1
觀天曆
The Guantian Calendar
2
步交會
Procedure for Lunar Nodes and Conjunctions
3
交終分：三十二萬七千三百六十一、秒九千九百四十四。
Convergence termination parts: 327,361; seconds 9,944.
4
交終日：二十七、餘二千五百五十一、秒九千九百四十四。
Convergence termination period: 27 days, remainder 2,551 parts, seconds 9,944.
5
交中日：一十三、餘七千二百九十、秒九千九百七十二。
Convergence middle period: 13 days, remainder 7,290 parts, seconds 9,972.
6
朔差日：二、餘三千八百三十一、秒五十六。
New-moon difference: 2 days, remainder 3,831 parts, seconds 56.
7
望策：一十四、餘九千二百六、秒五千。
Full-moon stride: 14 days, remainder 9,206 parts, seconds 5,000.
8
後限日：一、餘一千九百一十五、秒五千二十八。
Rear limit day: 1 day, remainder 1,915 parts, seconds 5,028.
9
前限日：一十二、餘五千三百七十五、秒四千九百四十四。
Front limit day: 12 days, remainder 5,375 parts, seconds 4,944.
10
以上秒母同一萬。
For the above constants, the seconds denominator is 10,000.
11
交率：一百八十三。
Convergence ratio: 183.
12
交數：二千三百三十一。
Convergence number: 2,331.
13
交終度：三百六十三分七十六。
Convergence termination arc: 363 parts 76.
14
交中度：一百八十一分八十八。
Convergence middle arc: 181 parts 88.
15
交象度：九十分九十四。
Convergence image arc: 90 parts 94.
16
半交象度：四十五分四十七。
Half convergence image arc: 45 parts 47.
17
陽曆食限：四千九百，定法四百九十。
Solar-path eclipse limit: 4,900; fixing divisor 490.
18
陰曆食限：七千九百，定法七百九十。
Lunar-path eclipse limit: 7,900; fixing divisor 790.
19
求天正十一月經朔加時入交泛日：置天正十一月經朔加時積分，以交終分及秒去之，不盡，滿統法為日，不滿為餘秒，即天正十一月經朔加時入交泛日及餘秒。
To find the eleventh civil month's mean new-moon hour of entry into general convergence: set the accumulated parts for that syzygy's hour, cast out full convergence termination parts and seconds, convert the remainder through the unification divisor into days and remainder-seconds, and obtain the general convergence day and remainder-seconds for the eleventh month's mean new moon.
20
求次朔及望加時入交泛日：置天正經朔加時入交泛日及餘秒，求次朔，以朔差加之。 求望，以望策加之，滿交終日及餘秒去之。 即次朔及望加時入交泛日及餘秒。 若以經朔小餘減之，餘為夜半入交泛日。
To find the next new and full moon hour general convergence entry: set the civil new year's mean new-moon hour general convergence day and remainder-seconds; for the next new moon, add the new-moon difference. For the full moon, add the full-moon stride and cast out full convergence termination days and remainder-seconds. That yields each next new or full moon hour general convergence day and remainder-seconds. Subtracting each mean syzygy's minor remainder yields the midnight general convergence day and remainder-seconds for that new or full moon.
21
求定朔望夜半入交泛日：置經朔、望夜半入交泛日，若定朔、望大餘有進退者，亦進退交日，否則因經為定，即定朔望夜半入交泛日及餘秒。
To find fixed syzygy midnight general convergence entry: start from mean midnight general convergence day and remainder-seconds; if the fixed syzygy's day-period advances or retreats, adjust the convergence day likewise, otherwise take the mean value as fixed, and obtain fixed syzygy midnight general convergence day and remainder-seconds.
22
求次朔夜半入交泛日：置定朔夜半入交泛日及餘秒，大月加二日，小月加一日，餘皆加九千四百七十八、秒五十六，求次日，累加一日，滿交終日及餘秒去之，即次定朔及每日夜半入交泛日及餘秒。
For the next fixed new-moon midnight entry: from each fixed new-moon midnight general convergence day and remainder-seconds, add two days in a long month and one in a short month, and always add 9,478 parts and 56 seconds to the remainder; to advance day by day, add one day cumulatively, cast out full convergence termination days and remainder-seconds, and obtain each next fixed new moon and each day's midnight general convergence day and remainder-seconds.
23
求朔望加時入交常日：置經朔、望入交泛日及餘秒，以其朔、望入盈縮限朏朒定數朏減朒加之，即朔、望加時入交常日及餘秒。
To find syzygy hour regular convergence day: set mean syzygy general convergence day and remainder-seconds, apply entry-into-expansion-contraction limit tiao–chuo fixed number (tiao subtract, chuo add), and obtain syzygy hour regular convergence day and remainder-seconds.
24
求朔望加時入交定日：置其朔、望入轉朏朒定數，以交率乘之，交數而一，所得，以朏減朒加入交常日及余秒，滿與不足，進退其日，即朔、望加時入交定日及餘秒。
To find syzygy hour fixed convergence day: set entry-into-rotation tiao–chuo fixed number, multiply by convergence ratio and divide by convergence number, tiao subtract and chuo add to the regular convergence remainder, adjust the day as needed, and obtain syzygy hour fixed convergence day and remainder-seconds.
25
求月行入陰陽曆：置其朔、望入交定日及餘秒，在交中已下為月行陽曆； 已上去之，餘為月行陰曆。
To find the moon's solar-path or lunar-path: if the syzygy's fixed convergence day and remainder-seconds are at or below the convergence middle, the moon is in the solar-path; if above, subtract the convergence middle; the moon is in the lunar-path.
26
求朔望加時月行入陰陽曆積度：置月行入陰陽曆日及餘秒，以統法通日，內餘，九而一為分，分滿百為度，即朔望加時月行入陰陽曆積度及分。
To find syzygy hour accumulated solar-path or lunar-path degrees: set the moon's entered yin-yang day and remainder-seconds, convert the day through the unification divisor with the remainder, divide by nine for parts, convert hundreds to degrees, and obtain syzygy hour accumulated solar-path or lunar-path degrees and parts.
27
求朔望加時月去黃道度：置入陰陽曆積度及分，如交象已下為入少象； 已上，覆減交中度，餘為入老象。 皆列於上，下列交中度，相減相乘，進位，如一百三十八而一，為泛差。 又視入老、少象度，如半交象已下為初； 已上去之，餘為末。 皆二因，退位，初減末加泛差，滿百為度，即朔、望加時月去黃道度及分。
To find syzygy hour lunar ecliptic latitude: set entered yin-yang accumulated degrees and parts; at or below the convergence image counts as the young image; if above, cover and subtract convergence middle; the remainder is entry into the old image. Array each above and convergence middle below, subtract and multiply, advance the place, divide by 138, and obtain the general difference. Again inspect entered young or old image degrees; at or below half the convergence image counts as initial; if above, subtract half the convergence image; the remainder is final. Multiply each by two and shift one place; subtract the general difference for initial and add for final, convert hundreds to degrees, and obtain syzygy hour lunar ecliptic latitude in degrees and parts.
28
求日月食甚定餘：置定朔小餘，如半統法已下，與半統法相減相乘，如三萬六千九十而一為時差，以減。 如半統法已上減去半統法，餘亦與半統法相減相乘，如一萬八千四十五而一為時差，午前以減，午後以加，皆加、減定朔小餘，為日食甚小餘。 與半法相減，餘為午前、後分。 其月食者，以定望小餘為月食甚小餘。
To find eclipse-greatest fixed remainder: set the fixed new-moon minor remainder; if at or below half the unification divisor, subtract and multiply against half the unification divisor, divide by 36,090 for the time difference, and subtract. If above half the unification divisor, subtract half the unification divisor, subtract and multiply the remainder against half the unification divisor, divide by 18,045 for the time difference, subtract before noon and add after noon to the fixed new-moon minor remainder, and obtain the solar eclipse-greatest minor remainder. Subtract the half method to obtain the before- and after-noon fractions. For a lunar eclipse, the fixed full-moon minor remainder itself serves as the lunar eclipse-greatest minor remainder.
29
求日月食甚辰刻：各置食甚小餘，倍之，以辰法除之為辰數，不滿，五因，滿刻法而一為刻，不滿為分。 其辰數命子正，算外，即食甚辰刻及分。 若加半辰，即命起子初。
To find eclipse maximum watch and quarter: double the eclipse-greatest minor remainder, divide by the watch divisor for watches, multiply the remainder by five and divide by the quarter divisor for quarters, and obtain parts. Count watches from zi-zheng, outside the count, to obtain eclipse maximum watch, quarter, and parts. If half a watch is added, count from the start of zi.
30
求氣差：置其朔盈、縮限度及分，自相乘，進二位，盈初、縮末一百九十七而一，盈末、縮初二百一十九而一，皆用減四千一十，為氣泛差。 以乘午前、後分，如半晝分而一，所得，以減泛差，為定差。 春分後，交初以減，交中以加； 秋分後，交初以加，交中以減。 如食在夜，反用之。
To find qi difference: set the new moon's expansion and contraction limit degrees and parts, square and shift two places, divide expansion initial and contraction final by 197 and expansion final and contraction initial by 219, subtract each from 4,010, and obtain the qi general difference. Multiply by the before- and after-noon parts, divide by half day-part, subtract from the general difference, and obtain the fixed difference. After the spring equinox: subtract at convergence beginning and add at convergence middle; after the autumn equinox: add at convergence beginning and subtract at convergence middle. If the eclipse occurs at night, reverse the application.
31
求刻差：置其朔盈、縮限度及分，與半周天相減相乘，進二位，二百九而一，為刻泛差。 以乘午前、後分，如三千七百半而一，為定差。 冬至後午前、夏至後午後，交初以加，交中以減。 冬至後午後、夏至後午前，交初以減，交中以加。
To find quarter difference: set the new moon's expansion and contraction limit degrees and parts, subtract and multiply against half the circuit of heaven, shift two places, divide by 209, and obtain the quarter general difference. Multiply by the before- and after-noon parts, divide by 3,700½, and obtain the fixed difference. After the winter solstice before noon and after the summer solstice after noon: add at convergence beginning and subtract at convergence middle. After the winter solstice after noon and after the summer solstice before noon: subtract at convergence beginning and add at convergence middle.
32
求日入食限交前後分：置朔入交定日及餘秒，以氣、刻、時三差各加減之，如交中日已下為不食； 已上去之，如後限已下為交後分； 前限已上覆減交中日，餘為交前分。
To find the sun's entry into the eclipse limit before- and after-convergence parts: set the new moon's fixed convergence day and remainder-seconds, apply the qi, quarter, and time differences, and if at or below the convergence middle day there is no eclipse; if above, subtract the convergence middle; if at or below the rear limit, it is the after-convergence part; if at or above the front limit, cover and subtract the convergence middle day; the remainder is the before-convergence part.
33
求日食分：置交前後分，如陽曆食限已下為陽曆食定分； 已上，用減一萬二千八百，餘為陰曆食定分。 如不足減者，日不食。
To find solar eclipse magnitude: set the before- and after-convergence parts; if at or below the solar-path eclipse limit, that is the solar-path eclipse fixed parts; if above, subtract from 12,800; the remainder is the lunar-path eclipse fixed parts. If the subtraction cannot be completed, there is no solar eclipse.
34
各如定法而一為大分，不盡，退除為小分。 小分半已上為半強，已下為半弱。 命大分以十為限，即得日食之分。
In each case divide by the fixing divisor for major parts; reduce any remainder to minor parts. Minor parts of half or more count as half-strong; below half count as half-weak. Take ten as the limit for major parts to obtain the solar eclipse magnitude.
35
求日食泛用分：置日食定分，退二位，列於上，在陽曆列九十八於下，在陰曆列一百五十八於下，各相減相乘，陽以二百五十而一，陰以六百五十而一，各為日食泛用分。
To find solar eclipse general application parts: set the solar eclipse fixed parts, shift two places, array above, place 98 below for the solar-path and 158 below for the lunar-path, subtract and multiply, divide by 250 for the solar-path and 650 for the lunar-path, and obtain the solar eclipse general application parts.
36
求月入食限交前後分：置望月行入陰陽曆日及餘秒，如後限已下為交後分。 前限已上覆減交中日，餘為交前分。
To find the moon's entry into the eclipse limit before- and after-convergence parts: set the full moon's entered yin-yang day and remainder-seconds; if at or below the rear limit, it is the after-convergence part. If at or above the front limit, cover and subtract the convergence middle day; the remainder is the before-convergence part.
37
求月食分：置交前後分，如三千七百已下，為食既； 已上，覆減一萬一千七百，不足減者為不食。
To find lunar eclipse magnitude: set the before- and after-convergence parts; if at or below 3,700, the eclipse is total; if above, cover and subtract 11,700; if the subtraction cannot be completed, there is no lunar eclipse.
38
餘以八百而一為大分，不盡，退除為小分。 小分半已上為半強，已下為半弱。 命大分以十為限，即得月食之分。
Divide the remainder by 800 for major parts; reduce any remainder to minor parts. Minor parts of half or more count as half-strong; below half count as half-weak. Take ten as the limit for major parts to obtain the lunar eclipse magnitude.
39
求月食泛用分：置望交前、後分，自相乘，退二位，交初以一千一百三十八而一，用減一千二百三，交中以一千二百六十四而一，用減一千八十三，各為月食泛用分。
To find lunar eclipse general application parts: set the full moon's before- and after-convergence parts, square and shift two places, divide by 1,138 and subtract from 1,203 at convergence beginning, divide by 1,264 and subtract from 1,083 at convergence middle, and obtain the lunar eclipse general application parts.
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，以定朔、望入轉算外轉定分而一，所得，為日月食定用分。
To find solar and lunar eclipse fixed application parts: set the eclipse general application parts, multiply by 1,337, divide by the fixed syzygy entry-into-rotation count-outside rotation fixed parts, and obtain the eclipse fixed application parts.
41
求日月食虧初復滿小餘：置日月食甚小餘，以定用分減之，為虧初； 加之，為復滿：即各得所求小餘。 若求辰刻，依食甚術入之。
To find eclipse first-contact and full-restoration minor remainders: set the eclipse-greatest minor remainder; subtract the fixed application parts for first contact; add the fixed application parts for full restoration, and obtain each sought minor remainder. To find watch and quarter, apply the eclipse-greatest procedure.
42
求月食更籌法：置望辰分，四因，退位，為更法； 五除之，為籌法。
To find the lunar eclipse geng and chou divisors: set the full-moon watch parts, multiply by four and shift one place to obtain the geng divisor; divide by five to obtain the chou divisor.
43
求月食入更籌：置虧初、食甚、復滿小餘，在晨分已下加晨分，昏分已上減去昏分，皆以更法除之為更數，不盡，以籌法除之為籌數。 其更、籌數命初更，算外，即各得所入更、籌。
To find lunar eclipse entry into geng and chou: set the first-contact, eclipse-greatest, and full-restoration minor remainders; add dawn parts if at or below dawn parts and subtract dusk parts if at or above dusk parts, divide by the geng divisor for geng count and by the chou divisor for the chou count. Count geng and chou from the first geng, outside the count, to obtain each entered geng and chou.
44
求日月食甚宿次：置朔、望之日晨前夜半黃道日度及分，以統法約日月食甚小餘，加之，內月食更加半周天，各依宿次去之，即日月食甚所在宿次。
To find the eclipse-greatest lodge position: set the syzygy day's dawn-before-midnight ecliptic solar degrees and parts, reduce the eclipse-greatest minor remainder through the unification divisor and add, add half the circuit of heaven for lunar eclipse, cast out by lodge position, and obtain the lodge where the eclipse is greatest.
45
求月食既內外刻分：置月食交前、後分，覆減三千七百，如不足減者，為食不既。
To find lunar eclipse total internal and external quarter parts: set the lunar eclipse before- and after-convergence parts, cover and subtract 3,700; if the subtraction cannot be completed, the eclipse is not total.
46
退二位，列於上，下列七十四，相減相乘，進位，如三十七而一，所得以定用分乘之，如泛用分而一，為既內分； 以減定用分，餘為既外分。
Shift two places and array above, place 74 below, subtract and multiply, advance the place, divide by 37, multiply by the fixed application parts and divide by the general application parts, and obtain the total internal parts; subtract from the fixed application parts; the remainder is the total external parts.
47
求日月帶食出入所見之分：各以食甚小餘與日出、入分相減，餘為帶食差。 其帶食差在定用分已上，為不帶食出入。
To find the magnitude visible when the sun or moon rises or sets with an eclipse: subtract the sunrise or sunset parts from the eclipse-greatest minor remainder in each case, and obtain the eclipse-accompanying difference. If the eclipse-accompanying difference is at or above the fixed application parts, the body rises or sets without an accompanying eclipse.
48
以乘所食之分，滿定用分而一，若月食既者，以既內分減帶食差，餘乘所食之分，如既外分而一，所得，以減既分，如不足減者，為帶食既出入。
Multiply by the eclipsed magnitude and divide by the fixed application parts; if the lunar eclipse is total, subtract the total internal parts from the eclipse-accompanying difference, multiply the remainder by the eclipsed magnitude and divide by the total external parts, subtract from the total magnitude, and if the subtraction cannot be completed, the body rises or sets with a total accompanying eclipse.
49
以減所食之分，餘為帶食出、入所見之分。
Subtract from the eclipsed magnitude; the remainder is the magnitude visible when rising or setting with an accompanying eclipse.
50
求日食所起：日在陽曆，初起西南，甚于正南，復滿東南； 日在陰曆，初起西北，甚於正北，復滿東北。 其食八分已上者，皆起正西，復滿正東。 此據午地而論之，當審黃道斜正可知。
To find where the solar eclipse begins: when the sun is in the solar-path, it first appears in the southwest, reaches maximum due south, and restores fullness in the southeast; when the sun is in the lunar-path, it first appears in the northwest, reaches maximum due north, and restores fullness in the northeast. If the eclipse is eight parts or more, it begins due west and restores fullness due east. This is discussed with reference to the meridian region; one must consider whether the ecliptic is oblique or direct.
51
求月食所起：月在陽曆，初起東北，甚於正北，復滿西北； 月在陰曆，初起東南，甚于正南，復滿西南。 其食八分已上者，皆起正東，復滿正西。 此據午地而論之，當審黃道斜正可知。
To find where the lunar eclipse begins: when the moon is in the solar-path, it first appears in the northeast, reaches maximum due north, and restores fullness in the northwest; when the moon is in the lunar-path, it first appears in the southeast, reaches maximum due south, and restores fullness in the southwest. If the eclipse is eight parts or more, it begins due east and restores fullness due west. This is discussed with reference to the meridian region; one must consider whether the ecliptic is oblique or direct.
52
步五星
Procedure for the Five Planets
53
五星曆策：一十五度、約分二十一、秒九十。
Five-planet calendar stride: 15 degrees, simplified parts 21, 90 seconds.
54
木星周率：四百七十九萬八千五百二十六、秒九十二。？ 周日：三百九十八、余一萬五百八十六、秒九十二。
Jupiter circuit rate: 4,798,526; seconds 92. Circuit day: 398, remainder 10,586, seconds 92.
55
歲差：一百一十六、秒七十二。
Annual precession: 116, seconds 72.
56
伏見度：一十三半。
Heliacal setting visibility: 13½ degrees.
57
木星盈縮曆
Jupiter excess-deficit calendar
58
火星周率：九百三十八萬二千五百六十、秒七十六。
Mars circuit rate: 9,382,560; seconds 76.
59
周日：七百七十九、余一萬一千一百九十、秒七十六。
Circuit day: 779, remainder 11,190, seconds 76.
60
歲差：一百一十六、秒一十三。
Annual precession: 116, seconds 13.
61
伏見度：一十八。
Heliacal setting visibility: 18 degrees.
62
火星盈縮曆
Mars excess-deficit calendar
63
土星周率：四百五十四萬八千四百三十一、秒八十五。
Saturn circuit rate: 4,548,431; seconds 85.
64
周日：三百七十八、餘一千九十一、秒八十五。
Circuit day: 378, remainder 1,091, seconds 85.
65
歲差：一百一十六、秒三十。
Annual precession: 116, seconds 30.
66
伏見度：一十六半。
Heliacal setting visibility: 16½ degrees.
67
土星盈縮曆
Saturn excess-deficit calendar
68
金星周率：七百二萬四千三百二十一、秒三十四。
Venus circuit rate: 7,024,321; seconds 34.
69
周日：五百八十三、余一萬八百三十一、秒三十四。
Circuit day: 583, remainder 10,831, seconds 34.
70
歲差：一百一十六、秒六十九。
Annual precession: 116, seconds 69.
71
伏見度：一十一半。 金星盈縮曆
Heliacal setting visibility: 11½ degrees. Venus excess-deficit calendar
72
水星周率：一百三十九萬四千二、秒七。
Mercury circuit rate: 1,394,002; seconds 7.
73
周日：一百一十五、余一萬五百五十二、秒七。
Circuit day: 115, remainder 10,052, seconds 7.
74
歲差：一百一十六、秒四十。
Annual precession: 116, seconds 40.
75
夕見晨伏度：一十五。
Evening appearance and morning setting visibility: 15 degrees.
76
晨見夕伏度：二十一。
Morning appearance and evening setting visibility: 21 degrees.
77
水星盈縮曆
Mercury excess-deficit calendar
78
求五星天正冬至後平合中積中星：置天正冬至氣積分，各以其星周率去之，不盡，用減周率，余滿統法約之為度，不滿，退除為分秒，命之為平合中積。 因而重列之為平合中星，各以前段變日加平合中積，又以前段變度加平合中星，其經退行者即減之，各得五星諸變中積中星。
To find each planet's mean-conjunction central accumulation and central star after civil new-year winter solstice: set the winter solstice qi accumulated parts, cast out each star's circuit rate, subtract the remainder from the circuit rate, reduce through the unification divisor for degrees and the remainder to parts and seconds, and name the mean-conjunction central accumulation. Then array it again as the mean-conjunction central star; add each prior segment's change in days to the mean-conjunction central accumulation and its change in degrees to the mean-conjunction central star, subtracting where retrograde motion applies, and obtain each planet's central accumulation and central star for every change.
79
求五星入曆：各以其星歲差乘所求積年，滿周天分去之，不盡，以統法約之為度，不滿，退除為分秒，以減平合中星，為平合入曆度及分秒。 求諸變者，各以前段限度累加之，為五星諸變入曆度及分秒。
To find each planet's entry into the calendar: multiply that star's annual precession by the sought accumulated years, cast out the circuit-of-heaven parts, reduce the remainder through the unification divisor for degrees and to parts and seconds, subtract from the mean-conjunction central star, and obtain mean-conjunction entry-into-calendar degree, parts, and seconds. For each change, cumulatively add prior segment limit degrees to obtain each planet's entry-into-calendar degree, parts, and seconds for every change.
80
求五星諸變盈縮定差：各置其星其變入曆度及分秒，如半周天已下為盈，已上去之為縮。 以五星曆策度除之為策數，不盡，為入策度及分秒。 以其策下損益率乘之，如曆策而一為分，分滿百為度，以損益其下盈縮積度，即五星諸段盈縮定差。
To find each planet's excess-and-deficit fixed difference for every change: set each star and change's entry-into-calendar degree, parts, and seconds; at or below half the circuit of heaven counts as excess, and above counts as deficit. Divide by the five-planet calendar stride degree to obtain the stride count; the remainder is entry-into-stride degree, parts, and seconds. Multiply the tabulated loss-and-gain rate for that stride, divide by the calendar stride for parts, convert parts filling 100 to degrees, apply to the underlying excess-and-deficit accumulation, and obtain each planet's segment excess-and-deficit fixed difference.
81
求五星平合及諸變定積：各置其星其變中積，以其段盈縮定差盈加縮減之，即其段定積日及分。 以天正冬至大餘及約分加之，滿統法去之，不盡，命甲子，算外，即定日辰及分。
To find each planet's mean conjunction and every change fixed accumulation: set each star and change's central accumulation, add or subtract that segment's excess-and-deficit fixed difference according to excess or deficit, and obtain the segment's fixed-accumulation days and parts. Add the civil new-year winter solstice major remainder and approximate parts, cast out the unification divisor, name from jiazi outside the count, and obtain the fixed day, double-hour, and parts.
82
求五星諸變入所在月日：各置其星其變定積，以天正閏日及約分加之，滿朔策及約分除之為月數，不盡，為入月已來日數。 命月數起天正十一月，算外，即其星其段入其月經朔日數及分。 乃以其朔日、辰相距，即所在月、日。
To find the month and day for each planet's every change: set each star and change's fixed accumulation, add civil new-year intercalary days and approximate parts, divide by the new-moon policy and approximate parts for month count, and take the remainder as days elapsed within the month. Name the month count from civil new-year month eleven outside the count to obtain the segment's days and parts since mean new moon. The interval of new-moon day and double-hour gives the month and day where it falls.
83
求五星平合及諸變加時定星：各置其星其變中星，以盈縮定差盈加縮減之，內金倍之，水三之，然後加減，即五星諸段定星。 以天正冬至加時黃道日度加時命之，即其星其段加時所在宿度及分秒。 五星皆因留為後段初日定星，餘依術算。
To find each planet's mean conjunction and every change hour-of-addition fixed star: set each star and change's central star, apply the excess-and-deficit fixed difference (double for Venus, triple for Mercury), then add or subtract according to excess or deficit to obtain each planet's segment fixed star. Add to the winter solstice ecliptic solar degree at hour-of-addition and name the lodges to obtain that star and segment's hour-of-addition lodge degree, parts, and seconds. For all five planets, station marks the next segment's first-day fixed star; compute the remainder by the procedure.
84
求五星諸變初日晨前夜半定星：各以其段初行率乘其段加時分，百約之，以順減退加其日加時定星，即為其星其段初日晨前夜半定星。 加命如前，即得所求。
To find each planet's first-day before-dawn midnight fixed star for every change: multiply the segment's initial motion rate by its hour-of-addition parts and hundred-reduce, then subtract for direct motion or add for retrograde from that day's hour-of-addition fixed star to obtain the segment's first-day before-dawn midnight fixed star. Add and name as before to obtain the result.
85
求諸變日率度率：各以其段日辰距至後段日辰為其段日率； 以其段夜半定星與後段夜半定星相減，余為其段度率。
To find each change's day rate and degree rate: take each segment's day-and-double-hour interval to the next segment as its day rate; subtract that segment's midnight fixed star from the next segment's midnight fixed star to obtain its degree rate.
86
求諸變平行分：各置其段度率，以其段日率除之，為其段平行度及分秒。
To find each change's parallel parts: set each segment's degree rate, divide by its day rate, and obtain the segment's parallel degree, parts, and seconds.
87
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，退一位，為總差。 若前段無平行分相減為泛差者，因後段初日行分與其段平行分相減，為半總差，倍之，為總差。 若後段無平行分相減為泛差者，因前段末日行分與其段平行分相減，為半總差，倍之，為總差。 其在再行者，以本段平行分十四乘之，十五而一，為總差。 內金星依順段術求之。
To find each change's total difference: subtract each segment's parallel parts from the next segment's parallel parts; the remainder is the general difference. Add the previous segment's general difference, multiply by four, and shift one decimal place to obtain the total difference. When the prior segment lacks parallel parts for subtraction, subtract that segment's parallel parts from the next segment's first-day motion parts; double the half-total difference to obtain the total difference. When the following segment lacks parallel parts for subtraction, subtract that segment's parallel parts from the previous segment's last-day motion parts; double the half-total difference to obtain the total difference. For renewed retrograde segments, multiply the segment's parallel parts by fourteen and divide by fifteen to obtain the total difference. For Venus, apply the direct-segment procedure.
88
求初末日行分：各半其段總差，加減其段平行分，後行分少，加之為初，減之為末； 後行分多，減之為初，加之為末。 退行者，前段減之為初，加之為末； 後段加之為初，減之為末。
To find first- and last-day motion parts: halve the segment's total difference and add or subtract its parallel parts; when the next segment's motion parts are fewer, add for the first day and subtract for the last; when the next segment's motion parts are greater, subtract for the first day and add for the last. For retrograde motion, subtract for the first day and add for the last in the prior segment; add for the first day and subtract for the last in the following segment.
89
為其星其段初、末日行分。
That yields the star and segment's first- and last-day motion parts.
90
求每日晨前夜半星行宿次：置其段總差，減日率一以除之，為日差； 累損益初日行分，後行分少，損之； 後行分多，益之。
To find each day's before-dawn midnight star lodge position: set the segment's total difference, subtract one from the day rate and divide by it to obtain the day difference; cumulatively adjust the first-day motion parts; when the next segment's motion parts are fewer, decrease them; when the next segment's motion parts are greater, increase them.
91
為每日行度及分秒； 乃順加退減其星其段初日晨前夜半定星，命之，即每日夜半星行所在宿次。
That yields each day's motion in degrees, parts, and seconds; then add for direct motion or subtract for retrograde from the segment's first-day before-dawn midnight fixed star, name the lodges, and obtain each day's midnight star lodge position.
92
徑求其日宿次：置所求日，減一，半之，以日差乘而加減初日行分，後行分少，減之； 後行分多，加之算。
To find a given day's lodge position directly: set the sought day, subtract one and halve it, multiply by the day difference and add or subtract to the first-day motion parts; when the next segment's motion parts are fewer, subtract; when the next segment's motion parts are greater, add and compute.
93
以所求日乘之，為積度； 以順加退減其星其段初日夜半宿次，即所求日夜半宿次。
Multiply by the sought day to obtain the accumulated degree; add for direct motion or subtract for retrograde from the segment's first-day midnight lodge position to obtain the sought day's midnight lodge position.
94
求五星合見伏行差：木、火、土三星，以其段初日星行分減太陽行分，為行差。 金、水二星順行者，以其段初日太陽行分減星行分，為行差。 金、水二星退行者，以其段初日星行分並太陽行分，為行差。 內水星夕伏、晨見，直以太陽行分為行差。
To find each planet's conjunction, appearance, and setting motion difference: for Jupiter, Mars, and Saturn, subtract the sun's motion parts from the segment's first-day star motion parts to obtain the motion difference. For Venus and Mercury in direct motion, subtract the star's motion parts from the segment's first-day solar motion parts to obtain the motion difference. For Venus and Mercury in retrograde motion, add the segment's first-day star motion parts to the solar motion parts to obtain the motion difference. For Mercury at evening setting and morning appearance, the solar motion parts themselves serve as the motion difference.
95
求五星定合見伏泛用積：木、火、土三星，各以平合晨疾、夕伏定積，便為定合見、伏泛用積。 金、水二星各置其段盈縮定差，內水星倍之，以其段行差除之為日，不滿，退除為分，在平合夕見、晨伏者，盈減縮加定積，為定合見、伏泛用積； 在退合夕伏、晨見者，盈加縮減定積，為定合見、伏泛用積。
To find each planet's fixed conjunction, appearance, and setting general application accumulation: for Jupiter, Mars, and Saturn, the mean-conjunction morning fast and evening setting fixed accumulation directly serves as the fixed conjunction, appearance, and setting general application accumulation. For Venus and Mercury, set each segment's excess-and-deficit fixed difference (double for Mercury), divide by the segment's motion difference for days and reduce the remainder to parts; at mean-conjunction evening appearance and morning setting, subtract for excess and add for deficit to the fixed accumulation to obtain the fixed conjunction, appearance, and setting general application accumulation; at retrograde-conjunction evening setting and morning appearance, add for excess and subtract for deficit from the fixed accumulation to obtain the fixed conjunction, appearance, and setting general application accumulation.
96
求五星定合積定星：木、火、土三星，以平合行差除其日盈縮分，為距合差日。 以盈縮分減之，為距合差度。 以差日、差度盈減縮加其星定合泛用積，為其星定合定積、定星。 金、水二星順合者，以平合行差除其日盈縮分，為距合差日。 以盈縮分加之，為距合差度； 以差日、差度盈加縮積其星定合泛用積，為其星定合定積、定星。 金、水二星退合者，以平合行差除其日盈縮分，為距合差日； 以減盈縮減之分，為距合差度； 以差日盈減縮加，以差度盈加縮減再定合泛用積，為其星再定合定積、定星。 各以天正冬至大餘及約分加定積，滿統法去之，命甲子，算外，即得定合日辰。 以天正冬至加時黃道日度加定星，依宿次去之，即得定合所在宿次。
To find each planet's fixed conjunction accumulation and fixed star: for Jupiter, Mars, and Saturn, divide that day's excess-and-deficit parts by the mean-conjunction motion difference to obtain the distance-from-conjunction difference in days. Subtract the excess-and-deficit parts to obtain the distance-from-conjunction difference in degrees. Apply the difference in days and degrees, subtracting for excess and adding for deficit to the star's fixed-conjunction general application accumulation, to obtain the star's fixed-conjunction fixed accumulation and fixed star. For Venus and Mercury in direct conjunction, divide that day's excess-and-deficit parts by the mean-conjunction motion difference to obtain the distance-from-conjunction difference in days. Add the excess-and-deficit parts to obtain the distance-from-conjunction difference in degrees; apply the difference in days and degrees, adding for excess and accumulating for deficit to the star's fixed-conjunction general application accumulation, to obtain the star's fixed-conjunction fixed accumulation and fixed star. For Venus and Mercury in retrograde conjunction, divide that day's excess-and-deficit parts by the mean-conjunction motion difference to obtain the distance-from-conjunction difference in days; subtract the excess-and-deficit subtracted parts to obtain the distance-from-conjunction difference in degrees; apply the difference in days, subtracting for excess and adding for deficit, and the difference in degrees, adding for excess and subtracting for deficit, to the renewed fixed-conjunction general application accumulation to obtain the star's renewed fixed-conjunction fixed accumulation and fixed star. Add the fixed accumulation to the civil new-year winter solstice major remainder and approximate parts, cast out the unification divisor, name from jiazi outside the count, and obtain the fixed-conjunction day and double-hour. Add the fixed star to the winter solstice ecliptic solar degree at hour-of-addition, cast out by lodge position, and obtain the lodge where the fixed conjunction falls.
97
求五星定見伏定積：木、火、土三星以泛用積晨加、夕減一象，如半周天已下自相乘，已上，覆減一周天，餘亦自相乘，七十五而一，所得，以其星伏見度乘之，十五而一為差，如其段行差除之為日，不滿，退除為分，見加伏減泛用積，為其星定見、伏定積。 金、水二星以行差除其日盈縮分為日，在夕見、晨伏，盈加縮減泛用積，為常用積； 夕伏、晨見，盈減縮加泛用積，為常用積； 如常用積在半周天已下為冬至後； 已上去之，餘為夏至後。 各在一象已下自相乘，已上，覆減一周天，餘亦自相乘，冬至後晨、夏至後夕，以十八而一； 冬至後夕、夏至後晨，以七十五而一，所得，以其星伏見度乘之，十五而一為差，如其段行差除之為日，不滿，退除為分，冬至後晨見、夕伏，夏至後夕見、晨伏，以加常用積，為其星定見、伏定積； 冬至後夕見、晨伏，夏至後晨見、夕伏，以減常用積，為其星定見、伏定積。 加命如前，即得定見、伏日辰。
To find each planet's fixed appearance and setting fixed accumulation: for Jupiter, Mars, and Saturn, add one image in the morning and subtract one in the evening from the general application accumulation; if at or below half the circuit of heaven, square it, otherwise cover and subtract one circuit of heaven and square the remainder, divide by 75, multiply by the star's heliacal setting visibility and divide by 15 for the difference, divide by the segment's motion difference for days and reduce the remainder to parts, add for appearance and subtract for setting from the general application accumulation, and obtain the star's fixed appearance and setting fixed accumulation. For Venus and Mercury, divide that day's excess-and-deficit parts by the motion difference for days; at evening appearance and morning setting, add for excess and subtract for deficit from the general application accumulation to obtain the regular application accumulation; at evening setting and morning appearance, subtract for excess and add for deficit to the general application accumulation to obtain the regular application accumulation; if the regular application accumulation is at or below half the circuit of heaven, it is after the winter solstice; if above, subtract half the circuit of heaven; the remainder is after the summer solstice. In each case, if at or below one image, square it; if above, cover and subtract one circuit of heaven and square the remainder; after the winter solstice in the morning and after the summer solstice in the evening, divide by 18; after the winter solstice in the evening and after the summer solstice in the morning, divide by 75, multiply by the star's heliacal setting visibility and divide by 15 for the difference, divide by the segment's motion difference for days and reduce the remainder to parts; after the winter solstice at morning appearance and evening setting and after the summer solstice at evening appearance and morning setting, add to the regular application accumulation to obtain the star's fixed appearance and setting fixed accumulation; After the winter solstice at evening appearance and morning setting and after the summer solstice at morning appearance and evening setting, subtract from the regular application accumulation to obtain the star's fixed appearance and setting fixed accumulation. Add and name as before to obtain the fixed appearance and setting day and double-hour.