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律曆二
Measures and Calendar, Part Two.
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○應天乾元儀天曆
○ The Yingtian, Qianyuan, and Yitian Calendars.
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步月離入先後曆 〈（《乾元》謂之月離。 《儀天》謂之步月離。）〉
Entering the Lunar Apogee Advance/Retardation Table. (The Qianyuan calendar calls this "lunar apogee." The Yitian calendar calls it "entering lunar apogee.")
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離總：五萬五千一百二十、秒一千二百四十二。 〈（《乾元》轉分一萬六千二百、秒一千二百四。 《儀天》曆終分二十七萬八千三百一、秒一百六十五。）〉
Apogee total: 55,120 parts, 1,242 seconds. (In Qianyuan the revolution parts are 16,200, 1,204 seconds. Yitian's calendar terminal parts are 278,301, 165 seconds.)
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轉日：二十七、五千五百四十六、秒六千二百一十。 〈（《乾元》轉曆二十七、一千六百三十、秒六千二十。 《儀天》曆週二十七、五千六百一、秒一百六十五。）〉
Revolution period: 27 days, 5,546 parts, 6,210 seconds. (Qianyuan's revolution cycle is 27 days, 1,630 parts, 6,020 seconds. Yitian's calendar cycle is 27 days, 5,601 parts, 165 seconds.)
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曆中日：一十三、七千七百七十四、秒三千一百五。 〈（《乾元》不立此法。 《儀天》曆中十三日、七千八百五十、秒五千八十二半。 《儀天》有象限六日、八千九百七十五、秒二千五百四十一少。）〉
Mid-cycle day: 13 days, 7,774 parts, 3,105 seconds. (The Qianyuan calendar does not use this procedure. Yitian's mid-cycle is 13 days, 7,850 parts, 5,082½ seconds. Yitian also has an image limit of 6 days, 8,975 parts, 2,541¼ seconds.)
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朔差日：一、九千七百六十二、秒三千七百九十。 〈（《乾元》轉差一、三千八百六十九、秒三千九百八十。 《儀天》會差日一、九千八百五十七、秒九千八百三十五。）〉
New-moon discrepancy: 1 day, 9,762 parts, 3,790 seconds. (Qianyuan's revolution discrepancy is 1 day, 3,869 parts, 3,980 seconds. Yitian's conjunction discrepancy is 1 day, 9,857 parts, 9,835 seconds.)
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〈（《儀天》又有象差日空、四千九百八十、秒四千九百五十八太； 望一百八十二度六千三百四十四、秒四千九百五十。）〉
(Yitian also has an image discrepancy of 0 days, 4,980 parts, 4,958¾ seconds; at full moon: 182° 6,344', 4,950".)
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度母：一萬一百。
Degree divisor: 10,100.
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秒法：一萬。 〈（二曆同）〉
Second divisor: 10,000. (Both calendars agree.)
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求天正十一月朔入先後曆： 〈（《乾元》謂之求月離入曆，求弦、望入曆。 《儀天》謂之推天正經朔入曆。）〉 以通餘減元積，餘以離總去之為總數； 不盡者，半而進位，以元法收為日，不滿為分。 如曆中日以下為入先曆； 以上者去之，為入後曆。 命日，算外，即得天正十一月朔入先後曆日分。 累加七日、三千八百二十七分、秒六，盈曆中日及分秒去之，各得次朔、望入先後曆日分。 〈（《乾元》以朔餘減歲積分，以轉分去之，餘以五因之，滿元率收之為度； 以弦策加之，即弦、望所入。 以轉差加之，得後朔曆； 累加之，即得弦、望入曆及分。 《儀天》以閏餘減歲積分，餘以曆終分去之，不滿，以宗法除之為日； 在象限以下為初限，以上去之，餘為末限，各為入遲疾曆初、末限。）〉
To find entry into the advance/retardation table at the Heavenly Standard eleventh-month new moon: (Qianyuan terms this finding lunar-apogee table entry and finding first-quarter and full-moon table entry. Yitian calls it computing the Heavenly Standard canonical new moon's table entry.) Subtract the communication remainder from the epoch accumulation; divide the remainder by the apogee total to obtain the cycle count; for the remainder, round at the half and carry, convert to days using the origin divisor; what does not fill a day counts as parts. If the value is at or below the mid-cycle day, entry is into the advance portion of the table; if above, subtract the mid-cycle day—entry is into the retardation portion. Name the day; the count exclusive yields the day and fractional parts for the Heavenly Standard eleventh-month new moon's table entry. Repeatedly add 7 days, 3,827 parts, and 6 seconds; when the sum exceeds the mid-cycle day and its parts and seconds, subtract the excess—obtaining the table entry in days and parts for each successive new moon and full moon. (In Qianyuan: subtract the new-moon remainder from the year's accumulated parts and remove revolution parts; multiply the remainder by five; when it fills the origin rate, convert to degrees; add the quarter-moon increment to obtain first-quarter and full-moon entries. Add the revolution discrepancy to obtain the following new moon's table entry; by repeated addition, obtain first-quarter and full-moon table entries and their fractional parts. In Yitian: subtract the intercalation remainder from the year's accumulated parts and remove calendar terminal parts; divide the remainder by the lineage divisor to obtain days; if below the image limit it is the initial limit; if above, subtract the image limit—the remainder is the terminal limit—each marking entry into the initial or terminal portion of the slow-fast table.)
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七日：初數八千八百八十八， 〈（《乾元》初二千六百一十二。）〉 末數一千一百一十四。 〈（末三百二十八。）〉
Day 7: initial value 8,888, (Qianyuan's initial value is 2,612.) Terminal value: 1,114. (Terminal value: 328.)
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十四日：初數七千七百七十四， 〈（《乾元》初二千二百八十五。）〉 末數二千二百二十八。 〈（末六百五十五。 《乾元》又有二十一日：初一千九百五十八，末九百八十二； 二十八日：初一千六百三十二，末一千三百九。）〉
Day 14: initial value 7,774, (Qianyuan's initial value is 2,285.) Terminal value: 2,228. (Terminal value: 655. Qianyuan also has day 21: initial 1,958, terminal 982; day 28: initial 1,632, terminal 1,309.)
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又《儀天》法月離先後度數： 〈（《乾元》謂之月離陰陽差。 《儀天》謂之求朔弦望昇平定數。）〉 以月朔、弦、望入曆先後分通減元法，餘進位，下以其日損益率展之，以元法收為分，所得，損益次日下先後積為定數。 其七日、十四日，如初數以下者，返減之，以上者去之，餘，返減末數，皆進位，下以損益率展之，各滿末數為分，損益次日下先後積為定數。 〈（《乾元》置入曆分，以其日損益率乘之，元率收為分，損益其下陰陽差為定數。 四七術，如初數已下者，以初率乘之，如初數而一，以損益陰陽差為定數； 若初數以上者，以初數減之，餘乘末率，末數除之，用減初率，餘加陰陽差，各為定數。）〉
The Yitian method for lunar-apogee advance/retardation in degrees: (Qianyuan calls this the lunar-apogee yin-yang discrepancy. Yitian calls it finding the fixed leveling values for new moon, first quarter, and full moon.) For each new moon, first quarter, and full moon table entry, subtract the advance/retardation parts from the origin divisor (carrying remainders); apply that day's increase-decrease rate below; convert to parts using the origin divisor; the result adjusts the next day's advance/retardation accumulation to obtain the fixed value. On days 7 and 14: if at or below the initial value, apply reverse subtraction; if above, subtract the initial value and reverse-subtract the remainder from the terminal value (carrying as needed); expand below by the increase-decrease rate; when the result fills the terminal value, convert to parts; adjust the next day's accumulation to obtain the fixed number. (In Qianyuan: take the table-entry parts, multiply by that day's increase-decrease rate, convert using the origin rate, and adjust the yin-yang discrepancy below to obtain the fixed value. By the four-seven method: if at or below the initial value, multiply by the initial rate and divide by the initial value; adjust the yin-yang discrepancy to obtain the fixed number; if above the initial value, subtract it; multiply the remainder by the terminal rate and divide by the terminal value; subtract from the initial rate and add the remainder to the yin-yang discrepancy—each yielding a fixed number.)
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朔弦望定日：以日躔、月離先後定數，先加後減朔、弦、望中日，為定日。 〈（二曆法同。）〉
Fixed new-moon, first-quarter, and full-moon days: apply the solar traverse and lunar-apogee fixed advance/retardation values—add for advance, subtract for retardation—to the mean conjunction days to obtain the fixed dates. (Both calendars use the same method.)
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推定朔弦望日辰七直：以天正所盈之日加定積， 〈（視朔、弦、望中日，如入大、小雪氣，即加去年天正所盈之日分； 若入冬至氣者，即加今年天正所盈之日分。）〉 日滿七十六去之，不滿者，命從金星甲子，算外，即得定朔、弦、望日辰星直也。 視朔幹名與後朔同者大，不同者小，其月無中氣者為閏。 又視朔所入辰分皆與二分相減，餘二收，用減八分之六，其朔定小餘如此； 以上者進一日； 朔或有交正見者，其朔不進。 定望小餘在日出分以下者，退一日，若有虧初在辰分以下亦如之。 〈（二曆法同。）〉
To determine the fixed new moon, first quarter, and full moon—day, chronology, and seven directors: add the Heavenly Standard surplus days to the fixed accumulation, (If the mean conjunction falls within Major or Minor Snow, add last year's Heavenly Standard surplus in days and parts; if within Winter Solstice, add this year's Heavenly Standard surplus.) Remove multiples of 76 days; name the remainder from the Venus jiazi cycle; the count exclusive yields the fixed new moon, first quarter, and full moon—day, chronology, and star director. If a new moon shares the same stem-branch day-name as the next new moon, the month is long; if not, short; any month lacking a mid-qi is intercalary. Also compare the new moon's chronology parts with the equinox parts; pair remainders and subtract six-eighths accordingly to obtain the fixed small remainder for the new moon; if above the threshold, advance the date by one day; if the new moon coincides with a visible eclipse at the proper moment, do not advance that new moon. If the fixed full moon's small remainder falls below the sunrise fraction, retreat one day; the same applies if the eclipse's first contact falls below the chronology parts. (Both calendars use the same method.)
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〈（《儀天》又有求朔弦望加時月度，置弦、望加時日度，其合朔加時月與太陽同度，其日、度便為月離所次； 餘加弦、望象度及餘秒，滿黃道宿次去之，即定朔、弦、望加時日、度也。）〉
(Yitian also computes the moon's position at the hour of each phase: set the solar degree at first quarter and full moon; at conjunction the moon at that hour shares the sun's degree—that day and degree mark the lunar apogee position; add the remainder to the quarter and full-moon image degrees and seconds; remove full cycles along the Yellow Path lodges—obtaining the fixed day and degree for each phase at its hour.)
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九道宿度： 〈（《乾元》、《儀天》皆謂之月行九道。）〉 凡合朔所交，冬在陰曆，夏在陽曆，月行青道； 〈（冬至、夏至後，青道半交在春分之宿，出黃道東； 立夏、立冬後，青道半交在立春之宿，出黃道東南：至所沖之宿亦如之。）〉 冬在陽曆，夏在陰曆，月行白道； 〈（冬至、夏至後，白道半交在秋分之宿，出黃道西； 立冬、立夏后，白道半交在立秋之宿，出黃道西北：至所沖之宿亦如之。）〉 春在陽曆，秋在陰曆，月行朱道； 〈（春分、秋分後，朱道半交在夏至之宿，出黃道南； 立春、立秋後，朱道半交在立夏之宿，出黃道西南：至所衝之宿亦如之。）〉 春在陰曆，秋在陽曆，月行黑道。 〈（春分、秋分後，黑道半交在冬至之宿，出黃道北； 立春、立秋後，黑道半交在立冬之宿，出黃道東北：至所沖之宿亦如之。）〉 四序月離為八節，九道斜正不同，所入七十二候，皆與黃道相會。 各距交初黃道宿度，每五度為限。 初限十二，每限減半，終九限又減盡，距二立之宿減一度少強，卻從減盡起，每限減半，九限終十二而至半交，乃去黃道六度； 又自十二，每限減半，終九限又減一度少強，更從減盡起，每限增半，九限終十二，復與日軌相會。 交初、交中、半交，各以限數，遇半倍使，乘限度為泛差。 其交中前後各九限，以距二至之宿前後候數乘之，半交前後各九限，各至二分之宿前後候數乘之，皆滿百而一為黃道差。 在冬至之宿後，交初前後各九限為減，交中前後各九限為加； 夏至之宿後，交初前後各九限為加，交中前後各九限為減。 大凡月交後為出黃道外，交中後為入黃道內。 半交前後各九限，在春分之宿後出黃道外，秋分之宿後入黃道內，皆以差為加； 在春分之宿後入黃道內，秋分之宿後出黃道外，皆以差為減。 倍泛差，退一位， 〈（遇減，身外除三； 遇加，身外除一。）〉 又以黃道差減，為赤道差。 交初、交中前後各九限，以差加； 半交前後各九限，皆以差減。 以黃赤道差減黃道宿度為九道宿度，有餘分就近收為太、半、少之數。 〈（《乾元》初數九，每限減一，終於一，限數並同，即八十四除之。 《儀天》初數一百一十七，每限減一十，終於二十七，以一百一除。 二曆皆不身外為法。 初中正交、春秋二分、冬夏二至前後各九限，加減並同《應天》。 又《儀天》即除法是九十乘黃道泛差，一百一收為度，乃得月與黃、赤道定差。 以上入交定月出入各六度相較之差，黃道隨其日行所向，斜正各異，餘皆同《應天》。 《儀天》有求定朔望加時入遲疾曆初末限，置經朔、望入遲疾初末限日及餘秒，如求定朔、弦、望法入之，即各得所求。 又求初中正交入曆，置其朔、望加時入遲疾曆初末限日及餘秒，視其日月行入陰陽曆日及餘秒，如近前交者即加，近後交者即返減交中日餘，乃如之，各得初、中、正交入遲疾曆初末限日及餘秒也。 其加減滿或不足，即進退象限及餘秒，各得所求。 又求朔望加時及初、中、正交入遲疾限日入曆積度，各置小餘，以其日曆定分乘之，宗法收之為分，一百一除之為度，以加其日下曆積度，各得所求。 又《乾元》、《儀天》有求正交黃道月度，《乾元》元率通定交度及分，以一百二十七乘之，滿九十五而一，進一等，復收為入交度，用減其朔加時日度，即朔前月離正交黃道宿度。 《儀天》置朔、望及正交曆積度，以少減多，餘為月行去交度及分； 乃視其朔望在交前者加、交後者減朔望加時黃道月度，為初、中、正交黃道月度也。）〉
Lodge degrees along the Nine Paths: (Both Qianyuan and Yitian call this the moon's Nine-Path motion.) At conjunction, when winter falls in the yin half of the year and summer in the yang half, the moon follows the Green Path; (After the solstices, the Green Path's midpoint crosses at the spring-equinox lodge, emerging east of the ecliptic; after Start of Summer and Start of Winter it crosses at Start of Spring, emerging southeast of the ecliptic; at the opposite lodge the pattern is the same.) when winter is in the yang half and summer in the yin half, the moon follows the White Path; (After the solstices, the White Path's midpoint crosses at the autumn-equinox lodge, emerging west of the ecliptic; after Start of Winter and Start of Summer it crosses at Start of Autumn, emerging northwest of the ecliptic; at the opposite lodge the pattern is the same.) When spring falls in the yang half and autumn in the yin half, the moon follows the Red Path; (After the equinoxes, the Red Path's midpoint crosses at the summer-solstice lodge, emerging south of the ecliptic; after Start of Spring and Start of Autumn it crosses at Start of Summer, emerging southwest of the ecliptic; at the opposite lodge the pattern is the same.) When spring falls in the yin half and autumn in the yang half, the moon follows the Black Path. (After the equinoxes, the Black Path's midpoint crosses at the winter-solstice lodge, emerging north of the ecliptic; after Start of Spring and Start of Autumn it crosses at Start of Winter, emerging northeast of the ecliptic; at the opposite lodge the pattern is the same.) Lunar apogee through the four seasons yields the eight seasonal nodes; the Nine Paths vary in obliquity; all seventy-two hou they traverse meet the ecliptic. Each is reckoned from the ecliptic lodge at initial conjunction, in limits of five degrees. The first limit is 12, each successive limit halving until the ninth reaches zero; subtract 1° plus a small fraction from the two establishment lodges; then from zero halve again through nine limits until 12 at half-crossing—six degrees from the ecliptic; Again from 12, halving through nine limits, then subtracting 1° plus a fraction; from zero, each limit increases by half through nine limits back to 12, reuniting with the solar path. At initial conjunction, mid-conjunction, and half-crossing, apply the limit number—doubling when half—and multiply by the limit in degrees to obtain the general difference. For nine limits before and after mid-conjunction, multiply by the hou counts near the solstice lodges; for nine limits before and after half-crossing, multiply by those near the equinox lodges—divide by 100 to obtain the ecliptic correction. After the winter-solstice lodge, subtract for the nine limits around initial conjunction and add for those around mid-conjunction; After the summer-solstice lodge, add for the nine limits around initial conjunction and subtract for those around mid-conjunction. Generally, after conjunction the moon lies outside the ecliptic; after mid-conjunction it lies within it. For nine limits around half-crossing: after the spring-equinox lodge, outside the ecliptic; after the autumn-equinox lodge, inside—all corrections are additive; after the spring-equinox lodge, inside the ecliptic; after the autumn-equinox lodge, outside—all corrections are subtractive. Double the general difference and shift one decimal place, (When subtracting, divide the exterior remainder by three; when adding, divide the exterior remainder by one.) Then convert the ecliptic correction to an equatorial correction by subtraction. For nine limits around initial and mid-conjunction, add the correction; for nine limits around half-crossing, subtract the correction. Subtract the ecliptic and equatorial corrections from the ecliptic lodge to obtain the Nine-Path lodge; round fractional remainders to large, half, or small units. (Qianyuan's initial value is 9, decreasing by 1 per limit to 1; with identical limit counts, divide by 84. Yitian's initial value is 117, decreasing by 10 per limit to 27, then divide by 101. Neither calendar uses exterior-remainder division. Additions and subtractions for nine limits around initial, mid, and true conjunction, both equinoxes, and both solstices match the Yingtian calendar. Yitian's direct division multiplies the ecliptic general difference by 90 and converts by 101 to degrees, yielding the moon's fixed ecliptic and equatorial corrections. The above compares six-degree ingress/egress differences at fixed conjunction; the ecliptic follows the sun's daily motion with varying obliquity; otherwise all matches Yingtian. Yitian provides a method for fixed-phase hours entering the slow-fast table's initial or terminal limits: set the mean-phase entry days and seconds, then apply the fixed new-moon/quarter/full-moon procedure. To find table entry at initial, mid, and true conjunction: set phase-hour entry into slow-fast limits; compare solar and lunar yin-yang calendar entries—add if nearer the prior conjunction, reverse-subtract mid-conjunction remainder if nearer the next—obtaining entry days and seconds for each conjunction type. If additions or subtractions overflow or fall short, adjust the image limit and seconds accordingly. To find phase-hour and conjunction accumulated degrees in the slow-fast limits: multiply each small remainder by that day's fixed table parts, convert via the lineage divisor and 101 to degrees, and add to the day's accumulated table degrees. Both calendars compute true-conjunction ecliptic lunar position. Qianyuan: convert the fixed conjunction degree through the origin rate, multiply by 127, divide by 95 (carrying), reconverting to conjunction-entry degrees; subtract from the new-moon-hour solar degree to obtain pre-conjunction ecliptic lunar apogee position. In Yitian: set accumulated degrees for new moon, full moon, and true conjunction; subtract the smaller from the larger to obtain the moon's distance from conjunction in degrees and parts; then add if the phase precedes conjunction or subtract if it follows, from the phase-hour ecliptic lunar degree—yielding initial, mid, and true conjunction ecliptic positions.)
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九道交初月度： 〈（《乾元》謂之月離入交九道正交月度、九道朔度。 《儀天》謂之求月離正交九道宿度。）〉 置月離交初黃道宿度，各以所入限數乘之， 〈（遇半倍使）〉 如百而一，為泛差； 用求黃、赤二道差，依前法加減之，即月離交初九道宿度。 〈（《乾元》以日躔陰陽差陽加陰減，為朔、望常分； 又以所入限率乘，正交黃道宿度相從之，以求黃、赤二道差，如前加減，為月離正交九道宿度； 以入交定度加而命之，即朔月離宿度。 《儀天》置正交月離黃道，以距度下月九道差，宗法乘之，以距度所入限數乘度，餘從之，為總差； 半而退位，一百一收之，又計冬、夏二至以求度數乘，滿九十而一為度差，依前法加減，為正交月離九道。）〉
Lunar degree at Nine-Path initial conjunction: (Qianyuan terms this lunar apogee's Nine-Path true-conjunction degree and Nine-Path new-moon degree. Yitian calls it finding the Nine-Path lodge at lunar apogee true conjunction.) Set the lunar apogee's ecliptic lodge at initial conjunction; multiply each by its limit number, (Double when the limit is half.) divide by 100 to obtain the general difference; compute ecliptic and equatorial corrections and apply the prior addition/subtraction rules to obtain the Nine-Path lodge at initial conjunction. (In Qianyuan: apply the solar traverse yin-yang discrepancy—add for yang, subtract for yin—as mean new-moon and full-moon values; multiply by the entered limit rate with the true-conjunction ecliptic lodge; compute ecliptic and equatorial corrections as before to obtain the Nine-Path lodge at true conjunction; add the fixed conjunction-entry degree and name the lodge to obtain the new-moon lunar apogee position. In Yitian: set true-conjunction ecliptic lunar apogee; multiply the Nine-Path difference below the distance degree by the lineage divisor; multiply the entered limit by degrees with remainder following to obtain the total difference; halve and shift back; convert by 101; compute solstice factors, multiply, divide by 90 for the degree correction; apply prior addition/subtraction rules for true-conjunction Nine-Path lunar apogee.)
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求九道朔月度：百約月離先後定數，後加先減四十二，用減中盈而從朔日，乃加交初九道宿次，即得所求。 〈（《乾元》置九道正交之度及分，以入交定度加之，命以九道宿次，即其朔加時月離宿度及分也。 《儀天》法見下。 《乾元》又有定交度，置月離陰陽定數，以七十一乘之，滿九百一除之為分，用陰減陽加常分為度及分。）〉
To find the Nine-Path new-moon position: scale the lunar-apogee fixed advance/retardation by 100; add 42 for retardation and subtract for advance; adjust the mid surplus and new-moon day; add the initial-conjunction Nine-Path lodge to obtain the result. (In Qianyuan: set the Nine-Path true-conjunction degree and parts, add fixed conjunction-entry, and name the Nine-Path lodge for the new-moon hour's lunar apogee position. The Yitian method appears below. Qianyuan also has a fixed conjunction degree: take the lunar-apogee yin-yang fixed value, multiply by 71, divide by 901 for parts, and subtract for yin or add for yang to the mean parts.)
21
求九道望月度： 〈（《儀天》謂之求定朔、望加時日月度。）〉 以象積加朔九道月度，命以其道，即得所求。 〈（《乾元》置朔、望加時日相距之度，以天中度及分加之，為加時象積； 用加九道朔月度，命以其道宿次去之，即望日月度及分也。 自望推朔亦如之。 《儀天》求定朔望加時九道日度，以其朔、望去交度，交前者減之，交後者加之，滿九道宿度去之，即定朔、望加時九道日度也。 求定朔望加時九道月度，置其日加時九道日度，其合朔者非正交，即日在黃道、月在九道各入宿度，多少不同，考其去極，若應繩準。 故云月與太陽同度也。 如求黃道月度法，盈九道宿次去之，各得其日加時九道宿度，自此以後，皆如求黃道月度法入之，依九道宿度行之，各得所求也。）〉
To find the Nine-Path full-moon position: (Yitian calls this finding sun and moon positions at the fixed new-moon and full-moon hours.) Add the image accumulation to the new-moon Nine-Path position and name the lodge along that path. (In Qianyuan: set the solar separation at new-moon and full-moon hours; add the celestial mid-degree and parts to obtain the hour image accumulation; add to the new-moon Nine-Path lunar position, name the lodge along that path, and remove full cycles to obtain full-moon solar and lunar degrees. Deriving new moon from full moon follows the same procedure. Yitian finds fixed-phase-hour Nine-Path solar positions: subtract or add the distance from conjunction before or after it, respectively, and remove full Nine-Path lodge cycles. To find fixed-phase-hour Nine-Path lunar position: set that hour's Nine-Path solar degree; at conjunction (not true conjunction) the sun lies on the ecliptic and the moon on a Nine Path at different lodge positions; compare polar distances as with a plumb line. Hence the text says the moon shares the sun's degree. As with the ecliptic lunar-degree method: remove full Nine-Path lodge cycles to obtain each hour's Nine-Path position; thereafter apply the ecliptic method along Nine-Path lodges to obtain all desired results.)
22
求晨昏月： 〈（《乾元》謂之月離晨昏度。 《儀天》謂之求晨昏月度。）〉 置後曆七日下離分，與其日離分相比較，取多者乘朔、望定分，取少者乘晨昏分，皆滿元法為分，百除為度分，仍相減之， 〈（朔、望度多者為後，少者為前。）〉 各得晨昏前後度分； 前加後減朔、望九道月度為晨昏月。 〈（《乾元》置其月離差，在三百九十三以上者，用乘朔、望定分，以下者，只用三百九十三乘，為加時分； 元率除之，進一位，二百九十四收為度； 又以離差乘晨昏分，亦如前收之為度，與加時度相減之，加時度多為後、少為前，即得晨昏前後度及分，加減如《應天》。 《儀天》以晨昏分減定朔、弦、望小餘為後，不足者，返減之為前，以乘入曆定分，宗法除之，一百一約之為度，乃以前加後減加時月度為晨昏月度。）〉
To find the morning and evening moon: (Qianyuan calls this the lunar-apogee morning-and-evening degree. Yitian calls it finding the morning-and-evening lunar position.) Set apogee parts below day 7 of the retardation table and compare with that day's apogee parts; multiply new-moon/full-moon fixed parts by the greater and morning/evening parts by the lesser; convert via the origin divisor and divide by 100 for degree-parts, then subtract, (The larger new-moon/full-moon value is after; the smaller is before.) each yields the before-and-after morning/evening degree-parts; add the 'before' and subtract the 'after' corrections from the new-moon/full-moon Nine-Path position to obtain the morning and evening moon. (In Qianyuan: set the lunar-apogee discrepancy; if above 393, multiply new-moon/full-moon fixed parts by it; if below, use 393—obtaining hour parts; divide by the origin rate, shift one place, and convert by 294 to degrees; multiply apogee discrepancy by morning/evening parts and convert as before; compare with the hour degree—greater means 'after,' lesser means 'before'—obtaining before/after degrees; apply additions and subtractions as in Yingtian. Yitian: subtract morning/evening parts from fixed-phase small remainders for 'after,' or reverse-subtract for 'before' if insufficient; multiply by table-entry fixed parts, convert via lineage divisor and 101, then adjust the hour lunar position to obtain morning/evening positions.)
23
晨昏象積： 〈（《儀天》謂之求晨昏程積度。）〉 置加時象積，以前象前後度前減後加，又以後象前後度前加後減，即得所求。 〈（《乾元》法同。 《儀天》以所求朔、弦、望加時日度減後朔、弦、望加時日度，餘加弦、望度及餘，為加時程積； 以所求前後分返其加減，又以後朔、弦、望前後度分依其加減，各為晨昏程積度及餘也。）〉
Morning and evening image accumulation: (Yitian calls this finding the morning-and-evening procedure accumulation.) Set the hour image accumulation; subtract 'before' and add 'after' using the prior image's corrections, then add 'before' and subtract 'after' using the following image's corrections. (The Qianyuan method is identical. In Yitian: subtract the next phase's hour solar degree from the sought phase's hour solar degree; add the remainder to the quarter/full-moon degree and remainder for the hour procedure accumulation; reverse the before/after additions and subtractions for the sought phase; apply the next phase's before/after corrections similarly—each yielding morning/evening procedure accumulation in degrees and remainder.)
24
求每日晨昏月： 〈（《儀天》謂之求每日入曆定度。）〉 累計距後象離分，百除為度分，用減晨昏象積為加，不足，返減，以距後象日數除之，為日差； 用加減每日離分，百除為度分，累加晨昏月，命以九道宿次，即得所求。 〈（《乾元》法同。 《儀天》從所求日累計距後曆每日曆度及分，以減程積為進，不足，返減之，餘為退，以距後朔、弦、望日數均之，進加退減每日曆定度及分，各為每日曆定度及分也。）〉
To find the daily morning and evening moon: (Yitian calls this finding the daily fixed table-entry degree.) Accumulate apogee parts to the following image, divide by 100 for degree-parts, and subtract from (or reverse-subtract from) the morning/evening image accumulation; divide by days to the following image for the daily difference; apply daily apogee parts (divided by 100 for degree-parts) to accumulate the morning/evening moon along the Nine-Path lodges. (The Qianyuan method is identical. In Yitian: from the sought day, accumulate daily table degrees and parts to the following table entry; subtract from procedure accumulation for advance (reverse-subtract if insufficient) for retreat; average over days to the next new moon, first quarter, or full moon; add advance and subtract retreat for each day's fixed table degree and parts.)
25
步晷漏
Gnomon Shadow and Clepsydra Procedure.
26
求每日晷景去極度晨分： 〈（《乾元》謂之晷景距中度晨分。 《儀天》別立法，具後。）〉 各以氣數相減為分，自雨水後法十六，霜降後法十五，除分為中率，二率相減，為合差； 半之，加減中率為初、末率。 〈（前多者，加為初、減為末； 前少者，減為初、加為末。）〉 又以元法除合差，為日差； 〈（後多者累益初率，後少者累減初率。）〉 為每日損益率； 以其數累積之，各得諸氣初數也。 〈（《乾元》法同。）〉
To find each day's morning parts for gnomon shadow departure from the pole: (The Qianyuan calendar calls this the morning parts for gnomon shadow distance from midheaven. Yitian uses a separate method, given below.) Subtract qi numbers pairwise for parts; use divisor 16 after Rain Water and 15 after Frost's Descent to get the mean rate; the difference between the two rates is the combined difference; Halve it and add/subtract from the mean rate to obtain initial and terminal rates. (If the prior rate is greater, add for the initial rate and subtract for the terminal; if lesser, subtract for initial and add for terminal.) Divide the combined difference by the origin divisor for the daily difference; (If the later rate is greater, cumulatively increase the initial rate; if lesser, cumulatively decrease it.) yielding each day's increase/decrease rate; Accumulate these values to obtain each qi term's initial number. (The Qianyuan method is identical.)
27
求昏分：以晨分減元法為昏分。 〈（《乾元》謂之元率，《儀天》謂之宗法。）〉
To find dusk parts: subtract morning parts from the origin divisor. (Qianyuan calls this the origin rate; Yitian calls it the lineage divisor.)
28
求每日距中度： 〈（《乾元》同。 《儀天》謂之求每日距子度。）〉 以百乘晨分，如二千七百三十八為度，不盡，退除為距子度，用減半週天度，餘為距中星度分； 倍距子度分，五等除，為每更度分。 〈（《乾元》百約晨分，進一位，以三千六百五十三乘，如元率收為度，餘同《應天》。 《儀天》置晷漏母，五因，進一位，以一千三百八十二、小分五十五、微分三十五除為度，不盡，以一千三百六十八、小分八十六退除，皆為距子度，餘同《應天》。）〉
To find each day's distance from midheaven: (Qianyuan is the same. Yitian calls this finding each day's distance from the zi meridian.) Multiply morning parts by 100; divide by 2,738 for degrees; retreat-divide the remainder for distance-from-zi; subtract from half the circuit-of-heaven degree for distance-from-midheaven star degree-parts; Double distance-from-zi degree-parts and divide by five for each watch's degree-parts. (In Qianyuan: reduce morning parts by 100, advance one decimal place, multiply by 3,653, and divide by the origin rate for degrees; remainder as in Yingtian. In Yitian: set the gnomon-clepsydra mother, multiply by 5 and advance one place; divide by 1,382.55.35 for degrees; retreat-divide remainder by 1,368.86 for distance-from-zi; remainder as in Yingtian.)
29
求每日昏明中星： 〈（《乾元》謂之昏曉率星。）〉 置其日赤道日躔宿次，以距南度分加而命之，即其日昏中星； 以距子度分加之，為夜半中星； 又加之，為曉中星。 〈（二曆法同。）〉
To find each day's dusk and dawn culminating stars: (Qianyuan calls this dusk-and-dawn culminating stars.) Set that day's equatorial solar progression lodge; add distance-south degree-parts and name for the dusk culminating star; Add distance-from-zi degree-parts for the midnight culminating star; Add again for the dawn culminating star. (Both calendars use the same method.)
30
求五更中星：置昏中星為初更中星； 以每更度分加之，得二更初中星； 又加之，得三更初中星； 累加之，各得五更初中星所臨。 〈（二曆法同。）〉
To find culminating stars for the five watches: set the dusk star as the first-watch star; Add each watch's degree-parts for the second watch's initial culminating star; Add again for the third watch's initial star; Continue accumulating to obtain the positions for all five watches' initial culminating stars. (Both calendars use the same method.)
31
求日出入時刻： 〈（《乾元》謂之求晝夜出入辰刻。 《儀天》謂之求日出入晨刻及分。）〉 以二百五十加晨減昏為出入分，以八百三十三半除為時，不滿，百除為刻分，命如前，即得所求。 〈（《乾元》以七十三半加晨減昏為出入分，各以辰法除之。 為辰數； 不盡，以五因之，滿刻法為刻，命辰數起子正，算外，即日出入辰刻也。 《儀天》置其日晷漏母，以加昏明，餘以三因，滿辰法除為辰數，餘以刻法除為刻，不滿為分，辰數命子正，算外，即日出辰刻及分。 乃置日出辰刻及分，以加晝刻及分，滿辰法及分除為辰數，不滿，為入時之刻及分。 乃置其辰數，命子正，算外，即得日入辰刻及分。）〉
To find the times of sunrise and sunset: (Qianyuan calls this finding day-and-night rising-setting chen and marks. Yitian calls this finding sunrise/sunset morning marks and parts.) Add 250 to morning parts and subtract from dusk for rising-setting parts; divide by 833½ for hours, then by 100 for clepsydra marks and parts; name as before. (In Qianyuan: add 73½ to morning and subtract from dusk for rising-setting parts; divide each by the chen divisor. This is the chen count; Multiply remainder by 5; what fills the clepsydra divisor gives marks; count chen from zi midnight, outside the count, for sunrise/sunset chen and marks. In Yitian: set that day's gnomon-clepsydra mother and add to dusk/dawn; triple the remainder; divide by chen divisor for chen, by clepsydra divisor for marks (remainder as parts); count from zi midnight for sunrise chen, marks, and parts. Set sunrise chen, marks, and parts; add day marks and parts; divide by chen divisor for chen count; remainder gives sunset marks and parts. Name chen from zi midnight, outside the count, for sunset chen, marks, and parts.)
32
晝夜分： 〈（《乾元》謂之晝夜刻。 《儀天》謂之求每日夜半定漏、求每日晝夜刻。）〉 倍日出分，為夜分； 減元法，為晝分； 百約，為盡夜分。 〈（《乾元》置日入分，以日出分減之為晝分，以減元率為夜分，以五因之，以刻法除為晝夜刻分。 《儀天》先求夜半定漏，置其日晷漏母，以刻法除之為刻，不滿，三因為分，為夜半定漏及分。 置夜半定漏刻及分，倍之，其分滿刻法為刻，不滿為分，即得夜刻及分。 以夜刻減一百刻，餘者為晝刻及分，減晝五刻，加夜刻，為日出沒刻之數。）〉
Day and night parts: (Qianyuan calls this day-and-night clepsydra marks. Yitian calls this finding each day's midnight fixed clepsydra and day-night marks.) Double sunrise parts for night parts; Subtract from the origin divisor for day parts; Reduce by 100 for total night parts. (In Qianyuan: set sunset parts; subtract sunrise for day parts; subtract from origin rate for night parts; multiply by 5 and divide by clepsydra divisor for day-night marks. In Yitian: first find midnight fixed clepsydra from that day's gnomon-clepsydra mother; divide by clepsydra divisor for marks, triple remainder for parts. Double midnight fixed clepsydra marks and parts; parts filling clepsydra divisor become marks—yielding night marks and parts. Subtract night marks from 100 for day marks; subtract 5 day marks and add night marks for sunrise-sunset clepsydra count.)
33
更籌： 〈（《乾元》謂之更點差分。）〉 倍晨分，以五收，為更差； 又五收，為籌差。 〈（《乾元》法同。 《儀天》不立此法。）〉
Watch tallies: (Qianyuan calls this watch-and-bell difference parts.) Double morning parts and collect by 5 for watch difference; Collect by 5 again for tally difference. (The Qianyuan method is identical. Yitian does not use this method.)
34
步晷漏
Gnomon Shadow and Clepsydra Procedure.
35
冬至後初夏至後次象：八十八日、小餘八千八百九十九半，約餘八千八百一十一分。
Winter-solstice-to-first-summer-solstice image, then to second summer-solstice image: 88 days, minor remainder 8,899½, reduced remainder 8,811 parts.
36
夏至後初冬至後次象：九十三日、小餘七千四百八十五，約餘七千四百一十二分。
Summer-solstice-to-first-winter-solstice image, then to second winter-solstice image: 93 days, minor remainder 7,485, reduced remainder 7,412 parts.
37
前限：一百八十八十一日、小餘六千二百八十五，約餘六千二百二十太。
Prior limit: 188 11 days, minor remainder 6,285, reduced remainder 6,220 and a fraction over.
38
辰法：八百四十一分三分之二。
Chen divisor: 841⅔ parts.
39
刻法：一百一分。
Clepsydra divisor: 101 parts.
40
辰：八刻三十三分三分之二。
Chen: 8 marks 33⅔ parts.
41
昏明：二百五十二分半。
Dusk-dawn: 252½ parts.
42
冬至後上限五十九日，下限一百二十三日、小餘六千二百八十五，約餘六千二百二十二太。
After winter solstice: upper limit 59 days; lower limit 123 days, minor remainder 6,285, reduced remainder 6,222 and a fraction over.
43
中晷：一丈二尺七寸一分半。
Mid-gnomon shadow: 1 zhang 2 chi 7 cun 1½ parts.
44
冬至後上差、夏至後下差：二千一百三十分。
Winter-solstice upper difference / summer-solstice lower difference: 2,130 parts.
45
升法：一十五萬六千四百二十八分。
Ascending divisor: 156,428 parts.
46
冬至後下差、夏至後上差：四千八百一十二分。
Winter-solstice lower difference / summer-solstice upper difference: 4,812 parts.
47
平法：一十七萬四千三分。
Leveling divisor: 174,003 parts.
48
夏至後上限同冬至後下限，夏至後下限同冬至後上限。
Summer-solstice upper limit equals winter-solstice lower limit; summer-solstice lower equals winter-solstice upper.
49
中晷：一尺四寸七分、小分八十四。
Mid-gnomon shadow: 1 chi 4 cun 7 parts, 84 minor parts.
50
《儀天》求每日陽城晷景常數：置入冬、夏二至後求日數及分，以所入象日數下盈縮分盈減縮加之為其日定積，又以減其象小餘為夜半定積及分。 又隔位除一，用若夜半定積及分在二至上限以下者，為入上限之數； 以上者，以返減前限日及約餘，為入下限日及分。 若冬至後上限、夏至後下限，以十四乘之，所得，以減上下限差分，為定差法； 以所入上下限日數再乘之，所得，滿一百萬為尺，不滿為寸及分，以減冬至晷影，餘為其日中晷景常數也。 若夏至後上限、冬至後下限，以三十五乘之，以上下差分為定法； 以入上下限日數再乘之，退一等，滿一百萬為尺，不滿尺為寸及分，用加夏至晷景，即得其日中晷景常數。
In Yitian, for each day's Yangcheng gnomon-shadow constant: set days and parts after solstices; apply surplus-deficit of the entered image for the day's fixed accumulation; subtract the image's minor remainder for midnight fixed accumulation. Divide by one at alternate positions; if midnight fixed accumulation is below the solstice upper limit, that is the upper-limit entry; If above, reverse-subtract from prior-limit days and reduced remainder for lower-limit entry. For winter-solstice upper or summer-solstice lower limit, multiply by 14 and subtract from upper-lower limit difference for the fixed-difference divisor; Multiply again by entered limit days; divide by 1,000,000 for chi (remainder cun and parts); subtract from winter-solstice shadow for that day's mid-gnomon constant. For summer-solstice upper or winter-solstice lower limit, multiply by 35 and use upper-lower difference as the fixed divisor; Multiply again by entered limit days, retreat one place; divide by 1,000,000 for chi (remainder cun and parts); add to summer-solstice shadow for that day mid-gnomon constant.
51
《儀天》求晷景每日損益差：以其日晷景與次日晷景相減，其日景長於次日晷影為損，短於次日晷景為益。
In Yitian, for each day gnomon-shadow increase-decrease difference: subtract next day shadow from this day; longer shadow than the next day is decrease, shorter is increase.
52
《儀天》求陽城中晷景定數：置五千分，以其日晷景定數損益差乘之，所得，以萬約之為分，冬至後用減，夏至後用加； 冬至一日有減無加，夏至一日有加無減。
In Yitian, for Yangcheng mid-gnomon fixed number: set 5,000 parts; multiply by that day shadow fixed-number difference; divide by 10,000 for parts; subtract after winter solstice, add after summer solstice; On winter-solstice day only decrease applies; on summer-solstice day only increase.
53
《儀天》求晷漏損益度入前後限數：置入冬至後來日數，在前限以下者為損； 以上者，減去前限，餘為入後限日數者為益。 若算立成，自冬至後一日，日加滿初象，即加象下約餘，為一象之數。
In Yitian, for gnomon-clepsydra increase-decrease entry into prior/posterior limits: set days after winter solstice; below the prior limit is decrease; If above, subtract the prior limit; the remainder as posterior-limit entry days is increase. With ready-made tables, from the day after winter solstice add daily until the first image fills, then add the image reduced remainder for one image count.
54
《儀天》求每日晷漏損益數：置入前後限損益日數及分，如初象以下為在上限； 以上者，返減前限，餘為下限，皆自相乘之，其分半以下乘，半以上收之； 以一百通日，內其分，乃乘之； 所得，在冬至後初象、夏至後次象，以升法除之。 若冬至後次象、夏至後初象，以平法除之； 皆為分，不滿，退除為小分； 所得，置於上位，又別置五百五分於下，以上減下，以下乘上； 用在升法者，以二千八百五十除之； 用在平法者，以五千五百五十二除之； 皆為分，不滿，退除為小分； 所得，以加上位，為其日損益數。
In Yitian, for each day gnomon-clepsydra increase-decrease number: set prior/posterior limit increase-decrease days and parts; below the first image is upper limit; If above, reverse-subtract the prior limit for lower limit; self-multiply each; multiply parts at half or below, collect at half or above; Convert the day with 100, include parts, then multiply; For winter-solstice first image or summer-solstice second image, divide the product by the ascending divisor. For winter-solstice second image or summer-solstice first image, divide by the leveling divisor; Yield parts; retreat-divide remainder for minor parts; Place the product above and 505 parts below; subtract below from above and multiply below by above; When using the ascending divisor, divide by 2,850; When using the leveling divisor, divide by 5,552; Yield parts; retreat-divide remainder for minor parts; Add the product to the upper position for that day increase-decrease number.
55
《儀天》求每日黃道去極度及赤道內外度分：若春分後置損益差，以五十乘之，以一千五十二除之為度，不滿，以一千四十二除之為分，以加六十七度三千八百四十五。 若秋分後，置損益差，以五十乘之，以一千六十除之為度，不滿，以一千五十退除為分，以減一百一十五度二千二百二十二分，即得黃道去極度。 置去極度分，與九十一度三千八百四十五相減，餘者為赤道內外度分。 若黃道去極度分在九十一度三千八百四十五以下者為內，若在以上者為外度及分。
In Yitian, for each day Yellow Path departure from the pole and equator inner/outer degree-parts: after spring equinox set increase-decrease difference, multiply by 50, divide by 1,052 for degrees (by 1,042 for parts), add to 67°3,845. After autumn equinox: set increase-decrease difference, multiply by 50, divide by 1,060 for degrees (retreat-divide by 1,050 for parts), subtract from 115°2,222 for Yellow Path departure from the pole. Set departure-from-pole parts; subtract from 91°3,845 for equator inner/outer degree-parts. If Yellow Path departure-from-pole parts are below 91°3,845, it is inner; if above, outer degrees and parts.
56
《儀天》求每日晷漏母：各以其日損益差，自春分初日以後加一千七百六十八，自秋分初日以後減二千七百七十七，各得其日晷漏母，又曰晨分。
In Yitian, for each day gnomon-clepsydra mother: apply that day increase-decrease difference; add 1,768 after spring equinox, subtract 2,777 after autumn equinox—yielding the gnomon-clepsydra mother, also called morning parts.
57
《儀天》求每日昏分及距午分：置日元分，以其日晷漏母減之，餘者為昏分。 又以其日晷漏母減五千五十分，餘者為其日距午分。
In Yitian, for dusk parts and distance-from-noon parts: set day origin parts; subtract that day gnomon-clepsydra mother for dusk parts. Subtract that day gnomon-clepsydra mother from 5,050 for distance-from-noon parts.
58
月離九道交會 〈（《乾元》謂之交會，《儀天》謂之步交會。）〉
Lunar Apogee: Nine-Path Conjunction. (Qianyuan calls this conjunction; Yitian calls it the conjunction procedure.)
59
交總：七十一萬七千八百一、秒八十二。
Conjunction total: 717,801, 82 seconds.
60
正交：三百六十三度、八千二百八十三、秒七。
True conjunction: 363°, 8,283, 7 seconds.
61
半交：一百八十一度、九千一百四十二、秒五十三半。
Half conjunction: 181°, 9,142, 53½ seconds.
62
少交：九十度、九千五百二十一、秒二十六太。
Minor conjunction: 90°, 9,521, 26 and a fraction over seconds.
63
平朔：一度、四千六百三十二。
Mean new moon: 1°, 4,632.
64
平望：空、七千三百一十六。
Mean full moon: 0°, 7,316.
65
朔差：二度、八千八百四十一。
New-moon difference: 2°, 8,841.
66
望差：二度、一千五百二十五。
Full-moon difference: 2°, 1,525.
67
初準：一萬六千六百四十一。
Initial standard: 16,641.
68
中準：一萬八千一百九十一。
Middle standard: 18,191.
69
末準：一千五百五十。
Terminal standard: 1,550.
70
《乾元》交會
Qianyuan Conjunction.
71
交率：一萬六千、秒七千八百九十一。
Conjunction rate: 16,000, 7,891 seconds.
72
交策：二十七、餘六百二十三、秒九千四百五十五。
Conjunction sequence: 27, remainder 623, 9,455 seconds.
73
朔準：二、九百三十六、秒五百四十五。
New-moon standard: 2, 936, 545 seconds.
74
望準：十四、二千二百五十。
Full-moon standard: 14, 2,250.
75
初限：三萬六千五百九十四。
Initial limit: 36,594.
76
中限：四萬二。
Middle limit: 42,000.
77
末限：三千四百八。
Terminal limit: 3,408.
78
《儀天》步交會
Yitian Conjunction Procedure.
79
交終分：二十七萬四千八百四十三、秒二千二百七十九。
Conjunction terminal parts: 274,843, 2,279 seconds.
80
交終日：二十七、餘二千一百四十三、秒二千二百七十九。
Conjunction terminal day: 27, remainder 2,143, 2,279 seconds.
81
交中日：一十三、餘六千一百二十一、秒六千一百二十一。
Conjunction mid-day: 13, remainder 6,121, 6,121 seconds.
82
交朔日：二、餘三千二百一十五、秒七千七百二十一。
Conjunction new-moon day: 2, remainder 3,215, 7,721 seconds.
83
交望日：一十四、餘七千七百二十九、秒五千。
Conjunction full-moon day: 14, remainder 7,729, 5,000 seconds.
84
前限日：一十二、餘四千五百一十三、秒七千二百七十九。
Prior-limit day: 12, remainder 4,513, 7,279 seconds.
85
後限日：一、餘一千六百七、秒八千八百六十半。
Posterior-limit day: 1, remainder 1,607, 8,860½ seconds.
86
交差：四十五。
Conjunction difference: 45.
87
交數：五百七十二。
Conjunction number: 572.
88
秒母：一萬。
Seconds mother: 10,000.
89
陰限：七千二百八十六。
Yin limit: 7,286.
90
交日：空、小餘六千一百四十六、秒三百七十三。
Conjunction day: 0, minor remainder 6,146, 373 seconds.
91
陽限：三千一百七十四。
Yang limit: 3,174.
92
月食既限：二千五百八十二。
Lunar eclipse totality limit: 2,582.
93
月食分法：九百一十二半。
Lunar eclipse parts divisor: 912½.
94
中盈度： 〈（《乾元》謂之求平交朔日。 《儀天》謂之求天正朔入交。）〉 以通餘減元積，七十五展之，以四百六十七除為分，滿交總去之，為總數； 不盡，半而進位，倍總數，百收為分，用減之，餘以元法收為度，不滿為分，命曰中盈度及分。 〈（《乾元》置朔分，以交率去之，餘以五因之，滿元率收為日，即得平交朔日及分； 次朔、望，以朔、望準加之，即得所求。 《儀天》置天正朔積分，以交終分去之，滿宗法為日，即得所求。）〉
Mid surplus degree: (Qianyuan calls this finding the mean-conjunction new-moon day. Yitian calls this finding the celestial-standard new moon entry into conjunction.) Subtract common remainder from origin accumulation, expand by 75, divide by 467 for parts, remove what fills conjunction total for the total number; Halve remainder and advance; double total number; collect by 100 and subtract; remainder to degrees by origin divisor (remainder as parts)—named mid surplus degree and parts. (In Qianyuan: set new-moon parts, remove by conjunction rate, multiply remainder by 5, collect by origin rate for mean-conjunction new-moon day and parts; for following new moon and full moon, add new-moon/full-moon standards. In Yitian: set celestial-standard new-moon accumulation, remove by conjunction terminal parts, divide by lineage divisor for days.)
95
求次朔望中盈： 〈（《儀天》謂之求次朔入交。）〉 各置天正經朔中盈度分，視十一月望，十二月朔、望中日，如二十九日五千三百七以下者，即加朔、望差度分秒，餘月即加平朔、望度分秒，即得所求。 〈（《乾元》法見上。 《儀天》置天正朔入交泛日餘秒，如交朔及交望餘秒皆滿交終日及餘秒即去之，各得朔、望入交泛日及餘秒。）〉
To find the following new moon and full moon mid surplus: (Yitian calls this finding the following new moon entry into conjunction.) Set celestial-standard canonical new-moon mid surplus; for November full moon and December new/full moons, if ≤29 days 5,307, add new/full-moon difference; other months add mean new/full-moon parts. (Qianyuan method as above. In Yitian: set new moon general entry-into-conjunction day and seconds; when new/full-moon parts fill conjunction terminal day, remove—yielding general entry days for new and full moon.)
96
月離朔交初度分： 〈（《乾元》謂之求朔望交分。 《儀天》謂之求入交常日。）〉 置其朔中盈度分， 〈（常與其朔常日度分合之，如正交以下者減半法，以上者倍而加之。）〉 加減訖為定，用減天正加時黃道宿度分，餘命起天正之宿初算，即得所求。 〈（《乾元》置平交朔、望日及分，以元率通之，以日躔陰陽差陽加陰減，為朔、望交分。 《儀天》以其日入盈朔限昇平定數，升加平減入交泛日，即為其朔、望入交常日也。 《儀天》又有求朔、望入定交日，置其日入遲疾限昇平定數，以交差乘之，如交數而一，升加平減入交常日，即為入定交日。）〉
Lunar apogee: new-moon conjunction initial degree-parts: (Qianyuan calls this finding new-moon and full-moon conjunction parts. Yitian calls this finding entry-into-conjunction regular day.) Set the new-moon mid surplus degree-parts, (Combine with the new-moon regular-day parts; if ≤ true conjunction subtract half divisor, if above double and add.) When fixed, subtract from celestial-standard hour Yellow Path lodge parts; name from celestial-standard lodge initial count. (In Qianyuan: set mean-conjunction new/full-moon day and parts, convert by origin rate, apply solar progression yin-yang difference (yang add, yin subtract) for conjunction parts. In Yitian: apply surplus-new-moon limit ascending-leveling fixed number (ascending add, leveling subtract) to general entry day for regular entry day. Yitian also finds fixed entry day: apply slow-fast limit ascending-leveling number × conjunction difference ÷ conjunction number to regular entry day.)
97
月入陰陽曆： 〈（《乾元》謂之求朔望陰陽定分，《儀天》謂之求月行陰陽曆。）〉 以月離先後定數，先加後減朔、望中盈，用加朔、望常日月分， 〈（分即百除，度即百通。）〉 如中準以下者為月出黃道外； 以上者去之，餘為月入黃道內。 〈（《乾元》以一百四十二乘陰陽差，一千八百二除，陽加陰減朔、望交分，為度定分； 中限以上為陽，以下為陰。 《儀天》視入交定日及餘秒，在交中日以下為陽，以上者去之，餘為月入陰曆。）〉
Moon Enters the Yin-Yang Table: (Qianyuan calls this new/full-moon yin-yang fixed parts; Yitian calls it finding the moon yin-yang table.) With lunar apogee advance-retard fixed number, advance add and retard subtract mid surplus; add to new/full-moon regular-day moon parts, (Divide parts by 100; convert degrees by 100.) If ≤ middle standard, the moon exits the Yellow Path outer; If above, remove it; remainder is the moon entering the Yellow Path inner. (In Qianyuan: multiply yin-yang difference by 142, divide by 1,802, yang add and yin subtract new/full-moon conjunction parts for fixed degree-parts; Above the middle limit is yang; below is yin. In Yitian: if fixed entry day and seconds are below conjunction mid-day, it is yang; if above, remove—remainder is moon entering the yin table.)
98
求食甚定餘：置朔定分，如半法以下者返減半法，餘為午前分； 前以上者減去半法，餘為午後分； 以乘三百，如半晝分而一，為差。 〈（午後加之，午前半而減之。）〉 加減定朔分，為食定餘。 以差皆加午前、後分，為距中分。 其望定分，便為食定餘。 〈（《乾元》以半晝刻約刻法為時差，乃視定朔小餘，在半法以下為用減半法為午前分； 以上者去之，為午後分； 以時差乘，五因之，如刻法而一，午前減，午後加，又皆加午前、後分，為距日分； 刻法而一，為距午刻分。 月只以定朔小餘為食定餘。 《儀天》置月行去交黃赤道差，視月道差，如黃赤道交者，依其加減； 不如黃赤道交者，返其加減； 定朔、望小餘為食甚餘，亦返其加減去交定分。 其日食，則又以其日晝刻，其三百五十四為時差，乃視食甚餘，如半法以下，返減半法，餘為初率； 半法以上者，半法去之，餘為末率； 滿一百一收之，為初率； 以減末率，倍之，以加食甚餘，為食定餘； 亦加減初、末率，為距午退分； 置之，皆如求發斂加時術入之，即日、月食甚辰刻及分也。）〉
To find greatest-eclipse fixed remainder: set new-moon fixed parts; if ≤ half divisor, reverse-subtract half divisor for before-noon parts; If above before-noon threshold, subtract half divisor for after-noon parts; Multiply by 300 and divide by half day-parts for the difference. (After noon add; before noon halve and subtract.) Add/subtract to fixed new-moon parts for eclipse fixed remainder. Add the difference to before- and after-noon parts for distance-from-midheaven parts. Full-moon fixed parts serve directly as eclipse fixed remainder. (In Qianyuan: reduce half-day clepsydra marks by clepsydra divisor for time difference; if fixed new-moon minor remainder ≤ half divisor, reverse-subtract half divisor for before-noon parts; If above, remove for after-noon parts; Multiply by time difference, multiply by 5, divide by clepsydra divisor; subtract before noon, add after noon; add to before/after-noon parts for distance-from-sun parts; Divide by clepsydra divisor for distance-from-noon clepsydra parts. For lunar eclipses, use only fixed new-moon minor remainder as eclipse fixed remainder. In Yitian: set moon motion departure-from-conjunction Yellow Path–equator difference; if like Yellow Path–equator conjunction, follow its add/subtract; If not like Yellow Path–equator conjunction, reverse add/subtract; Use fixed new/full-moon minor remainder as greatest-eclipse remainder; also reverse add/subtract from conjunction fixed parts. For solar eclipse: use that that day's clepsydra marks, 354 as time difference; if greatest-eclipse remainder ≤ half divisor, reverse-subtract half divisor for initial rate; If ≥ half divisor, remove half divisor for terminal rate; Collect by 101 for initial rate; Subtract from terminal rate, double, add to greatest-eclipse remainder for eclipse fixed remainder; Also add/subtract initial and terminal rates for distance-from-noon retreat parts; Set them and apply the emission-collection hour-addition procedure for solar/lunar greatest-eclipse chen, marks, and parts.)
99
入食限：置黃道內、外分，如初準已上、末準已下為入食限。 望入食限則月食，朔入食限則日食。 月在黃道內則日食，在外則不食，望則無問內、外皆食。 末準已下為交後分； 初準以上者，返減中準，為交前分。 〈（《乾元》置陰陽定分，在初限以上、末限以下，為入食限，餘同《應天》。 《儀天》置朔、望入交月行陰陽曆日及餘秒，如前限以上、後限以下者，為入食限。 望入食限則月食，朔入食限、月入陰曆則日食。 如後限以下為交後限，以上以減交中日及餘秒為交前限，各得所求。）〉
Entry into eclipse limit: set Yellow Path inner/outer parts; between initial standard (inclusive) and terminal standard (inclusive) is eclipse limit entry. Full moon in eclipse limit yields lunar eclipse; new moon in eclipse limit yields solar eclipse. Moon inside the Yellow Path: solar eclipse; outside: none; at full moon, eclipses regardless of inner/outer. At or below terminal standard: post-conjunction parts; At or above initial standard: reverse-subtract middle standard for pre-conjunction parts. (In Qianyuan: set yin-yang fixed parts; between initial and terminal limits is eclipse limit entry; remainder as in Yingtian. In Yitian: set new/full-moon entry yin-yang table day and seconds; between prior and posterior limits is eclipse limit entry. Full moon in eclipse limit: lunar eclipse; new moon in eclipse limit with moon in yin table: solar eclipse. At or below posterior limit: post-conjunction limit; above: subtract from conjunction mid-day for pre-conjunction limit.)
100
入盈縮曆： 〈（《乾元》、《儀天》不立此法。）〉 置朔定積，如一百八十二日、六千二百二十三以下為入盈日分； 以上者去之，餘為入縮日分。
Entry into the Surplus-Deficit Table: (Qianyuan and Yitian omit this method.) Set new-moon fixed accumulation; if ≤182 days 6,223, entry into surplus day-parts; If above, remove for entry into deficit day-parts.
101
黃道差： 〈（《乾元》謂之求晷差。 《儀天》謂之求黃道食差。）〉 置其朔入曆盈、縮日及分，如四十五日以上、一百三十七日以下，皆以一千五百乘，為泛差； 如四十五日以下，返減之，餘為初限日，一百三十七日以上者減去之，餘為末限日及分，以六十七乘，半之，用減泛差，以乘距午分，以元法收為黃道定分； 入盈，以定分午前內減外加、午後內加外減； 入縮，以定分午前內加外減，午後內減外加。 〈（《乾元》置入氣日，以距冬至之氣，以十五乘之，以所入氣日通之，以一百八十二日以下為入陽曆，以上者去之，為入陰曆。 置入曆分，在四十五日以下，以三十七乘，五除，退一等，為泛差； 在四十五日以上、一百三十七日以下，只用三十三、秒三十為泛差； 一百三十七以上者去之，餘以三十七乘，五除，退一位，用減三十三、秒三十為泛差； 皆以距午分乘為晷差。 《儀天》二至後日益差至立春、立秋，得一百一十三、小分六十二半，立夏、立冬後每日損，以宗法乘之； 冬至、立冬後三氣用四十四萬二千三百八十四，夏至、立夏后各三氣用二十七萬九千八百五十八除，為食差； 以食甚距午正刻乘其日食差，為定差； 冬至後，甚在午正東，陰減陽加； 甚在午正西，陰加陽減； 夏至後即返此； 立冬初日後，每氣益差二十、秒四十四，至冬至初日加六十二、秒三十二； 自後每氣損差二十、秒四十四，終於大寒，甚在午正西，即每刻累益其差，陰曆加，陽曆減。）〉
Yellow Path Difference: (Qianyuan calls this finding gnomon difference. Yitian calls this finding Yellow Path eclipse difference.) Set new-moon surplus/deficit table entry days and parts; if 45–137 days, multiply by 1,500 for general difference; If ≤45 days, reverse-subtract for initial-limit days; if ≥137 days, subtract for terminal-limit days; ×67, halve, subtract from general difference; × distance-from-noon parts; collect by origin divisor for Yellow Path fixed parts; Entering surplus: before noon inner subtract/outer add; after noon inner add/outer subtract; Entering deficit: before noon inner add/outer subtract; after noon inner subtract/outer add. (In Qianyuan: set entry-into-qi day, multiply qi distance from winter solstice by 15, convert with entered qi day; ≤182 days is yang table entry, above is yin table entry. Set table-entry parts; if ≤45 days, ×37, ÷5, retreat one place for general difference; If 45–137 days, use only 33, 30 seconds for general difference; If ≥137 days, remove; remainder ×37, ÷5, retreat one place; subtract from 33, 30 seconds for general difference; Each × distance-from-noon parts for gnomon difference. In Yitian: from solstice daily increase to spring/autumn establishment reaches 113, 62½ minor parts; after summer/winter establishment daily decrease; multiply by lineage divisor; After winter solstice/winter establishment three qi divide by 442,384; after summer solstice/summer establishment three qi divide by 279,858 for eclipse difference; Multiply that day eclipse difference by greatest-eclipse distance-from-noon true marks for fixed difference; After winter solstice: if greatest is east of true noon, yin subtract and yang add; If west of true noon, yin add and yang subtract; After summer solstice reverse these rules; After winter establishment first day, each qi increases difference by 20, 44 seconds, until winter solstice first day adds 62, 32 seconds; Thereafter each qi decreases difference by 20, 44 seconds until Great Cold; if greatest is west of true noon, cumulatively increase difference per mark; yin table add, yang table subtract.)
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赤道差： 〈（《乾元》謂之求離差，《儀天》謂之求赤道食差。）〉 置入盈縮曆日及分，如九十一日以下，返減之，為初限日； 以上者，用減一百八十二日半，餘為末限日及分； 四因之，用減三百七十四，為泛差； 以乘距中分，如半晝分而一，用減泛差，為赤道定分； 盈初縮末內減外加、縮初盈末內加外減。 〈（《乾元》計春、秋二分後日加入氣日，以十五乘，在九十以下，以九十一乘，退為泛差； 九十一以上去之，餘以九十一乘，退一等，以減八百一十九，為泛差； 二分氣內置入氣日，以九十一乘，退為泛差； 以半晝刻而一，以乘距午分，用加減泛差，為離差； 食甚在出沒以前者，不用求離差，只用泛差，春分後陰加陽減，秋分後陰減陽加。 《儀天》二分後益差至二至，積差皆二千八百二十六，自後累減至二分空，冬至後日損三十一、小分八十，夏至後日益三十、小分十五，又以宗法乘積差，各以盈縮初末限分除之，為日差； 乃以末限累增、初限累損，各為每日食差； 又以半晝刻數約其日食差，以乘食甚距午正刻，所得以減食差，餘為定數。 餘同《乾元》。）〉
Equator Difference: (Qianyuan calls this apogee difference; Yitian calls it equator eclipse difference.) Set surplus/deficit table entry days and parts; if ≤91 days, reverse-subtract for initial-limit days; If above, subtract from 182½ days for terminal-limit days and parts; Multiply by 4 and subtract from 374 for general difference; Multiply by distance-from-midheaven parts, divide by half day-parts, subtract from general difference for equator fixed parts; Surplus-initial/deficit-terminal: inner subtract, outer add; deficit-initial/surplus-terminal: inner add, outer subtract. (In Qianyuan: count post-equinox days plus entry-into-qi day, ×15; if ≤90, ×91 and retreat for general difference; If ≥91, remove; remainder ×91, retreat one place; subtract from 819 for general difference; Within equinox qi, set entry-into-qi day, ×91 and retreat for general difference; Divide by half-day clepsydra marks, × distance-from-noon parts, add/subtract general difference for apogee difference; If greatest eclipse is before rising/setting, use only general difference; after spring equinox yin add/yang subtract, after autumn equinox yin subtract/yang add. In Yitian: post-equinox increase to solstices reaches accumulated difference 2,826; then cumulative decrease to equinox zero; winter-solstice daily −31, 80 minor parts; summer-solstice daily +30, 15 minor parts; × accumulated difference by lineage divisor, ÷ surplus/deficit initial/terminal limits for daily difference; Terminal limit cumulative increase and initial limit cumulative decrease yield daily eclipse difference; Reduce daily eclipse difference by half-day clepsydra count, × greatest-eclipse distance-from-noon true marks, subtract from eclipse difference for fixed number. Remainder as in Qianyuan.)
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日食差：依黃、赤二差，同名相從，異名相消，為食差。 〈（二曆法同。）〉
Solar eclipse difference: combine Yellow Path and equator differences—same signs add, opposite signs cancel. (Both calendars use the same method.)
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距交分： 〈（《乾元》謂之去交分。 《儀天》謂之去交定分。）〉 置交前後分，以黃、赤二差加減之，為距交分。 如月在內道不足減者，返減入外道，不食； 如月在外道不足減，返減食差，為返減入內道即有食。 〈（《乾元》置陰陽曆去交前後分，以食差合加減者，依其加減，所得為去交前後定分。 月在陰曆，去交前後分不足減者，即返減食差，交前減之，餘者為得陽曆交後得減之，餘者為陽曆交前定分，並不入食限。 月在陽曆，去交前後分不足減者，亦返減食差，交前減之，餘者為陰曆交後定分，交後減之，餘者為陰曆交前定分，併入食限。 《儀天》應食差，同名相從，異名相消，餘同《乾元》法。）〉
Distance from Conjunction: (Qianyuan calls this departure-from-conjunction parts. Yitian calls it departure-from-conjunction fixed parts.) Set pre/post-conjunction parts; apply Yellow Path and equator differences for distance-from-conjunction parts. If the moon is on the inner path and subtraction is insufficient, reverse-subtract to the outer path—no eclipse; If on the outer path and subtraction is insufficient, reverse-subtract eclipse difference—reverse-subtract to inner path and there is eclipse. (In Qianyuan: set yin-yang table pre/post-conjunction parts; apply combined eclipse-difference add/subtract for fixed pre/post-conjunction parts. Moon in yin table: if pre/post-conjunction parts insufficient, reverse-subtract eclipse difference; pre-conjunction subtract—remainder to yang table post-conjunction; both outside eclipse limit. Moon in yang table: if insufficient, reverse-subtract eclipse difference; pre-conjunction subtract for yin post-conjunction fixed parts; post-conjunction subtract for yin pre-conjunction fixed parts; both in eclipse limit. In Yitian: apply eclipse difference—same signs add, opposite cancel; remainder as in Qianyuan.)
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日食分：置距交分，如四百二十以下者類同陽曆分； 以上者去之，為陰曆分； 又以食定餘減四分之三， 〈（午前倍之，午後半之。）〉 皆退一等，用減陰陽曆分，為食定分； 如不足減，即返減之，餘進一位，加陰曆分，為食定分； 陽以四十二除，為食之大分； 陰九百六十以下返減之，如九十六而一，為食之大分，命十為限。 〈（《乾元》置交前後分，以食差加減之，為定交分； 在九百二十以下為陽，以上去之為陰。 在陽以九十四、在陰以二百一十三除為大分，餘同《應天》。 《儀天》置入限去交定分，減七百二十八，陽限以上為陰曆食，以陽限去之，餘減陰限為陰曆食分，以下者為陽曆食分，亦減三百一十七，如限除之，皆進一位，各命十為限，餘同《應天》。）〉
Solar eclipse magnitude: set distance-from-conjunction parts; if ≤420, treat as yang-table parts; If above, remove for yin-table parts; Again subtract three-fourths from eclipse fixed remainder, (Before noon double; after noon halve.) Each retreat one place; subtract from yin-yang table parts for eclipse fixed parts; If insufficient, reverse-subtract; advance remainder one place; add yin-table parts for eclipse fixed parts; Yang: divide by 42 for eclipse major parts; Yin: if ≤960 reverse-subtract; divide by 96 for eclipse major parts; name 10 as limit. (In Qianyuan: set pre/post-conjunction parts; apply eclipse difference for fixed conjunction parts; If ≤920 it is yang; if above, remove for yin. In yang divide by 94; in yin by 213 for major parts; remainder as in Yingtian. In Yitian: set limit-entry departure-from-conjunction fixed parts, subtract 728; above yang limit is yin-table eclipse; remove yang limit, subtract yin limit for yin-table eclipse parts; below is yang-table eclipse parts; also subtract 317; divide by limit, advance one place, name 10 as limit; remainder as in Yingtian.)
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月食分：置黃道內外前後分，如食限三百四十以下者，食既； 以上者，返減末準，餘以一百二十一除，為月食之大分。 〈（其食五分以下，在子正前後八刻內，以二百四十二除為食之大分，命十為限。）〉 其前後分，以九百以上入或食或不食之限， 〈（乾元交定分在七百五十二以下，食既； 以上，返減末限，以二百六十四除之為大分。 儀天陽減陰加前後定分九百一十二半，在既限以下、食既以上，以去交分減之，以月食法除之為大分。）〉
Lunar eclipse magnitude: set Yellow Path inner/outer pre/post parts; if ≤ eclipse limit 340, totality; If above, reverse-subtract terminal standard; divide remainder by 121 for lunar eclipse major parts. (If eclipse ≤5 parts, within 8 clepsydra marks of zi midnight, divide by 242 for major parts; name 10 as limit.) Pre/post parts ≥900 enter the uncertain-eclipse limit, (In Qianyuan: conjunction fixed parts ≤752, totality; If above, reverse-subtract terminal limit; divide by 264 for major parts. In Yitian: yang subtract/yin add pre/post fixed parts 912½; between totality and eclipse limits, subtract departure-from-conjunction parts; divide by lunar eclipse divisor for major parts.)
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日月食虧初复末： 〈（《乾元》謂之求定用刻，《儀天》謂之求日月泛用分、求虧初复末。）〉 百通日月食之大小分，以一千三百三十七乘之，各如其日離分，為定用分； 加食定餘，為復末定分； 減之，為虧初定分。 其月食，以食限減定用分，用減食甚，為虧初定分； 如不足減者，即以食限分如望定餘為食定分，餘卻依日食加減，各得月食虧初、复末定分也。 〈（《儀天》月以五百八十八，日以五百二十九、秒二十乘所食分，退一等，半之，為定用刻。 《儀天》日以五百四十五、秒四十，月以六百六，皆乘所食分，其小分以本母除，從之，為泛用分； 其食又視去交定分在一千七百二十六以下增半刻，八百五十六以下又增半刻，以一千三百五十乘，以辰定分除，為定用刻； 皆減定朔、望小餘為虧初，加之為復末。）〉
Solar and Lunar Eclipse: First Loss and Last Return: (Qianyuan calls this finding fixed-use clepsydra marks; Yitian calls it finding general-use parts and first loss/last return.) Convert eclipse major/minor parts by 100; multiply by 1,337; each by that day apogee parts for fixed-use parts; Add eclipse fixed remainder for last-return fixed parts; Subtract for first-loss fixed parts. For lunar eclipse: subtract fixed-use parts from eclipse limit and from greatest eclipse for first-loss fixed parts; If insufficient, use eclipse limit parts like full-moon fixed remainder; remainder follows solar eclipse add/subtract for lunar first-loss and last-return fixed parts. (In Yitian: moon use 588, sun use 529, 20 seconds × eclipse parts; retreat one place, halve for fixed-use clepsydra marks. In Yitian: sun 545, 40 seconds; moon 606; each × eclipse parts; minor parts ÷ original mother for general-use parts; Also if departure-from-conjunction fixed parts ≤1,726 add half mark; if ≤856 add another half mark; ×1,350, ÷ chen fixed parts for fixed-use clepsydra marks; Subtract from fixed new/full-moon minor remainder for first loss; add for last return.)
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日食起虧： 〈（《乾元》謂之求日食初起。）〉 視距交分如四百二十以上者，初起西北，甚於正北，復於東北； 如以下者，初起西南，甚於正南，復於東南。 凡食八分以上者，皆初起正西，復於正東。 〈（《儀天》、《乾元》日在陰曆，初起西北； 在陽曆，初起西南，餘並同《應天》。）〉
Solar Eclipse First-Loss Direction: (Qianyuan calls this finding solar eclipse initial direction.) If distance-from-conjunction parts ≥420: first loss northwest, greatest at true north, return northeast; If below: first loss southwest, greatest at true south, return southeast. For eclipses ≥8 parts: first loss due west, return due east. (In Yitian and Qianyuan: sun in yin table, first loss northwest; in yang table, first loss southwest; remainder as in Yingtian.)
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月食起虧： 〈（《乾元》謂之月食初定，《儀天》謂之月食初起。）〉 月在內道，初起東南，甚於正南，復於西南； 月在外道，初起東北，甚於正北，復於西北。 凡食八分以上者，初起正東，復於正西。 〈（《乾元》《儀天》以內道為陰曆，外道為陽曆，餘皆同《應天》。 而《儀天》又法雲，此法據古經所載，以究天體，食在午中前後一辰之內，其餘方若要的驗，當視日月食時所在方位高下，審祥黃道斜正、月行所向，起虧、复滿皆可知也。）〉
Lunar Eclipse First-Loss Direction: (Qianyuan calls this lunar eclipse initial fixed; Yitian calls it lunar eclipse initial direction.) Moon on inner path: first loss southeast, greatest at true south, return southwest; Moon on outer path: first loss northeast, greatest at true north, return northwest. For eclipses ≥8 parts: first loss due east, return due west. (In Qianyuan and Yitian, inner path is yin table and outer path is yang table; remainder as in Yingtian. Yitian also notes: this method follows ancient classics to investigate the heavens; within one chen of noon, for other directions observe eclipse-time location and altitude, examine Yellow Path obliquity and lunar motion—first loss and last return can all be determined.)
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帶食出入： 〈（《儀天》謂之求帶食出入見食分數。）〉 視其日出入分，如在虧初定分以上、复末定分以下，即帶食出入。 食甚在出入分以下，以出入分減复末定分，為帶食差； 食甚在出入分以上者，以虧初定分減出入分，為帶食差； 以乘食定分，滿定用而一，日陽以四十二、陰以九十六、月一百二十一除之，為帶食之大分，餘為小分。 〈（《乾元》各以食甚餘與其日晨昏分相減，餘為帶食差； 其帶食差在定用刻以下者，即帶食出入； 以上者，即不帶食出入也。 以帶食差乘所食之分，滿定用刻而一，所得以減所食之分，即帶食出入所見之分也。 其朔日食甚在晝者，晨為已食之分，昏為所殘之分； 若食甚在夜，昏為已食之分，晨為所殘之分。 其月食，見此可以知之也。 《儀天》以食甚餘減晨昏分，餘為出入前分，不足者，返減食甚，餘為出入後分，以乘所食之分，其食分以本母通之，從其小分，滿定用分除之，所得以本母約之，不滿者，半以上為半強，半以下為半弱，即得帶食出入之分數也。 其日、月食甚在出入前者，為所殘之分，在出入後者，為已退之分。）〉
Eclipse at Rising or Setting: (Yitian calls this finding visible eclipse magnitude at rising or setting.) If rising-setting parts fall between first-loss and last-return fixed parts, it is an eclipse at rising or setting. If greatest eclipse is below rising-setting parts, subtract rising-setting parts from last-return fixed parts for the difference; If above rising-setting parts, subtract rising-setting parts from first-loss fixed parts for the difference; Multiply by eclipse fixed parts, divide by fixed-use; sun yang ÷42, yin ÷96, moon ÷121 for major parts (remainder minor parts). (In Qianyuan: subtract dawn/dusk parts from greatest-eclipse remainder for the difference; If the difference is below fixed-use clepsydra marks, it is eclipse at rising/setting; If above, it is not eclipse at rising/setting. Multiply difference by eclipsed parts, divide by fixed-use clepsydra marks, subtract from eclipsed parts for visible magnitude at rising/setting. For new-moon solar eclipse in daytime: dawn is already-eclipsed portion, dusk is remaining portion; If at night: dusk is already-eclipsed, dawn is remaining. For lunar eclipses, the same principle applies. In Yitian: subtract dawn/dusk from greatest-eclipse remainder for before-rising-setting parts (reverse-subtract if insufficient for after-rising-setting); × eclipsed parts; convert and divide by fixed-use; reduce; round half-strong/half-weak for visible magnitude at rising/setting. Greatest eclipse before rising/setting is remaining portion; after rising/setting is already-retreated portion.)
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更點： 〈（《乾元》、《儀天》謂之月食入定點。）〉 各置虧初、食甚、复末定分，如晨分以下者加晨分，昏分以上者減去昏分，皆以更分除為更數，不盡，以點分除之為點數。 命初更，算外，即得所求。 〈（《乾元》法同。 《儀天》倍其日晨分，以五除之為更分，又以五除之為點分。 乃視所求小餘，如晨分以下加晨分，昏分以上減去昏分，求更點並同《應天》。）〉
Watch Bells: (Qianyuan and Yitian call this lunar eclipse fixed bell-points.) Set first-loss, greatest, and last-return fixed parts; add morning parts if ≤ morning parts, subtract dusk parts if ≥ dusk parts; divide by watch parts for watch count, by bell parts for bell count. Name from the first watch, outside the count. (The Qianyuan method is identical. In Yitian: double that day morning parts; ÷5 for watch parts, again ÷5 for bell parts. Observe sought minor remainder; add morning or subtract dusk parts as above; watch bells as in Yingtian.)
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日月食宿分： 〈（《乾元》謂之日月食宿。）〉 以天正冬至黃道日度加朔望常日月度，命起鬥初，算外，即日月食在宿分也。 〈（《乾元》以距日沒辰至食甚辰之數，約其日離差，用加昏度。 《儀天》用加時定月度也。）〉
Solar and Lunar Eclipse Lodge Positions: (Qianyuan calls this solar-lunar eclipse lodge position.) Add celestial-standard winter-solstice Yellow Path solar degree to new/full-moon regular lunar degree; name from Dipper initial, outside the count, for eclipse lodge position. (In Qianyuan: count from sunset chen to greatest-eclipse chen, reduce that day apogee difference, add to dusk degree. Yitian uses hour-added fixed lunar degree.)