，去交四時已上入曆，盈加、縮減二百八十； 夏，盈加、縮減二百八十； 秋，去交十一時已下，惟盈加二百八十，已上者，盈加五百五十，縮加二百八十； 冬，去交五時已下，惟盈加二百八十：皆為定餘。 十二乘之，時法而一，命子半算外； 不盡，為時餘，副之。 仲辰半前，以副減法為差率； 半後，退半辰，以法加餘，以副為差率。 季辰半前，以法加副為差率； 半後，退半辰，以法加餘，倍法加副，為差率。 孟辰半前，三因其法，以副減之，餘為差率； 半後，退半辰，以法加餘，又以法加副，乃三因其法，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； 十二已上，從十二。 若季辰半後，孟辰半前，去交六時已上者，皆從其六。 六時已下，依數不加。 皆乘差率，十四除，為時差。 子午半後，以加時餘； 卯酉半後，以減時餘； 加之滿若不足，進退時法：孟謂寅、巳、申，仲謂午、卯、酉，季謂辰、未、戌。 得日蝕加時。
Set aside the fixed minor remainder for the eclipse new moon. On sequence day 1, subtract 280; On day 15, add instead; On day 14, add 550; On day 28, subtract instead; This gives the fixed value. After this, do not apply the four-seasons add-subtract limits. On the inner path in spring, if the node distance is 4 double-hours or more and the sequence is entered, add 280 when waxing and subtract when waning; In summer, add 280 when waxing and subtract when waning; In autumn, if node distance is 11 double-hours or below, add only 280 when waxing; if above, add 550 when waxing and add 280 when waning; In winter, if node distance is 5 double-hours or below, add only 280 when waxing—all as the fixed remainder. Multiply by 12, divide by the time-factor, and count from midnight of zi; The remainder is the time remainder—set it aside. In the first double-hour before the half-period, subtract the aside from the factor to get the difference-rate; After the half-period, retreat half a double-hour, add the remainder to the factor, and use the aside as the difference-rate. In the intermediate double-hour before the half-period, add the aside to the factor for the difference-rate; After the half-period, retreat half a double-hour, add the remainder to the factor, double the factor and add the aside—that is the difference-rate. In the first double-hour before the half-period, triple the factor, subtract the aside—the remainder is the difference-rate; After the half-period, retreat half a double-hour, add the remainder to the factor, add the aside to the factor, then triple the factor and subtract the aside—that is the difference-rate. Also set the node-distance in double-hours: if 3 or below, add 3; if 6 or below, add 2; if 9 or below, add 1; if above 9, use the number as is; if 12 or above, cap at 12. If in the intermediate double-hour after the half-period or the first double-hour before the half-period the node distance is 6 double-hours or more, use 6 in all cases. At 6 double-hours or below, use the number without adding. Multiply all by the difference-rate and divide by 14 to get the time-difference. After the half-period of zi and wu, add to the time remainder; After the half-period of mao and you, subtract from the time remainder; When addition overflows or subtraction underflows, adjust by the time-factor: first double-hours are yin, si, shen; intermediate are wu, mao, you; last are chen, wei, xu. This yields the solar eclipse time.
55
望去交分，冬先後交，皆去二時； 春先交，秋後交，去半時； 春後交，秋先交，去二時； 夏則依定。 不足去者，既。 乃以三萬六千一百八十三為法而一，以減十五，餘為月蝕分。
For node distance: in winter, whether before or after the node, subtract 2 double-hours; in spring before the node and autumn after the node, subtract half a double-hour; in spring after the node and autumn before the node, subtract 2 double-hours; in summer, use the fixed value. If subtraction would go negative, the eclipse is total. Divide by 36,183; subtract the quotient from 15—the remainder is the lunar eclipse magnitude.
56
朔去交，在內道，五月朔，加時在南方，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，加時在未西； 處暑畢寒露，後交十三時外，值盈，加時在巳東，皆不蝕。 交在外道，先後去交一時內者，皆蝕。 若二時內，及先交值盈、後交值縮二時外者，亦蝕。 夏去交二時內，加時在南方者，亦蝕。 若去分、至十二時內，去交六時內者，亦蝕。 若去春分三日內，後交二時； 秋分三日內，先交二時內者，亦蝕。 諸去交三時內有星伏，土、木去見十日外，火去見四十日外，金晨伏去見二十二日外，有一星者，不蝕。 各置去交分。 秋分後，畢立春，均減二十二萬八百分。 啟蟄初日，畢芒種，日損所減千八百一十分。 夏至後，畢白露，日增所減二千四百分。 以減去交分，餘為不蝕分。 不足減，反相減為不蝕分。 亦以減望差為定法。 後交值縮者，直以望差為定法。 其不蝕分，大寒畢立春，後交五時外，皆去一時。 時差值減者，先交減之，後交加之。 時差值加者，先交加之，後交減之。 不足減者，皆既。 十五乘之，定法而一，以減十五，餘為日蝕分。
For new-moon node distance on the inner path: on a fifth-month new moon with eclipse time in the south and prior node beyond 13 double-hours; on a sixth-month new moon with subsequent node beyond 13 double-hours—no eclipse. Awakening of Insects through Clear and Bright, prior node beyond 13 double-hours, moon waning, eclipse time in wei west; End Heat through Cold Dew, subsequent node beyond 13 double-hours, moon waxing, eclipse time in si east—in all cases, no eclipse. If the node is on the outer path and the distance before or after the node is within 1 double-hour, there is an eclipse. If within 2 double-hours, or prior node with waxing moon or subsequent node with waning moon beyond 2 double-hours, there is also an eclipse. In summer, within 2 double-hours of the node with eclipse time in the south, there is also an eclipse. If within 12 double-hours of an equinox or solstice and within 6 double-hours of the node, there is also an eclipse. If within 3 days of the spring equinox and subsequent node within 2 double-hours; or within 3 days of the autumn equinox and prior node within 2 double-hours, there is also an eclipse. If within 3 double-hours of the node a planet is invisible—Saturn or Jupiter more than 10 days from appearance, Mars more than 40 days, Venus morning invisibility more than 22 days—and one such planet is present, no eclipse. Set the distance-from-node parts for each case. After the autumn equinox through Start of Spring, subtract an even 228,800 parts. From the first day of Awakening of Insects through Grain in Ear, reduce the subtraction by 1,810 parts each day. After summer solstice through White Dew, add back 2,400 parts of the subtraction each day. Subtract this from the distance-from-node parts—the remainder is the non-eclipse threshold. If subtraction would go negative, reverse the subtraction to get the non-eclipse threshold. Also subtract this from the full-moon difference to get the fixed divisor. If the subsequent node occurs during waning, use the full-moon difference directly as the fixed divisor. For the non-eclipse threshold: Great Cold through Start of Spring, subsequent node beyond 5 double-hours—subtract 1 double-hour in all cases. If the time-difference calls for subtraction, subtract for the prior node and add for the subsequent node. If the time-difference calls for addition, add for the prior node and subtract for the subsequent node. If subtraction would go negative, the eclipse is total. Multiply by 15, divide by the fixed divisor, subtract the quotient from 15—the remainder is the solar eclipse magnitude.
57
置日月蝕分，四已下，因增二； 五已下，因增三； 六已上，因增五； 各為刻率，副之。 以乘所入曆損益率，四千五十七為法而一。 值盈，反其損益； 值縮，依其損益。 皆損益其副，為定用刻。 乃六乘之，十而一，以減蝕甚辰刻，為虧初。 又四乘之，十而一。 以加食甚辰刻，為復滿。
For eclipse magnitude: if 4 or below, add 2; if 5 or below, add 3; if 6 or above, add 5; each becomes the quarter-rate—set it aside. Multiply by the sequence increase-decrease rate and divide by 4,057. If waxing, reverse the increase-decrease; If waning, follow the increase-decrease. Apply the increase-decrease to the aside value—this is the fixed usable quarter-count. Multiply by 6, divide by 10, subtract from the maximum-eclipse double-hour quarters—this is the beginning of obscuration. Multiply by 4 and divide by 10 again. Add to the maximum-eclipse double-hour quarters—this is full restoration.