1
志第十一歷五
Treatise Eleven: Calendrics, Part Five.
2
大統曆法三上推步
▲ Grand Concordance Calendar Method, Part Three (Upper): Stepwise Computation.
3
大統推步，悉本《授時》，惟去消長而已。 然《通軌》諸捷法，實爲布算所須，其間次序，亦有與《曆經》微別者。 如氣朔發斂，《授時》原分二章，今古合爲一。 《授時》盈縮差在日躔，遲疾差在月離，定朔、經朔離爲二處。 今則經朔後，即求定朔，於用殊便。 其目七：曰氣朔，曰日躔，曰月離，曰中星，曰交食，曰五星，曰四餘。
Grand Concordance stepwise computation follows the Season Granting Calendar throughout, omitting only the solar expansion–contraction tables. Yet the shortcut procedures in the Comprehensive Track are indispensable for tabular computation, and their ordering also diverges slightly from the Calendar Classic. Solar terms, lunations, and the issuing-and-gathering sequence, for example, formed two chapters in the Season Granting; here the older and newer treatments are combined into one. In the Season Granting, the equation of center belonged to solar motion and the anomaly to lunar motion, while mean and true conjunction were computed separately. Here, once the mean conjunction is found, the true conjunction follows at once—a considerable practical advantage. There are seven sections: qi and lunation; solar motion; lunar motion; culminating stars; nodes and eclipses; the five planets; and the four residues.
4
▲步氣朔發斂附
▲ Stepwise Computation: Qi, Lunation, and Issuing-and-Gathering (attached)
5
洪武十七年甲子歲爲元。 上距至元辛巳一百零四算。
The epoch is the jiazi year, Hongwu 17. The interval back to the Zhiyuan xinsi epoch is 104 counts.
6
歲週三百六十五萬二千四百二十五分，實測無消長。 半之爲歲周，四分之爲氣象限，二十四分之爲氣策。
Tropical year length: 3,652,425 parts; by observation, without solar equation correction. Halve it for the half-year circuit; quarter it for the qi-image limit; divide by twenty-four for the qi divisor.
7
日週一萬。 即一百刻，刻有百分，分有百秒，以下微纖，皆以百遞析。
Day unit: 10,000. That is one hundred quarter-hours; each quarter-hour has one hundred minutes, each minute one hundred seconds; finer subdivisions below all descend by hundreds.
8
氣應五十五萬零三百七十五分。
Qi offset: 550,375 parts.
9
置距算一百零四，求得中積三億七千六百一十九萬九千七百七十五分，加辛巳氣應五十五萬零六百分，得通積三億七千六百七十五萬零三百七十五分，滿紀法六十去之，餘爲《大統》氣應。
Set the interval count at 104 to obtain a middle accumulation of 376,199,775 parts; add the xinsi qi offset of 550,600 parts for a universal accumulation of 376,750,375 parts; divide by the era divisor of 60 and take the remainder as the Grand Concordance qi offset.
10
開應一十八萬二千零百七十零分一十八秒。
Intercalation offset: 182,070 parts 18 seconds.
11
置中積，加辛巳閏應二十零萬二千零五十分，得閏積三億七千六百四十零萬一千八百二十五分，滿朔實去之，餘爲《大統》閏應。
Set the middle accumulation and add the xinsi intercalation offset of 202,050 parts to obtain an intercalation accumulation of 376,401,825 parts; divide by the synodic-month divisor and take the remainder as the Grand Concordance intercalation offset.
12
轉應二十零萬九千六百九十零分。
Rotation offset: 209,690 parts.
13
置中積，加辛巳轉應一十三萬零二百零五分，共得三億七千六百三十二萬九千九百八十分，滿轉終去之，餘爲《大統》轉應。
Set the middle accumulation and add the xinsi rotation offset of 130,205 parts, totaling 376,329,980 parts; divide by the rotation cycle and take the remainder as the Grand Concordance rotation offset.
14
交應一十一萬五千一百零五分零八秒。
Node offset: 115,105 parts 08 seconds.
15
置中積加辛巳交應二十六萬零三百八十八分，共得三億七千六百四十六萬零一百六十三分，滿交終去之，餘爲《大統》交應。
Set the middle accumulation and add the xinsi node offset of 260,388 parts, totaling 376,460,163 parts; divide by the nodal cycle and take the remainder as the Grand Concordance node offset.
16
按《授時曆》既成之後，閏轉交三應數，旋有改定，故《元志》、《曆經》閏應二十零萬一千八百五十分，而《通軌》載閏應二十零萬二千零五十分，實加二百分，是當時經朔改早二刻也。 《曆經》轉應一十三萬一千九百零四分，《通軌》載轉應一十三萬零二百零五分，實減一千六百九十九分，是入轉改遲一十七刻弱也。 《曆經》交應二十六萬零一百八十七分八十六秒，《通軌》交應二十六萬零三百八十八分，實加二百分一十四秒，是正交改早二刻強也。 或以《通軌》辛巳三應，與《元志》互異，目爲元統所定，非也。 夫改憲必由測驗，即當具詳始末，何反追改《授時曆》，自沒其勤乎？ 是故《通軌》所述者，乃《授時》續定之數，而《曆經》所存，則其未定之初藁也。
After the Season Granting Calendar was completed, its intercalation, rotation, and node offsets were soon revised. The Yuan History and Calendar Classic give an intercalation offset of 201,850 parts, whereas the Comprehensive Track records 202,050 parts—an increase of 200 parts that then advanced the mean conjunction by two quarter-hours. The Calendar Classic gives a rotation offset of 131,904 parts; the Comprehensive Track records 130,205 parts—a reduction of 1,699 parts that delayed nodal entry by seventeen quarter-hours and a fraction. The Calendar Classic gives a node offset of 260,187 parts 86 seconds; the Comprehensive Track records 260,388 parts—an increase of 200 parts 14 seconds that advanced true conjunction by two quarter-hours and a fraction. Some treat the Comprehensive Track's three xinsi offsets, which differ from the Yuan History, as Yuan–Ming revisions—this is mistaken. Calendar reform must rest on measurement and verification, and should then record the full course of events—why instead retroactively rewrite the Season Granting Calendar and efface their labor? Accordingly, the Comprehensive Track preserves the Season Granting's subsequently fixed constants, while the Calendar Classic retains its earlier, unfixed draft.
17
通餘五萬二千四百二十五分。
Universal remainder: 52,425 parts.
18
朔策二十九萬五千三百零五分九十三秒，一名朔寶。 半之爲望策，一名交望。 又半之爲弦策。
Synodic-month divisor: 295,305 parts 93 seconds, also called the synodic-month constant. Halve it for the full-moon divisor, also called the nodal full moon. Halve again for the quarter-moon divisor.
19
通閏一十零萬八千七百五十三分八十四秒。
Universal intercalation: 108,753 parts 84 seconds.
20
月閏九千零百六十二分八十二秒。
Monthly intercalation unit: 9,062 parts 82 seconds.
21
閏限一十八萬六千五百五十二分零九秒。 一名閏準。
Intercalation threshold: 186,552 parts 09 seconds. Also called the intercalation criterion.
22
盈初縮末限八十八萬九千零百九十二分二十五秒。
Expansion-initial and contraction-final interval: 889,092 parts 25 seconds.
23
縮初盈末限九十三萬七千一百二十零分二十五秒。
Contraction-initial and expansion-final interval: 937,120 parts 25 seconds.
24
轉終二十七萬五千五百四十六分，半之爲轉中。
Rotation cycle: 275,546 parts; halve it for rotation midpoint.
25
朔轉差一萬九千七百五十九分九十三秒。
New-moon rotation increment: 19,759 parts 93 seconds.
26
日轉限一十二限二十。
Daily rotation limit: 12 limits 20.
27
轉中限一百六十八限零八三零六零。 以日轉限乘轉中。 一名限總。
Rotation midpoint limit: 168 limits 083060. Multiply the rotation midpoint by the daily rotation limit. Also called the limit aggregate.
28
朔轉限二十四限一零七一一四六。 以日轉限乘朔轉差。
New-moon rotation limit: 24 limits 1071146. Multiply the new-moon rotation increment by the daily rotation limit.
29
弦轉限九十零限零六八三零八六五。 以日轉限乘弦策。 一名限策。
Quarter-moon rotation limit: 90 limits 06830865. Multiply the quarter-moon divisor by the daily rotation limit. Also called the limit divisor.
30
交終二十七萬二千一百二十二分二十四秒。
Nodal cycle: 272,122 parts 24 seconds.
31
朔交差二萬三千一百八十三分六十九秒。
New-moon node increment: 23,183 parts 69 seconds.
32
氣盈二千一百八十四分三十七秒五十微。
Qi expansion: 2,184 parts 37 seconds 50 micro-parts.
33
朔虛四千六百九十四分零七秒。
New-moon void: 4,694 parts 07 seconds.
34
沒限七千八百一十五分六十二秒五十微。
Submergence threshold: 7,815 parts 62 seconds 50 micro-parts.
35
盈策九萬六千六百九十五分二十八秒。
Expansion divisor: 96,695 parts 28 seconds.
36
虛策二萬九千一百零四分二十二秒。
Void divisor: 29,104 parts 22 seconds.
37
土王策三萬零四百三十六分八十七秒五十微。
Earth-king divisor: 30,436 parts 87 seconds 50 micro-parts.
38
宿策一萬五千三百零五分九十三秒。
Lodging divisor: 15,305 parts 93 seconds.
39
紀法六十萬。 即旬週六十日。
Era divisor: 600,000. That is the sixty-day decad cycle.
40
推天正冬至置距洪武甲子積年減一，以歲周乘之爲中積，加氣應爲通積，滿紀法去之，至不滿之數，爲天正冬至。 以萬爲日，命甲子算外，爲冬至日辰。 累加通餘，即得次年天正冬至。
To find the winter solstice of the civil year: set the interval from the Hongwu jiazi epoch, subtract one, multiply by the tropical year length for the middle accumulation, add the qi offset for the universal accumulation, and divide by the era divisor; the remainder not reaching a full cycle is the winter solstice of the civil year. Take 10,000 as one day; count off from the jiazi origin to obtain the winter solstice date and double-hour. Add the universal remainder repeatedly to obtain the next year's civil winter solstice.
41
推天正閏餘置中積，加閏應，滿朔策去之，至不滿之數，爲天正閏餘。 累加通閏，即得次年天正閏餘。
To find the civil-year intercalation remainder: set the middle accumulation, add the intercalation offset, divide by the synodic-month divisor, and take the remainder not reaching a full month as the civil intercalation remainder. Add the universal intercalation repeatedly to obtain the next year's civil intercalation remainder.
42
推天正經朔置冬至，減閏餘，遇不及減，加紀法減之，爲天正經朔。 無閏加五十四萬三六七一一六。 十二朔策紀法。 有閏，加二十三萬八九七七零九。 十三朔實去紀法。 滿紀法仍去之，即得次年天正經朔視天正閏餘在閏限已上，其年有閏月。
To find the civil mean conjunction: set the winter solstice and subtract the intercalation remainder; if subtraction fails, add the era divisor and subtract again—this is the civil mean conjunction. In a common year, add 543,671,116. Twelve synodic-month divisors equal the era divisor. In an intercalary year, add 238,977,709. Thirteen synodic-month constants subtract the era divisor. When the era divisor is full, subtract it again to obtain the next year's civil mean conjunction. If the civil intercalation remainder lies above the intercalation threshold, that year has an intercalary month.
43
推天正盈縮置半歲周，內減其年閏餘全分，餘爲所求天正縮歷。 如徑求次年者，於天正縮歷內減通閏，即得。 減後，視在一百五十三日零九已下者，復加朔實，爲次年天正縮歷。
To find the civil-year solar equation at the winter solstice: set half the tropical year and subtract that year's full intercalation remainder; the remainder is the sought contraction sequence for the civil year. To proceed directly to the next year, subtract the universal intercalation from the civil contraction sequence. After subtraction, if the value falls below 153 days 09, add the synodic-month constant again for the next year's civil contraction sequence.
44
推天正遲疾置中積，加轉應，減去其年閏餘全分，餘滿轉終去之，即天正入轉。 視在轉中已下爲疾歷，已上去之爲遲歷。 如徑求次年者，加二十三萬七一一九一六，十二轉差之積。 經閏再加轉差，皆滿轉終去之，遲疾各仍其舊。 若滿轉中去之，爲遲疾相代。
To find the civil lunar anomaly: set the middle accumulation, add the rotation offset, subtract that year's full intercalation remainder, and divide the remainder by the rotation cycle—this is the civil entry into rotation. If below the rotation midpoint, it is the rapid sequence; if above, subtract the midpoint for the slow sequence. To proceed directly to the next year, add 237,191,916—the accumulated twelve rotation increments. In an intercalary year add the rotation increment again; divide by the rotation cycle whenever full; slow and rapid each retain their prior state. When the rotation cycle is full, subtract it and slow and rapid alternate in turn.
45
推天正入交置中積，減閏餘，加交應，滿交終去之，即天正入交凡日。 如徑求次年者，加六千零八十二分零四秒，十二交差內去交終。 經閏加二萬九千二百六十五分七十三秒，十三交差內去交終。 皆滿交終仍去之，即得。
To find the civil nodal entry: set the middle accumulation, subtract the intercalation remainder, add the node offset, and divide by the nodal cycle—this is the civil nodal entry in days. To proceed directly to the next year, add 6,082 parts 04 seconds—the twelve node increments, subtracting the nodal cycle within. In an intercalary year add 29,265 parts 73 seconds—the thirteen node increments, subtracting the nodal cycle within. Whenever the nodal cycle is full, subtract it again to obtain the result.
46
推各月經朔及弦望置天正經朔策，滿紀法去之，即得正月經朔。 以弦策累加之，去紀法，即得弦望及次朔。
To find each month's mean conjunction and quarter- and full moons: set the civil mean conjunction divisor; divide by the era divisor; the remainder is the first month's mean conjunction. Add the quarter-moon divisor repeatedly and subtract the era divisor to obtain the quarter- and full moons and successive conjunctions.
47
推各恆氣置天正冬至，加三氣策，滿紀法去之，即得立春恆日。 以氣策累加之，去紀法，即得二十四氣恆日。
To find each mean solar term: set the civil winter solstice, add three qi divisors, divide by the era divisor, and obtain the constant day of Beginning of Spring. Add the qi divisor repeatedly and subtract the era divisor to obtain the constant days of all twenty-four terms.
48
推閏在何月置朔策，以有閏之年之閏餘減之，餘爲實，以月閏爲法而一，得數命起天正次月算外，即得所閏之月。 閏有進退，仍以定朔無中氣爲定。 如減餘不及月閏，或僅及一月閏者，爲閏在年前。
To find which month receives the intercalation: set the synodic-month divisor and subtract that intercalary year's intercalation remainder; take the remainder as dividend and the monthly intercalation unit as divisor; the quotient, counted from the month after the civil new year, names the intercalary month. Intercalation may advance or retreat; the true conjunction lacking a mid-term qi remains the final criterion. If the remainder after subtraction falls short of one monthly intercalation unit, or only reaches one such unit, the intercalation belongs to the preceding year.
49
推各月盈縮歷置天正縮歷，加二朔策，去半歲周，即得正月經朔下盈歷。 累加弦策，各得弦望及次朔，如滿半歲周去之交縮，滿半周又去之即復交盈。
To find each month's solar equation sequence: set the civil contraction sequence, add two synodic-month divisors, and subtract half the tropical year to obtain the expansion sequence below the first month's mean conjunction. Add the quarter-moon divisor repeatedly for each quarter- and full moon and successive conjunction; when half the tropical year is full, subtract it for nodal contraction; when half a cycle is full again, subtract once more to return to nodal expansion.
50
推初末限視盈歷在盈初縮末限已下，縮歷在縮初盈末限已下，各爲初。 已上用減半歲周爲末。
To find the initial and final limits: if the expansion sequence lies below the expansion-initial and contraction-final interval, or the contraction sequence below the contraction-initial and expansion-final interval, each is initial. Above those thresholds, subtract half the tropical year for the final interval.
51
推盈縮差置初末歷小余，以立成內所有盈縮加之乘之爲實，日週一萬爲法除之，得婁數以加其下盈縮積，即盈縮差。
To find the solar equation correction: set the small remainder of the initial or final interval, add the expansion–contraction entry from the ready reckoners and multiply for the dividend; divide by the day unit of 10,000; add the resulting lou number to the underlying equation accumulation to obtain the equation correction.
52
推各月遲疾歷置天正經朔遲疾歷，加二轉差，得正月經朔下遲疾歷。 累加弦策，得弦望及次朔，皆滿轉中去之，爲遲疾相代。
To find each month's lunar anomaly sequence: set the civil mean-conjunction anomaly sequence and add two rotation increments to obtain the anomaly sequence below the first month's mean conjunction. Add the quarter-moon divisor repeatedly for quarter- and full moons and successive conjunctions; whenever the rotation cycle is full, subtract it and slow and rapid alternate in turn.
53
推遲疾限各置遲次歷，以日轉限乘之，即得限數。 以弦轉限累加之，滿轉中限去之，即各弦望及次朔限。 如徑求次月，以朔轉限加之，亦滿轉中去之，即得。 又法：視立成中日率，有與遲疾歷較小布相近者以減之，餘在八百二十已下，即所用限。
To find the anomaly limits: for each slow-interval count, multiply by the daily rotation limit to obtain the limit number. Add the quarter-moon rotation limit repeatedly; when the rotation midpoint limit is full, subtract it to obtain the limit for each quarter- and full moon and successive conjunction. To find the next month directly, add the new-moon revolution limit; when it fills the revolution midpoint, discard the excess to obtain the result. Alternate method: take the day-rate from the ready reckoners; where it is close to the small cloth value in the slow-rapid ephemeris, subtract it; a remainder of 820 or less gives the applicable limit.
54
求遲疾差置遲疾歷，以立成日率減之，如不及減，則退一位。 餘以其下損益分乘之爲實，八百二十分爲法除之，得數以加其下遲疾積，即遲疾差。
To find the slow-rapid difference: set the slow-rapid ephemeris and subtract the ready-reckoner day-rate; if subtraction is impossible, shift back one decimal place. Multiply the remainder by the increase-decrease increment below as dividend; divide by 820 parts as divisor; add the quotient to the underlying slow-rapid accumulation to obtain the slow-rapid difference.
55
推加減差視經朔弦望下所得盈縮差、遲疾差，以盈遇遲、縮遇疾爲同相併，盈遇疾、縮遇遲爲異相較，各以八百二十分乘之爲實，再以遲疾限行度內減去八百於二十分，爲定限度爲法，法除實爲加減差。 盈遲爲加，縮疾爲減，異名相較者，盈多疾爲加，疾多於盈爲減，縮多於遲減，遲多於縮加。
To find the additive-subtractive difference: take the expansion-contraction and slow-rapid differences below each mean new moon, first quarter, full moon, and last quarter; combine like cases (surplus with slow, deficit with rapid) and compare unlike cases (surplus with rapid, deficit with slow); multiply each by 820 parts as dividend; subtract 820 from the degrees within the slow-rapid limit to form the fixed-limit divisor; divide to obtain the additive-subtractive difference. Surplus paired with slow yields addition; deficit paired with rapid yields subtraction. In cross-comparison: if surplus exceeds slow, add; if rapid exceeds surplus, subtract; if deficit exceeds slow, subtract; if slow exceeds deficit, add.
56
推定朔望各置經朔弦望，以加減差加減之，即爲定日。 視定朔幹名，與後朔同者月大，不同者月小，內無中氣者爲閏月。 其弦望在立成相同日日出分已下者，則退一日命之。
To find the true new and full moons: set each mean new moon, first quarter, full moon, and last quarter, then add or subtract the additive-subtractive difference to obtain the true date. Check the sexagenary day of the true new moon: if it matches the next new moon the month is long; if not, short; if no mean qi falls within the month, it is intercalary. If a quarter or full moon shares the same ready-reckoner day but sunrise parts have already elapsed, name it one day earlier.
57
推各月入交置天正經朔入交凡日加二交差，得正月經朔下入交凡日。 累加交望，滿交終去之，即得各月下入交凡日。 徑求次月，加交差即得。
To find each month's nodal entry: set the mean new moon's nodal entry at the celestial first month; add twice the nodal difference to obtain the first civil month's mean new moon nodal entry. Accumulate nodal quarter intervals; when full, discard the nodal cycle to obtain each month's nodal entry. To find the next month directly, add the nodal difference.
58
推土王用事置穀雨、大暑、霜降、大寒恆氣日，減土王策，如不及減，加紀法減之，即各得土王用事日。
To find the Earth-king dominion: set the mean-qi days of Grain Rain, Great Heat, Frost's Descent, and Great Cold; subtract the Earth-king parameter; if insufficient, add the era cycle and subtract again to obtain each dominion day.
59
推發斂加時各置所推定朔弦望及恆氣之小余，以十二乘之，滿萬爲時，命起子正。 滿五千，又進一時，命起子初。 算外得時不滿者，以一千二百除之爲刻，命起初刻。 初正時之刻，皆以初一二三四爲好，於算外命之。 其第四刻爲畸零，得刻法三之一，凡三時成一刻，以足十二時百刻之數。
To find emission-contraction with time-of-day: for each true new moon, quarter, full moon, and mean qi, take the lesser remainder, multiply by twelve, and divide by 10,000 for the double-hour, counting from zi midnight. At 5,000 parts, advance one further double-hour, counting from the first quarter of zi. After the double-hour is fixed, divide the remainder by 1,200 for quarter-hours, counting from the first quarter-hour. For the first quarter-hour at midnight, only the first through fourth quarter-hours are used in naming; the remainder is named beyond the count. The fourth quarter-hour is fractional, taking one-third of a quarter-hour unit; three such double-hours complete one quarter-hour, filling out twelve double-hours and one hundred quarter-hours.
60
按古因及《授時》，皆以發斂爲一章。 發斂去者，日道發南斂北之細數也，而加時附焉，則又所以紀發斂之辰刻，故曰發斂加時也。 《大統》取其便算，故合發斂與氣朔共爲一章，或以乘除疏發斂，非其質矣。
In antiquity and in the Season Granting Calendar, emission-contraction formed its own chapter. Emission-contraction records the fine gradations of the sun's path opening southward and closing northward; attaching time-of-day further fixes the double-hours and quarter-hours of that process—hence emission-contraction with time-of-day. The Grand Concordance favored ease of calculation and merged emission-contraction with qi and new moon into one chapter, or thinned emission-contraction through multiplication and division—neither preserves its true substance.
61
推盈日視恆氣小余，在沒限已上，爲有盈之氣。 置策餘一萬零一四五六二五，以十五日除氣策。 以有盈之氣小余減之，餘以六十八分六六以氣盈除十五日。 乘之，得數以加恆氣大餘，滿紀法去之，命甲子算外，得盈日。 求盈日及分秒，以盈策加之，又去紀法，即得。
To find surplus days: inspect the mean qi's lesser remainder; if it lies above the extinction limit, that qi has surplus. Set the parameter remainder at 10,014,562.5 and divide the qi parameter by fifteen days. Subtract the surplus qi's lesser remainder; with the remainder, apportion the qi surplus over fifteen days at 68 parts 66. Multiply to obtain a value, add it to the mean qi's greater remainder, discard the era cycle when full, and count from jiazi beyond the count for the surplus day. To find the surplus day with parts and seconds, add the surplus parameter and again discard the era cycle.
62
推虛日視經朔小余在朔虛已下，爲有虛之朔。 置有虛之朔小余，以六十三分九一以朔虛除三十日。 乘之，得數以加經朔大餘，滿紀法去之，命甲子算外爲虛日。 求次虛。 置日及分秒，以虛策加之，又去紀法，即得。
To find void days: inspect the mean new moon's lesser remainder; if below the new-moon void limit, that new moon carries void. Set the void new moon's lesser remainder; apportion the new-moon void over thirty days at 63 parts 91. Multiply to obtain a value, add it to the mean new moon's greater remainder, discard the era cycle when full, and count from jiazi beyond the count for the void day. To find the next void day. Set the day with parts and seconds; add the void parameter and again discard the era cycle.
63
推直宿置通積，以氣應加中積。 減閏應，以宿會二十八萬累去之，餘命起翼宿算外，得天正經朔直宿。 置天正經宿直宿，加兩宿策，爲正月經朔直宿。 以宿策累加，得各月經朔直宿。 再以各月朔下加減差加減之，爲定朔直宿。
To find the lodge directly below: set the general accumulation and add the qi correspondence to the central accumulation. Subtract the intercalation correspondence; repeatedly remove the lodge conjunction of 280,000; with the remainder count from Wings beyond the count for the celestial first month's mean new moon lodge. Set the celestial first month's lodge; add two lodge parameters for the first civil month's mean new moon lodge. Accumulate by adding the lodge parameter to obtain each month's mean new moon lodge. Then add or subtract each month's new-moon additive-subtractive difference to obtain the true new moon's lodge.
64
▲步日躔
▲ Solar motion.
65
周天三百六十五度二十五分七十五秒，半之爲半周天，又半之爲象限。
The celestial circuit is 365°25′75″; half is half the celestial circuit; half again is the quadrant limit.
66
歲差一分五十秒。
Precession: 1′50″ per year.
67
周應三百一十五度一十分七十五秒。
Circuit correspondence: 315°10′75″.
68
按此係至元辛巳之周應，乃自虛七度至箕十之度數也。 洪武甲子相距一百四年，歲差已退天五十四分五十秒，而周應仍用舊數，殆傳習之誤耳。
This is the Zhiyuan xinsi circuit correspondence—the arc from Void 7° to Winnowing Basket 10°. Hongwu jiazi lies one hundred four years later; precession has already shifted heaven by 54′50″, yet the circuit correspondence still uses the old value—likely a transmission error.
69
推天正冬至日躔赤道宿次置中積，加周應，應減距曆元甲子以來歲差。 滿周天去之，不盡，起虛七度，依各宿次去之，即冬至加時赤道日度。 如求次年，累減歲差，即得。
To find the winter solstice's solar position in equatorial lodges: set the central accumulation, add the circuit correspondence, and subtract precession accumulated since the epoch jiazi. Discard full celestial circuits; with the remainder start at Void 7° and subtract each lodge in turn to obtain the equatorial solar degree at winter solstice with time-of-day. For the following year, subtract precession cumulatively.
70
表格略
[Table omitted.]
71
推天正冬至日躔黃道宿次置冬至加時赤道日度，以至後赤道積度減之，餘以黃道率乘之。 如赤道率而一，得數以加黃道積度，即冬至加時黃道日度。 黃赤道積度及度率，俱見《法原》。
To find the winter solstice solar position in ecliptic lodges: set the equatorial solar degree at winter solstice with time-of-day, subtract the post-solstice equatorial accumulation, and multiply the remainder by the ecliptic rate. Divide by the equatorial rate, add the quotient to the ecliptic accumulation, and obtain the ecliptic solar degree at winter solstice with time-of-day. Ecliptic and equatorial accumulations and degree-rates all appear in Origins of the Method.
72
表格略
[Table omitted.]
73
推定象限度以冬至加時赤道日度，與冬至加時黃道日度相減，爲黃赤道差。 以本年黃赤道差，與次年黃赤道相減，餘以四而一，加入氣象限內，爲定象限度。
To find the fixed quadrant limit: subtract the ecliptic winter-solstice solar degree from the equatorial to obtain the ecliptic-equatorial difference. Subtract the following year's ecliptic-equatorial difference from this year's; divide the remainder by four and add it within the qi image limit to obtain the fixed quadrant limit.
74
推四正定氣日置所推冬至分，即爲冬正定氣，加盈初縮末限，滿紀法去之，餘爲人正定氣。 加縮初盈末限，去紀法，餘爲秋正定氣。 加縮初盈末限，去紀法，餘爲次年冬正定氣。
To find the four fixed qi days: set the computed winter solstice parts as the winter fixed qi; add the expansion-initial and contraction-final limit; discard the era cycle when full; the remainder is the spring fixed qi. Add the contraction-initial and expansion-final limit, discard the era cycle, and obtain the autumnal fixed qi. Add the contraction-initial and expansion-final limit again, discard the era cycle, and obtain the next year's winter fixed qi.
75
推四正相距日以前正定氣大餘，減次正定氣大餘，加六十日，得相距日。 如次正氣不及減者，加六十日減之，再加六十日，爲相距日。
To find the four fixed intervals in days: subtract the next fixed qi's greater remainder from the previous and add sixty days. If the next fixed qi is smaller, add sixty days before subtracting; add sixty days again for the interval.
76
推四正加時黃道積度置冬至加時黃道日度，累加定象限，各得四正加時黃道積度。
To find the four fixed ecliptic accumulations at time-of-day: set the winter solstice ecliptic solar degree and add the fixed quadrant limit cumulatively for each fixed time.
77
推四正加時減分置四正定氣小余，以其初日行度乘之，如日周而一，爲各正加時減分。
To find the four fixed reduction parts at time-of-day: take each fixed qi's lesser remainder, multiply by its first-day travel degree, and divide by the solar circuit.
78
冬正行一度零五一零八五。 春正距夏正九十三日者，行零度九九九七零三，距九十四日者行一度。 夏正行零度九五一五一六。 秋正距冬正八十八日者，行一度零零零五零五，距八十九日者行一度。
The winter fixed qi travels 1°0.51085 per day. From the spring fixed qi, at ninety-three days to the summer fixed qi the travel is 0°0.999703 per day; at ninety-four days, 1° per day. The summer fixed qi travels 0°0.951516 per day. From the autumn fixed qi, at eighty-eight days to the winter fixed qi the travel is 1°0.000505 per day; at eighty-nine days, 1° per day.
79
推四正夜半積度置四正加時黃道積芭，減去其加時減分，即得。
To find the four fixed midnight ecliptic accumulations: set each fixed time's ecliptic accumulation and subtract its time-of-day reduction.
80
推四正夜半黃道宿次置四正夜半黃道積度，滿黃道宿度去之，即得。
To find the four fixed midnight ecliptic lodges: set each midnight ecliptic accumulation and discard full ecliptic lodge degrees.
81
推四正夜半相距度置次正夜半黃道積度，以前正夜半黃道積度減之，餘爲兩正相距度，遇不及減者，加周天減之。
To find the four fixed midnight interval degrees: subtract the previous midnight ecliptic accumulation from the next; if insufficient, add the celestial circuit before subtracting.
82
推四正行度加減日差雙相距度與相距日下行積度相減，餘如相距日而一，爲日差。 從相距度人減去行積度者爲加，從積度內減去相距度者爲減。
To find the four fixed travel day-differences: subtract the travel accumulation below the interval from twice the interval degree; divide the remainder by the interval in days for the day-difference. If the travel accumulation is subtracted from within the interval degree, the day-difference is additive; if the interval degree is subtracted from within the travel accumulation, it is subtractive.
83
秋正距冬至，冬至距春正八十八日，行積度九十度四零零九，八十九日行積度九十一度四零一四。 春正距夏至，夏至距秋秋正九十三日，行積度九十度五九九零，九十四日行積十五九八七。
From the autumn fixed qi to the winter solstice, and from the winter solstice to the spring fixed qi: at eighty-eight days the travel accumulation is 90°4009; at eighty-nine days, 91°4014. From the spring fixed qi to the summer solstice, and from the summer solstice to the autumn fixed qi: at ninety-three days the travel accumulation is 90°5990; at ninety-four days, 91°5987.
84
推每日夜度置四正後每日行度，在立成。 以日差加減之，爲每日行定度。 置四正夜半日度，以行定度每日加之，滿黃道宿度去之，即每日夜半日度。
To find each day's midnight solar degree: set the daily travel after each fixed time from the ready reckoners. Add or subtract the day-difference to obtain the fixed daily travel. Set the four fixed midnight solar degrees; add the fixed daily travel each day; discard full ecliptic lodge degrees to obtain each midnight solar degree.
85
黃道十二次宿度
Ecliptic degrees of the twelve stations.
86
危十二度六四九一，入娵訾，辰在亥。
Rooftop (Wei): 12°6491, entering Maidens' Mantle (Ziju zi); chronogram hai.
87
奎一度七三六二，入降婁，辰在戍。
Striding Legs (Kui): 1°7362, entering Sinking Barrens (Jiang lou); chronogram xu.
88
奎度四五六，入大梁，辰在酉。
Striding Legs: 4°56, entering Great Bridge (Daliang); chronogram you.
89
胃三七度七四五六，入大梁，辰在酉。
Stomach (Wei): 37°7456, entering Great Bridge; chronogram you.
90
畢六度八八零五，入實沈，辰在申。
Net (Bi): 6°8805, entering Established Quiescence (Shishen); chronogram shen.
91
井八度三四九四，入鶉首，辰在未。
Well (Jing): 8°3494, entering Quail's Head (Chunshou); chronogram wei.
92
柳三度八六八零，入鶉火，辰在午。
Willow (Liu): 3°8680, entering Quail's Fire (Chunhuo); chronogram wu.
93
張十五度二六零六，入鶉尾，辰在巳。
Extended Net (Zhang): 15°2606, entering Quail's Tail (Chunwei); chronogram si.
94
軫十度零七九七，入壽星，辰在辰。
Chariot Shaft (Zhen): 10°0797, entering Longevity Star (Shouxing); chronogram chen.
95
氐一度一四五二，入大火，辰在卯。
Root (Di): 1°1452, entering Great Fire (Dahuo); chronogram mao.
96
尾三度一一五，入析木，辰在寅。
Tail (Wei): 3°115, entering Split Wood (Ximu); chronogram yin.
97
鬥三度七六八五，入星紀，辰在醜。
Dipper (Dou): 3°7685, entering Star Chronicle (Xingji); chronogram chou.
98
女二度零六三八，入玄枵，辰在子。
Girl (Nü): 2°0638, entering Dark Axletree (Xuanxiao); chronogram zi.
99
推日躔黃道入十二次時刻置入次宿度，以入次日夜，以入次日夜半日度減之，餘以日周乘之，一分作百分。 爲實。 以入次日夜半日度，與明日夜半日度相減，餘爲法。 實如法而一，各數，以發斂加時求之，即入次時刻。
To find the time when solar motion enters each of the twelve stations: set the entry lodge degree; subtract the midnight solar degree on the entry night; multiply the remainder by the solar circuit, treating one part as one hundred. That is the dividend. Subtract the entry night's midnight solar degree from the next night's midnight solar degree for the divisor. Divide dividend by divisor; apply emission-contraction with time-of-day to obtain the station-entry moment.
100
▲步月離
▲ Lunar motion.
101
月平行度一十三度三十六分八十七秒半。
Mean lunar travel: 13°36′87.5″ per day.
102
周限三百三十六、半之爲中限，又半之爲初限。
Cycle limit: 336; half is the middle limit; half again is the initial limit.
103
限平行度零九分六十二秒。
Limit mean travel: 0°0.962 per day.
104
太陽限行八分二十秒。
Solar limit travel: 0°8′20″ per day.
105
上弦九十一度三十一發四十三秒太。
First quarter: 91°31′43″ excess.
106
望一百八十二度六十二分八十七秒半。
Full moon: 182°62′87.5″.
107
下弦二百七十三度九十四分三十一秒少。
Last quarter: 273°94′31″ deficit.
108
交終度三百六十三度七十九分三十四秒一九六。
Nodal cycle: 363°79′34.196.
109
朔平行度三百九十四度七八七一一五一六八七五。
Mean new-moon parallel motion: 394.78711516875°.
110
推朔後平交日置交終分，風氣朔歷。 減天正經朔交凡分，爲朔後平交日。 如推次月，累減交差二日三一八六九，得次月朔平交日。 不及減交差者，加交終減之，其交又在本月，爲重交月朔後平交日。 每歲必有重交之月。
To find the mean nodal day after new moon: set the nodal-cycle remainder and apply the solar-term and conjunction sequence. Subtract the civil first-month mean new-moon nodal-entry parts to obtain the mean nodal day after new moon. For the following month, subtract the nodal difference of 2 days 31869 repeatedly to obtain each month's mean nodal day after new moon. If subtraction fails, add the nodal cycle and subtract again; when the node again falls within the month, this is the double-nodal month's mean nodal day after new moon. Every year necessarily contains one double-nodal month.
111
推平交入轉遲疾歷置經朔遲疾歷，加入朔後平交日爲平交入轉。 在轉中已下，其遲疾與經朔同，已上減去轉中疾交遲，遲交疾。 如推次月，累減交轉差三千四百二十三分七六，交差內減轉差數。 即得。 如不及減，加轉中減之，亦遲疾相代。
To find mean nodal entry into the rotation anomaly sequence: set the mean new-moon anomaly sequence and add the mean nodal day after new moon to obtain mean nodal rotation entry. Below the rotation midpoint the anomaly matches the mean new moon; above it, subtract the midpoint—fast at the node becomes slow, slow becomes fast. For the following month, subtract the nodal-rotation difference of 3423.76 parts repeatedly, deducting the rotation difference from within the nodal difference. The result follows. If subtraction fails, add the rotation midpoint and subtract again; slow and rapid alternate as before.
112
推平交入限遲疾差置平交入轉遲疾歷，依步氣朔內，推遲疾差，那得。
To find the mean nodal limit anomaly difference: set the mean nodal rotation-entry anomaly sequence and derive the anomaly difference by the solar-term and conjunction procedure.
113
推平交加減定差置平交入限遲疾差，雙日率八百二十分乘之，以所入遲疾限下行度而一，即得。 在遲爲加，在疾爲減。
To find the mean nodal crossing fixed correction: set the mean nodal limit anomaly difference, multiply by the paired-day rate of 820 parts, and divide by the travel degrees within the applicable anomaly limit. Add in the slow interval; subtract in the fast interval.
114
推經朔加時積置經朔盈縮歷，見步氣朔內。 在盈歷即爲加時中積，在縮歷加半歲周。 如推次月，累加朔策，滿歲周去之，即各朔加時中積，命日爲度。 若月內有二交，後交即注前交經朔加時中積。
To find the mean new-moon time-of-day accumulation: set the mean new-moon expansion–contraction sequence, as in the solar-term and conjunction section. In the expansion sequence it is the time-of-day central accumulation; in the contraction sequence add half the tropical year. For the following month, add the synodic-month divisor repeatedly and discard the tropical year when full to obtain each conjunction's time-of-day central accumulation, naming days as degrees. When a month contains two nodes, the later node uses the former node's mean new-moon time-of-day central accumulation.
115
推正交距冬至加時黃道積度及宿次置朔後平交日，以月平行乘之爲距後度，加以經朔加時中積，爲各月正交距冬至加時黃道積度。 加冬至加時黃道日度，見日躔。 以黃道積度鈐減之，至不滿宿次，即正交月離。 如推次月，累減月平交朔差一度四六三一零二。 以交終度減天周，其數宜爲一度四六四零八零。 遇重交月，同次朔。 後仿此。
To find the true node's ecliptic accumulation and lodge from the winter solstice time-of-day: set the mean nodal day after new moon, multiply by mean lunar travel for the post-node degrees, and add the mean new-moon time-of-day central accumulation to obtain each month's true nodal ecliptic accumulation from the winter solstice moment. Add the winter solstice ecliptic solar degree at time-of-day to inspect solar motion. Subtract successive entries from the ecliptic accumulation tally until the remainder falls within one lodge to obtain the true nodal month's lodge. For the following month, subtract the mean nodal new-moon difference of 1.463102° repeatedly. Subtract the nodal-cycle arc from the celestial circuit; the value should be 1.464080°. In a double-nodal month, proceed as for the next mean new moon. Apply the same rule thereafter.
116
▲黃道積度鈐
▲ Ecliptic accumulation tally.
117
表格略
[Table omitted.]
118
推正交日辰時刻置朔後症交日，加經朔，去紀法，以平交定差加減之，其日命甲子算外，小余依發斂加時求之，即得正交日辰時刻。 如推次月，累加交終，滿紀去之。 如遇重交，再加交終。
To seek true nodal day double-hour and quarter-hour: set the mean nodal day after new moon (text reads symptom-node day), add the mean new moon, remove the era cycle, add or subtract the mean nodal fixed difference; for the day count from jiazi beyond the count, for the small remainder seek by emission-contraction with time-of-day—then obtain the true nodal day, double-hour, and quarter-hour. For the following month, add the nodal cycle repeatedly and discard the era cycle when full. In a double-nodal month, add the nodal cycle once more.
119
推四正赤道宿次置冬至赤道日度，以氣象限累加之，滿赤道積度去之，爲四正加時赤道日度。
To find the four cardinal equatorial lodges: set the winter solstice equatorial solar degree, add the qi quadrant repeatedly, and discard the equatorial accumulation when full to obtain the four cardinal equatorial degrees at time-of-day.
120
▲赤道積度鈐
▲ Equatorial accumulation tally.
121
表格略
[Table omitted.]
122
推正交黃道在二至後初末限置正交距冬至加時黃道積度，在半歲周已下爲冬至後，已上減去半歲周，餘爲夏至後。 又視二至後度分，在氣象限已下爲初限，已上用減半歲周，餘爲末限。 推次月者，若本月初限，則累減月平交朔差，餘爲次月初限。 不及減者，反減月平交朔差，餘爲次月末限。 若本月末限則累加月平交朔差，爲次月天限，至滿氣象限，以減半歲周，餘爲次月初限。
To find the initial and final limits of the true nodal ecliptic arc after the solstices: set the true nodal ecliptic accumulation from the winter solstice moment; below half the tropical year counts as post-winter-solstice, above that subtract half the year for post-summer-solstice. Inspect the arc after each solstice: below the qi quadrant is the initial limit; above it, subtract half the tropical year for the final limit. For the next month: if the current month is in the initial limit, subtract the mean nodal new-moon difference repeatedly; the remainder is the next month's initial limit. If subtraction fails, subtract the mean nodal new-moon difference in reverse; the remainder is the next month's final limit. If the current month is in the final limit, add the mean nodal new-moon difference repeatedly for the next month's final interval; when the qi quadrant is full, subtract half the tropical year—the remainder is the next month's initial limit.
123
推定差度置初末限，以象極總差一分六零五五零八乘之，即爲定差度。 象極總差，是以象限除極差，其數宜爲一十六分零五四四二。 如推次月初限則累減，末限則累加，俱以極平差二十三分四九零二加減之。 極平差，是以月平交朔差，乘象極總差，其數宜爲二十三分五零四九。
To find the fixed-difference degree: set the initial or final limit and multiply by the quadrant-to-pole total difference of 1.605508 to obtain the fixed difference in degrees. The quadrant-to-pole total difference divides the pole difference by the quadrant; the value should be 16.05442 parts. For the next month's initial limit subtract the pole mean difference of 23.4902 parts repeatedly; for the final limit add it repeatedly. The pole mean difference is the mean nodal new-moon difference multiplied by the quadrant-to-pole total difference; the value should be 23.5049 parts.
124
推距差度置極差十四度六六，減去定差度，即得。 求次月，以極平差加減之。 初限加，末限減。
To find the distance-difference degree: set the pole difference at 14.66° and subtract the fixed-difference degree. For the following month, add or subtract the pole mean difference. Add in the initial limit; subtract in the final limit.
125
推定限度置定差度，以定極總差一分六三七一零七乘之，定極總差，是以極差除二十四度，其數宜爲一度六三七一零七。 所得視正交在冬至後爲減，夏至後爲加，皆置九十八度加減之，即得。
To find the fixed-limit degree: set the fixed-difference degree and multiply by the fixed-pole total difference of 1.637107; this is the pole difference divided by 24°, which should equal 1.637107°. Apply the product as subtraction after the winter solstice and addition after the summer solstice, in each case against the base of 98°.
126
推月道與赤道正交宿度正交在冬至後，置春正赤道積度，以距差度初限加末限減之，在夏至後，置秋正赤道積度，以距差初限減末限加之。 得數，滿赤道積度鈐去之，即得。
To find the lunar-path and equator true-nodal lodge: after the winter solstice set the spring equinox accumulation and add the distance difference in the initial limit while subtracting it in the final limit; after the summer solstice set the autumn equinox accumulation and subtract in the initial limit while adding in the final limit. Discard full cycles from the equatorial accumulation tally to obtain the lodge.
127
推月道與赤道正交後積度併入初末限視月道與赤道正交所入某宿次，即置本宿赤道全度，減去月道與赤道正交宿度，差爲正後積度。 以赤道各宿全度累中之，滿氣象限去之，爲半交後。 又滿去之，爲中交後。 再滿去之，爲半交後。 視各交積度，在半象限以焉爲初限，以上覆減象限，餘爲末限。
To find post-true-nodal accumulation with initial and final limits: identify the lodge of true nodal crossing, set that lodge's full equatorial arc, subtract the true-nodal lodge degree, and the remainder is the post-true accumulation. Accumulate each lodge's full equatorial arc and discard the qi quadrant when full to obtain the post-half-node interval. Discard the qi quadrant again when full to obtain the post-middle-node interval. Discard once more when full to obtain the second post-half-node interval. For each crossing accumulation: below half the quadrant is the initial limit; above it, subtract the quadrant from above for the final limit.
128
推定差置每交定限度，與初末限相乘，得數，千約之爲度，即得。 正交、中交後爲加，半交後爲減。
To find the fixed difference: multiply each crossing's fixed-limit degree by the initial or final limit, scale by a thousand, and express the result in degrees. Add after true and middle nodal crossings; subtract after half-node crossings.
129
推月道定積度及宿次置月道與赤道各交後每宿積度，以定差加減之，爲各交月道積度。 加月道與赤道正交定宿度，共爲正交後宿度。 以前宿定積度減之，即得各交月道宿次。
To find the lunar-path fixed accumulation and lodge: set the per-lodge accumulation after each crossing, apply the fixed difference, and obtain each crossing's lunar-path accumulation. Add the true-nodal fixed lodge degree on the lunar path and equator to obtain the post-true-nodal lodge degree. Subtract the preceding lodge's fixed accumulation to obtain each crossing's lunar-path lodge.
130
▲活象限例
▲ Examples for the active quadrant limit.
131
置正交後宿次，加前交後半交末宿定積度。 爲活象限。 如正交後宿次度少，加前交不及數，卻置正交後宿次加氣象限即是。 如遇換交之月，置正交後宿次，以前交前半交末宿定積度加之，爲換交活象限。 假如前交正交是軫，後交正交是角，其前交欠一軫。 求活象限者，置正交後宿次，不從翼下取定積度加之，仍於軫下取定積度也。 又如前交、正交是軫，後交、正交是翼，其前交多一翼。 求活象限者，置正交後宿次，不從翼下取定積度加之，仍於張下取定積度也。
Set the post-true-nodal lodge and add the previous crossing's post-half-node final-lodge fixed accumulation. This yields the active quadrant limit. If the post-true-nodal lodge degree is too small to absorb the previous crossing's addition, set the post-true-nodal lodge and add the qi quadrant instead. In a node-shift month, set the post-true-nodal lodge and add the previous crossing's first-half final-lodge fixed accumulation to obtain the node-shift active quadrant limit. Suppose the previous true node lies in Axletree (Zhen) and the later in Horn (Jiao), with the previous crossing one Axletree lodge short. To find the active quadrant limit: set the post-true-nodal lodge, do not add the fixed accumulation from below Wings (Yi), but still add that from below Axletree (Zhen). Again, if the previous true node is Axletree and the later is Wings (Yi), the previous crossing exceeds by one Wings lodge. To find the active quadrant limit: set the post-true-nodal lodge, do not add from below Wings, but add the fixed accumulation from below Extended Net (Zhang).
132
推相距日置定上弦大餘，減去定朔大餘，即得。 上弦至望，望至下弦，下弦至朔仿此。 不及減者，加紀法減之。
To find the interval in days: subtract the fixed new-moon large remainder from the fixed first-quarter large remainder. Apply the same method from first quarter to full moon, full moon to last quarter, and last quarter to new moon. If subtraction fails, add the era cycle and subtract again.
133
推定朔弦望入盈歷及盈縮定差置各月朔弦望入盈縮歷，以朔弦望加減差加減之，並在步氣朔內。 爲定盈縮歷。 視盈歷在盈初限下爲盈初已上用減半歲周，餘爲盈末限。 縮歷在縮初限已下爲縮初限，已上用減半歲周，餘爲縮末限。 依步氣朔內求盈縮差，爲盈縮定差。
To find true new moon, quarters, and full moons entering the equation sequence and fixed equation correction: set each month's equation sequence at conjunction and quarters, apply the conjunction–quarter correction, all by the solar-term and conjunction procedure. This is the true equation sequence. In the expansion sequence, below the expansion-initial interval is expansion-initial; above it, subtract half the tropical year for expansion-final. In the contraction sequence, at or below the contraction-initial interval is contraction-initial; above it, subtract half the tropical year for contraction-final. Derive the equation correction by the solar-term and conjunction procedure to obtain the fixed equation correction.
134
推定朔弦望加時中積置定盈縮歷，如是盈歷在朔，便爲加時中積，在上弦加氣象限，在望加半歲周，在下弦加三象限。 如是縮歷在朔，加半歲周。 在上弦加三象限，在望便爲加時中積，在下弦加氣象限，加後滿周天去之。
To find the true conjunction and quarters' time-of-day central accumulation: set the true equation sequence; at new moon in expansion it is the central accumulation; add the qi quadrant at first quarter, half the tropical year at full moon, and three quadrants at last quarter. At new moon in the contraction sequence, add half the tropical year. Add three quadrants at first quarter; at full moon use the central accumulation directly; add the qi quadrant at last quarter; discard the full celestial circuit when the sum is full.
135
推黃朔弦望加時中定積度置定朔弦望加時中積，以其下盈縮定差盈加縮之，即得。
To find the ecliptic central fixed accumulation at true new moon, quarters, and full moon: set the time-of-day central accumulation and add or subtract the underlying fixed equation correction—add for expansion, subtract for contraction.
136
推赤道加時積度及宿次置黃道加時定積度，在周天象限已下爲至後，已上去之爲分後，滿兩象限去之爲至後，滿三象限去之爲分後。 置分至後黃道積度，以立成內分至後積度減之，餘以其下赤道度率乘之，如黃道度率而一，得數加入分至後積度，次以所去象限合之，爲赤道加時定積度。 置赤度加時定積度，加入天正冬至加時赤道日度，滿赤道積度鈐去之，得定朔弦望赤道加時宿次。
To find the equatorial accumulation and lodge at time-of-day: set the ecliptic fixed accumulation at time-of-day; below one celestial quadrant is post-solstice, above is post-equinox; discard two quadrants for post-solstice and three for post-equinox. Set the post-solstice or post-equinox ecliptic accumulation, subtract the tabulated value, multiply the remainder by the equatorial rate and divide by the ecliptic rate, add to the interval accumulation, then restore the discarded quadrants to obtain the equatorial fixed accumulation at time-of-day. Add the civil winter solstice equatorial degree at time-of-day to the equatorial fixed accumulation and discard full cycles from the equatorial tally to obtain the true conjunction and quarters' equatorial lodges at time-of-day.
137
推正半合交後積度置定朔弦望加時赤道宿次，視朔弦望在何交後，正半、中半。 即以交生積度，在朔望加時赤道宿前一宿者加之，即爲正半中交後積度，滿氣象限去之，爲正半中換交。
To find the post-true, post-half, and combined-crossing accumulation: set the equatorial lodge at true conjunction or quarters and determine whether the event falls after the true-half or middle-half crossing. Add the crossing-generated accumulation from the preceding lodge to obtain the post-true or post-middle accumulation, discarding the qi quadrant when full to mark a node shift.
138
推初末限視正半中交後積度，在半象已下爲初限，已上覆減氣象限，餘爲末限。
To find the initial and final limits: if the post-true or post-middle accumulation is below half the quadrant it is initial; above, subtract the qi quadrant for the final limit.
139
推月道與赤道定差置其交定限度，與初末限相減相乘，所得，千約之爲度，即定差。 在正交、中交爲加。 在半交爲減。
To find the lunar-path and equator fixed difference: multiply the crossing's fixed-limit degree by the initial or final limit, scale by a thousand, and express in degrees. Add at true and middle nodal crossings. Subtract at half-node crossings.
140
推定朔弦望加時月道宿次置定朔弦望加時月道定積度，取交後月道定積度，取交後月道定積度，在所置罕前一宿者減之，即得。 遇轉交則前積度多，所置積度少爲不及減。 從半轉正，加其交活象限減之。 從正轉半，從半轉中，從中轉半，皆加氣象限減之。
To find the lunar-path lodge at true new moon, quarters, and full moon at time-of-day: set the lunar-path fixed accumulation at time-of-day, take the post-crossing lunar-path fixed accumulation, and subtract that of the lodge before the one set. At a node shift the prior accumulation is larger and the current value smaller, so subtraction fails. When shifting from half-node to true-node, add the active quadrant limit for that crossing and subtract. From true to half, half to middle, or middle to half, add the qi quadrant and subtract in each case.
141
推夜半入轉日置經朔弦望遲疾歷，以定朔弦望加減差加減之。 大疾歷，便爲定朔弦望加時入轉日。 在遲歷，用加轉中置定朔弦望加時入轉日，以定朔弦望小余減之，爲夜半入轉日，遇入轉日少不及減者，加轉終減之。
To find midnight rotation-entry day: set the mean conjunction and quarter anomaly sequences and apply the true conjunction–quarter correction. In the fast sequence this is the rotation-entry day at true conjunction or quarter at time-of-day. In the slow sequence add the rotation midpoint to the time-of-day rotation-entry day and subtract the small remainder for midnight entry; if the entry day is too small, add the rotation cycle and subtract again.
142
推加時入轉度置定朔弦望小余，去秒，取夜半入轉日下轉定度乘之，萬約之爲分，即得。
To find rotation degrees at time-of-day: take the small remainder without seconds, multiply by the fixed rotation degree under the midnight rotation-entry day, and scale by ten thousand to obtain fen.
143
▲遲疾轉定度鈐
▲ Slow–rapid rotation fixed-degree tally.
144
表格略
[Table omitted.]
145
推定朔弦望夜半入轉積度及宿次置定朔弦望加時月道定積度，減去加時入轉度，爲夜半積度。 如朔弦望加時定積度初換交，則不及減，半正相接，用活象限，正半、中半相接，用氣象限加之，然後減加時入轉度，則正者爲後年，後年爲中，中爲前半，前半爲正。 置朔弦望夜半月道定積度，依推定朔弦望加時月道宿次法減之，爲夜半宿次。
To find the midnight rotation accumulation and lodge for true new moon, quarters, and full moon: subtract the time-of-day rotation degrees from the lunar-path fixed accumulation at time-of-day to obtain the midnight accumulation. If the fixed accumulation at new moon, first quarter, full moon, or last quarter first crosses a node at time-of-event, subtraction falls short. Where half and major adjoin, apply the variable quadrant; where major-half and middle-half adjoin, add the qi quadrant; then subtract the rotation degrees entered at time-of-event. Major becomes later half, later half middle, middle first half, and first half major. Set the midnight lunar-path fixed accumulation for new moon, first quarter, full moon, or last quarter; subtract by the procedure for time-of-event lunar-path lodge positions to obtain the midnight lodge.
146
推晨昏入轉日及轉度置夜半入轉日，以定盈縮歷檢立成日下晨分加之，爲晨入轉日滿轉終去之。 置其日晨分，取夜半入轉日下轉定度乘之，萬約爲分，爲晨轉度。 如求昏轉日轉度，依法檢日下昏分，即得。
Procedure to derive morning-and-evening rotation-entry day and rotation degrees: Set the midnight rotation-entry day, look up the morning fraction for that day in the fixed expansion–contraction ready reckoner and add it to obtain the morning rotation-entry day, discarding the rotation period when full. Set the day's morning fraction, multiply by the fixed rotation degree under the midnight rotation-entry day, scale by ten thousand to obtain fen, and the product is the morning rotation degree. To find the evening rotation day and rotation degree, look up the evening fraction for that day by the same procedure.
147
推晨昏轉積度及宿次置朔弦望夜半月道定積度，加晨轉度，爲晨轉積度。 如求昏轉積度，則加昏轉度，滿氣象限去之，則換交。 若推夜半積度之時，因朔弦望加時定積不及減轉度，以半正相接，而加活象限之者，今復換正交，則以活象限減之。 置晨轉積度，依前法減之，爲晨分宿次。 置昏轉積度，依法減之，爲昏分宿次。
Procedure to derive morning-and-evening rotation accumulated degrees and lodge positions: Set the midnight lunar-path fixed accumulation for new moon, first quarter, full moon, or last quarter and add the morning rotation degree to obtain the morning rotation accumulation. For the evening rotation accumulation, add the evening rotation degree; when the sum reaches the qi quadrant, discard it—this marks a node change. If, when deriving the midnight accumulation, the time-of-event fixed accumulation could not subtract the rotation degree and half and major adjoined so that the variable quadrant was added, then on a further change to the major node subtract the variable quadrant instead. Set the morning rotation accumulation and subtract by the preceding method to obtain the morning-fraction lodge. Set the evening rotation accumulation and subtract by the same method to obtain the evening-fraction lodge.
148
推相距度朔與上弦相距，上弦與望相距，用昏轉積度。 望與下弦相距，下弦與朔相距，用晨轉積度。 置後段晨昏轉積度，視與前段同交者，竟以前段晨昏轉積度減之，餘爲相距度。 若後段與前段接兩交者，從正入半，從半入中，從中入半，加氣象限。 從半入正，加活象限。 然後以前段晨昏轉積度減之。 若後段與前段接三交者，其內無從半入正，則加二氣象限，其內有從半入正，則加一活象限，一氣象限，以前段晨昏轉積度減之。
Procedure to derive separation degrees: From new moon to first quarter and from first quarter to full moon, use the evening rotation accumulation. From full moon to last quarter and from last quarter to new moon, use the morning rotation accumulation. Set the later segment's morning-or-evening rotation accumulation; if it lies on the same node as the earlier segment, subtract the earlier segment's morning-or-evening rotation accumulation in full—the remainder is the separation degree. If the later and earlier segments span two nodes—from major to half, half to middle, or middle to half—add the qi quadrant. From half to major, add the variable quadrant. Then subtract the earlier segment's morning-or-evening rotation accumulation. If the later and earlier segments span three nodes: when no half-to-major transition occurs within, add two qi quadrants; when a half-to-major transition does occur within, add one variable quadrant and one qi quadrant; then subtract the earlier segment's morning-or-evening rotation accumulation.
149
推轉定積度置晨昏入轉日，朔至弦，弦至望，用昏。 望至弦，弦至朔，用晨。 以前段減後段，不及減者，加二十八日減之，爲晨昏相距日。 從前段下，於鈐內驗晨昏相距日同者，取其轉定積度。 若朔弦望相距日少晨昏相距日一日者，則於晨昏相距日同者，取其轉積度，減去轉定極差一十四度七一五四，餘爲前段至後段轉定積度。
Procedure to derive the fixed rotation accumulation: Set the morning-or-evening rotation-entry day. From new moon to quarter and from quarter to full moon, use evening. From full moon to quarter and from quarter to new moon, use morning. Subtract the earlier segment from the later; if subtraction fails, add twenty-eight days and subtract again to obtain the morning-and-evening separation in days. Under the earlier segment, look up the matching morning-and-evening separation day in the register and take its fixed rotation accumulation. If the new-moon-to-quarter separation in days is one day shorter than the morning-and-evening separation, find the matching morning-and-evening separation day, take its rotation accumulation, subtract the fixed rotation extreme difference of 14°7154, and the remainder is the fixed rotation accumulation from the earlier to the later segment.
150
▲轉定積度鈐
▲ Register of Fixed Rotation Accumulations
151
以下表格略
[Table below omitted.]
152
推加減差以相距度與轉定積度相減爲實，以其朔弦望相距目爲法除之，所得視相距度多爲加差，少爲減差。
Procedure to derive the addition-and-subtraction difference: Subtract the fixed rotation accumulation from the separation degree as dividend and divide by the number of days from new moon to quarter (or the corresponding interval) as divisor. If the separation exceeds the quotient's basis, the result is an addition difference; if less, a subtraction difference.
153
推每日太陰行定度置朔弦望晨昏入轉日，視遲疾轉定度鈐日下轉定度，累日以加減差加減之，至所距日而止，即得。
Procedure to derive the Moon's fixed daily motion: Set the morning-or-evening rotation-entry day at new moon, first quarter, full moon, or last quarter; take the fixed rotation degree under that day from the slow–rapid fixed-rotation register; accumulate day by day, adding or subtracting the addition-and-subtraction difference, until the separation day—the result is obtained.
154
推每日月離晨昏宿次置朔弦望晨昏宿次，以每日太陰行度加之，滿月道宿次減之，即得。
Procedure to derive the lunar-departure lodge at morning and evening each day: Set the morning-or-evening lodge at new moon, first quarter, full moon, or last quarter; add the fixed daily lunar motion each day and, when the sum fills the lunar-path lodges, subtract—the lodge is obtained.
155
▲赤道十二宮界宿次
▲ Equatorial Twelve-Palace Boundary Lodges
156
表格略
[Table omitted.]
157
推月與赤道正交後宮界積度視月道與赤道正交後，各宿積度宮界，某宿次在後，即以加之，便爲某宮正交後宮界積度。 求次宮者，累加宮率二十度四三八一，滿氣象限去之，各得某宮下半產交後宮界積度。
Procedure to derive the accumulated degree after a palace boundary following the Moon's major equatorial node crossing: After the major crossing of the lunar path and equator, identify each lodge's accumulated degree at a palace boundary; add the lodge that lies next behind to obtain that palace's post-crossing boundary accumulation. For the next palace, add the palace rate of 20°4381 cumulatively; discard the qi quadrant when full—each sum gives a palace's lower-half post-crossing boundary accumulation.
158
推宮界定積度視宮界度在半象限已下爲初限，已上覆減氣象限，餘爲末限。 置某交定限度，與初末限相減、相乘，所得，千約之爲度，在正交、中交爲加差，在半交爲減差。 置宮界正半中交後積度，以定差加減之，爲宮界定積度。
Procedure to derive the fixed palace-boundary accumulation: If the boundary degree lies below the half-quadrant, it is the initial limit; if above, subtract the qi quadrant in reverse—the remainder is the terminal limit. Set the fixed limit degree for the node in question; subtract and multiply against the initial and terminal limits; scale the product by a thousand to degrees. At major and middle nodes this yields an addition difference; at half nodes, a subtraction difference. Set the accumulated degree after the major, half, or middle palace boundary and add or subtract the fixed difference to obtain the fixed palace-boundary accumulation.
159
推宮界宿次置宮界定積度，於月道內取其在所置前一宿者減之之不及減者，加氣象限減之。
Procedure to derive the palace-boundary lodge: Set the fixed palace-boundary accumulation; within the lunar path subtract the lodge immediately preceding the one placed—if subtraction fails, add the qi quadrant and subtract again.
160
推每月每日下交宮時刻置每月宮界宿次，減入交宮日下月離晨昏宿次。 如不及減者，加宮界宿次前宿減之，餘以日周乘之，以其日太陰行定度而一，得數，又視定盈縮歷取立成日下晨昏分加之。 晨加晨分，昏加昏分。
Procedure to derive the time of palace entry below the node for each day of each month: Set the month's palace-boundary lodge and subtract the morning-or-evening lunar-departure lodge on the node-palace-entry day. If subtraction fails, add the lodge before the palace-boundary lodge and subtract; multiply the remainder by the day circuit and divide by that day's fixed lunar motion; then look up the morning-and-evening fraction for the day in the fixed expansion–contraction ready reckoner and add it. For morning, add the morning fraction; for evening, add the evening fraction.
161
如滿日周交宮在次日，不滿在本日，依發斂推之，即交宮時刻。
If the sum fills the day circuit, palace entry falls on the next day; if not, on the current day. Apply the emission-and-collection procedure to obtain the time of node-palace entry.
162
▲步中星
▲ Stepping the Culminating Stars
163
推每日夜半赤道置推到每日夜半黃道，見日躔。 依法以黃道積度減之，餘如黃道率而一，以加赤道積度。 又以天正科至赤道加之，如在春正後，再加一象限，夏至後加半周天，秋正後加三象限，爲每日夜半赤道積度。
Procedure to derive the midnight equator for each day: Take the midnight ecliptic derived for each day (see Solar Progression). By the prescribed method subtract the ecliptic accumulation; divide the remainder by the ecliptic ratio and add the quotient to the equatorial accumulation. Add the equator at the celestial origin's seasonal nodes. After the spring standard add one quadrant; after the summer solstice add half a circumpolar circuit; after the autumn standard add three quadrants—the sum is each day's midnight equatorial accumulation.
164
推夜半赤道宿度置夜半赤道度，以赤道宿度挨次減之，爲本日夜半赤道宿度。
Procedure to derive the midnight equatorial lodge: Set the midnight equatorial degree and subtract successive equatorial lodge widths to obtain this day's midnight equatorial lodge.
165
推晨距度及更差度置立成內每日晨分，以三百六十六度二十五分七十五秒乘之爲實，如日周而一，爲晨距度。 倍晨距度，以五除之，爲更差度。
Procedure to derive the morning separation degree and watch-interval difference: Take each day's morning fraction from the ready reckoner, multiply by 366°25′75″ as dividend, and divide by the day circuit to obtain the morning separation degree. Double the morning separation degree and divide by five to obtain the watch-interval difference degree.
166
推每日夜半中星置推到每日夜半赤道宿度，加半周天，即夜半中唾積度。 以赤道度挨次減之，爲夜半中星宿度。
Procedure to derive the culminating stars at midnight each day: Set the midnight equatorial lodge derived for each day, add half a circumpolar circuit, and the sum is the midnight culminating accumulation. Subtract successive equatorial lodge widths to obtain the midnight culminating-star lodge.
167
推昏旦中星置夜半中星積度，減晨距度，爲昏中星積度。 以更差度累加之，爲遂更及旦中星積度。 俱滿赤道宿度去之，即得。 以晨分五之一，加們爲更率。 更率五而一爲點率。 凡昏分，即一更一點，累加更率爲各更。 凡交更即爲一點，累加點率爲各點。
Procedure to derive the dusk and dawn culminating stars: Set the midnight culminating accumulation and subtract the morning separation degree to obtain the dusk culminating accumulation. Add the watch-interval difference cumulatively to obtain the culminating accumulations for successive watches and for dawn. Whenever any sum fills the equatorial lodges, discard the circuit—the lodges are obtained. Take one-fifth of the morning fraction and add it to the watches to obtain the watch rate. Divide the watch rate by five to obtain the point rate. Each dusk fraction equals one watch and one point; add the watch rate cumulatively for successive watches. Each change of watch equals one point; add the point rate cumulatively for successive points.