1
○授時曆經下△步中星第五大都北極，出地四十度太強。 冬至，去極一百一十五度二十一分七十三秒。
From the Shoushi Calendar Canon, lower volume. Section 5: Computing Culminating Stars. At the Great Capital, the north celestial pole stands a little more than forty degrees above the horizon. At the winter solstice, the polar distance is 115° 21′ 73″.
2
夏至，去極六十七度四十一分一十三秒。 冬至晝，夏至夜，三千八百一十五分九十二秒。 夏至晝，冬至夜，六千一百八十四分八秒。 昏明，二百五十分。
At the summer solstice, the polar distance is 67° 41′ 13″. The length of the winter-solstice day and the corresponding summer-solstice night is 3,815 parts and 92 seconds. The length of the summer-solstice day and the corresponding winter-solstice night is 6,184 parts and 8 seconds. Twilight lasts 250 parts.
3
黃道出入赤道內外去極度及半晝夜分 〈（表略）〉 求每日黃道出入赤道內外去極度
Polar distances as the ecliptic crosses inside and outside the equator, together with half day-and-night divisions Table omitted.〉 To find, for each day, the ecliptic's polar distance as it lies inside or outside the equator
4
置所求日晨前夜半黃道積度，滿半歲周，去之，在象限已下，為初限； 已上，復減半歲周，余為入末限； 滿積度，去之，余以其段內外差乘之，百約之，所得，用減內外度，為出入赤道內外度； 內減外加象限，即所求去極度及分秒。
Take the accumulated ecliptic longitude at midnight before dawn on the date in question. Divide by half the tropical year circumference and discard the quotient. If the remainder is below the quadrant limit, it is the initial limit; if it is above that limit, subtract half the tropical year circumference again; the remainder is the entry into the final limit; divide by the accumulated-degree period and take the remainder. Multiply the remainder by that segment's inside-outside difference, reduce by a factor of one hundred, and subtract the result from the tabulated inside-outside degree to obtain the degrees inside or outside the equator; subtract inside the equator and add outside it to the quadrant limit, and the result is the polar distance sought, in degrees, parts, and seconds.
5
求每日半晝夜及日出入晨昏分
To find, for each day, the half day-and-night division and the parts for sunrise, sunset, dawn, and dusk
6
置所求入初末限，滿積度，去之，余以晝夜差乘之，百約之，所得，加減其段半晝夜分，為所求日半晝夜分；） 前多後少為減，前少後多為加。 以半夜分便為日出分，用減日周，余為日入分； 以昏明分減日出分，余為晨分； 加日入分，為昏分。
Take the entry into the initial or final limit for the date sought. Divide by the accumulated-degree period and discard the quotient. Multiply the remainder by the day-night difference, reduce by one hundred, and add or subtract the result to that segment's half day-and-night value to obtain the half day-and-night division for the day in question;) Subtract when the earlier value is larger and the later smaller; add when the earlier is smaller and the later larger. Treat the midnight division directly as the sunrise division. Subtract it from the day circumference; the remainder is the sunset division; subtract the twilight division from the sunrise division; the remainder is the dawn division; add the twilight division to the sunset division to obtain the dusk division.
7
求晝夜刻及日出入辰刻
To find the day-and-night divisions in ke and the double-hour ke of sunrise and sunset
8
置半夜分，倍之，百約，為夜刻； 以減百刻，余為晝刻； 以日出入分依發斂求之，即得所求辰刻。 求更點率置晨分，倍之，五約，為更率； 又五約更率，為點率。
Take the midnight division, double it, and reduce by one hundred to obtain the night ke; subtract that from one hundred ke; the remainder is the day ke; Apply the sunrise and sunset divisions according to the expansion-and-contraction procedure to obtain the sought double-hour ke. To find the watch and bell-stroke rates: take the dawn division, double it, and reduce by five to obtain the watch rate; reduce the watch rate by five again to obtain the bell-stroke rate.
9
求更點所在辰刻
To find the double-hour ke for each watch and bell-stroke
10
置所求更點數，以更點率乘之，加其日昏分，依發斂求之，即得所求辰刻。 求距中度及更差度
Take the watch or bell-stroke number sought, multiply it by the watch-and-bell rate, add the day's dusk division, and apply the expansion-and-contraction procedure to obtain the corresponding double-hour ke. To find the median-distance degree and the watch-interval degree
11
置半日周，以其日晨分減之，余為距中分； 以三百六十六度二十五分七十五秒乘之，如日周而一，所得，為距中度； 用減一百八十三度一十二分八十七秒半，倍之，五除，為更差度及分。
Take half the day circumference and subtract that day's dawn division; the remainder is the median-distance division; multiply by 366° 25′ 75″, divide by the day circumference, and the quotient is the median-distance in degrees; subtract that from 183° 12′ 87½″, double the result, and divide by five to obtain the watch-interval in degrees and parts.
12
求昏明五更中星
To find the culminating stars at dusk, dawn, and each of the five night watches
13
置距中度，以其日午中赤道日度加而命之，即昏中星所臨宿次，命為初更中星； 以更差度累加之，滿赤道宿次去之，為逐更及曉中星宿度及分秒。 其九服所在晝夜刻分及中星諸率，並準隨處北極出地度數推之。 〈（已上諸率，與晷漏所推自相符契。）〉
Take the median-distance degree, add the sun's equatorial longitude at noon that day, and assign the lodge; that lodge is where the dusk culminating star stands—take it as the first-watch culminating star; add the watch-interval degree cumulatively for each watch; whenever the sum fills a full circuit of equatorial lodges, discard one circuit. This gives the lodge, degree, and parts for each watch and for dawn. For each of the nine domains, the day-and-night ke, parts, and all culminating-star rates are likewise derived from the local altitude of the north celestial pole. (The rates above agree of themselves with the results obtained from the gnomon-shadow and water-clock procedures.)
14
求九服所在漏刻
To find the clepsydra divisions for each of the nine domains
15
各於所在以儀測驗，或下水漏，以定其處冬至或夏至夜刻，與五十刻相減，余為至差刻。 置所求日黃道，去赤道內外度及分，以至差刻乘之，進一位，如二百三十九而一，所得內減外加五十刻，即所求夜刻； 以減百刻，余為晝刻。 〈（其日出入辰刻及更點等率，依術求之。）〉
At each locality determine by instrument observation, or by setting up a water clock, the night ke at the winter or summer solstice for that place. Subtract it from fifty ke; the remainder is the solstice-difference ke. Take the ecliptic's degrees inside or outside the equator for the day sought, multiply by the solstice-difference ke, advance one decimal place, and divide by 239. Subtract the result inside the equator and add it outside, then apply that to fifty ke to obtain the night ke for the date in question; subtract the night ke from one hundred ke; the remainder is the day ke. (The double-hour ke of sunrise and sunset and the watch-and-bell rates are obtained by the procedures above.)
16
△步交會第六交終分，二十七萬二千一百二十二分二十四秒。 交終，二十七日二千一百二十二分二十四秒。 交中，十三日六千六十一分一十二秒。
Section 6: Computing Conjunctions. Nodal period in parts: 272,122 parts and 24 seconds. Nodal period: 27 days, 2,122 parts, and 24 seconds. Half nodal period: 13 days, 6,061 parts, and 12 seconds.
17
交差，二日三千一百八十三分六十九秒。 交望，十四日七千六百五十二分九十六秒半。 交應，二十六萬一百八十七分八十六秒。 交終，三百六十三度七十九分三十四秒。
Nodal inequality: 2 days, 3,183 parts, and 69 seconds. Synodic nodal interval from new moon to full moon: 14 days, 7,652 parts, and 96½ seconds. Nodal epoch constant: 260,187 parts and 86 seconds. Nodal period in degrees: 363° 79′ 34″.
18
交中，一百八十一度八十九分六十七秒。 正交，三百五十七度六十四分。 中交，一百八十八度五分。 日食陽歷限，六度。 定法，六十。
Half nodal period in degrees: 181° 89′ 67″. Primary node longitude: 357° 64′. Median node longitude: 188° 05′. Solar eclipse limit on the solar side: 6°. Correction divisor: 60.
19
陰歷限，八度。 定法，八十。 月食限，十三度五分。 定法，八十七。 推天正經朔入交
Lunar-side eclipse limit: 8°. Correction divisor: 80. Lunar eclipse limit: 13° 05′. Correction divisor: 87. To compute entry into the nodes at the Heavenly Standard month's canonical new moon
20
置中積，加交應，減閏余，滿交終分，去之； 不盡，以日周約之為日，不滿為分秒，即天正經朔入交泛日及分秒。） 上考者，中積內加所求閏余，減交應，滿交終去之，不盡，以減交終，余如上。
Take the median accumulation, add the nodal epoch constant, and subtract the intercalary remainder. Divide by the nodal period in parts and discard the quotient; convert the remainder by the day circumference into days, with the leftover as parts and seconds. The result is the provisional day, parts, and seconds of nodal entry at the Heavenly Standard canonical new moon.) For retrospective calculation, add the intercalary remainder sought to the median accumulation and subtract the nodal epoch. Divide by the nodal period and discard the quotient; subtract the remainder from one full nodal period and proceed as above.
21
求次朔望入交
To find nodal entry for successive new and full moons
22
置天正經朔入交泛日及分秒，以交望累加之，滿交終日，去之，即為次朔望入交泛日及分秒。 求定朔望及每日夜半入交
Take the provisional nodal-entry day, parts, and seconds at the Heavenly Standard new moon and add the nodal synodic interval cumulatively for each subsequent syzygy. Whenever the sum fills a nodal period in days, discard one period. The remainder gives the provisional nodal entry for each following new or full moon. To find nodal entry at the fixed new and full moons and at midnight for each day
23
各置入交泛日及分秒，減去經朔望小余，即為定朔望夜半入交。 若定日有增損者，亦如之。 否則因經為定，大月加二日，小月加一日，餘皆加七千八百七十七分七十六秒，即次朔夜半入交； 累加一日，滿交終日，去之，即每日夜半入交泛日及分秒。
For each syzygy, take the provisional nodal-entry day, parts, and seconds and subtract the canonical syzygy's small remainder to obtain nodal entry at the fixed syzygy's midnight. If the fixed date has been adjusted up or down, apply the same adjustment. Otherwise derive the fixed value from the canonical one: add two days after a long month and one day after a short month, and always add 7,877 parts and 76 seconds to the remainder to obtain nodal entry at the next new moon's midnight; add one day cumulatively for each following day; whenever the sum fills a nodal period in days, discard one period. This yields the provisional nodal entry at midnight for each day.
24
求定朔望加時入交
To find nodal entry at the true hour of each fixed new or full moon
25
置經朔望入交泛日及分秒，以定朔望加減差加減之，即定朔望加時入交日及分秒。 求交常交定度
Take the canonical syzygy's provisional nodal entry in days, parts, and seconds and add or subtract the fixed syzygy's time-correction difference to obtain nodal entry at the true hour of the fixed syzygy. To find the mean and true nodal longitudes
26
置經朔望入交泛日及分秒，以月平行度乘之，為交常度； 以盈縮差盈加縮減之，為交定度。 求日月食甚定分
Multiply the canonical syzygy's provisional nodal entry in days, parts, and seconds by the moon's mean daily motion to obtain the mean nodal longitude; add the equation-of-center correction when in excess and subtract when in deficiency to obtain the true nodal longitude. To find the fixed time-parts for greatest eclipse of the sun or moon
27
日食：視定朔分在半日周已下，去減半周，為中前； 已上，減去半周，為中後； 與半周相減、相乘，退二位，如九十六而一，為時差； 中前以減，中後以加，皆加減定朔分，為食甚定分； 以中前後分各加時差，為距午定分。
For a solar eclipse: if the fixed new-moon division is below half the day circumference, subtract it from half the circumference to obtain the pre-noon interval; if it is above that, subtract half the circumference from it to obtain the post-noon interval; subtract the interval from half the circumference and multiply the two quantities, shift two decimal places, and divide by 96 to obtain the time correction; subtract the correction before noon and add it after noon to the fixed new-moon division to obtain the fixed division for greatest eclipse; add the time correction to the pre- or post-noon interval respectively to obtain the fixed distance-from-noon division.
28
月食：視定望分在日周四分之一已下，為卯前； 已上，覆減半周，為卯後； 在四分之三已下，減去半周，為酉前； 已上，覆減日周，為酉後。 以卯酉前後分自乘，退二位，如四百七十八而一，為時差； 子前以減，子後以加，皆加減定望分，為食甚定分； 各依發斂求之，即食甚辰刻。
For a lunar eclipse: if the fixed full-moon division is below one quarter of the day circumference, it is before mao, in early morning; if above that but still within the first half of the day, subtract from half the circumference to obtain the interval after mao; if below three quarters of the day circumference, subtract half the circumference to obtain the interval before you, in the evening; if above three quarters, subtract from the full day circumference to obtain the interval after you. Square the mao- or you-interval, shift two decimal places, and divide by 478 to obtain the time correction; subtract the correction before zi, at midnight, and add it after zi to the fixed full-moon division to obtain the fixed division for greatest eclipse; apply the expansion-and-contraction procedure to each to obtain the double-hour ke of greatest eclipse.
29
求日月食甚入盈縮歷及日行定度
To find, at greatest eclipse, entry into the solar anomaly sequence and the true solar longitude
30
置經朔望入盈縮歷日及分，以食甚日及定分加之，以經朔望日及分減之，即為食甚入盈縮歷； 依日躔術求盈縮差，盈加縮減之，為食甚入盈縮歷定度。
Take the canonical syzygy's entry into the anomaly sequence in days and parts, add the day and fixed parts to greatest eclipse, and subtract the canonical syzygy's day and parts to obtain the anomaly-sequence entry at greatest eclipse; by the solar-motion procedure find the equation-of-center correction, add it when in excess and subtract when in deficiency, to obtain the true solar longitude at greatest eclipse.
31
求南北差
To find the north-south correction
32
視日食甚入盈縮歷定度，在象限已下，為初限； 已上，用減半歲周，為末限； 以初末限度自相乘，如一千八百七十而一，為度，不滿，退除為分秒； 用減四度四十六分，余為南北泛差； 以距午定分乘之，以半晝分除之，所得，以減泛差，為定差。 泛差不及減者，反減之為定差，應加者減之，應減者加之。 在盈初縮末者，交前陰歷減，陽歷加，交後陰歷加，陽歷減； 在縮初盈末者，交前陰歷加，陽歷減，交後陰歷減，陽歷加。
For a solar eclipse, take the true solar longitude at greatest eclipse; if it is below the quadrant limit, it is the initial limit; if above the quadrant limit, subtract from half the tropical year circumference to obtain the final limit; square the initial or final limit, divide by 1,870 to obtain degrees, and convert any remainder into parts and seconds; subtract the result from 4° 46′ to obtain the provisional north-south correction; multiply the provisional correction by the fixed distance-from-noon division and divide by the half-day division; subtract the quotient from the provisional correction to obtain the fixed correction. If the provisional correction is smaller than the amount to be subtracted, reverse the subtraction to obtain the fixed correction, reversing the sign: add where subtraction was indicated and subtract where addition was indicated. When the sun is in excess at the initial limit and in deficiency at the final limit: before the node subtract on the lunar side and add on the solar side; after the node add on the lunar side and subtract on the solar side; When the sun is in deficiency at the initial limit and in excess at the final limit: before the node add on the lunar side and subtract on the solar side; after the node subtract on the lunar side and add on the solar side.
33
求東西差
To find the east-west correction
34
視日食甚入盈縮歷定度，與半歲周相減相乘，如一千八百七十而一，為度，不滿，退除為分秒，為東西泛差； 以距午定分乘之，以日周四分之一除之，為定差。 若在泛差已上者，倍泛差減之，余為定差，依其加減。 在盈中前者，交前陰歷減，陽歷加； 交後陰歷加，陽歷減； 中後者，交前陰歷加，陽歷減； 交後陰歷減，陽歷加。 在縮中前者，交前陰歷加，陽歷減； 交後陰歷減，陽歷加； 中後者，交前陰歷減，陽歷加； 交後陰歷加，陽歷減。
Take the true solar longitude at greatest solar eclipse, subtract it from half the tropical year circumference and multiply the two quantities, divide by 1,870 to obtain degrees, and convert any remainder into parts and seconds. The result is the provisional east-west correction; multiply by the fixed distance-from-noon division and divide by one quarter of the day circumference to obtain the fixed correction. If the provisional correction has already been exceeded, double the provisional correction and subtract it; the remainder is the fixed correction, to be added or subtracted accordingly. When in excess before the median point: before the node subtract on the lunar side and add on the solar side; after the node add on the lunar side and subtract on the solar side; after the median point: before the node add on the lunar side and subtract on the solar side; after the node subtract on the lunar side and add on the solar side. When in deficiency before the median point: before the node add on the lunar side and subtract on the solar side; after the node subtract on the lunar side and add on the solar side; after the median point: before the node subtract on the lunar side and add on the solar side; after the node add on the lunar side and subtract on the solar side.
35
求日食正交中交限度
To find the limit degrees for a solar eclipse at the primary and median nodes
36
置正交、中交度，以南北東西差加減之，為正交、中交限度及分秒。 求日食入陰陽歷去交前後度
Take the primary-node and median-node longitudes, add or subtract the north-south and east-west corrections, and obtain the limit degrees, parts, and seconds for the primary and median nodes. To find, for a solar eclipse, entry into the solar or lunar side and the degrees before or after the node
37
視交定度，在中交限已下，以減中交限，為陽歷交前度； 已上，減去中交限，為陰歷交後度； 在正交限已下，以減正交限，為陰歷交前度； 已上，減去正交限，為陽歷交後度。
Take the true nodal longitude; if it is below the median-node limit, subtract it from that limit to obtain the solar-side degrees before the node; if above the limit, subtract the median-node limit to obtain the lunar-side degrees after the node; if below the primary-node limit, subtract from that limit to obtain the lunar-side degrees before the node; if above the primary-node limit, subtract that limit to obtain the solar-side degrees after the node.
38
求月食入陰陽歷去交前後度
To find, for a lunar eclipse, entry into the solar or lunar side and the degrees before or after the node
39
視交定度，在交中度已下，為陽歷； 已上，減去交中，為陰歷。 視入陰陽歷，在後準十五度半已下，為交後度； 前準一百六十六度三十九分六十八秒已上，覆減交中，余為交前度及分。
Take the true nodal longitude; if it is below the median-node degree, it is on the solar side; if above the median-node degree, subtract the nodal median to obtain the lunar side. For entry into the solar or lunar side, if below the rear standard of 15°30′, it is after the node; if above the front standard of 166°39′68″, subtract from the nodal median to obtain the degrees and parts before the node.
40
求日食分秒
To find the magnitude of a solar eclipse in parts and seconds
41
視去交前後度，各減陰陽歷食限， 〈（不及減者不食。）〉 余如定法而一，各為日食之分秒。 求月食分秒
Take the degrees before or after the node and subtract the eclipse limit for the solar or lunar side, If the remainder is insufficient to subtract, there is no eclipse.〉 divide each remainder by the correction divisor to obtain the parts and seconds of the solar eclipse. To find the magnitude of a lunar eclipse in parts and seconds
42
視去交前後度，） 不用南北東西差者。 用減食限， 〈（不及減者不食。）〉 余如定法而一，為月食之分秒。 求日食定用及三限辰刻
Take the degrees before or after the node,) without applying the north-south or east-west corrections. Subtract the eclipse limit, If the remainder is insufficient to subtract, there is no eclipse.〉 divide the remainder by the correction divisor to obtain the parts and seconds of the lunar eclipse. To find the fixed duration of a solar eclipse and the three contact times in double-hour ke
43
置日食分秒，與二十分相減、相乘，平方開之，所得，以五千七百四十乘之，如入定限行度而一，為定用分； 以減食甚定分，為初虧； 加食甚定分，為復圓； 依發斂求之，為日食三限辰刻。
Take the parts and seconds of the solar eclipse, subtract them from twenty parts and multiply the two quantities, extract the square root, multiply by 5,740, and divide by the limit-entry motion in degrees to obtain the fixed-duration division; subtract from the fixed division at greatest eclipse to obtain first contact; add to the fixed division at greatest eclipse to obtain last contact; apply the expansion-and-contraction procedure to obtain the three contact times of the solar eclipse in double-hour ke.
44
求月食定用及三限五限辰刻
To find the fixed duration of a lunar eclipse and the three or five contact times in double-hour ke
45
置月食分秒，與三十分相減、相乘，平方開之； 所得，以五千七百四十乘之，如入定限行度而一，為定用分； 以減食甚定分，為初虧； 加食甚定分，為復圓； 依發斂求之，即月食三限辰刻。
Take the parts and seconds of the lunar eclipse, subtract them from thirty parts and multiply the two quantities, and extract the square root; multiply the result by 5,740 and divide by the limit-entry motion in degrees to obtain the fixed-duration division; subtract from the fixed division at greatest eclipse to obtain first contact; add to the fixed division at greatest eclipse to obtain last contact; apply the expansion-and-contraction procedure to obtain the three contact times of the lunar eclipse in double-hour ke.
46
月食既者，以既內分與一十分相減、相乘，平方開之，所得，以五千七百四十乘之，如入定限行度而一，為既內分； 用減定用分，為既外分； 以定用分減食甚定分，為初虧； 加既外，為食既； 又加既內，為食甚； 再加既內，為生光； 復加既外，為復圓； 依發斂求之，即月食五限辰刻。
For a total lunar eclipse, subtract the within-totality division from ten parts and multiply the two quantities, extract the square root, multiply by 5,740, and divide by the limit-entry motion in degrees to obtain the within-totality division; subtract that from the fixed-duration division to obtain the outside-totality division; subtract the fixed-duration division from the fixed division at greatest eclipse to obtain first contact; add the outside-totality division to obtain the onset of totality; add the within-totality division again to obtain greatest eclipse; add the within-totality division once more to obtain emergence from totality; add the outside-totality division again to obtain last contact; apply the expansion-and-contraction procedure to obtain the five contact times of the lunar eclipse in double-hour ke.
47
求月食入更點
To find which night watch and bell-stroke a lunar eclipse falls in
48
置食甚所入日晨分，倍之，五約，為更法； 又五約更法，為點法。 乃置初末諸分，昏分已上，減去昏分，晨分已下，加晨分，以更法除之，為更數； 不滿，以點法收之，為點數； 其更點數，命初更初點算外，各得所入更點。
Take the dawn division of the day on which greatest eclipse occurs, double it, and reduce by five to obtain the watch divisor; reduce the watch divisor by five again to obtain the bell-stroke divisor. Then take the initial and final divisions: if above the dusk division, subtract the dusk division; if below the dawn division, add the dawn division. Divide by the watch divisor to obtain the watch number; convert any remainder by the bell-stroke divisor to obtain the bell-stroke number; Assign the watch and bell-stroke numbers starting from the first watch and first bell-stroke outside the count to determine which watch and bell-stroke each contact falls in.
49
求日食所起
To find the direction in which a solar eclipse begins
50
食在陽歷，初起西南，甚於正南，復於東南； 食在陰歷，初起西北，甚於正北，復於東北； 食八分已上，初起正西，復於正東。 〈（此據午地而論之。）〉
If the eclipse is on the solar side, it first appears in the southwest, reaches maximum due south, and ends in the southeast; if on the lunar side, it first appears in the northwest, reaches maximum due north, and ends in the northeast; if the eclipse magnitude is eight-tenths or greater, it begins due west and ends due east. This is stated with reference to the noon meridian.〉
51
求月食所起
To find the direction in which a lunar eclipse begins
52
食在陽歷，初起東北，甚於正北，復於西北； 食在陰歷，初起東南，甚於正南，復於西南； 食八分已上，初起正東，復於正西。 〈（此亦據午地而論之。）〉
If the eclipse is on the solar side, it first appears in the northeast, reaches maximum due north, and ends in the northwest; if on the lunar side, it first appears in the southeast, reaches maximum due south, and ends in the southwest; if the eclipse magnitude is eight-tenths or greater, it begins due east and ends due west. This too is stated with reference to the noon meridian.〉
53
求日月出入帶食所見分數
To find the magnitude visible when the sun or moon rises or sets during an eclipse
54
視其日日出入分，在初虧已上、食甚已下者，為帶食。 各以食甚分與日出入分相減，余為帶食差； 以乘所食之分，滿定用分而一， 〈（如月食既者，以既內分減帶食差，余進一位，如既外分而一，所得，以減既分，即月帶食出入所見之分； 不及減者，為帶食既出入。）〉 以減所食分，即日月出入帶食所見之分。 〈（其食甚在晝，晨為漸進，昏為已退； 其食甚在夜，晨為已退，昏為漸進。）〉
If that day's sunrise or sunset division falls after first contact and before greatest eclipse, the eclipse is visible at rising or setting. For each case subtract the sunrise or sunset division from the greatest-eclipse division; the remainder is the rising-or-setting eclipse difference; multiply by the eclipse magnitude and divide by the fixed-duration division, For a total lunar eclipse, subtract the within-totality division from the rising-or-setting difference, advance the remainder one decimal place, and divide by the outside-totality division. Subtract the quotient from the totality division to obtain the magnitude seen when the moon rises or sets in eclipse; if the subtraction cannot be completed, the eclipse is total at rising or setting.〉 subtract that from the eclipse magnitude to obtain the magnitude visible at sunrise or sunset. If greatest eclipse occurs in daytime, at dawn the eclipse is still advancing and at dusk it has already retreated; if greatest eclipse occurs at night, at dawn it has already retreated and at dusk it is still advancing.〉
55
求日月食甚宿次
To find the lodge and degree at greatest solar or lunar eclipse
56
置日月食甚入盈縮歷定度，在盈，便為定積； 在縮，加半歲周，為定積。 望即更加半周天度。 以天正冬至加時黃道日度，加而命之，各得日月食甚宿次及分秒。
Take the true solar longitude at greatest eclipse; if in excess, use it directly as the fixed accumulation; if in deficiency, add half the tropical year circumference to obtain the fixed accumulation. For a full moon, add half the celestial circumference in degrees as well. Add the fixed accumulation to the ecliptic solar longitude at the hour of the Heavenly Standard winter solstice and assign the lodge to obtain the lodge, degree, and parts at greatest eclipse for the sun or moon.
57
△步五星第七歷度三百六十五度二十五分七十五秒。 歷中
Section 7: Computing the Five Planets. Circumference: 365° 25′ 75″. Half circumference:
58
一百八十二度六十二分八十七秒半。 歷策一十五度二十一分九十秒六十二微半。 木星
182° 62′ 87½″. Circumference stride: 15° 21′ 90″ 62½ microparts. Jupiter
59
周率，三百九十八萬八千八百分。 周日，三百九十八日八十八分。 曆率，四千三百三十一萬二千九百六十四分八十六秒半。 度率，一十一萬八千五百八十二分。
Orbital period in parts: 3,988,800 parts. Sidereal period: 398 days and 88 parts. Epicycle rate: 43,312,964 parts and 86½ seconds. Degree rate: 118,582 parts.
60
合應，一百一十七萬九千七百二十六分。 歷應，一千八百九十九萬九千四百八十一分。 盈縮立差，二百三十六加。 平差，二萬五千九百一十二減。
Conjunction epoch constant: 1,179,726 parts. Sequence epoch constant: 18,999,481 parts. Established excess-deficiency difference: add 236. Mean difference: subtract 25,912.
61
定差，一千八十九萬七千。 伏見，一十三度。 〈（表略）〉 火星
Fixed difference: 10,897,000. Arc of visibility at heliacal setting/rising: 13°. Table omitted.〉 Mars
62
周率，七百七十九萬九千二百九十分。 周日，七百七十九日九十二分九十秒。 曆率，六百八十六萬九千五百八十分四十三秒。 度率，一萬八千八百七分半。
Orbital period in parts: 7,799,290 parts. Sidereal period: 779 days, 92 parts, 90 seconds. Epicycle rate: 6,869,580 parts and 43 seconds. Degree rate: 18,807½ parts.
63
合應，五十六萬七千五百四十五分。 歷應，五百四十七萬二千九百三十八分。 盈初縮末立差，一千一百三十五減。 平差，八十三萬一千一百八十九減。
Conjunction epoch constant: 567,545 parts. Sequence epoch constant: 5,472,938 parts. Established difference for the excess-initial, deficiency-final arc: subtract 1,135. Mean difference: subtract 831,189.
64
定差，八千八百四十七萬八千四百。 縮初盈末立差，八百五十一加。 平差，三萬二百三十五負減。 定差，二千九百九十七萬六千三百。
Fixed difference: 88,478,400. Established difference for the deficiency-initial, excess-final arc: add 851. Mean difference: negative subtraction of 30,235. Fixed difference: 29,976,300.
65
伏見，一十九度。 〈（表略）〉 土星周率，三百七十八萬九百一十六分。
Arc of visibility at heliacal setting/rising: 19°. Table omitted.〉 Saturn orbital period in parts: 3,780,916 parts.
66
周日，三百七十八日九分一十六秒。 曆率，一億七百四十七萬八千八百四十五分六十六秒。 度率，二十九萬四千二百五十五分。 合應，一十七萬五千六百四十三分。
Sidereal period: 378 days, 9 parts, 16 seconds. Epicycle rate: 174,788,845 parts and 66 seconds. Degree rate: 294,255 parts. Conjunction epoch constant: 175,643 parts.
67
歷應，五千二百二十四萬五百六十一分。 盈立差，二百八十三加。 平差，四萬一千二十二減。 定差，一千五百一十四萬六千一百。
Sequence epoch constant: 52,240,561 parts. Established excess difference: add 283. Mean difference: subtract 41,022. Fixed difference: 15,146,100.
68
縮立差，三百三十一加。 平差，一萬五千一百二十六減。 定差，一千一百一萬七千五百。 伏見，一十八度。
Established deficiency difference: add 331. Mean difference: subtract 15,126. Fixed difference: 11,117,500. Arc of visibility at heliacal setting/rising: 18°.
69
〈（表略）〉 金星周率，五百八十三萬九千二十六分。 周日，五百八十三日九十分二十六秒。
Table omitted.〉 Venus orbital period in parts: 5,839,026 parts. Sidereal period: 583 days, 90 parts, 26 seconds.
70
曆率，三百六十五萬二千五百七十五分。 度率，一萬。 合應，五百七十一萬六千三百三十分。 歷應，一十一萬九千六百三十九分。
Epicycle rate: 3,652,575 parts. Degree rate: 10,000 parts. Conjunction epoch constant: 5,716,330 parts. Sequence epoch constant: 119,639 parts.
71
盈縮立差，一百四十一加。 平差，三減。 定差，三百五十一萬五千五百。 伏見，一十度半。
Established excess-deficiency difference: add 141. Mean difference: subtract 3. Fixed difference: 3,515,500. Arc of visibility at heliacal setting/rising: 10½°.
72
〈（表略）〉 水星周率，一百一十五萬八千七百六十分。 周日，一百一十五日八十七分六十秒。
Table omitted.〉 Mercury orbital period in parts: 1,158,760 parts. Sidereal period: 115 days, 87 parts, 60 seconds.
73
曆率，三百六十五萬二千五百七十五分。 度率，一萬。 合應，七十萬四百三十七分。 歷應，二百五萬五千一百六十一分。
Epicycle rate: 3,652,575 parts. Degree rate: 10,000 parts. Conjunction epoch constant: 700,437 parts. Sequence epoch constant: 2,055,161 parts.
74
盈縮立差，一百四十一加。 平差，二千一百六十五減。 定差，三百八十七萬七千。 晨伏夕見，一十六度半。
Established excess-deficiency difference: add 141. Mean difference: subtract 2,165. Fixed difference: 3,877,000. Arc of visibility at morning hiding and evening appearance: 16½°.
75
夕伏晨見，一十九度。 〈（表略）〉 推天正冬至後五星平合及諸段中積中星
Arc of visibility at evening hiding and morning appearance: 19°. Table omitted.〉 To compute, after the Heavenly Standard winter solstice, the median accumulation and median star for each planet's mean conjunction and for every phase.
76
置中積，加合應，以其星周率去之，不盡，為前合； 復減周率，余為後合； 以日周約之，得其星天正冬至後平合中積中星。 〈（命為日，日中積； 命為度，日中星。）〉 以段日累加中積，即諸段中積； 以平度累加中星，經退則減之，即為諸段中星。 〈（上考者，中積內減合應，滿周率去之，不盡，便為所求後合分。）〉
Take the median accumulation, add the conjunction epoch, divide by that planet's orbital period in parts, and the remainder is the prior conjunction interval; then subtract the orbital period once more; the remainder is the later conjunction interval; Reduce by the day circumference to obtain the median accumulation and median star for that planet's mean conjunction after the Heavenly Standard winter solstice. When named as days, this is the median accumulation; when named as degrees, it is the median star.〉 Accumulate the phase-day intervals onto the median accumulation to obtain each phase's median accumulation; Accumulate mean degrees onto the median star, subtracting where retrograde motion occurs, to obtain each phase's median star. For backward calculation, subtract the conjunction epoch from the median accumulation, divide by the orbital period, and the remainder is the later-conjunction fraction sought.〉
77
推五星平合及諸段入歷
To compute entry into the epicycle for each planet's mean conjunction and for every phase.
78
各置中積，加歷應及所求後合分，滿曆率，去之； 不盡，如度率而一為度，不滿，退除為分秒，即其星平合入歷度及分秒； 以諸段限度累加之，即諸段入歷。 〈（上考者，中積內減歷應，滿曆率去之，不盡，反減曆率，余加其年後合，余同上。）〉
For each planet, take the median accumulation, add the sequence epoch and the later-conjunction fraction sought, and divide by the epicycle rate; The remainder, divided by the degree rate, gives degrees and parts for that planet's mean-conjunction entry into the epicycle; Add the phase limits successively to obtain each phase's entry into the epicycle. For backward calculation, subtract the sequence epoch from the median accumulation, divide by the epicycle rate, subtract the remainder from the epicycle rate, add the result to that year's later conjunction, and proceed as above.〉
79
求盈縮差
To find the excess-deficiency correction.
80
置入歷度及分秒，在歷中已下，為盈； 已上，減去歷中，余為縮。 視盈縮歷，在九十一度三十一分四十三秒太已下，為初限； 已上，用減歷中，余為末限。
Take the entry into the epicycle in degrees and parts; if it is at or below half the epicycle, the motion is in excess; If above, subtract half the epicycle; the remainder is the deficiency arc. On the excess-deficiency epicycle, if the excess arc is at or below 91° 31′ 43″ excessive, this is the initial limit; If above, subtract from half the epicycle; the remainder is the final limit.
81
其火星，盈歷在六十度八十七分六十二秒半已下，為初限； 已上，用減歷中，余為末限。 縮歷在一百二十一度七十五分二十五秒已下，為初限； 已上，用減歷中，餘為末限。
For Mars, on the excess arc, if at or below 60° 87′ 62½″, this is the initial limit; If above, subtract from half the epicycle; the remainder is the final limit. On the deficiency arc, if at or below 121° 75′ 25″, this is the initial limit; If above, subtract from half the epicycle; the remainder is the final limit.
82
置各星立差，以初末限乘之，去加減平差，得，又以初末限乘之，去加減定差，再以初末限乘之，滿億為度，不滿退除為分秒，即所求盈縮差。
Take each planet's established difference, multiply by the initial or final limit, and use the result to add or subtract the mean difference; repeat with the fixed difference; multiply again by the initial or final limit, divide by 100 million to obtain degrees and parts, and the result is the excess-deficiency correction sought.
83
又術：置盈縮歷，以歷策除之，為策數，不盡為策余； 以其下損益率乘之，歷策除之，所得，益加損減其下盈縮積，亦為所求盈縮差。 求平合諸段定積
Alternate method: take the position on the excess-deficiency epicycle, divide by the sequence stride to get the stride count and remainder; Multiply the remainder by the tabulated rate for that stride, divide by the sequence stride, and add or subtract from the accumulated correction at the lower stride to obtain the same excess-deficiency correction. To find the true accumulation for mean conjunction and for each phase.
84
各置其星其段中積，以其盈縮差盈加縮減之，即其段定積日及分秒； 以天正冬至日分加之，滿紀法去之，不滿，命甲子算外，即得日辰。 求平合及諸段所在月日
For each planet and phase, take the median accumulation and apply the excess-deficiency correction—adding if in excess, subtracting if in deficiency—to obtain the true accumulation in days and parts; Add the Heavenly Standard winter solstice date and time, reduce modulo the era divisor, and assign the day and hour from the sexagenary cycle. To find the month and day on which mean conjunction and each phase occur.
85
各置其段定積，以天正閏日及分加之，滿朔策，除之為月數，不盡，為入月已來日數及分秒。 其月數，命天正十一月算外，即其段入月經朔日數及分秒； 以日辰相距，為所在定朔月日。
Take the true accumulation for each phase, add the intercalary offset from the Heavenly Standard new year, divide by the new-moon stride to get the month count and the elapsed days within the month. Count months forward from the Heavenly Standard eleventh month to obtain the days and parts within the month since the new moon for that phase; Combine with the assigned day and hour to obtain the calendar month and day of occurrence.
86
求平合及諸段加時定星
To find the true planetary longitude at the hour of mean conjunction and for each phase.
87
各置其段中星，以盈縮差盈加縮減之， 〈（金星倍之，水星三之。）〉 即諸段定星； 以天正冬至加時黃道日度加而命之，即其星其段加時所在宿度及分秒。
For each phase, take the median star and apply the excess-deficiency correction—adding when in excess and subtracting when in deficiency— (Double the correction for Venus; triple it for Mercury.) to obtain the true star for every phase; Add the Heavenly Standard winter solstice ecliptic longitude at the hour of occurrence and assign the lodge position; the result is that planet's ecliptic lodge-degree and parts at the hour of that phase.
88
求諸段初日晨前夜半定星
To find the true planetary longitude at midnight before dawn on the first day of each phase.
89
各以其段初行率，乘其段加時分，百約之，乃順減退加其日加時定星，即其段初日晨前夜半定星； 加命如前，即得所求。 求諸段日率度率
For each phase, multiply the initial motion rate by the fractional hour of occurrence, divide by one hundred, then subtract for prograde motion or add for retrograde motion from that day's true longitude at the hour of occurrence; the result is the true longitude at midnight before dawn on the first day of the phase; Add and assign the lodge position as before to obtain the value sought. To find the day rate and degree rate for each phase.
90
各以其段日辰距後段日辰為日率，以其段夜半宿次與後段夜半宿次相減，余為度率。 求諸段平行分各置其段度率，以其段日率除之，即其段平行度及分秒。
For each phase, take the elapsed days and hours to the next phase as the day rate, and the difference between midnight lodge positions of the two phases as the degree rate. To find the uniform daily motion for each phase, divide the degree rate by the day rate to obtain the mean daily motion in degrees and parts.
91
求諸段增減差及日差
To find the variation correction and daily increment for each phase.
92
以本段前後平行分相減，為其段泛差； 倍而退位，為增減差； 以加減其段平行分，為初末日行分。） 前多後少者，加為初，減為末； 前少後多者，減為初，加為末。 倍增減差，為總差； 以日率減一，除之，為日差。
Subtract the uniform motion rates at the boundaries of the phase; the result is the phase's general difference; Double the result and shift the decimal place; this is the variation correction; Add or subtract it from the phase's uniform motion rate to obtain the daily motion on the first and last days.) If the earlier rate exceeds the later, add the correction at the start and subtract at the end; If the earlier rate is less than the later, subtract at the start and add at the end. Double the variation correction to obtain the total difference; Divide by the day rate minus one to obtain the daily increment.
93
求前後伏遲退段增減差前伏者，置後段初日行分，加其日差之半，為末日行分。
To find the variation correction for stationary-hidden, slow, and retrograde phases: for a prior stationary-hidden phase, take the next phase's first-day motion rate, add half the daily increment, and the result is the last day's motion rate.
94
後伏者，置前段末日行分，加其日差之半，為初日行分； 以減伏段平行分，余為增減差。 前遲者，置前段末日行分，倍其日差，減之，為初日行分。
For a later stationary-hidden phase, take the prior phase's last-day motion rate, add half the daily increment, and the result is the first day's motion rate; Subtract this from the stationary-hidden phase's uniform motion rate; the remainder is the variation correction. For a prior slow-motion phase, take the prior phase's last-day motion rate, subtract twice the daily increment, and the result is the first day's motion rate.
95
後遲者，置後段初日行分，倍其日差，減之，為末日行分； 以遲段平行分減之，余為增減差。） 前後近留之遲段。 木火土三星，退行者，六因平行分，退一位，為增減差。
For a later slow-motion phase, take the next phase's first-day motion rate, subtract twice the daily increment, and the result is the last day's motion rate; Subtract this from the slow-motion phase's uniform motion rate; the remainder is the variation correction.) This applies to slow-motion segments adjacent to a station. For Jupiter, Mars, and Saturn in retrograde, multiply the uniform motion rate by six and shift the decimal place; the result is the variation correction.
96
金星，前後退伏者，三因平行分，半而退位，為增減差。 前退者，置後段初日行分，以其日差減之，為末日行分。
For Venus in retrograde stationary-hidden phases before and after, multiply the uniform motion rate by three, halve, and shift the decimal place to obtain the variation correction. For a prior retrograde phase, take the next phase's first-day motion rate, subtract the daily increment, and the result is the last day's motion rate.
97
後退者，置前段末日行分，以其日差減之，為初日行分； 乃以本段平行分減之，余為增減差。
For a later retrograde phase, take the prior phase's last-day motion rate, subtract the daily increment, and the result is the first day's motion rate; Then subtract this from the phase's uniform motion rate; the remainder is the variation correction.
98
水星，退行者，半平行分，為增減差； 皆以增減差加減平行分，為初末日行分。 前多後少者，加為初，減為末； 前少後多者，減為初，加為末。 又倍增減差，為總差； 以日率減一，除之，為日差。
For Mercury in retrograde, take half the uniform motion rate as the variation correction; In every case, add or subtract the variation correction from the uniform motion rate to obtain the daily motion on the first and last days. If the earlier rate exceeds the later, add the correction at the start and subtract at the end; If the earlier rate is less than the later, subtract at the start and add at the end. Again double the variation correction to obtain the total difference; Divide by the day rate minus one to obtain the daily increment.
99
求每日晨前夜半星行宿次
To find each day's planetary lodge position at midnight before dawn.
100
各置其段初日行分，以日差累損益之，後少則損之，後多則益之，為每日行度及分秒； 乃順加退減，滿宿次去之，即每日晨前夜半星行宿次。 求五星平合見伏入盈縮歷
For each phase, start from the first-day motion rate and adjust cumulatively by the daily increment—subtracting when the later rate is smaller, adding when it is larger—to obtain each day's motion in degrees and parts; Then add for prograde motion and subtract for retrograde, removing one lodge span whenever the sum is full; the result is the planetary lodge position at midnight before dawn for each day. To find entry into the excess-deficiency epicycle for each planet's mean conjunction, appearance, and stationary-hidden phases.
101
置其星其段定積日及分秒， 〈（若滿歲周日及分秒，去之，余在次年天正冬至後。）〉 如在半歲周已下，為入盈歷； 滿半歲周，去之，為入縮歷； 各在初限已下，為初限； 已上，反減半歲周，余為末限； 即得五星平合見伏入盈縮歷日及分秒。
Take the true accumulation in days and parts for that planet and phase, (If the value fills a full tropical year in days and parts, discard one year; the remainder falls after the next year's Heavenly Standard winter solstice.) If it is at or below half a tropical year, the planet is on the excess arc; When it fills half a tropical year, discard that amount; the planet is on the deficiency arc; If the position is at or below the initial limit, it falls within the initial limit; If above, subtract half a tropical year; the remainder gives the position within the final limit; This yields the days and parts of entry into the excess-deficiency epicycle for each planet's mean conjunction, appearance, and stationary-hidden phases.
102
求五星平合見伏行差
To find the relative motion for each planet's mean conjunction, appearance, and stationary-hidden phases.
103
各以其星其段初日星行分，與其段初日太陽行分相減，余為行差。 若金、水二星退行在退合者，以其段初日星行分，並其段初日太陽行分，為行差； 內水星夕伏晨見者，直以其段初日太陽行分為行差。
For each planet and phase, subtract the Sun's first-day motion rate from the planet's first-day motion rate; the remainder is the relative motion. If Venus or Mercury is retrograde at retrograde conjunction, add the planet's first-day motion rate to the Sun's first-day motion rate to obtain the relative motion; For Mercury in its evening stationary-hidden, morning-appearance configuration, take the Sun's first-day motion rate directly as the relative motion.
104
求五星定合定見定伏泛積
To find the provisional true accumulation for fixed conjunction, appearance, and stationary-hidden phases of the five planets.
105
木火土三星，以平合晨見夕伏定積日，便為定合伏見泛積日及分秒。
For Jupiter, Mars, and Saturn, the true accumulation day for mean conjunction with morning appearance and evening stationary hidden serves directly as the provisional true accumulation for conjunction, stationary hidden, and appearance.
106
金水二星，置其段盈縮差度及分秒， 〈（水星倍之。）〉 各以其段行差除之，為日，不滿，退除為分秒。 在平合夕見晨伏者，盈減縮加； 在退合夕伏晨見者，盈加縮減； 各以加減定積為定合伏見泛積日及分秒。
For Venus and Mercury, take the excess-deficiency correction in degrees and parts for that phase, (Double the value for Mercury.) Divide by the relative motion for that phase to convert to days, with the remainder as parts and seconds. For mean conjunction with evening appearance and morning stationary hidden, subtract when in excess and add when in deficiency; For retrograde conjunction with evening stationary hidden and morning appearance, add when in excess and subtract when in deficiency; Apply the correction to the true accumulation to obtain the provisional true accumulation for fixed conjunction, stationary hidden, and appearance in days and parts.
107
求五星定合定積定星
To find the true accumulation and true longitude for fixed conjunction of the five planets.
108
木火土三星，各以平合行差除其段初日太陽盈縮積，為距合差日； 不滿，退除為分秒，以太陽盈縮積減之，為距合差度。 各置其星定合泛積，以距合差日盈減縮加之，為其星定合定積日及分秒； 以距合差度盈減縮加之，為其星定合定星度及分秒。
For Jupiter, Mars, and Saturn, divide the first-day solar excess-deficiency accumulation by the mean-conjunction relative motion to obtain the days of separation-from-conjunction correction; Convert the remainder to parts and seconds, then subtract from the solar excess-deficiency accumulation to obtain the degrees of separation-from-conjunction correction. Take the provisional true accumulation for fixed conjunction and apply the days of separation-from-conjunction correction—subtracting when in excess and adding when in deficiency—to obtain the true accumulation in days and parts; Apply the degrees of separation-from-conjunction correction—subtracting when in excess and adding when in deficiency—to obtain the true longitude at fixed conjunction in degrees and parts.
109
金水二星，順合退合者，各以平合退合行差，除其日太陽盈縮積，為距合差日； 不滿，退除為分秒，順加退減太陽盈縮積，為距合差度。 順合者，盈加縮減其星定合泛積，為其星定合定積日及分秒； 退合者，以距合差日盈減縮加、距合差度盈加縮減其星退定合泛積，為其星退定合定積日及分秒； 命之，為退定合定星度及分秒。 以天正冬至日及分秒，加其星定合定積日及分秒，滿旬周，去之，命甲子算外，即得定合日辰及分秒。 以天正冬至加時黃道日度及分秒，加其星定合定星度及分秒，滿黃道宿次，去之，即得定合所躔黃道宿度及分秒。 〈（徑求五星合伏定日：木、火、土三星，以夜半黃道日度，減其星夜半黃道宿次，余在其日太陽行分已下，為其日伏合； 金、水二星，以其星夜半黃道宿次，減夜半黃道日度，余在其日金、水二星行分已下者，為其日伏合。 金、水二星伏退合者，視其日太陽夜半黃道宿次，未行到金、水二星宿次，又視次日太陽行過金、水二星宿次，金、水二星退行過太陽宿次，為其日定合伏退定日。）〉
For Venus and Mercury at direct or retrograde conjunction, divide that day's solar excess-deficiency accumulation by the appropriate relative motion to obtain the days of separation-from-conjunction correction; Convert the remainder to parts and seconds, then add for direct motion or subtract for retrograde motion from the solar excess-deficiency accumulation to obtain the degrees of separation-from-conjunction correction. For direct conjunction, add when in excess and subtract when in deficiency from the provisional true accumulation to obtain the true accumulation in days and parts; For retrograde conjunction, apply the days correction by subtracting when in excess and adding when in deficiency, and the degrees correction by adding when in excess and subtracting when in deficiency, to the provisional retrograde true accumulation to obtain the true accumulation in days and parts; Assign the lodge position to obtain the true longitude at retrograde fixed conjunction in degrees and parts. Add the Heavenly Standard winter solstice date and time to the true accumulation for fixed conjunction, reduce modulo the ten-day cycle, and assign the day and double-hour from the sexagenary cycle to obtain the calendar date and time of fixed conjunction. Add the Heavenly Standard winter solstice ecliptic longitude at the hour of occurrence to the true longitude at fixed conjunction, reducing modulo one lodge span, to obtain the ecliptic lodge-degree and parts where the fixed conjunction occurs. (To determine fixed conjunction and stationary-hidden days directly: for Jupiter, Mars, and Saturn, subtract the planet's midnight ecliptic lodge position from the midnight ecliptic solar longitude; if the remainder is at or below the Sun's daily motion, that day is the stationary-hidden conjunction; For Venus and Mercury, subtract the midnight ecliptic solar longitude from the planet's midnight ecliptic lodge position; if the remainder is at or below the planet's daily motion, that day is the stationary-hidden conjunction. (For Venus and Mercury in stationary-hidden retrograde conjunction, check whether at midnight the Sun's ecliptic lodge position has not yet reached the planet's lodge position; then on the next day, when the Sun passes the planet's lodge position and the planet in retrograde passes the Sun's lodge position, that day is the fixed retrograde conjunction and stationary-hidden date.)
110
求木火土三星定見伏定積日
To find the true accumulation for fixed appearance and stationary hidden of Jupiter, Mars, and Saturn.
111
各置其星定見定伏泛積日及分秒，晨加夕減九十一日三十一分六秒，如在半歲周已下，自相乘，已上，反減歲周，余亦自相乘，滿七十五，除之為分，滿百為度，不滿，退除為秒； 以其星見伏度乘之，一十五除之； 所得，以其段行差除之，為日，不滿，退除為分秒； 見加伏減泛積，為其星定見伏定積日及分秒； 加命如前，即得定見定伏日辰及分秒。
Take the provisional true accumulation for fixed appearance and stationary hidden; add ninety-one days 31 parts 6 seconds for morning appearance and subtract for evening stationary hidden; if at or below half a tropical year, square the value, otherwise subtract the tropical year and square the remainder; divide by seventy-five for parts and by one hundred for degrees, with the remainder as seconds; Multiply by the planet's appearance-and-stationary-hidden arc and divide by fifteen; Divide the result by the relative motion for that phase to convert to days, with the remainder as parts and seconds; Add for appearance and subtract for stationary hidden from the provisional accumulation to obtain the true accumulation for fixed appearance and stationary hidden in days and parts; Add and assign the day and hour as before to obtain the calendar date and time of fixed appearance and stationary hidden.
112
求金水二星定見伏定積日
To find the true accumulation for fixed appearance and stationary hidden of Venus and Mercury.
113
各以伏見日行差，除其段初日太陽盈縮積，為日，不滿，退除為分秒； 若夕見晨伏，盈加縮減； 如晨見夕伏，盈減縮加； 以加減其星定見定伏泛積日及分秒，為常積。 如在半歲周已下，為冬至後； 已上，去之，余為夏至後。 各在九十一日三十一分六秒已下，自相乘，已上，反減半歲周，亦自相乘。 冬至後晨，夏至後夕，一十八而一，為分； 冬至後夕，夏至後晨，七十五而一，為分； 又以其星見伏度乘之，一十五除之； 所得，滿行差，除之，為日，不滿，退除為分秒，加減常積，為定積。 在晨見夕伏者，冬至後加之，夏至後減之； 夕見晨伏者，冬至後減之，夏至後加之； 為其星定見定伏定積日及分秒； 加命如前，即得定見定伏日晨及分秒。
For each phase, divide the first-day solar excess-deficiency accumulation by the daily relative motion for stationary hidden and appearance to obtain days, with the remainder as parts and seconds; For evening appearance and morning stationary hidden, add when in excess and subtract when in deficiency; For morning appearance and evening stationary hidden, subtract when in excess and add when in deficiency; Apply the correction to the provisional true accumulation for fixed appearance and stationary hidden to obtain the regular accumulation. If at or below half a tropical year, the date falls after the winter solstice; If above, subtract half a tropical year; the remainder falls after the summer solstice. If the value is at or below ninety-one days 31 parts 6 seconds, square it; if above, subtract half a tropical year and square the remainder. For morning appearances after the winter solstice and evening appearances after the summer solstice, divide by eighteen to obtain parts; For evening appearances after the winter solstice and morning appearances after the summer solstice, divide by seventy-five to obtain parts; Multiply again by the planet's appearance-and-stationary-hidden arc and divide by fifteen; Divide the result by the relative motion to convert to days, with the remainder as parts and seconds, then add or subtract from the regular accumulation to obtain the true accumulation. For morning appearance and evening stationary hidden, add the correction after the winter solstice and subtract after the summer solstice; For evening appearance and morning stationary hidden, apply the correction by subtracting after the winter solstice and adding after the summer solstice; this gives the true accumulation in days and parts for fixed appearance and stationary hidden; Add and assign the day and hour as before to obtain the calendar date and time of fixed appearance and stationary hidden.