1
崇天曆
Chongtian Calendar
2
步交會
Procedure for Conjunction and Eclipse
3
交終分：二十八萬八千一百七十七、秒四千二百七十七。
Conjunction terminal parts: 288,177, 4,277 seconds.
4
交終日：二十七、餘二千二百四十七、秒四千二百七十七。
Conjunction terminal day: 27, remainder 2,247, 4,277 seconds.
5
交中日：一十三、餘六千四百一十八、秒七百三十八半。
Conjunction mid-day: 13, remainder 6,418, 738½ seconds.
6
朔差日：二、餘三千三百七十一、秒五千七百二十三。
New-moon discrepancy day: 2, remainder 3,371, 5,723 seconds.
7
後限日：一、餘一千六百八十五、秒七千八百六十一半。
Rear limit day: 1, remainder 1,685, 7,861½ seconds.
8
望策：十四、餘八千一百四、秒五十。
Full-moon stride: 14, remainder 8,104, 50 seconds.
9
前限日：十二、餘四千七百三十二、秒九千二百七十七。 $交率：一百四十一。
Front limit day: 12, remainder 4,732, 9,277 seconds. Crossing rate: 141.
10
交數：一千七百九十六。
Crossing number: 1,796.
11
交終度：三百六十三度七十六分。
Conjunction terminal degree: 363° 76'.
12
交象：九十度九十四。
Conjunction image: 90° 94'.
13
半交：一百八十一度八十八。
Half-conjunction: 181° 88'.
14
陽曆食限：四千二百。
Solar-path eclipse limit: 4,200.
15
陽曆定法：四百二十。
Solar-path fixed divisor: 420.
16
陰曆食限：七千。
Lunar-path eclipse limit: 7,000.
17
陰曆定法：七百。
Lunar-path fixed divisor: 700.
18
推天正十一月經朔加時入交：置天正十一月朔積分，以交終分秒去之，不盡，滿樞法為日，不滿為餘秒，即天正經朔加時入交泛日及餘秒。
To derive entry into conjunction at hour-of-addition for the eleventh-month beginning at civil new year: set the accumulated parts for that month-beginning, discard full conjunction terminal parts and seconds, divide the remainder by the pivot divisor for days with the remainder as parts and seconds, and obtain the generalized conjunction day and remainder in seconds at hour-of-addition for the civil new-year month-beginning.
19
求次朔及望入交：因天正經朔加時入交泛日及餘秒，求次朔，以朔差日及餘秒加之； 求望，以望策及餘秒加之：滿交終日及餘秒皆去之，即次朔及望加時所入。 若以經朔、望小餘減之，即各得朔、望夜半入交泛日及餘秒。
To find the next month-beginning and full moon entering conjunction: starting from the civil new-year month-beginning generalized conjunction day and remainder seconds at hour-of-addition, for the next month-beginning add the new-moon discrepancy in days and remainder seconds; To find the full moon, add the full-moon stride and remainder-seconds; cast out full crossing-termination days and remainder-seconds to obtain the hour-of-addition entry for the next month-beginning and full moon. Subtracting each mean syzygy's minor remainder yields the general crossing day and remainder-seconds at midnight for that month-beginning or full moon.
20
求定朔夜半入交：因經朔、望夜半入交，若定朔、望大餘有進退者，亦進退交日，不則因經為定，各得所求。
To find fixed month-beginning midnight crossing entry: start from mean syzygy midnight crossing; if the fixed syzygy's major remainder advances or retreats, adjust the crossing day accordingly, otherwise use the mean value as fixed.
21
求次定朔夜半入交：各因前定朔夜半入交，大月加日二，小月加日一，餘皆加八千三百四十二、秒五千七百二十三； 若求次日，累加一日：滿交終日及餘秒皆去之，即得次定朔及每日夜半入交泛日及餘秒。
For the next fixed month-beginning midnight crossing entry: from each prior fixed month-beginning midnight entry, add two days in a long month and one in a short month, and always add 8,342 remainder and 5,723 seconds. To advance day by day, add one day cumulatively, cast out full crossing-termination days and remainder-seconds, and obtain the next fixed month-beginning and each day's midnight general crossing day and remainder-seconds.
22
求朔望加時入交常日：置經朔、望入交泛日及餘秒，以其朔、望入氣朏朒定數，朏減朒加之，即朔、望入交常日及餘秒。
To find syzygy hour-of-addition regular crossing day: set mean syzygy general crossing day and remainder-seconds, apply entry-into-qi waxing-waning fixed numbers (waxing subtract, waning add), and obtain syzygy regular crossing day and remainder-seconds.
23
求朔望加時入交定日：置其朔、望入轉朏朒定數，以交率乘之，如交數而一，所得，以朏減朒加入交常日余，滿若不足，進退其日，即朔、望加時入交定日及餘秒。
To find syzygy hour-of-addition fixed crossing day: set entry-into-rotation waxing-waning fixed number, multiply by crossing rate and divide by crossing number, waxing subtract and waning add to the regular crossing remainder, adjust the day as needed, and obtain fixed crossing day and remainder-seconds at hour-of-addition.
24
求月行入陰陽曆：視其朔、望入交定日及餘秒，在中日及餘秒以下者為月在陽曆； 如中日及餘秒已上者，減去之，為月在陰曆。 凡入交定日，陽初陰末為交初，陰初陽末為交中。
To find the moon's solar-path or lunar-path: if the syzygy's fixed crossing day and remainder-seconds are at or below the middle day and remainder-seconds, the moon is in the solar-path; if at or above the middle day and remainder-seconds, subtract the middle day; the moon is in the lunar-path. For fixed crossing days: solar-path at the start and lunar-path at the end marks crossing beginning; lunar-path at the start and solar-path at the end marks crossing middle.
25
求朔望加時月入陰陽曆積度：置其月入陰陽曆日及余，其餘，先以一百乘之，樞法除為約分。
To find syzygy hour accumulated solar-path or lunar-path degrees: set the moon's entered yin-yang day and remainder, multiply the remainder by 100, and divide by the pivot divisor to obtain simplified parts.
26
以九百九乘之，六十八除為度，不盡，退除為分，即朔、望加時月入陰陽曆積度及分。 其月在陽曆，即為入陽曆積度； 月在陰曆，即為入陰曆積度。
Multiply by 999, divide by 68 for degrees, reduce any remainder to parts, and obtain syzygy hour accumulated solar-path or lunar-path degrees and parts. If the moon is in the solar-path, this is solar-path accumulated degrees; if the moon is in the lunar-path, this is lunar-path accumulated degrees.
27
求朔望加時月去黃道度：置入陰陽曆積度及分，如交象以下為在少象； 已上，覆減半交，餘為入老象。 置所入老少象度及分，以五因之，用減一千一十，餘，以老少象度及分乘之，八十四而一，列於上位； 又置所入老少象度及分，如半象以下為在初限； 已上，減去半象，餘為入末限。 置初、末限底及分於上，列半象度及分於下，以上減下，餘以乘上，四十而一，所得，初限以減，末限以加，上位滿百為度，不滿為分，即朔、望加時月去黃道度數及分。
To find syzygy hour lunar latitude: set entered yin-yang accumulated degrees and parts; at or below the crossing image counts as the young image; if above, cover and subtract half-crossing; the remainder is entry into the old image. Set entered young or old image degrees and parts, multiply by five, subtract from 1,010, multiply the remainder by the image degrees and parts, divide by 84, and place in the upper row; Again set entered young or old image degrees and parts; at or below half an image counts as the initial limit; if above, subtract half an image; the remainder is entry into the final limit. Place initial and final limit bases above and half-image degrees below, subtract and multiply (divide by 40), subtract for the initial limit and add for the final limit, convert hundreds to degrees, and obtain syzygy hour lunar latitude in degrees and parts.
28
求食定餘：置定朔小餘，如半法以下覆加半法，餘為午前分； 已上，減去半法，餘為午後分。 置午前、後分於上，列半法於下，以上減下，以下乘上，午前以三萬一千七百七十除，午後以一萬三千八百八十五除之，各為時差。 午前以減、午後以加定朔小餘，各為食定小餘。 以時差加午前、後分，為午前、後定分。 其月食，直以定望小餘便為食定小餘。
To find the eclipse-fixed remainder: set the fixed month-beginning minor remainder; if at or below the half method, add back the half method to obtain the before-noon fraction; if above, subtract the half method to obtain the after-noon fraction. Place before- and after-noon parts above and the half method below, subtract and multiply, divide before-noon by 31,770 and after-noon by 13,885, and obtain the time difference for each. Subtract the time difference before noon and add it after noon to the fixed month-beginning minor remainder to obtain the eclipse-fixed minor remainder. Add the time difference to the before- and after-noon parts to obtain the before- and after-noon fixed parts. For a lunar eclipse, the fixed full-moon minor remainder itself serves as the eclipse-fixed minor remainder.
29
求日月食甚辰刻：置食定小餘，以辰法除之為辰數，不滿，進一位，刻法除之為刻，不滿為刻分。 其辰數命子正，算外，即食甚辰、刻及分。
To find eclipse maximum watch and quarter: divide the eclipse-fixed minor remainder by the watch divisor for watches, advance the remainder one place and divide by the quarter divisor for quarters, and obtain quarter parts. Count watches from zi-zheng, outside the count, to obtain eclipse maximum watch, quarter, and parts.
30
求氣差：置其朔中積，滿二至限去之，餘在一象以下為在初； 已上，覆減二至限，餘為在末。 皆自相乘，進二位，滿二百三十六除之，用減三千五百三十三，為氣差。 以乘距午定分，半晝分而一，所得以減氣差，為定數。 春分後，交初以減，交中以加； 秋分後，交初以加，交中以減。
To find qi difference: set the new-moon middle accumulation, cast out the two-solstice limit, and if the remainder is at or below one image it is initial; if above, cover and subtract the two-solstice limit; the remainder is final. Square each case, shift two places, divide by 236, subtract from 3,533, and obtain qi difference. Multiply by the noon-distance fixed parts, divide by half day-part, subtract from qi difference, and obtain the fixed number. After the spring equinox: subtract at crossing beginning and add at crossing middle; after the autumn equinox: add at crossing beginning and subtract at crossing middle.
31
求刻差：置其朔中積，滿二至限去之，餘，列二至限於下，以上減下，餘以乘上，進二位，滿二百三十六除之，為刻差以乘距午定分，四因之，樞法而一，為定數。 冬至後食甚在午前，夏至後食甚在午後。 交初以加，交中以減。 冬至後食甚在午後，夏至後食甚在午前。 交初以加，交中以減。
To find quarter difference: set new-moon middle accumulation, cast out two-solstice limit, subtract and multiply against the limit (divide by 236), multiply by noon-distance fixed parts, multiply by four, divide by the pivot divisor, and obtain the fixed number. After the winter solstice, eclipse maximum falls before noon; after the summer solstice, it falls after noon. Add at crossing beginning and subtract at crossing middle. After the winter solstice, eclipse maximum falls after noon; after the summer solstice, it falls before noon. Add at crossing beginning and subtract at crossing middle.
32
求日入食限：置入交定日及餘秒，以氣、刻、時三差定數各加減之，如中日及餘秒以下為不食； 已上者，減去中日及餘秒，如後限以下、前限已上為入食限； 後限以下為交後分； 前限以上覆減中日，餘為交前分。
To find solar entry into the eclipse limit: set fixed crossing day and remainder-seconds, apply qi-, quarter-, and time-difference fixed numbers, and if at or below the middle day and remainder-seconds there is no eclipse; if above, subtract the middle day and remainder-seconds; if at or below the rear limit and at or above the forward limit, the syzygy has entered the eclipse limit; at or below the rear limit is the after-crossing part; at or above the forward limit, cover and subtract the middle day; the remainder is the before-crossing part.
33
求日食分：置入交前後分，如陽曆食限以下者為陽曆食定分； 已上者，覆減一萬一千二百，餘為陰曆食定分； 不足減者，不食。
To find solar eclipse magnitude: set before- and after-crossing parts; if at or below the solar-path eclipse limit, that is the solar-path eclipse fixed part; if above, cover and subtract 11,200; the remainder is the lunar-path eclipse fixed part; if subtraction cannot be completed, there is no eclipse.
34
各如限陽曆定法而一，為食之大分，不盡，退除為小分，半已上為半強，半以下為半弱。 命大分以十為限，得日食之分。
Divide each by the solar-path fixed divisor to obtain the eclipse major part, reduce any remainder to minor parts, and treat half or above as half-strong and below half as half-weak. Take the major part on a scale of ten to obtain solar eclipse magnitude.
35
求日食泛用分：置朔入陰陽曆食定分，一百約之，在陽曆者列入十四於下，在陰曆者列一百四十於下，各以上減下，餘以乘上，進二位，陽曆以一百八十五除，陰曆以五百一十四除，各為日食泛用分。
To find solar eclipse general use-parts: set month-beginning yin-yang eclipse fixed part and hundred-reduce; for solar-path place 14 below and for lunar-path 140 below, subtract and multiply (shift two places), divide by 185 (solar-path) or 514 (lunar-path), and obtain solar eclipse general use-parts.
36
求月入食限：視月入陰陽曆日及餘，如後限以下為交後分； 前限已上覆減中日，為交前分。
To find lunar entry into the eclipse limit: inspect the moon's yin-yang day and remainder; if at or below the rear limit, that is the after-crossing part; at or above the forward limit, cover and subtract the middle day to obtain the before-crossing part.
37
求月食分：置交前後分，如三千二百以下者，食既； 已上，用減一萬二百，不足減者不食； 餘以七百除之為大分，不盡，退除為小分，小分半已上為半強，半已下為半弱。 命大分以十為限，得月食之分。
To find lunar eclipse magnitude: set before- and after-crossing parts; if at or below 3,200, the eclipse is total; if above, subtract 10,200; if subtraction cannot be completed, there is no eclipse; divide the remainder by 700 for the major part, reduce any remainder to minor parts, and treat half or above as half-strong and half or below as half-weak. Take the major part on a scale of ten to obtain lunar eclipse magnitude.
38
求月食泛用分：置望入交前後分，退一等，自相乘，交初以九百三十五除，交中以一千一百五十六除之，得數用減刻率，交初以一千一百一十一為刻率，交中以九百為刻率。
To find lunar eclipse general use-parts: set full-moon before- and after-crossing parts, shift one place and square, divide by 935 (crossing beginning) or 1,156 (crossing middle), and subtract from the quarter rate of 1,111 (crossing beginning) or 900 (crossing middle).
39
各得所求。
Each procedure yields the value sought.
40
求日月食定用分：置日月食泛用分，以一千三百三十七乘之，以所食日轉定分除之，即得所求。
To find eclipse fixed use-parts: multiply general use-parts by 1,337 and divide by the rotation fixed parts of the eclipse day to obtain the fixed use-part.
41
求日月食虧初復滿小餘：各以定用分減食甚小餘，為虧初； 加食甚小餘，為復滿：即各得虧初復滿小餘。 若求時刻者，依食甚術入之。
To find eclipse first diminishment, restoration fullness, and minor remainders: subtract the fixed use-part from the eclipse-greatest minor remainder for first diminishment; add the fixed use-part to the eclipse-greatest minor remainder for restoration fullness; each case then yields first diminishment and restoration fullness minor remainders. To find clock time, apply the eclipse-greatest procedure.
42
求月食更籌定法：置其望晨分，四因之，退一等，為更法； 倍之，退一等，為籌法。
To find lunar eclipse watch and tally divisors: quadruple the full-moon dawn parts and shift one place to obtain the watch method; double that value and shift one place to obtain the tally method.
43
求月食入更籌：置虧初、食甚、復滿小餘，在晨分以下加晨分，昏分已上減去昏分，餘以更法除之為更數，不滿，以籌法除之為籌數。 其更數命初更，算外，即各得所入更、籌。
To find lunar eclipse watch and tally entry: set first diminishment, eclipse maximum, and restoration fullness minor remainders; below dawn parts add dawn parts, above dusk parts subtract dusk parts; divide by the watch method for watch count, then by the tally method for tally count. Name from the first watch outside the count to obtain each entered watch and tally.
44
求朔、望食甚宿次：置其經朔、望入氣小餘，以入氣、入轉朏朒定數朏減朒加之，乘其日升降分，樞法而一，加減其日盈縮分，至後、分前以加，分後、至前以減。
To find syzygy eclipse-greatest lodge position: set mean syzygy entry-into-qi minor remainder, apply entry-into-qi and entry-into-rotation waxing-waning fixed numbers (waxing subtract, waning add), multiply by that day's ascending-descending parts and divide by the pivot divisor, add or subtract excess-deficit parts (add after solstice and before parts, subtract after parts and before solstice).
45
一百約之為分，分滿百為度，以盈加縮減其定朔、望加時中積，以天正冬至加時黃道日度及分加而命之，即定朔、望加時日躔宿次。 其望加半周天，命如前，即朔、望食甚宿次。
Hundred-reduce to parts and convert hundreds to degrees, apply excess-add and deficit-subtract to fixed syzygy hour-of-addition middle accumulation, add to the civil new-year winter solstice ecliptic solar degree at hour-of-addition, name the lodges, and obtain fixed syzygy hour-of-addition solar lodge position. At full moon add half a circuit of heaven and name as before to obtain month-beginning and full-moon eclipse-greatest lodge positions.
46
求月食既內外刻分：置月食交前、後分，覆減三千二百，不及減者，為食下既。
To find lunar eclipse total inner and outer quarter parts: set lunar eclipse before- and after-crossing parts, cover and subtract 3,200; if subtraction cannot be completed, the eclipse falls short of totality.
47
一百約之，列六十四於下，以上減下，餘以乘上，進二位，交初以二百九十三除，交中以三百六十五除，所得，以定用分乘之，如泛用分而一，為月食既內刻分； 覆減定用分，即既外刻分。
Hundred-reduce, place 64 below, subtract and multiply (shift two places), divide by 293 (crossing beginning) or 365 (crossing middle), multiply by fixed use-parts and divide by general use-parts, and obtain lunar eclipse total inner quarter parts; cover and subtract the fixed use-part to obtain the total outer quarter part.
48
求日月帶食出入分數：各以食定小餘與日出、入分相減，餘為帶食差； 其帶食差滿定用分已上者，不帶食出入也。
To find the visible fraction when an eclipse is caught at sunrise or sunset: subtract rise or set parts from the eclipse-fixed minor remainder to obtain the horizon-partial difference; if the horizon-partial difference meets or exceeds the fixed use-part, no horizon-partial eclipse occurs at rise or set.
49
以帶食差乘所食分，滿定用分而一，若月食既者，以既內刻分減帶食差，餘所食分，以既外刻分而一，不及減者，為帶食既出入也。
Multiply the horizon-partial difference by the eclipsed fraction and divide by the fixed use-part; for a total lunar eclipse, subtract the total inner quarter part from the horizon-partial difference, divide the remainder and eclipsed fraction by the total outer quarter part; if subtraction cannot be completed, the eclipse is partially total at rise or set.
50
各以減所食分，即帶出、入所見之分。 其朔日食甚在晝者，晨為漸進之分，昏為已退之分； 若食甚在夜者，晨為已退之分，昏為漸進之分。 其月食者，見此可知也。
Subtract the result from the eclipsed fraction to obtain the visible fraction at rise or set with horizon partial eclipse. When a month-beginning solar eclipse maximum falls in daylight, at dawn the eclipse is still advancing and at dusk it has already retreated; if eclipse maximum falls at night, at dawn it has already retreated and at dusk it is still advancing. For a lunar eclipse, the same principle applies.
51
求日食所起：日在陰曆，初起西北，甚於正北，復于東北； 日在陽曆，初起西南，甚于正南，復于東南。 其食八分已上者，皆起正西，復於正東。 此據午地而論之，其餘方位，審黃道斜正、月行所向，可知方向。
To find solar eclipse direction of first contact: with the sun in the lunar-path, first contact is northwest, maximum due north, and recovery northeast; with the sun in the solar-path, first contact is southwest, maximum due south, and recovery southeast. For eclipses of eight-tenths or greater, first contact is always due west and recovery due east. This assumes an observer on the local meridian; for other latitudes, determine direction from the ecliptic's tilt and the moon's course.
52
求月食所起：月在陰曆，初起東南，甚于正南，復於西南； 月在陽曆，初起東北，甚於正北，復於西北。 其食八分已上，皆起正東，復於正西。 此亦據午地而論之，其餘方位，依日食所向，即知既虧、復滿。
To find lunar eclipse direction of first contact: with the moon in the lunar-path, first contact is southeast, maximum due south, and recovery southwest; with the moon in the solar-path, first contact is northeast, maximum due north, and recovery northwest. For eclipses of eight-tenths or greater, first contact is always due east and recovery due west. This also assumes an observer on the local meridian; for other latitudes, follow the solar eclipse directions to determine first diminishment and restoration to full.
53
步五星
Step Five Planets
54
五星會策：十五度二十一分、秒九十。
Five-planet conjunction stride: 15° 21', 90".
55
木星周率：四百二十二萬四千五十八、秒三十二。
Jupiter circuit rate: 4,224,058, 32 seconds.
56
周日：三百九十八、餘九千二百三十八、秒三十二。
Circuit day: 398, remainder 9,238, 32 seconds.
57
歲差：一百三、秒六。
Precession: 103, 6 seconds.
58
伏見度：一十三。
Heliacal visibility: 13.
59
木星盈縮曆火星周率：八百二十五萬九千三百六十六、秒五十九。
Jupiter excess-deficit calendar. Mars circuit rate: 8,259,366, 59 seconds.
60
周日：七百七十九、餘九千七百五十六、秒五十九。
Circuit day: 779, remainder 9,756, 59 seconds.
61
歲差：一百三、秒五十三。
Precession: 103, 53 seconds.
62
伏見度：二十。
Heliacal visibility: 20.
63
火星盈縮曆
Mars excess-deficit calendar
64
土星周率：四百萬三千八百七十二、秒三十九。
Saturn circuit rate: 4,003,872, 39 seconds.
65
周日：三百七十八、餘八百五十二、秒三十九。
Circuit day: 378, remainder 852, 39 seconds.
66
歲差：一百三、秒七十八。
Precession: 103, 78 seconds.
67
伏見度：一十六。
Heliacal visibility: 16.
68
土星盈縮曆金星周率：六百一十八萬三千五百九十九、秒一十六。
Saturn excess-deficit calendar. Venus circuit rate: 6,183,599, 16 seconds.
69
周日：五百八十三、餘九千六百二十九、秒一十六。
Circuit day: 583, remainder 9,629, 16 seconds.
70
歲差：一百三十、秒八十。
Precession: 130, 80 seconds.
71
夕見晨伏度：一十一。
Evening visibility morning concealment degree: 11.
72
晨見夕伏度：九。
Morning visibility evening concealment degree: 9.
73
金星盈縮曆
Venus excess-deficit calendar
74
水星周率：一百二十二萬七千一百七十、秒二十八。
Mercury circuit rate: 1,227,170, 28 seconds.
75
周日：一百一十五、餘九千三百二十、秒二十八。
Circuit day: 115, remainder 9,320, 28 seconds.
76
歲差：一百三、秒九十四。
Precession: 103, 94 seconds.
77
夕見晨伏度：一十四。
Evening visibility morning concealment degree: 14.
78
晨見夕伏度：二十一。
Morning visibility evening concealment degree: 21.
79
水星盈縮曆
Mercury excess-deficit calendar
80
推五星天正冬至後諸變中積中星：置氣積分，各以其星周率去之，不盡，覆減周率，余滿樞法除之為日，不滿，退除為分，即天正冬至後平合中積； 命之，積平合中星，以諸段變日、變度累加之，即諸變中積中星。 其經退行者，即其變度； 累減之，即其星其變中星。
To find each planet's central accumulation and central star for all variations after civil new-year winter solstice: set the qi accumulated parts, cast out each star's circuit rate, subtract the remainder again from the circuit rate, divide by the pivot divisor for days with the remainder as parts, and obtain the mean-conjunction central accumulation after civil new-year winter solstice; Name the result and accumulate the mean-conjunction central star; add each segment's variation days and degrees cumulatively to obtain the central accumulation and central star for all variations. For segments in retrograde motion, use the variation degree; subtract cumulatively to obtain that star's variation central star.
81
求五星諸變入曆：以其星歲差乘積年，滿周天分去之，不盡，以樞法除之為度，不滿，退除為分，以減其星平合中星，即平合入曆； 以其星其變限度依次加之，各得其星諸變入曆度分。
To find entry into the calendar cycle for all five-planet variations: multiply that star's precession by accumulated years, cast out full circuit-of-heaven parts, convert the remainder to degrees and parts by the pivot divisor, subtract from that star's mean-conjunction central star, and obtain mean-conjunction entry into the calendar cycle; Add that star's variation limit degrees in order to obtain entry-into-calendar degree and parts for every variation.
82
求五星諸變盈縮定差：各置其星其變入曆度分，半周天以下為在盈； 以上，減去半周天，餘為在縮。 置盈縮限度及分，以五星會策除之為會數，不盡，為入會度及分； 以其會下損益率乘之，會策除之為分，分滿百為度，以損益其下盈縮積度，即其星其變盈縮定差。 若用立成者，以其所入會度下差而用之。
To find the excess-and-deficit fixed difference for all five-planet variations: set each star and variation's entry-into-calendar degree and parts; at or below half circuit-of-heaven counts as excess; if above half circuit-of-heaven, subtract it; the remainder is in deficit. Set the excess-and-deficit limit degree and parts, divide by the five-planet conjunction stride to obtain the conjunction count, and take the remainder as entry-into-conjunction degree and parts; Multiply the tabulated loss-and-gain rate for that conjunction, divide by the conjunction stride for parts, convert parts filling 100 to degrees, apply to the underlying excess-and-deficit accumulated degree, and obtain that star and variation's excess-and-deficit fixed difference. If using precomputed tables, take the tabulated difference below the conjunction degree entered.
83
其木火土三星後退、後留者，置盈縮差，各列其星盈縮極度於下，皆以上減下，餘以乘上，八十七除之，所得，木、土三因，火直用之； 在盈益減損加、在縮益加損減其段盈縮差，為後退、後留定差，因為後遲初段定差。 各須類會前留定差，觀其盈縮初末，審察降殺，皆裒多益少而用之。
For Jupiter, Mars, and Saturn in posterior retreat and posterior station: set the excess-and-deficit difference, array each star's excess-and-deficit extreme degree below, subtract below from above, multiply the remainder by the upper value and divide by 87; triple the result for Jupiter and Saturn, and use it directly for Mars; When in excess, subtract where beneficial and add where deficient; when in deficit, add where beneficial and subtract where deficient — apply to the segment excess-and-deficit difference to obtain the posterior retreat and posterior station fixed difference, and thereby the posterior slow initial segment fixed difference. Each case must be matched to the prior station fixed difference, observe excess and deficit at beginning and end, examine the gradation, and apply by taking from the greater and giving to the lesser.
84
求五星諸變定積：各置其星其變中積，以其變盈縮定差盈加縮減之，即其星其變定積及分； 以天正冬至大餘及分加之，即其星其變定日及分； 以紀法去定日，不盡，命甲子，算外，即得日辰。
To find the fixed accumulation for all five-planet variations: set each star and variation's central accumulation, add or subtract the variation's excess-and-deficit fixed difference according to excess or deficit, and obtain that star and variation's fixed accumulation and parts; Add the civil new-year winter solstice greater remainder and parts to obtain that star and variation's fixed day and parts; Cast out full era rules from the fixed day, name from jiazi outside the count, and obtain the day and double-hour.
85
求五星諸變在何月日：各置諸變定日，以其年天正經朔大餘及分減之，若冬至大餘少，加經朔大餘者，加紀法乃減之。
To find which month and day each of the five planets' variations falls on: set each variation's fixed day and subtract the year's civil new-year mean month-beginning major remainder and parts; if the winter solstice major remainder is smaller and the mean month-beginning major remainder must be added first, add the era method before subtracting.
86
餘以朔策及分除之為月數，不滿，為入月日數及分。 其月數命以天正十一月，算外，即其星其變入其月經朔日數及分。 若置定積，以天正閏月及分加之，朔策除為月數，亦得所求。
Divide the remainder by the new-moon stride and parts to obtain the month count; the remainder gives the days and parts elapsed within the month. Count months forward from the eleventh month at civil new year, outside the count, to obtain that planet's variation days and parts since that month's mean month-beginning. Alternatively, set the fixed accumulation, add the civil new-year intercalary month and parts, divide by the new-moon stride for the month count, and obtain the same result.
87
求五星諸變入何氣日：置定積，以氣策及約分除之為氣數，不盡，為入氣已來日數及分。 其氣數命起天正冬至，算外，即五星諸變入其氣日及分。 其定積滿歲周日及分即去之，餘在來年天正冬至後。
To find which qi-period day each planetary variation enters: set the fixed accumulation, divide by the qi stride and approximate parts for the qi count, and take the remainder as days and parts elapsed within the qi-period. Count qi-periods forward from the civil new-year winter solstice, outside the count, to obtain the qi day and parts at which the five planets' variations enter. When the fixed accumulation reaches a full year-circuit day and parts, cast it out; the remainder falls after the following year's civil new-year winter solstice.
88
求五星諸變定星：各置其變中星，以其變盈縮定差盈加縮減之，其金、水二星，金以倍之，水以三之，乃可加減。
To find each planetary variation's fixed star: set the variation mid-star and adjust by its excess-deficit fixed difference, adding for excess and subtracting for deficit; for Venus and Mercury, double the difference for Venus and triple it for Mercury before applying the adjustment.
89
即五星諸變定星； 以天正冬至加時黃道日度加而命之，即其星其變加時定星宿次及分。 五星皆以前留為前退初日定星，後留為後遲初日定星。
This yields the fixed star for each of the five planets' variations; Add to the civil new-year winter solstice ecliptic solar degree at hour-of-addition and assign the lodge to obtain that planet's variation fixed star lodge sequence and parts at hour-of-addition. For all five planets, the prior stationary point serves as the first-day fixed star of prior retrograde motion, and the subsequent stationary point serves as the first-day fixed star of subsequent slow motion.
90
求五星諸變初日晨前夜半定星：以其星其變盈縮所入會度下盈縮積度與次度下盈縮積度相減，餘為其度損益分； 乘其變初行率，一百約之，所得，以加減其日初行率，在盈，益加損減； 在縮，益減損加。
To find each variation's first-day dawn-before-midnight fixed star: subtract the excess-deficit accumulated degrees at the next degree from those at the entered-conjunction degree of the planet's variation excess-deficit; the remainder is the degree's increment-decrement parts; Multiply by the variation's initial motion rate and hundred-reduce; use the result to adjust that day's initial motion rate — in the excess cycle, add for increment and subtract for decrement; In the deficit cycle, subtract for increment and add for decrement.
91
為初行積率； 又置一百分，亦依其數加減之，以除初行積率，為初日定行率； 以乘其率初日約分，一百約之，順減退加其日加時定星，為其變晨前夜半定星； 加冬至時日度命之，即所在宿次。
This gives the initial motion accumulated rate; Also set one hundred parts, adjust it by the same amount, divide the initial motion accumulated rate by the result, and obtain the first-day fixed motion rate; Multiply by that rate's first-day approximate parts and hundred-reduce; subtract in direct motion and add in retrograde motion to that day's hour-of-addition fixed star to obtain the variation's dawn-before-midnight fixed star; Add to the winter solstice solar degree at that time and assign the lodge to obtain its lodge position.
92
求諸變日度率：置後變定日，以其變定日減之，餘為其變日率； 又置後變夜半定星，以其變夜半定星及分減之，餘為其變度率及分。
To find each variation's day and degree rates: set the subsequent variation's fixed day and subtract this variation's fixed day; the remainder is that variation's day rate; Also set the subsequent variation's midnight fixed star and subtract this variation's midnight fixed star and parts; the remainder is that variation's degree rate and parts.
93
求諸變平行分：各置其變度率及分，以其變日率除之為平行分，不滿，退除為秒，即各得平行度及分秒。
To find each variation's parallel motion: set its degree rate and parts, divide by its day rate to obtain parallel motion; convert any remainder by retreat-division into seconds, yielding parallel degrees and parts-seconds for each.
94
求諸變總差：各以其段平行分與後段平行分相減，餘為泛差； 並前段泛差，四因之，九而一，為總差。 若前段無平行分相減為泛差者，各因後段初日行分與其段平行分相減，為半總差； 倍之，為總差。
To find each variation's total difference: subtract this segment's parallel motion from the subsequent segment's parallel motion; the remainder is the generalized difference; Add the prior segment's generalized difference, multiply by four, divide by nine, and obtain the total difference. If the prior segment lacks a parallel motion for subtraction, subtract this segment's parallel motion from the subsequent segment's first-day motion parts for each case; the result is the half total difference; Double the result to obtain the total difference.
95
若後段無平行分相減為泛差者，各因前段末日行分與其段平行分相減，為半總差。
If the subsequent segment lacks a parallel motion for subtraction, subtract this segment's parallel motion from its last-day motion parts for each case; the result is the half total difference.
96
其前後退行者，各置本段平行分，十四乘，十五除，為總差。 其金星夕退、夕伏、再合、晨退，各依順段術入之，即得所求。
For segments in prior and subsequent retrograde motion, set each segment's parallel motion, multiply by fourteen, divide by fifteen, and obtain the total difference. For Venus in evening retrograde, evening concealment, second conjunction, and morning retrograde, apply the direct-segment method in each case to obtain the desired values.
97
求諸段初末日行分：各半其段總差，加減其段平行分，後段行分多者，減之為初，加之為末； 後段行分少者，加之為初，減之為末。
To find each segment's first- and last-day motion parts: halve the segment's total difference and add or subtract it from the segment's parallel motion — if the subsequent segment's motion parts are greater, subtract for the first day and add for the last; If the subsequent segment's motion parts are lesser, add for the first day and subtract for the last.
98
即各得其星其段初、末日行度及分秒。 凡前後段平行分俱多或俱少，乃平注之； 及本段總差不滿大分者，亦平注之。 其退行段，各以半總差前變減之為初，加之為末； 後變加之為初，減之為末。
Each planet's segment then yields first- and last-day motion in degrees and parts-seconds. When consecutive segments' parallel motion is both greater or both lesser, distribute the correction evenly; Likewise, when a segment's total difference is less than one major part, distribute the correction evenly. For retrograde segments, subtract half the total difference before the variation for the first day and add it for the last; For the subsequent variation, add for the first day and subtract for the last.
99
求每日晨前夜半星行宿次：置其段總差，減其段日率，以除之，為日差； 以日差累損益初日行分，後段行分少，日損之； 後段行分多，日益之。
To find each day's dawn-before-midnight star lodge position: set the segment's total difference, subtract the segment's day rate, and divide to obtain the day difference; Apply the day difference cumulatively to the first-day motion parts — if the subsequent segment's motion parts are lesser, decrement daily; If the subsequent segment's motion parts are greater, increment daily.
100
為每日行度及分； 以每日行度及分累加其星其段初日晨前夜半宿次，命之，即每日星行宿次。 遇退行者，以每日行分累減之，即得所求。
This yields each day's motion in degrees and parts; Cumulatively add each day's motion degrees and parts to the planet's segment first-day dawn-before-midnight lodge position and assign lodges to obtain each day's star motion lodge sequence. In retrograde motion, cumulatively subtract each day's motion parts to obtain the desired lodge sequence.
101
徑求其日宿次：置所求日，減一，日差乘之，加減初日行分，後行分少，即減之； 後行分多，即加之。
To find a given day's lodge position directly: set the sought day, subtract one, multiply by the day difference, and adjust the first-day motion parts — subtract if the subsequent motion parts are lesser; Add if the subsequent motion parts are greater.
102
為所求日行分； 加日行分而半之； 以所求日乘之，為徑求積度； 加減其星初日宿次； 命之，即其日星行宿次。
This gives the sought day's motion parts; Add the daily motion parts and halve the sum; Multiply by the sought day to obtain the direct-seek accumulated degree; Add or subtract this from the star's first-day lodge position; Assign the lodge to obtain that day's star motion lodge sequence.
103
求五星定合日定星：以其星平合初日行分減一百分，餘以約其日太陽盈縮分為分，分滿百為日，不滿為分，命為距合差日； 以盈縮分減之，為距合差度； 以差日、差度縮加盈減平合定積、定星，為其星定合日定積、定星。 其金、水二星，以一百分減初日行分，餘以除其日太陽盈縮分，為距合差日； 以盈縮分加之，為距合差度； 以差日、差度盈加縮減之。
To find each planet's fixed conjunction day and fixed star: subtract one hundred parts from the mean-conjunction first-day motion parts; use the remainder to reduce that day's solar excess-deficit into parts, converting hundreds of parts to days with the remainder as parts, and name the result the distance-from-conjunction difference day; Subtract the excess-deficit parts to obtain the distance-from-conjunction difference degree; Apply the difference day and difference degree to the mean-conjunction fixed accumulation and fixed star, adding when in deficit and subtracting when in excess, to obtain that star's fixed-conjunction day, fixed accumulation, and fixed star. For Venus and Mercury, subtract one hundred parts from the first-day motion parts, divide that day's solar excess-deficit parts by the remainder, and obtain the distance-from-conjunction difference day; Add the excess-deficit parts to obtain the distance-from-conjunction difference degree; Apply the difference day and difference degree, adding when in excess and subtracting when in deficit.
104
金、水二星退合者，以初日行分加一百分，以除太陽盈縮分，為距合差日； 以距合差日減盈縮分，為距合差度； 以差日、差度盈減縮加再合定積定星為其星再合定日定積定星。
For Venus and Mercury in retrograde conjunction, add one hundred parts to the first-day motion parts, divide the solar excess-deficit parts by the sum, and obtain the distance-from-conjunction difference day; Subtract the distance-from-conjunction difference day from the excess-deficit parts to obtain the distance-from-conjunction difference degree; Apply the difference day and difference degree to the second-conjunction fixed accumulation and fixed star, subtracting when in excess and adding when in deficit, to obtain that star's second-conjunction fixed day, fixed accumulation, and fixed star.
105
其金、水二星定積，各依見伏術，先以盈縮差求其加減訖，然後以距合差日、差度加減之。
For Venus and Mercury fixed accumulations, follow the visibility-and-concealment procedure in each case: first apply the excess-deficit difference, then adjust by the distance-from-conjunction difference day and difference degree.
106
求木火土三星晨見夕伏定日：各置其星其段定積，乃加減一象度，晨見加之，夕伏減之。
To find Jupiter, Mars, and Saturn morning appearance and evening concealment fixed days: set each star's segment fixed accumulation, then add or subtract one-quadrant degree — add for morning appearance and subtract for evening concealment.
107
半周天已下自相乘，半周天已上，覆減周天度及分，餘亦自相乘，一百約為分，以其星伏見度乘之，十五除之，為差； 乃以其段初日行分覆減一百分，餘以除其差為日，不滿，退除為分，所得，以加減定積，晨見加之，夕伏減之。 各得晨見、夕伏定積； 加天正冬至大餘及分，命甲子，算外，即得日辰。
If at or below half circuit-of-heaven, square the value; if above, subtract from circuit-of-heaven degrees and parts and square the remainder; hundred-reduce to parts, multiply by that star's heliacal visibility degree, divide by fifteen, and obtain the difference; Then subtract one hundred parts from that segment's first-day motion parts, divide the difference by the remainder for days with the remainder as parts, and add or subtract the result from the fixed accumulation — add for morning appearance and subtract for evening concealment. Each then yields morning appearance and evening concealment fixed accumulation; Add the civil new-year winter solstice major remainder and parts, count from jiazi, outside the count, and obtain the day and time.
108
求金水二星夕見晨伏定日：各置其星其段定積，其定積先倍其段盈縮差，縮加盈減之，乃加減一象度，夕見減之，晨伏加之。
To find Venus and Mercury evening appearance and morning concealment fixed days: set each star's segment fixed accumulation, double the segment's excess-deficit difference and apply it to the fixed accumulation (add when in deficit, subtract when in excess), then add or subtract one-quadrant degree — subtract for evening appearance and add for morning concealment.
109
半周天已下自相乘，已上，覆減周天度，餘亦自相乘，一百約為分，以其星伏見度乘之，十五除為差； 乃置其段初日行分，減去一百分，餘以除其差為日，不滿，退除為分，所得，以加減定積，夕見加之，晨伏減之。
If at or below half circuit-of-heaven, square the value; if above, subtract from circuit-of-heaven degrees and square the remainder; hundred-reduce to parts, multiply by that star's heliacal visibility degree, divide by fifteen, and obtain the difference; Then subtract one hundred parts from that segment's first-day motion parts, divide the difference by the remainder for days with the remainder as parts, and add or subtract the result from the fixed accumulation — add for evening appearance and subtract for morning concealment.
110
各得夕見、晨伏定積。
Each then yields evening appearance and morning concealment fixed accumulation.
111
求金水二星晨見夕伏定日：置其星其段定積，其定積先以一百乘其段盈縮差，乃以一百分加其日行分，以除其差，所得，盈加縮減之，加減一象度，晨見加之，夕伏減之。
To find Venus and Mercury morning appearance and evening concealment fixed days: set the star's segment fixed accumulation, multiply the segment's excess-deficit difference by one hundred and apply it to the fixed accumulation, add one hundred parts to the daily motion parts and divide the difference, apply the result (add when in excess, subtract when in deficit), then add or subtract one-quadrant degree — add for morning appearance and subtract for evening concealment.
112
半周天已下自相乘，已上，覆減周天度，餘亦自相乘，一百約為分，以其星伏見度乘之，十五除，為差； 乃置其段初日行分，如一百，以除其差為日，不滿，退除為分，所得，以加減定積，晨見加之，夕伏減之。
If at or below half circuit-of-heaven, square the value; if above, subtract from circuit-of-heaven degrees and square the remainder; hundred-reduce to parts, multiply by that star's heliacal visibility degree, divide by fifteen, and obtain the difference; Then set that segment's first-day motion parts to one hundred, divide the difference for days with the remainder as parts, and add or subtract the result from the fixed accumulation — add for morning appearance and subtract for evening concealment.
113
各為其星晨見、夕伏定積。
Each yields that star's morning appearance and evening concealment fixed accumulation.
114
曆既成，以來年甲子歲用之，是年五月丁亥朔，日食不效，算食二分半，候之不食。
When the calendar was finished it was adopted for the following jiazi year; on the dinghai new moon of the fifth month that year the predicted solar eclipse failed — the calculation showed an eclipse of two and a half parts, but observation found none.
115
詔候驗。 至七年，命入內都知江德明集曆官用渾儀較測。 時周琮言：「古之造曆，必使千百年間星度交食，若應繩准，今曆成而不驗，則曆法為未密。」 又有楊皞、於淵者，與琮求較驗，而皞術於木為得，淵于金為得，琮於月、土為得，詔增入《崇天曆》，其改用率數如後：
An edict ordered observational verification. In the seventh year Jiang Deming, Director of the Inner Palace, was ordered to assemble calendar officials and compare measurements with the armillary sphere. At the time Zhou Cong said: "Ancient calendar-makers made star positions and eclipses over a thousand years align like a plumb line; now the calendar is finished yet fails verification, so the calendar method is not yet precise." Yang Hao and Yu Yuan also joined Cong in seeking comparative verification; Hao's method proved correct for Jupiter, Yuan's for Venus, and Cong's for the Moon and Saturn. An edict ordered these incorporated into the Chongtian Calendar. The revised rate constants are as follows:
116
周天分：三百八十六萬八千六十六、秒一十七。
Circuit-of-heaven parts: 3,868,066, 17 seconds.
117
周天：三百六十五度。 虛分二千七百一十六、秒十七，約分二十五、秒六十一。
Circuit-of-heaven: 365 degrees. Void fraction: 2,716, 17 seconds; approximate fraction: 25, 61 seconds.
118
歲差：一百二十六、秒一十七。
Precession: 126, 17 seconds.
119
求諸變總差：各以其段平行分與後段平行分相減，餘為泛差； 並前段泛差，四因之，退一等，為總差。 若前段無平行分相減為泛差，各因後段初日行分與其段平行分相減，為半總差； 倍之，為總差。
To find each variation's total difference: subtract this segment's parallel motion from the subsequent segment's parallel motion; the remainder is the generalized difference; Add the prior segment's generalized difference, multiply by four, retreat one place, and obtain the total difference. If the prior segment lacks a parallel motion for subtraction, subtract this segment's parallel motion from the subsequent segment's first-day motion parts for each case; the result is the half total difference; Double the result to obtain the total difference.
120
若後段無平行分相減為泛差者，各因前段末日行分與其段平行分相減，為半總差； 倍之，為總差。
If the subsequent segment lacks a parallel motion for subtraction, subtract this segment's parallel motion from its last-day motion parts for each case; the result is the half total difference; Double the result to obtain the total difference.
121
其前後退行者，各置本段平行分，十四乘，十五除，為總差。 其金星夕退、夕伏、再合、晨退，各依順段術入之，即得所求。
For segments in prior and subsequent retrograde motion, set each segment's parallel motion, multiply by fourteen, divide by fifteen, and obtain the total difference. For Venus in evening retrograde, evening concealment, second conjunction, and morning retrograde, apply the direct-segment method in each case to obtain the desired values.
122
求五星定合及見伏泛用積：其木、火、土三星，各以平合及前疾、後伏定積為泛用積，金、水二星平合及夕見、晨伏者，置其星其段盈縮差，金以倍之，水以三之，列於上位； 又置盈縮差，以其段初行率乘之，退二等，以減上位； 又置初行率，減去一百分，餘以除之為日，不滿，退除為分，乃盈減縮加中積，為其星其變泛用積。
To find each planet's fixed conjunction and visibility-and-concealment general-use accumulation: for Jupiter, Mars, and Saturn, use the mean-conjunction and prior-swift and posterior-concealment fixed accumulations as the general-use accumulation; for Venus and Mercury at mean-conjunction, evening appearance, and morning concealment, set the segment's excess-deficit difference (double for Venus, triple for Mercury) in the upper row; Set the excess-deficit difference again, multiply by that segment's first-motion rate, retreat two places, and subtract from the upper row; Set the first-motion rate, subtract one hundred parts, divide by the remainder for days with the remainder as parts, then subtract when in excess and add when in deficit to the middle accumulation, and obtain that star and variation's general-use accumulation.
123
金、水二星再合及夕伏、晨見者，其星其段盈縮差，金星直用，水以倍之，進二位，以其段初行率加一百分以除之，所得，並盈縮差，以盈加縮減中積，為其星其段泛用積。
For Venus and Mercury at second conjunction, evening concealment, and morning appearance, set the segment's excess-deficit difference (use directly for Venus, double for Mercury), advance two places, add one hundred parts to the first-motion rate and divide, combine with the excess-deficit difference, add or subtract to the middle accumulation according to excess or deficit, and obtain that star and segment's general-use accumulation.
124
求五星定合定積定星：其木、火、土三星平合者，以平合初日行分減一百分，餘以約其日太陽盈縮分為分，滿百為日，不滿為分，命為距合差日； 以盈縮分減之，為距合差度； 以差日、差度縮加盈減其星平合泛用積，為其星定合日定積定星。 金、水二星平合者，以一百分減初日行分，餘以除其日太陽盈縮分，為距合差日； 以盈縮分加之，為距合差度； 以差日、差度盈加縮減平合泛用積，為其星定合日定積定星也。
To find each planet's fixed-conjunction fixed accumulation and fixed star: for Jupiter, Mars, and Saturn at mean-conjunction, subtract one hundred parts from the mean-conjunction first-day motion parts, use the remainder to reduce that day's solar excess-deficit into parts, convert hundreds of parts to days with the remainder as parts, and name the result the distance-from-conjunction difference day; Subtract the excess-deficit parts to obtain the distance-from-conjunction difference degree; Apply the difference day and difference degree to the mean-conjunction general-use accumulation, adding when in deficit and subtracting when in excess, to obtain that star's fixed-conjunction day, fixed accumulation, and fixed star. For Venus and Mercury at mean-conjunction, subtract one hundred parts from the first-day motion parts, divide that day's solar excess-deficit parts by the remainder, and obtain the distance-from-conjunction difference day; Add the excess-deficit parts to obtain the distance-from-conjunction difference degree; Apply the difference day and difference degree to the mean-conjunction general-use accumulation, adding when in excess and subtracting when in deficit, to obtain that star's fixed-conjunction day, fixed accumulation, and fixed star.
125
金、水二星退合者，以初日行分一百分，以除太陽盈縮分，為距合差日； 以距合差日減盈縮分，為距合差度； 以差日盈減縮加再合泛用積，為其星再合定日定積差度； 盈加縮減再合泛用積，為其星再合日定星； 各加冬至大、小余及黃道加時日躔宿次命之，即得其日日辰及宿次。
For Venus and Mercury in retrograde conjunction, add one hundred parts to the first-day motion parts, divide the solar excess-deficit parts by the sum, and obtain the distance-from-conjunction difference day; Subtract the distance-from-conjunction difference day from the excess-deficit parts to obtain the distance-from-conjunction difference degree; Apply the difference day to the second-conjunction general-use accumulation, subtracting when in excess and adding when in deficit, to obtain that star's second-conjunction fixed day, fixed accumulation, and difference degree; Add when in excess and subtract when in deficit to the second-conjunction general-use accumulation to obtain that star's second-conjunction fixed day and fixed star; Add the winter solstice major and minor remainders for each case and assign lodges from the ecliptic hour-of-addition solar position to obtain that day's date, time, and lodge sequence.
126
求木火土星晨見夕伏定用積：各置其星其段泛用積，乃加減一象度，晨見加之，夕伏減之。
To find Jupiter, Mars, and Saturn morning appearance and evening concealment fixed-use accumulation: set each star's segment general-use accumulation, then add or subtract one-quadrant degree — add for morning appearance and subtract for evening concealment.
127
半周天已下自相乘，已上，覆減周天度，餘亦自相乘，各二因百約之，在一百六十七已上，以一百約其日太陽盈縮分減之，不滿一百六十七者即加之，以其星本伏見度乘之，十五除，為差； 乃置其段初日行分，覆減一百分，餘以除其差為日，不滿，退除為分所得，以加減泛用積，晨見加之，夕伏減之。
If at or below half circuit-of-heaven, square the value; if above, subtract from circuit-of-heaven degrees and square the remainder; double and hundred-reduce each; if at or above 167 subtract one hundred of that day's solar excess-deficit parts, otherwise add; multiply by that star's base heliacal visibility degree, divide by fifteen, and obtain the difference; Then subtract one hundred parts from that segment's first-day motion parts, divide the difference by the remainder for days with the remainder as parts, and add or subtract the result from the general-use accumulation — add for morning appearance and subtract for evening concealment.
128
各得其星晨見、夕伏定用積； 加天正冬至大餘，命甲子，算外，即得日辰。
Each then yields that star's morning appearance and evening concealment fixed-use accumulation; Add the civil new-year winter solstice major remainder, count from jiazi, outside the count, and obtain the day and time.
129
求金水二星夕見晨伏定用積：各置其星其段泛用積，乃加減一象度，夕見減之，晨伏加之。
To find Venus and Mercury evening appearance and morning concealment fixed-use accumulation: set each star's segment general-use accumulation, then add or subtract one-quadrant degree — subtract for evening appearance and add for morning concealment.
130
半周天已下自相乘，已上，覆減周天度，餘亦自相乘，二因百約之，滿一百六十七已上，以一百約太陽盈縮分減之，不滿一百六十七者即加之，以其星本伏見度乘之，十五除，為差； 乃置其段初日行分，減去一百分，餘。
If at or below half circuit-of-heaven, square the value; if above, subtract from circuit-of-heaven degrees and square the remainder; double and hundred-reduce; if at or above 167 subtract one hundred of the solar excess-deficit parts, otherwise add; multiply by that star's base heliacal visibility degree, divide by fifteen, and obtain the difference; Then subtract one hundred parts from that segment's first-day motion parts, divide the difference by the remainder for days with the remainder as parts, and add or subtract the result from the general-use accumulation — add for evening appearance and subtract for morning concealment.