1
曆象誌上
Celestial Phenomena Treatise I
2
遼以幽、營立國，禮樂制度，規模日完，授歷頒朔二百餘年。 今奉詔修遼史，體與宋、金似，其《大明曆》不可少也。 歷書法禁不可得，求《大明曆》元，得祖沖之法於外史。 沖之之法，遼歷之所從出也歟？ 國朝亦嘗因之。 以沖之法算，而至於遼更歷之年，以起元數，是蓋遼《大明曆》。 遼歷因是固可補，然弗之補，史貴闕文也。 外史紀其法，司天存其職，《遼史》誌是足矣。 作《歷象志》。
The Liao rose from You and Ying; as ritual and institutions matured, they issued calendars and set the new moon for over two hundred years. Ordered to compile the History of Liao on the model of the Song and Jin histories, we cannot omit the Great Brightness Calendar. Official calendrical methods were barred from circulation; in seeking the Great Brightness epoch we found Zu Chongzhi's procedures in an outside history. Did the Liao calendar ultimately derive from Chongzhi's methods? The present dynasty too once adopted them. Working forward with Chongzhi's arithmetic to the Liao calendar reform and deriving the epoch from that point yields the Liao Great Brightness Calendar. The Liao calendar could be reconstructed from this material, but we leave it unreconstructed, for histories prize deliberate gaps. An outside history records the procedures, the Directorate of Astronomy keeps the office, and this treatise in the History of Liao is enough. Hence the Calendar and Celestial Phenomena Treatise.
3
宋武帝大明六年，祖沖之上《甲子元曆》法，未及施用，因名《大明曆》。 上元甲子至宋大明七年癸卯，五萬一千九百三十九年算外。 元法：五十九萬二千三百六十五。
In Song Daming year 6 (462 CE) Zu Chongzhi presented the Jiazi Epoch Calendar methods; before adoption they were called the Great Brightness Calendar. From upper-origin jiazi to Song Daming year 7 (guimao) is 51,939 years beyond the epoch count. Origin factor (yuan fa): 592,365.
4
紀法：三萬九千四百九十一。 章歲：三百九十一。 章月：四千八百三十六。 章閏：一百四十四。
Era factor (ji fa): 39,491. Rule years (zhang sui): 391. Rule months (zhang yue): 4,836. Rule intercalations (zhang run): 144.
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閏法：十二。 月法：十一萬六千三百二十一。 日法：三千九百三十九。 餘數：二十萬七千四十四。
Intercalation factor (run fa): 12. Month factor (yue fa): 116,321. Day factor (ri fa): 3,939. Remainder number (yu shu): 207,044.
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歲餘：九千五百八十九。 沒分：三百六十萬五千九百五十一。 沒法：五萬一千七百六十一。 周天：一千四百四十二萬四千六百六十四。
Year remainder (sui yu): 9,589. Submergence parts (mo fen): 3,605,951. Submergence factor (mo fa): 51,761. Circumference of heaven (zhou tian): 14,424,664.
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虛分：萬四百四十九。 行分法：二十三。 小分法：一千七百一十七。 通周：七十二萬六千八百一十。
Void parts (xu fen): 10,449. Motion-part factor (xing fen fa): 23. Small-part factor (xiao fen fa): 1,717. Communication cycle (tong zhou): 726,810.
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會周：七十一萬七千七百七十七。 通法：二萬六千三百七十七。 差率：三十九。 推朔術：
Conjunction cycle (hui zhou): 717,777. Communication factor (tong fa): 26,377. Difference rate (cha lu): 39. Procedure to compute new moons:
9
置入上元年數算外，以章月乘之，滿章歲為積月，不盡為閏餘。 閏餘二百四十七以上，其年有閏。 以月法乘積月，滿日法為積日，不盡為小餘。 六旬去積日，不盡為大餘。 大餘命以甲子，算外，所求年天正十一月朔也。 小餘千八百四十九以上，其月大。
Take the years since upper origin (outside the count), multiply by rule months, divide by rule years for accumulated months; the remainder is intercalary remainder. If intercalary remainder is 247 or above, the year has a leap month. Multiply accumulated months by the month factor; divide by the day factor for accumulated days; the remainder is fractional day remainder (xiao yu). Subtract full sixty-day cycles from accumulated days; the remainder is greater remainder (da yu). Count greater remainder from jiazi (outside the count) to obtain the new moon of civil month 11 for the target year. If fractional day remainder is 1,849 or more, the month is a long month (30 days).
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求次月：
To find the next month:
11
加大餘二十九，小餘二千九十。 小餘滿日法從大餘，大餘滿六旬去之，命如前，次月朔也。 求弦望：
Add 29 to greater remainder and 2,090 to fractional day remainder. When fractional remainder fills the day factor, carry to greater remainder; when greater remainder fills sixty days, drop it and name as before for the next new moon. To find first quarter and full moon:
12
加朔大餘七，小餘千五百七，小分一。 小分滿四從小餘，小餘滿日法從大餘，命如前，上弦日也。 又加得望，又加得下弦，又加得後月朔也。 推閏術：
Add 7 to new-moon greater remainder, 1,507 to small remainder, and 1 to small parts. When small parts reach 4, carry to small remainder; when small remainder fills the day factor, carry to greater remainder; name as before for first quarter. Add again for full moon, again for last quarter, and again for the following new moon. Procedure for computing intercalation:
13
以閏餘減章歲，余滿閏法得一月，命以天正，算外，閏所在也。 閏有進退，以無中氣為正。 推二十四氣：
Subtract intercalary remainder from rule years; each time the remainder fills the intercalation factor you gain one month; count from civil new year (outside the count) to locate the leap month. Leap placement may shift; the rule is that a month without a mid-climate is intercalary. Procedure to compute the twenty-four qi:
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置入上元年數算外，以餘數乘之，滿紀法為積日，不盡為小餘。 六旬去積日，不盡為大餘。 大餘命以甲子，算外，天正十一月冬至日也。 求次氣：
Take years since upper origin (outside the count), multiply by remainder number, divide by era factor for accumulated days; the remainder is fractional day remainder. Subtract sixty-day cycles from accumulated days; the remainder is greater remainder. Name greater remainder from jiazi (outside the count) for the winter solstice of civil month 11. To find the next qi:
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加大餘十五，小餘八千六百二十六，小分五。 小分滿六從小餘，小餘滿紀法從大餘，命如前，次氣日也。 求土王用事：
Add 15 to greater remainder, 8,626 to small remainder, and 5 to small parts. When small parts reach 6, carry to small remainder; when small remainder fills era factor, carry to greater remainder; name as before for the next qi. To find the earth phase (tu wang) period:
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加冬至大餘二十七，小餘萬五千五百二十八，季冬土用事日也。 又加大餘九十一，小餘萬二千二百七十，次土用事日也。 推沒術：
Add 27 to winter-solstice greater remainder and 15,528 to small remainder for the late-winter earth tenure day. Add again 91 to greater remainder and 12,270 to small remainder for the next earth tenure day. Procedure for computing submergence (mo):
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以九十乘冬至小餘，以減沒分，滿沒法為日，不盡為日餘，命日以冬至，算外，沒日也。 求次沒：
Multiply winter-solstice fractional remainder by 90, subtract from submergence parts; divide by submergence factor for days; the remainder is day remainder; count from winter solstice (outside the count) for the submergence day. To find the next submergence:
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加日六十九，日餘三萬四千四百四十二，余滿沒法從日，次沒日也。 日餘盡為滅。 推日所在度術：
Add 69 days and 34,442 to day remainder; when remainder fills submergence factor, carry to days for the next submergence. When day remainder is exhausted, the cycle ends (mie). Procedure for the sun's daily lodge:
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以紀法乘朔積日為度實，周天去之，余滿紀法為積度，不盡為度餘。 命以虛一，次宿除之，算外，天正十一月朔夜半日所在度也。 求次月：
Multiply new-moon accumulated days by era factor for degree dividend; subtract circumference of heaven; quotient by era factor gives accumulated degrees; remainder is degree remainder. Name from lodge Void I, subtract successive lodges (outside the count) for the sun's lodge at midnight on month-11 new moon. To find the next month:
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大月加度三十，小月加度二十九，入虛去度分。 求行分：
Add 30 degrees for a long month, 29 for a short month; when entering Void, subtract degree parts. To find motion parts:
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以小分法除度餘，所得為行分，不盡為小分，小分滿法從行分，行分滿法從度。 求次日：加一度。 入虛去行分六，小分百四十七。
Divide degree remainder by small-part factor for motion parts; the remainder is small parts; carry when parts fill their factors. For the next day: add one degree. On entering Void, subtract 6 motion parts and 147 small parts.
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推月所在度術：
Procedure for the moon's lodge:
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以朔小餘乘百二十四為度餘，又以朔小餘乘八百六十為微分，微分滿月法從度餘，度餘滿紀法為度。 以減朔夜半日所在，則月所在度。 求次月：
Multiply new-moon fractional remainder by 124 for degree remainder and by 860 for minute parts; carry minute parts into degree remainder, then degree remainder into degrees by era factor. Subtract this from the new-moon midnight solar position to obtain the moon's lodge. To find the next month:
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大月加度三十五，度餘三萬一千八百三十四，微分七萬七千九百六十七，小月加度二十二，度餘萬七千二百六十一，微分六萬二千七百三十六，入虛去度也。
Long month: add 35 degrees plus the listed remainders and minute parts; short month: add 22 degrees plus its listed parts; on entering Void, subtract degrees.
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遲疾歷： 〈（見書522－524頁有表）〉 推入遲疾歷術：
Slow-Fast (chi ji) sequence: (See tables on pp. 522–524 of the source edition.)〉 Procedure to enter the Slow-Fast sequence:
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以通法乘朔積日為通實，通周去之，余滿通法為日，不盡為日餘。 命日算外，天正十一月朔夜半入歷日也。 求次月：
Multiply new-moon accumulated days by communication factor for communication dividend; subtract communication cycle; divide by communication factor for days; remainder is day remainder. Name the day (outside the count) for midnight entry into the sequence on month-11 new moon. To find the next month:
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大月加二日，小月加一日，日餘皆萬一千七百四十六。 歷滿二十七日，日餘萬四千六百三十一，則去之。 求次日：加一日。 求日所在定度：
Long month: add 2 days; short month: add 1 day; day remainder always 11,746. When the sequence reaches 27 days with day remainder 14,631, drop that cycle. For the next day: add one day. To find the sun's true (fixed) daily lodge:
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以夜半入歷日餘乘損益率，以損益盈縮積分，如差率而一，所得滿紀法為度，不盡為度餘，以盈加縮減平行度及余為定度。 益之或滿法，損之或不足，以紀法進退。 求度行分如上法。 求次日，如所入遲疾加之。 虛去分，如上法。
Multiply midnight sequence day remainder by the adjustment rate, apply surplus-deficit to accumulated parts, scale by difference rate; divide by era factor for degrees; add surplus or subtract deficit from mean solar degree for true position. If increase overflows the factor or decrease underflows, adjust by one era factor. Find degree motion parts by the same method. For the next day, add according to whether the sequence entry is slow or fast. On entering Void, subtract parts as above.
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陰陽歷：
Yin-Yang sequence:
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損益率
Adjustment rates (decrease-increase)
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益十六
Plus 16
32
益十五
Plus 15
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益十四
Plus 14
34
三十一
31
35
益十二
Plus 12
36
四十五
45
37
五十七
57
38
六十六
66
39
七十一
71
40
七十二
72
41
六十四
64
42
十一日
Day 11
43
損十三
Minus 13
44
五十四
54
45
十二日
Day 12
46
損十五
Minus 15
47
四十一
41
48
十三日
Day 13
49
損十六
Minus 16
50
二十六
26
51
十四日
Day 14
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損十六
Minus 16
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推入陰陽歷術：置通實以會周去之，不滿交數三十五萬八千八百八十八半為朔入陽歷分，各去之，為朔入陰歷分，各滿通法得一日，不盡為日餘。 命日算外，天正十一月朔夜半入歷日也。
To enter the Yin-Yang sequence: take communication dividend, subtract conjunction cycle; remainder under conjunction number 358,888.5 is yang-sequence entry for the new moon; subtract each portion for yin entry; divide by communication factor for days; remainder is day remainder. Name the day (outside the count) for midnight sequence entry on month-11 new moon.
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求次月：大月加二日，小月加一日，日餘皆二萬七百七十九。 歷滿十三日，日餘萬五千九百八十七半，則去之。 陽竟入陰，陰竟入陽。 求次日：加一日。
To find the next month: Long month: add 2 days; short month: add 1 day; day remainder always 20,779. When the sequence reaches 13 days with day remainder 15,987.5, drop that cycle. At yang's end one enters yin; at yin's end one enters yang.
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求朔望差：以二千二十九乘朔小餘，滿三百三為日餘，不盡倍之為小分，則朔差數也。 加一十四日，日餘二萬一百八十六，小分百二十五。 小分滿六百六從日餘，日餘滿通法為日，即望差數也。 又加之，後月朔也。
For the next day: add one day. For new/full moon correction: multiply new-moon fractional remainder by 2,029; divide by 303 for day remainder; double the remainder for small parts to get new-moon difference. Add 14 days, day remainder 21,186, and small parts 125. When small parts reach 606, carry to day remainder; when day remainder fills communication factor, convert to days for full-moon difference.
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求合朔月食：置朔望夜半入陰陽歷及余，有半者去之，置小分三百三，以差數加之。 小分滿六百六從日餘，日餘滿通法從日，日滿一歷去之。 命日算外，則朔望加時入歷也。 朔望加時入歷一日，日餘四千一百九十八，小分四百二十八以下，十二日，日餘萬一千七百八十八，小分四百八十一以上，朔則交會，望則月食。
Add again for the following new moon. For conjunction and eclipse: take midnight yin-yang sequence entry and remainder at new/full moon, drop halves, set small parts 303, and add the difference number. Carry small parts into day remainder and day remainder into days; when days complete one sequence, subtract the cycle. Name the day (outside the count) for sequence entry at the instant of new or full moon.
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求合朔月食定大小餘：令差數日餘加夜半入遲疾歷余，日餘滿通法從日，則朔望加時入歷也。 以入歷余乘損益率，以損益盈縮積分，如差法而一，以盈減縮加本朔望小餘為定小餘。 益之或滿法，損之或不足，以日法進退日。
If hour-entry is 1 day with remainder ≤4,198 and small parts ≤428, or 12 days with remainder ≥11,788 and small parts ≥481, new moon yields conjunction and full moon yields eclipse. For fixed remainders at conjunction/eclipse: add difference day remainder to midnight slow-fast remainder; carry into days when remainder fills communication factor. Multiply sequence remainder by adjustment rate, adjust surplus-deficit parts, scale by difference factor, then subtract surplus or add deficit from base new/full fractional remainder.
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求合朔月食加時：以十二乘定小餘，滿日法得一辰，命以子，算外，加時所在辰也。 有餘者四之，滿日法得一為少，二為半，三為太。 又有餘者三之，滿日法得一為強，以強並少為少強，並半為半強，並太為太強。 得二者為少弱，以並少為半弱，並半為太弱，並太為一辰弱，以前辰名之。
On overflow or underflow in adjustment, carry or borrow using the day factor. For event hour: multiply fixed fractional remainder by 12, divide by day factor for double-hours; count from zi (outside the count). If remainder remains, multiply by four; each day-factor quotient gives minor (shao), half (ban), or major (tai) subdivisions. Further remainder: multiply by three; quotients are strong (qiang); combine with prior subdivisions for compound labels.
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求月去日道度：置入陰陽歷余乘損益率，如通法而一，以損益兼數為定。 定數十二而一為度。 不盡四而一，為少、半、太。 又不盡者三而一，一為強，二為少弱，則月去日道數也。 陽歷在表，陰歷在裏。
Two quotients yield minor-weak; combine upward through half-weak and major-weak to a full weak double-hour named from the prior hour. For lunar latitude: take yin-yang sequence remainder, multiply by adjustment rate, divide by communication factor, and adjust the combined number. Divide the fixed number by twelve for degrees. The remainder in quarters yields minor, half, and major subdivisions. Further remainder in thirds: one gives strong, two minor-weak, completing lunar distance from the ecliptic.
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測景漏刻中星數： 〈（見書529－531頁有表）〉 求昏明中星：各以度數如夜半日所在，則中星度。 推五星術：
Yang sequence is outside the path; yin sequence inside. (See tables on pp. 529–531 of the source edition.)〉 For dusk, dawn, and culmination stars, apply the same degrees as at midnight solar position. Procedure to compute the five planets:
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木率：千五百七十五萬三千八十二。 火率：三千八十萬四千一百九十六。 土率：千四百九十三萬三百五十四。 金率：二千三百六萬一十四。
Jupiter rate (mu lu): 15,753,082. Mars rate (huo lu): 30,804,196. Saturn rate (tu lu): 14,930,354. Venus rate (jin lu): 23,060,014.
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水率：四百五十七萬六千二百四。
Mercury rate (shui lu): 4,576,204.
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推五星術：置度實各以率去之，余以減率，其餘，如紀法而一，為入歲日，不盡為日餘，命以天正朔，算外，星合日。
For each planet: take degree dividend, subtract its rate repeatedly; treat the remainder over era factor as days since civil new-year new moon (outside the count) for conjunction day.
64
求星合度：以入歲日及余從天正朔日積度及余，滿紀法從度，滿三百六十餘度分則去之，命以虛一，算外，星合所在度也。
For conjunction degree: add year-entry days and remainder to new-year solar accumulated degrees; carry at era factor; drop full circles; name from Void I (outside the count).
65
求星日見：以術伏日及余加星合日及余，余滿紀法從日，命如前，見日也。
For first visibility: add tabulated morning/evening concealment days and remainder to conjunction; carry remainder into days; name as before.
66
求星見度：以術伏度及余加星合度及余，余滿紀法從度，入虛去度分，命如前，星見度也。
For appearance degree: add concealment degrees and remainder to conjunction position; carry; subtract on entering Void; name as before.
67
行五星法：以小分法除度餘，所得為行分，不盡為小分，及日加所行分，滿法從度，留者因前，逆則減之，伏不盡度。 從行入虛，去行分六，小分百四十七，逆往出虛，則加之。
Planetary motion: convert degree remainder to motion and small parts; add daily motion; carry at factor; hold in station, subtract in retrograde; concealment may leave degrees unused. On direct motion entering Void, subtract 6 motion parts and 147 small parts; add them when retrograde leaves Void.
68
木星：
Jupiter:
69
初與日合，伏，十六日，日餘萬七千八百三十二，行二度，度餘三萬七千五百四，晨見東方。 從，日行四分，百一十二日行十九度十一分。 留，二十八日。 逆，日行三分，八十六日退十一度五分。 又留二十八日。 從，日行四分，百一十二日，夕伏西方，日度餘如初。 一終三百九十八日，日餘三萬五千六百六十四，行三十三度，度餘二萬五千二百一十五。
First conjunction with the sun: 16 days concealed (remainder 17,832), motion 2 degrees (remainder 37,504), then morning visibility in the east. Prograde at four parts per day; in 112 days advances 19 degrees 11 parts. Stationary for 28 days. Retrograde at three parts per day; in 86 days retreats 11 degrees 5 parts. Stationary again for 28 days. Prograde at four parts per day; after 112 days, evening concealment in the west with initial day and degree remainders. One synodic cycle: 398 days (remainder 35,564), motion 33 degrees (remainder 25,215).
70
火星：
Mars:
71
初與日合，伏，七十二日，日餘六百八，行五十五度，度餘二萬八千八百六十五，晨見東方。 從，疾，日行十七分，九十二日行六十八度。 小遲，日行十四分，九十二日行五十六度。 大遲，日行九分，九十二日行三十六度。 留，十日。 逆，日行六分，六十四日退十六度十六分。 又留，十日。 從，遲，日行九分，九十二日。 小疾，日行十四分，九十二日。 大疾，日行十七分，九十二日。 夕伏西方，日度餘如初。 一終七百八十日，日餘千二百一十六，行四百一十四度，度餘三萬二百五十八，除一周，定行四十九度，度餘萬九千八百九。
First conjunction: 72 days concealed (remainder 608), motion 55 degrees (remainder 28,865), morning visibility east. Prograde fast: seventeen parts per day; 92 days, 68 degrees. Slightly slow: fourteen parts per day; 92 days, 56 degrees. Very slow: nine parts per day; 92 days, 36 degrees. Stationary 10 days. Retrograde six parts per day; 64 days, retreat 16 degrees 16 parts. Stationary again 10 days. Prograde slow: nine parts per day for 92 days. Slightly fast: fourteen parts per day for 92 days. Very fast: seventeen parts per day for 92 days. Evening concealment in the west; day and degree remainders as at first. One cycle: 780 days (remainder 1,216), motion 414 degrees (remainder 30,258); subtract one circuit; net motion 49 degrees (remainder 19,809).
72
土星：
Saturn:
73
初與日合，伏，十七日，日餘千三百七十八，行一度，度餘萬九千三百三十三，晨見東方，行順，日行二分，八十四日行七度七分。 留，三十三日。 行逆，日行一分，百一十日退四度十八分。 又留，三十三日。 從，日行二分，八十四日，夕伏西方，日度餘如初。 一終三百七十八日，日餘二千七百五十六，行十二度，度餘三萬一千七百九十八。
First conjunction: 17 days concealed (remainder 1,378), 1 degree motion (remainder 19,333), morning east; then direct at two parts per day, 84 days for 7 degrees 7 parts. Stationary 33 days. Retrograde one part per day; 110 days, retreat 4 degrees 18 parts. Stationary again 33 days. Prograde two parts per day for 84 days; evening concealment west with initial remainders. One cycle: 378 days (remainder 2,756), motion 12 degrees (remainder 31,798).
74
金星：
Venus:
75
初與日合，伏，三十九日，日餘三萬八千一百二十六，行四十九度，度餘三萬八千一百二十六，夕見西方。 從，疾，日行一度五分，九十二日行百十二度。 小遲，日行一度四分，九十二日行百八度。 大遲，日行十七分，四十五日行三十三度六分。 留，九日。 遲，日行十六分，九日退六度六分，夕伏西方。 伏五日，退五度，而與日合。 又五日退五度，而晨見東方，逆，日行十六分，九日，留，九日。 從，遲，日行十七分，四十五日。 小疾，日行一度四分，九十二日。 大疾，日行一度五分，九十二日。 晨伏東方，日度餘如初。 一終五百八十三日，日餘三萬六千七百六十一，行星如之。 除一周，定行二百十八度，度餘二萬六千三百一十三。 合二百九十一日，日餘三萬八千一百二十六，行星亦如之。
First conjunction: 39 days concealed (remainder 38,126), motion 49 degrees (same remainder), evening visibility west. Prograde fast: 1 degree 5 parts per day; 92 days, 112 degrees. Slightly slow: 1 degree 4 parts per day; 92 days, 108 degrees. Very slow: seventeen parts per day; 45 days, 33 degrees 6 parts. Stationary 9 days. Slow retrograde: sixteen parts per day; 9 days, retreat 6 degrees 6 parts; evening concealment west. Concealed 5 days, retreat 5 degrees, then conjunction with the sun. Another 5 days retreat 5 degrees, morning east; retrograde sixteen parts per day for 9 days; stationary 9 days. Prograde slow: seventeen parts per day for 45 days. Slightly fast: 1 degree 4 parts per day for 92 days. Very fast: 1 degree 5 parts per day for 92 days. Morning concealment east; day and degree remainders as at first. One cycle: 583 days (remainder 36,761); planetary motion matches. Subtract one circuit; net motion 218 degrees (remainder 26,313). Conjunction half-cycle: 291 days (remainder 38,126); motion likewise.
76
水星：
Mercury:
77
初與日合，伏，十四日，日餘三萬七千一百一十五，行三十度，度餘三萬七千一百一十五，夕見西方。 從，疾，日行一度六分，二十三日行二十九度遲，日行二十分，八日行六度二十二分。 留，二日。 遲，日行十一分，二日退二十二分，夕伏西方。 伏八日，退八度，而與日合。 又八日退八度，晨見東方。 逆，日行十一分，二日。 留，二日。 從，遲，日行二十分，八日。 疾，日行一度六分，二十三日。 晨伏東方，日度餘如初。 一終百一十五日，日餘三萬四千七百三十九，行星如之。 一合五十七日，日餘三萬七千一百一十五，行星亦如之。
First conjunction: 14 days concealed (remainder 37,115), motion 30 degrees (same remainder), evening west. Prograde fast: 1 degree 6 parts per day, 23 days for 29 degrees; slow: twenty parts per day, 8 days for 6 degrees 22 parts. Stationary 2 days. Slow: eleven parts per day; 2 days, retreat 22 parts; evening concealment west. Concealed 8 days, retreat 8 degrees, then conjunction. Another 8 days retreat 8 degrees, then morning visibility east. Retrograde eleven parts per day for 2 days. Stationary 2 days. Fast: 1 degree 6 parts per day for 23 days. Morning concealment east; day and degree remainders as at first. One cycle: 115 days (remainder 34,739); motion matches. One conjunction: 57 days (remainder 37,115); motion likewise. At upper-origin year jiazi: civil new year jiazi new moon, winter solstice at midnight; sun, moon, and five planets at the start of Void; yin-yang and slow-fast sequences also begin here.
78
上元之歲，歲在甲子，天正甲子朔夜半冬至，日月五星聚於虛度之初，陰陽遲疾亦自此始。
At upper-origin year jiazi: civil new year jiazi new moon, winter solstice at midnight; sun, moon, and five planets at the start of Void; yin-yang and slow-fast sequences also begin here.
79
梁武帝天監三年，沖之子恒上疏，論何承天歷乖謬不可用。 九年正月，詔用祖沖之所造《甲子元曆》頒朔。 陳氏因梁，亦用祖沖之歷。 至遼，聖宗以賈俊所進新歷，因宋《大明》舊號行之。 金曰《重修大明曆》。 傳至皇元亦曰《重修大明曆》。 及改《授時曆》，別立司天監存肄之，每歲甲子冬至重修其法。 書在太史院，禁莫得聞。
Liang Tianjian year 3: Zu Chongzhi's son Heng memorialized that He Chengtian's calendar was erroneous and unusable. In year 9, month 1, an edict adopted Zu Chongzhi's Jiazi Epoch Calendar for issuing the new moon. Chen continued Liang practice and used Chongzhi's calendar. Under Liao, Emperor Shengzong adopted Jia Jun's new calendar while retaining the Song name Great Brightness. Jin titled it the Revised Great Brightness Calendar. In the Yuan it was likewise the Revised Great Brightness Calendar. After the Season-Granting Calendar replaced it, a Directorate of Astronomy kept the old methods, revised at each jiazi winter solstice. The text remained in the Grand Astrologer's archive, barred from circulation.