1
卷一十八志第八
Volume 18, Treatise 8 (chapter heading).
2
律曆志下
Treatise on Harmonics and Calendrics, part three (section title).
3
魏尚書郎楊偉表曰：「臣覽載籍，斷考歷數，時以紀農，月以紀事，其所由來，遐而尚矣。 乃自少昊，則玄鳥司分，顓頊、帝嚳，則重黎司天； 唐帝、虞舜，則羲和掌日，三代因之，則世有日官。 日官司曆，則頒之諸侯，諸侯受之，則頒於境內。 夏後之世，羲和湎淫，廢時亂日，則《書》載《胤征》。 由此觀之，審農時而重人事，歷代然之也。 逮至周室既衰，戰國橫鶩，告朔之羊，廢而不紹，登臺之禮，滅而不遵，閏分乖次而不識，孟陬失紀而莫悟，大火猶西流，而怪蟄蟲之不藏也。 是時也，天子不協時，司曆不書日。 諸侯不受職，日禦不分朔，人事不恤，廢棄農時。 仲尼之撥亂於《春秋》，托褒貶糾正，司曆失閏，則譏而書之，登臺頒朔，則謂之有禮。 自此以降，暨于秦漢，乃復以孟冬為歲首，閏為後九月，中節乖錯，時月紕繆。 加時後天，蝕不在朔，累載相襲，久而不革也。 至武帝元封七年，始乃悟其繆焉，於是改正朔，更歷數，使大才通人，更造《太初曆》，校中朔所差，以正閏分，課中星得度，以考疏密，以建寅之月為正朔，以黃鐘之月為曆初。 其曆斗分太多，後遂疏闊。 至元和二年，復用《四分曆》，施而行之，至於今日，考察日蝕，率常在晦，是則斗分太多，故先密後疏而不可用也。 是以臣前以制典餘日，推考天路，稽之前典，驗之以蝕朔，詳而精之，更建密曆，則不先不後，古今中天。 以昔在唐帝，協日正時，允厘百工，咸熙庶績也。 欲使當今國之典禮，凡百制度，皆韜合往古，鬱然備足，乃改正朔，更歷數，以大呂之月為歲首，以建子之月為曆初。 臣以為昔在帝代，則法曰《顓頊》，曩自軒轅，則曆曰《黃帝》，暨至漢之孝武，革正朔，更歷數，改元曰太初，因名《太初曆》。 今改元為景初，宜曰《景初曆》。 臣之所建《景初曆》，法數則約要，施用則近密，治之則省功，學之則易知。 雖復使研桑心算，隸首運籌，重黎司晷，羲和察景，以考天路，步驗日月，究極精微，盡術數之極者，皆未能並臣如此之妙也。 是以累代歷數，皆疏而不密，自黃帝以來，常改革不已。」
Yang Wei, a Wei Secretariat official, wrote: “I have combed the sources and worked through the calendar’s numbers: seasons organize agriculture, months organize administration—and that arrangement goes back to remote antiquity. Already under Shao Hao the swallow asterism marked the equinoxes; under Zhuan Xu and Di Ku, Chong and Li were charged with celestial regulation. The legendary rulers Yao and Shun placed Xi and He in charge of the solar calendar; the three early dynasties kept the office, so every reign maintained “sun” specialists. Those officers set the calendar and issued it to the regional lords, who in turn published it inside their own territories. Later in the Xia era, Xi and He neglected their duty, deranged the calendar, which is why the Book of Documents preserves the “Yin” punitive campaign against them. From this it is clear that every dynasty has treated careful timing for agriculture and attention to civic order as a standing principle. By late Zhou the realm fractured: monthly temple announcements lapsed, observatory ceremonies died out, leap months drifted out of place, and the civil year’s first month wandered—while Antares still hung in the western sky, courtiers marveled at insects that failed to burrow, blind to the calendar’s chaos. The king no longer coordinated the seasons, and official scribes stopped reliably dating events. Regional rulers ignored their responsibilities, astronomers lost track of true conjunctions, and governance neglected both people and the agricultural timetable. Confucius used the Spring and Autumn to restore order through moral judgment: he mocked missed intercalations, yet praised rulers who climbed the terrace to proclaim the civil calendar as behaving ritually. Down through Qin and Han the year again began in early winter, leap months were tacked after the ninth month, and solar terms and lunar months slipped badly out of alignment. Ephemeris corrections ran behind the sky, eclipses missed conjunction, and the mistake persisted reign after reign without overhaul. Only in 104 BCE did Emperor Wu recognize the fault: he reformed the civil calendar, commissioned the Taichu system, adjusted the leap fraction against true conjunctions, checked stellar longitudes, adopted jianyin (roughly late winter) as the first month of the year, and anchored the epoch in the “yellow bell” month. The Taichu system’s fractional remainder for the Dipper was too big, so in later reigns it grew increasingly imprecise. From 85 CE the Sifen (quarter-day) calendar returned; eclipse records show conjunctions drifting to month-end—proof the fractional excess was too large, yielding an apparently tight fit that soon loosened into uselessness. Hence I reworked the official surplus-day reckoning, checked it against eclipses and prior canons, and drew up a denser calendar that tracks the sky evenly—neither ahead nor behind—between antiquity and today. This matches Yao’s age, when aligning the days and seasons allowed every office to function and every undertaking to thrive. He wants current state ritual and every institution to echo ancient models; so he resets the civil year, revises the calendrical constants, begins the year in the great-lü month, and sets the computational epoch in the jianzi month. I note that each great reform named its system—Zhuanxu, the Yellow Emperor’s calendar, and Han Wudi’s Taichu calendar after his new era name. Since the reign era is now Jingchu, the new system ought to be called the Jingchu calendar. The Jingchu scheme I propose keeps the parameters lean yet numerically tight, runs efficiently in practice, and is straightforward to learn. No amount of legendary arithmetical skill—merchant Yan’s mental sums, Lishou’s rods, Chong and Li at the sundial, Xi and He at the solstice shadow—could match the precision I have reached. That is why earlier dynasties’ calendars always erred on the loose side, and why reform has followed reform ever since the Yellow Emperor.”
4
壬辰以來，至景初元年丁已歲，積四千四十六，算上。
From the Gengchen origin through the Jingchu era’s first year (a Dingsi year in the cycle), the accumulated count is 4046, inclusive on the upper stem count.
5
此元以天正建子黃鐘之月為曆初，元首之歲，夜半甲子朔旦冬至。
The epoch uses the true winter month (jianzi) aligned with the yellow-bell pitch month as calendar zero, so at the origin year’s start a Jiazi day begins at midnight exactly at winter solstice.
6
元法，萬一千五十八。
The fundamental divisor (yuanfa) is 11,058.
7
紀法，千八百四十三。
The cycle divisor (jifa) is 1,843.
8
紀月，二萬二千七百九十五。
Months per full cycle constant (jiyue): 22,795.
9
章歲。 十九。
Intercalation-cycle length in years (heading). The number nineteen.
10
章月，二百三十五。
There are 235 months in the cycle (zhang yue).
11
章閏，七。
Seven intercalary months in each nineteen-year cycle.
12
通數，十三萬四千六百三十。
The common multiple constant is 134,630.
13
日法，四千五百五十九。
The day divisor is 4,559.
14
餘數，九千六百七十。
The remainder numerator is 9,670.
15
周天，六十七萬三千一百五十。
The full-revolution constant is 673,150.
16
紀歲中，十二。
Twelve months in a standard civil year.
17
氣法，十二。
The solar-term divisor is twelve.
18
沒分，六萬七千三百一十五。
The “hidden-month” numerator is 67,315.
19
沒法，九百六十七。
The corresponding divisor is 967.
20
月周，二萬四千六百三十八。
The lunar revolution constant is 24,638.
21
通法，四十七。
The passage divisor is forty-seven.
22
會通，七十九萬百一十。
The conjunction accumulated product is 790,110.
23
朔望合數，六萬七千三百一十五。
The lunation conjunction–opposition combined constant is 67,315.
24
入交限數，七十二萬二千七百九十五。
The lunar-node crossing limit constant is 722,795.
25
通周，十二萬五千六百二十一。
The passage-cycle modulus is 125,621.
26
周日日餘，二千五百二十八。
The weekly day remainder is 2,528.
27
周虛，二千三十一。
The complement to complete the cycle (zhou xu) is 2,031.
28
斗分，四百五十五。
The Big Dipper fractional part is 455.
29
甲子紀第一
Cycle series I: Jiazi.
30
紀首合朔，月在日道里。
At the initial conjunction of the cycle the moon is north of the ecliptic.
31
交會差率四十一萬二千九百一十九。
The eclipse-node difference rate for this cycle is 412,919.
32
遲疾差率，十萬三千九百四十七。
The lunar anomaly difference rate is 103,947.
33
甲戌紀第二
Cycle series II: Jiaxu.
34
紀首合朔，月在日道里。
At this cycle’s starting conjunction the moon is inside the solar track.
35
交會差率，五十一萬六千五百二十九。
The conjunction–node difference rate is 516,529.
36
遲疾差率，七萬三千七百六十七。
The anomaly difference rate is 73,767.
37
甲申紀第三
Cycle series III: Jiashen.
38
紀首合朔，月在日道里。
At cycle head the moon lies inside the ecliptic at conjunction.
39
交會差率，六十二萬一百三十九。
The conjunction–node difference rate is 621,139.
40
遲疾差率，四萬三千五百八十七。
The anomaly difference rate is 43,587.
41
甲午紀第四
Cycle series IV: Jiawu.
42
紀首合朔，月在日道里。
At the cycle’s first conjunction the moon is on the inner track.
43
交會差率，七十二萬三千七百四十九。
The conjunction–node difference rate is 723,749.
44
遲疾差率，一萬三千四百七。
The anomaly difference rate is 13,407.
45
甲辰紀第五
Cycle series V: Jiachen.
46
紀首合朔，月在日道里。
At cycle start the moon at conjunction is inside the solar path.
47
交會差率，三萬七千二百四十九。
The conjunction–node difference rate is 37,249.
48
遲疾差率，十萬八千八百四十八。
The anomaly difference rate is 108,848.
49
甲寅紀第六
Cycle series VI: Jiayin.
50
紀首合朔，月在日道里。
At the Jiayin cycle head the moon lies inward of the ecliptic at new moon.
51
交會差率，十四萬八百五十九。
The conjunction–node difference rate is 140,859.
52
遲疾差率，七萬八千六百六十八。
The anomaly difference rate is 78,668.
53
交會紀差十萬三千六百一十。 求其數之所生者，置一紀積月，以通數乘之，會通去之，所去之余，紀差之數也。 以之轉加前紀，則得後紀。 加之未滿會通者，則紀首之歲天正合朔月在日道里； 滿去之，則月在日道表。 加表，滿在裏； 加里，滿在表。
The step from one cycle’s conjunction–node rate to the next is 103,610. To derive it: take the months in one cycle, multiply by the common multiple, discard full multiples of the conjunction product; what remains defines the cycle increment. Add that increment to the previous cycle’s rate to get the next cycle’s rate. If the sum stays below the conjunction product, the new moon at the cycle’s first civil year lies inside the solar track. If it crosses and you subtract the product, the moon at conjunction lies outside the solar track. Stepping from the outer side, a full wrap lands the moon inside. Stepping from the inner side, a full wrap lands it outside.
54
遲疾紀差三萬一百八十。 求其數之所生者，置一紀積月，以通數乘之，通周去之，餘以減通周，所減之余，紀差之數也。 以之轉減前紀，則得後紀。 不足減者，加通周。 求次元紀差率，轉減前元甲寅紀差率，餘則次元甲子紀差率也。 求次紀，如上法也。
The lunar-anomaly cycle difference between epochs is 30,180. To derive it: multiply cycle months by the common multiple, reduce modulo the passage cycle, then subtract from the passage cycle to get the decrement. Subtract that decrement from the prior cycle’s anomaly rate to obtain the next. If you cannot subtract, add the passage cycle first. For the next grand origin, subtract the previous Jiayin-cycle rate from the Jiazi-cycle rate as prescribed to get the new Jiazi increment. To advance to the following cycle, repeat the same rule.
55
推朔積月術曰：置壬辰元以來，盡所求年，外所求，以紀法除之，所得算外，所入紀第也，余則入紀年數也。 以章月乘之，如章歲而一，為積月，不盡為閏餘。 閏餘十二以上，其年有閏。 閏月以無中氣為正。
To count lunations from the Gengchen epoch up to (but not including) the target year, divide by 1,843: the quotient (plus one) gives the cycle index, the remainder the year within that cycle. Multiply that year-count by 22,795 and divide by 19 for total months; the remainder is the leap fraction. A leap year occurs when the intercalary remainder reaches twelve or higher. The leap month is the lunar month that contains no major solar term.
56
推朔術曰：以通數乘積月，為朔積分。 如日法而一，為積日，不盡為小餘。 以六十去積日，余為大餘。 大餘命以紀，算外，所求年天正十一月朔日也。
To compute conjunction instants, multiply accumulated months by 134,630. Divide that product by 4,559 for whole days; the residue is the fractional day. Reduce the day count modulo sixty for the sexagenary stem-branch index. Index that remainder against the sexagenary cycle to name the day of the eleventh month’s conjunction in the target civil year.
57
求次月，加大餘二十九，小餘二千四百一十九，小餘滿日法從大餘，命如前，次月朔日也。 小餘二千一百四十以上，其月大也。
For the following month add 29 to the day count and 2,419 to the fraction, carrying overflows by 4,559 into the day count, then read off the next conjunction. If the fractional part is 2,140 or greater, the month is a long (30-day) month.
58
推弦望，加朔大餘七，小餘千七百四十四，小分一，小分滿二從小餘，小餘滿日法從大餘，大餘滿六十去之，余命以紀，算外，上弦日也。 又加，得望、下弦、後月朔。 其月蝕望者，定小餘如在中節者定小餘如所近中節間限數、限數以下者，算上為日。 望在中節前後各四日以還者，視限數； 望在中節前後各五日以上者，視間限。
For quarters: add 7 days, 1,744 parts, and one small part to the new-moon line, propagating carries through the small-part, fractional-day, and sexagenary levels; the result names the first-quarter day. Repeat the same addition chain to reach full moon, third quarter, and the next conjunction. When a lunar eclipse falls on the full moon, adjust the fractional day so that, if the value sits on the major solar term, you align it with that term’s margin limits; anything at or under the limit rounds up to count as a full day. When full moon lies within four days before or after the major term, use the standard limit figure. When full moon sits five or more days away from the major term, switch to the wider gap-limit criterion instead.
59
推二十四氣術曰：置所入紀年，外所求，以餘數乘之，滿紀法為大餘，不盡為小餘。 大餘滿六十去之，余命以紀，算外，天正十一月冬至日也。
To place the twenty-four solar terms: take years elapsed within the current cycle (not counting the target year), multiply by 9,670, divide by 1,843 for the day count, and keep the fractional remainder. Reduce the day line modulo sixty and read it against the sexagenary cycle to name the winter solstice of the eleventh month.
60
求次氣，加大餘十五，小餘四百二，小分十一，小分滿氣法從小餘，小餘滿紀法從大餘，命如前，次氣日也。
For each successive term add 15 days, 402 parts, and 11 minute-fractions, carrying overflows through the fractional hierarchy as prescribed.
61
推閏月術曰：以閏余減章歲，餘以歲中乘之，滿章閏得一月，余滿半法以上，亦得一月。 數從天正十一月起，算外，閏月也。 閏有進退，以無中氣禦之。
To find leap months: subtract the leap fraction from 19, multiply the residue by 12, divide by 7, and add an extra leap month whenever the remainder reaches half the divisor. Count forward from the eleventh month; the slot so reached is the intercalary month. Leap placement can shift forward or back, but the governing rule remains “no major solar term within the leap month.”
62
大雪十一月節限數千二百四十二間限千二百四十八
Major Snow (11th month, minor term): margin 1242, gap margin 1248.
63
冬至十一月中限數千二百五十四間限千二百四十五
Winter solstice (11th month, major term): margin 1254, gap margin 1245.
64
小寒十二月節限數千二百三十五間限千二百二十四
Minor Cold (12th month, node): margin 1235, gap margin 1224.
65
大寒十二月中限數千二百一十三間限千一百九十二
Major Cold (12th month, major term): margin 1213, gap margin 1192.
66
立春正月節限數千一百七十二間限千一百四十七
Beginning of spring (1st month, node): margin 1172, gap margin 1147.
67
雨水正月中限數千一百二十二間限千九十三
Rain water (1st month, major term): margin 1122, gap margin 1093.
68
驚蟄二月節限數千六十五間限千三十六
Waking of insects (2nd month, node): margin 1065, gap margin 1036.
69
春分二月中限數千八間限九百七十九
Spring equinox (2nd month, major term): margin 1008, gap margin 979.
70
清明三月節限數九百五十一間限九百二十五
Clear and bright (3rd month, node): margin 951, gap margin 925.
71
穀雨三月中限數九百間限八百七十九
Grain rain (3rd month, major term): margin 900, gap margin 879.
72
立夏四月節限數八百五十七間限八百四十
Beginning of summer (4th month, node): margin 857, gap margin 840.
73
小滿四月中限數八百二十三間限八百一十二
Grain fills (4th month, major term): margin 823, gap margin 812.
74
芒種五月節限數八百間限七百九十九
Grain in ear (5th month, node): margin 800, gap margin 799.
75
夏至五月中限數七百九十八間限八百一
Summer solstice (5th month, major term): margin 798, gap margin 801.
76
小暑六月節限數八百五間限八百一十五
Minor heat (6th month, node): margin 805, gap margin 815.
77
大暑六月中限數八百二十五間限八百四十二
Major heat (6th month, major term): margin 825, gap margin 842.
78
立秋七月節限數八百五十九間限八百八十三
Beginning of autumn (7th month, node): margin 859, gap margin 883.
79
處暑七月中限數九百七間限九百三十五
End of heat (7th month, major term): margin 907, gap margin 935.
80
白露八月節限數九百六十二間限九百九十二
White dew (8th month, node): margin 962, gap margin 992.
81
秋分八月中限數千二十一間限千五十一
Autumn equinox (8th month, major term): margin 1021, gap margin 1051.
82
寒露九月節限數千八十間即千一百七
Cold dew (9th month, node): margin 1080, gap margin 1107.
83
霜降九月中限數千一百三十三間限千一百五十七
Frost descends (9th month, major term): margin 1133, gap margin 1157.
84
立冬十月節限數千一百八十一間限千一百九十八
Beginning of winter (10th month, node): margin 1181, gap margin 1198.
85
小雪十月中限數千二百一十五間限千二百二十九
Minor snow (10th month, major term): margin 1215, gap margin 1229.
86
推沒滅術曰：因冬至積日有小餘者，加積一，以沒分乘之，以沒法除之，所得為大餘，不盡為小餘。 大餘滿六十去之，余命以紀，算外，即去年冬至後日也。
To locate hidden days after the solstice line: if a fractional day remains, increment the day count, multiply by 67,315, divide by 967, and split quotient and remainder. The sexagenary index so obtained dates the day immediately following the previous winter solstice.
87
求次沒，加大餘六十九，小餘五百九十二，小餘滿沒法得一，從大餘，命如前。 小餘盡，為滅也。
For each successive hidden day add 69 days and 592 parts, carrying a full 967 parts into the day count. A zero fractional part marks the extinction point of the sequence.
88
推五行用事日：立春、立夏、立秋、立冬者，即木、火、金、水始用事日也。 各減其大餘十八，小餘四百八十三，小分六，余命以紀，算外，各四立之前，土用事日也。 大餘不足減者，加六十； 小餘不足者，減大餘一，加紀法； 小分不足減者，減小餘一，加氣法。
The four “begins” mark the days when wood, fire, metal, and water each assume seasonal authority. Subtract 18 days, 483 parts, and 6 minute-fractions from each term line to reach the earth-phase days before the solstices and equinoxes. If the day line underflows, borrow sixty days. If the fractional parts underflow, decrement the day count and add 1,843 to the fraction. If minute-fractions underflow, borrow one small-remainder unit and add twelve minute-fractions.
89
推卦用事日：因冬至大餘，六其小餘，即《坎卦》用事日也。 加小餘萬九十一，滿元法從大餘，即《中孚》用事日也。
Starting at the winter solstice line, multiply the fractional day by six to place the Kan hexagram’s governing day. Then add 10,091 fractional parts (carrying at 11,058) to reach the Zhong fu governing day.
90
求次卦，各加大餘六，小餘九百六十七。 其四正各因其中日，六其小餘。 推日度術曰：以紀法朔積日，滿周天去之，余以紀法除之，所得為度，不盡為分。 命度從牛前五起，宿次除之，不滿宿，則天正十一月朔夜半日所在度及分也。
Each following hexagram advances the line by six days and 967 parts. For the four cardinal hexagrams, multiply that term’s fractional day by six instead. To find the sun’s degree: reduce the lunation day count modulo 673,150, then divide by 1,843 for degrees and minutes of arc. Begin counting five degrees before the Ox lodge and step through the twenty-eight mansions to locate the sun at midnight on the eleventh month’s conjunction.
91
求次日，日加一度，分不加，經鬥除斗分，分少，退一度。
Advance one degree per civil day, drop the dipper correction when crossing that mansion, and borrow a degree if minutes run short.
92
推月度術曰：以月周乘朔積日，滿周天去之，余以紀法除之，所得為度，不盡為分，命如上法，則天正十一月朔夜半月所在度及分也。
For the moon, multiply the same day count by 24,638, reduce modulo the full circle, divide by 1,843, and read mansions as for the sun.
93
求次月，小月加度二十二，分八百六，大月又加一日，度十三，分六百七十九； 分滿紀法得一度，則並月朔夜半月所在度分及也。 其冬下旬，月在張、心署之。
Short months advance the moon 22°806′; long months add another day plus 13°679′. Carry 1,843 arc-minutes into one degree to finish the moon’s position at conjunction midnight. During late winter the text records the moon among the Zhang and Heart lodges.
94
推合朔度術曰：以章歲乘朔小餘，滿通法為大分，不盡為小分。 以大分從朔夜半日度分，分滿紀法從度，命如前，則天正十一月合朔日月所共合度也。
To combine sun–moon longitude at conjunction: multiply the conjunction fraction by 19 and divide by 47 for large and small arc-parts. Add those parts to the solar position at conjunction midnight, carrying arc-units into degrees to get the common ecliptic longitude.
95
求次月，加度二十九，大分九百七十七，小分四十二，小分滿通法從大分，大分滿紀法從度，經鬥除其分，則次月合朔日月所共合度也。
Each lunation adds 29°977′42″ of arc, with carries through the passage and cycle divisors and a dipper correction.
96
推弦望日所在度：加合朔度七，大分七百五，小分十，微分一，微分滿二從小分，小分滿通法從大分，大分滿紀法從度，命如前，則上弦日所在度也。 又加，得望，下弦、後月合也。
Quarters add fixed degree increments with micro-fraction carries to place the sun at first quarter. Repeat the addition to reach full moon, third quarter, and the following new moon.
97
推弦望月所在度：加合朔度九十八，大分千二百七十九，小分三十四，數滿命如前，即上弦月所在度也。 又加，得望，下弦、後月合也。
The moon’s first-quarter position adds 98°1279′34″ to the conjunction longitude with the same carry rules. Iterate to obtain full moon, last quarter, and the next conjunction line for the moon.
98
推日月昏明度術曰：日以紀法，月以月周，乘所近節氣夜漏，二百而一，為明分。 日以減紀法，月以減月周，餘為昏分。 各以分加夜半，如法為度。
Scale the night water-clock reading of the neighboring qi by 1,843 for the sun or 24,638 for the moon, then divide by 200 for the daylight segment. Subtract those products from the respective divisors to isolate the twilight segments. Add each segment to the midnight longitude and reduce by the usual degree rule.
99
推合朔交會月蝕術曰：置所入紀朔積分，以所入紀下交會差率之數加之，以會通去之，餘則所求年天正十一月合朔去交度分也。 以通數加之，滿會通去之，餘則次月合朔去交度分也。 以朔望合數各加其月合朔去交度分，滿會通去之，餘則各其月望去交度分也。 朔望去交分，如朔望合數以下，入交限數以上者，朔則交會，望則月蝕。
Combine the lunation fraction with the tabulated node-difference rate and reduce modulo 790,110 to find nodal distance at civil-year conjunction. Add 134,630 repeatedly modulo 790,110 for successive new moons. Add the conjunction–opposition offsets to get full-moon nodal distances similarly. When the nodal residue lies between the tabulated inner and outer thresholds, expect a solar eclipse at conjunction or a lunar eclipse at opposition.
100
推合朔交會月蝕月在日道表裏術曰：置所入紀朔積分，以所入紀下交會差率之數加之，倍會通去之，餘不滿會通者，紀首表，天正合朔月在表； 紀首裏，天正合朔月在裏。 滿會通去之，表滿在裏，裏滿在表。
Halve the modulo arithmetic on the nodal sum: a remainder under half the product marks the moon outside the ecliptic at the year’s first conjunction. If the half-modulo test marks the cycle head as inside, the new moon lies inside the solar track. Each full wrap of the conjunction product toggles the moon from outside to inside the ecliptic or the reverse.
101
求次月，以通數加之，滿會通去之，加里滿在表，加表滿在裏。 先交會後月蝕者，朔在表則望在表，朔在裹則望在裏。 先月蝕後交會者，看蝕月朔在裏則望在表，朔在表則望在裏。 交會月蝕如朔望合數以下，則前交後會； 如入交限數以上，則前會後交。 其前交後會近於限數者，則豫伺之； 前會後交近於限數者，則後伺之。
For later months add 134,630 modulo 790,110 and flip inside/outside whenever the running sum crosses the product boundary. If the node is crossed before opposition, inside and outside states match at new and full moons. If the eclipse occurs before the node, new moon and full moon sit on opposite sides of the ecliptic. Residuals below the conjunction–opposition constant mean the node is crossed before alignment. Values above the outer limit reverse the order: alignment happens before the node. Near-threshold cases with node-first geometry require advance observation. When conjunction leads but sits near the limit, schedule a later watch instead.
102
求去交度術曰：其前交後會者，今去交度分如日法而一，所得則卻交度分也。 其前會後交者，以去交度分減會通，餘如日法而一，所得則前去交度也。 餘皆度分也。 去交度十五以上，雖交不蝕也，十以下是蝕，十以上，虧蝕微少，光晷相及而已。 虧之多少，以十五為法。
For node-first geometry, divide the nodal residue by the day divisor to invert the crossing measure. For conjunction-first geometry, subtract from 790,110 before the same division. All such results are expressed in degrees and arc-minutes. Nodal distances of fifteen parts or more preclude eclipse; ten or below guarantee one; between ten and fifteen yields only a grazing penumbra. Eclipse magnitude scales proportionally to fifteen as the full unit.
103
求日蝕虧起角術曰：其月在外道，先交後會者，虧蝕西南角起； 先會後交者，虧蝕東南角起。 其月在內道，先交後會者，虧蝕西北角起； 先會後交者，虧蝕東北角起。 虧蝕分多少，如上以十五為法。 會交中者，蝕盡。 月蝕在日之沖，虧角與上反也。
With the moon north of the ecliptic and the node crossed before alignment, the solar bite opens in the southwest. If alignment leads the node while outside, the bite starts southeast. Inside the ecliptic with node first, the darkening starts in the northwest. Inside with conjunction first, the bite opens northeast. Magnitude rules mirror the fifteen-part scale used for lunar eclipses. A central crossing yields totality. Lunar eclipses occur at the anti-sun point, so the umbral direction mirrors the solar case.
104
月行遲疾度損益率盈縮積分月行分
Table heading: lunar speed columns for daily excess and deficit.
105
一日十四度十四分益二十六盈初二百八十
Day 1: mean motion 14°14′, +26 benefit, initial surplus 280.
106
二日十四度十一分益二十三盈積分118534二百七十七
Day 2: 14°11′, +23 benefit, running surplus 118,534, line index 277.
107
三日十四度八分益二十盈積分223391二百七十四
Day 3: 14°08′, +20 benefit, surplus integral 223,391, index 274.
108
四日十四度五分益十七盈積分314571二百七十一
Day 4: 14°05′, +17 benefit, surplus sum 314,571, index 271.
109
五日十四度一分益十三盈積分392714二百六十七
Day 5: 14°01′, +13 benefit, surplus sum 392,714, index 267.
110
六日十三度十四分益七盈積分451341二百六十一
Day 6: 13°14′, +7 benefit, surplus sum 451,341, index 261.
111
七日十三度七分損盈積分483254二百五十四
Day 7: 13°07′, surplus begins to shrink, running total 483,254, index 254.
112
八日十三度一分損六盈積分483254二百四十八
Day 8: 13°01′, −6 to surplus, total 483,254, index 248.
113
九日十二度十六分損十盈積分455900二百四十四
Day 9: 12°16′, −10 damage, surplus 455,900, index 244.
114
十日十二度十三分損十三盈積分410310二百四十一
Day 10: 12°13′, −13 damage, surplus 410,310, index 241.
115
十一日十二度十一分損十五盈積分351413二百三十九
Day 11: 12°11′, −15 damage, running surplus 351,413, index 239.
116
十二日十二度八分損十八盈積分282658二百三十六
Day 12: 12°08′, −18 damage, surplus sum 282,658, index 236.
117
十三日十二度五分損二十一盈積分200596二百三十三
Day 13: 12°05′, −21 damage, surplus 200,596, index 233.
118
十四日十二度三分損二十三盈積分104857二百三十一
Day 14: 12°03′, −23 damage, surplus 104,857, index 231.
119
十五日十二度五分益二十一縮初二百三十三
Day 15: 12°05′, +21 benefit, deficit phase opens at index 233.
120
十六日十二度七分益十九縮積分95739二百三十五
Day 16: 12°07′, +19 benefit, deficit sum 95,739, index 235.
121
十七日十二度九分益十七縮積分182336二百三十七
Day 17: 12°09′, +17 benefit, deficit integral 182,336, index 237.
122
十八日十二度十二分益十四縮積分259863二百四十
Day 18: 12°12′, +14 benefit, deficit 259,863, index 240.
123
十九日十二度十五分益十一縮積分323689二百四十三
Day 19: 12°15′, +11 benefit, deficit 323,689, index 243.
124
二十日十二度十八分益八縮積分373838二百四十六
Day 20: 12°18′, +8 benefit, deficit 373,838, index 246.
125
二十一日十三度三分益四縮積分410311二百五十
Day 21: 13°03′, +4 benefit, deficit 410,311, index 250.
126
二十二日十三度七分損縮積分428546二百五十四
Day 22: 13°07′, deficit correction applies, running deficit 428,546, index 254.
127
二十三日十三度十二分損五縮積分428546二百五十九
Day 23: 13°12′, −5 damage, deficit total 428,546, index 259.
128
二十四日十三度十八分損十一縮積分405751二百六十五
Day 24: 13°18′, −11 damage, deficit 405,751, index 265.
129
二十五日十四度五分損十七縮積分355602二百七十一
Day 25: 14°05′, −17 damage, deficit 355,602, index 271.
130
二十六日十四度十一分損二十三縮積分278099二百七十七
Day 26: 14°11′, −23 damage, deficit 278,099, index 277.
131
二十七日十四度十二分損二十四縮積分173242二百七十八
Day 27: 14°12′, −24 damage, deficit 173,242, index 278.
132
周日十四度十三分有小分六百二十六分損二十五有小分六百二十六
Anomalistic week line: 14°13′ plus 626 micro-parts, with matching −25 correction.
133
縮積分63826二百七十九
Deficit running total 63,826 at index 279.
134
有小分六百二十六
Micro-fraction component 626.
135
推合朔交會月蝕入遲疾曆術曰：置所入紀朔積分，以所入紀下遲疾差率數加之，以通周去之，余滿日法得一日，不盡為日餘，命日算外，則所求年天正十一月合朔入曆日也。
To place a lunation inside the lunar speed table: add the tabulated anomaly increment to the lunation fraction, reduce modulo 125,621, divide by 4,559 for whole days plus remainder.
136
求次月，加一日，日餘四千四百五十。 求望，加十四日，日餘三千四百八十九。 日余滿日法成日，日滿二十七去之。 又除余如周日余，日餘不足除者，減一日，加周虛。
For the following month add one day and 4,450 day-fraction parts. For opposition add fourteen days and 3,489 fractional parts. Carry fractions into days at 4,559, then reduce full weeks of 27 days. If the fractional division underflows, borrow one day and add the weekly complement 2,031.
137
推合朔交會月蝕定大小餘：以入曆日餘乘所入曆損益率，以損益盈縮積分，為定積分。 以章歲減所入曆月行分，餘以除之，所得以盈減縮加本小餘。 加之滿日法者，交會加時在後日； 減之不足者，交會加時在前日。 月蝕者，隨定大小餘為日加時。 入曆在周日者，以周日日餘乘縮積分，為定積分。 以損率乘入曆日餘，又以周日日餘乘之，以周日日度小分並之，以損定積分，餘為後定積分。 以章歲減周日月行分，余以周日日餘乘之，以周日度小分並之，以除後定積分，所得以加本小餘，如上法。
Correct conjunction times by multiplying the anomalistic day fraction by the tabulated rate and applying it to the surplus/deficit column. Divide the adjusted integral by the difference between 19 and the tabulated motion to update the fractional day. A carry past the day divisor pushes the event into the next civil day. A borrow moves the corrected instant onto the previous day. Lunar eclipses take their clock time directly from the corrected remainder line. At the anomalistic week boundary, multiply the deficit column by the weekly fractional day. Combine rate products and weekly micro-parts to obtain the post-boundary correction. Finish the week-crossing correction by dividing the adjusted stack with the prescribed divisor mix, then add to the base fraction.
138
推加時：以十二乘定小餘，滿日法得一辰，數從子起，算外，則朔望加時所在辰也。 有餘不盡者四之，如日法而一為少，二為半，三為太。 又有餘者三之，如日法而一為強，半法以上排成之，不滿半法廢棄之。 以強並少為少強，並半為半強，並太為太強。 得二強者為少弱，以之並少為半弱，以之並半為太弱，以之並太為一辰弱。 以所在辰命之，則各得其少、太、半及強，弱也。 其月蝕望在中節前後四日以還者，視限數； 在中節前後五日以上者，視間限。 定小餘如間限、限數以下者，以算上為日。
Multiply the corrected fraction by twelve and divide by 4,559 to name the twelve double-hours from midnight. Split the leftover into quarters of the divisor for the shao/ban/tai fine subdivisions. Further triple the tail to reach the “strong” step, rounding up at half-divisor. Add strong units to weak, half, or full subdivisions per the classical clepsydra notation. Two strongs collapse into a weak grade, stepping through the ladder to a full “weak” double-hour mark. Read the final label against the stem hour to recover shao/tai/ban/qiang/ruo. Eclipse full moons within four days of a major term use the tight limit table. Beyond five days from the major term, apply the wider gap-limit rule. When the adjusted fraction falls under both margin thresholds, promote the count to the next day.
139
鬥二十六分四百五十五牛八女十二虛十危十七室十六壁九
Northern lodge widths: Dipper 26°455′, Ox 8°, Maiden 12°, Emptiness 10°, Rooftop 17°, House 16°, Wall 9′ (catalogue line).
140
北方九十八度分四百五十五
Northern sky arc: 98° with fractional extension 455.
141
奎十六婁十二胃十四昴十一畢十六觜二參九
Western lodges: Straddles 16°, Harvest 12°, Stomach 14°, Hairy Head 11°, Net 16°, Turtle Beak 2°, Triaster 9°.
142
西方八十度
Western arc total: 80°.
143
井三十三鬼四柳十五星七張十八翼十八軫十七
Southern lodges: Well 33°, Ghost 4°, Willow 15°, Star 7°, Extended Net 18°, Wings 18°, Chariot Platform 17°.
144
南方百十二度
Southern arc total: 112°.
145
角十二亢九氐十五房五心五尾十八箕十一
Eastern lodges: Horn 12°, Gullet 9°, Base 15°, Chamber 5°, Heart 5°, Tail 18°, Winnowing Basket 11°.
146
東方七十五度
Eastern arc total: 75°.
147
右中節二十四氣，如術求之，得冬至十一月中也。 加之得次月節，加節得其月中。 中星以日所在為正，置所求年二十四氣小餘，四之，如法得一為少； 不盡少，三之，如法為強； 所得以減其節氣昏明中星各定。
The tabulated twenty-four terms yield winter solstice as the eleventh month’s major qi. Step forward with node increments to reach minor terms, then add nodes to land on major terms. Meridian stars follow solar longitude; quadruple each term’s fractional remainder for the fine shao split. Triple the leftover after the shao step to reach the qiang subdivision. Subtract those fine parts from the tabulated dusk and dawn culminations to fix each term’s reference star.
148
推五星術
Heading: planetary computation.
149
五星者，木曰歲星，火曰熒惑星，土曰填星，金曰太白星，水曰辰星。 凡五星之行，有遲有疾，有留有逆。 曩自開闢，清濁始分，則日月五星聚于星紀。 發自星紀，並而行天，遲疾留逆，互相逮及。 星與日會，同宿共度，則謂之合。 從合至合之日，則謂之終。 各一以終之日與一歲之日通分相約，終而率之，歲數歲則謂之合終歲數，歲終則謂之合終合數。 二率既定，則法數生焉。 以章歲乘合數，為合月法。 以紀法乘合數，為日度法。 以章月乘歲數，為合月分； 如合月法為合月數，合月之余為月餘。 以通數乘合月數，如日法而一，為大餘。 以六十去大餘，餘為星合朔大餘。 大餘之餘為朔小餘。 以通數乘月餘，以合月法乘朔小餘，並之，以日法乘合月法除之，所得星合入月日數也。 餘以通法約之，為入月日餘。 以朔小餘減日法，餘為朔虛分。 以曆斗分乘合數，為星度斗分。 木、火、土各以合數減歲數，余以周天乘之，如日度法而一，所得則行星度數也，余則度餘。 金、水以周天乘歲數，如日度法而一，所得則行星度數也，余則度餘也。
The five wanderers are Jupiter, Mars, Saturn, Venus, and Mercury under their classical names. Each planet alternates between slow direct motion, fast motion, halts, and retrogrades. At creation’s dawn, when heaven and earth cleaved apart, the luminaries stacked together in the Xingji asterism. Leaving that knot, they share the sky, overtaking one another through every kinematic phase. A planetary conjunction means the planet shares the sun’s lodge and longitude. One synodic period runs from one conjunction to the next. Reduce synodic days against the civil year to derive the mean conjunction rate and cycle count. Those two ratios generate every divisor used below. The conjunction-month divisor equals 19 times the cycle conjunction count. The day-degree divisor is 1,843 times the conjunction count. Multiply 235 by the synodic year count for the fractional-month numerator. Divide that product by the conjunction-month divisor for whole months plus a remainder. Scale month count by 134,630 and divide by 4,559 for the sexagenary day line. Modulo sixty leaves the planetary conjunction day index. What remains after removing sexagenary cycles is the fractional new moon. Combine fractional month and fractional day terms to learn how deep into the month the conjunction falls. Reduce the leftover with divisor 47 for the intramonth fraction. The gap to a full day divisor is the void fraction at conjunction. Multiply the tabulated dipper part 455 by the conjunction count for arc fine structure. For superior planets subtract synodic cycles from years, scale by 673,150, divide by the day-degree divisor. For inferior planets multiply years by the full revolution and divide by the same day-degree divisor.
150
木：合終歲數，一千二百五十五。
Jupiter mean synodic constant: 1,255 years per cycle.
151
合終合數，一千一百四十九。
Jupiter completes 1,149 conjunctions in that period.
152
合月法，二萬一千八百三十一。
Jupiter conjunction-month divisor: 21,831.
153
日度法，二百一十一萬七千六百七。
Jupiter day-degree divisor: 2,117,607.
154
合月數。 一十三。
Label: whole conjunction months. The integer part is thirteen.
155
月餘，一萬一千一百二十二。
Fractional month remainder: 11,122.
156
朔大餘，二十三。
Sexagenary day index twenty-three at conjunction.
157
朔小餘，四千九十三。
Fractional day 4,093 at conjunction.
158
入月日，一十五。
Conjunction falls on the fifteenth civil day of the lunation.
159
日餘，一百九十九萬五千六百六十四。
Intramonth fractional remainder 1,995,664.
160
朔虛分，四百六十六。
Void gap to full divisor: 466.
161
斗分，五十二萬二千七百九十五。
Dipper fine arc: 522,795.
162
行星度，三十三。
Mean motion: thirty-three degrees per cycle segment.
163
度餘，一百四十七萬二千八百六十九。
Arc remainder: 1,472,869.
164
火：合終歲數，五千一百五。
Mars synodic year constant: 5,105.
165
合終合數，二千三百八十八。
Mars conjunction count per grand cycle: 2,388.
166
合月法，四萬五千三百七十二。
Mars conjunction-month divisor: 45,372.
167
日度法，四百四十萬一千八十四。
Mars day-degree divisor: 4,401,084.
168
合月數，二十六。
Whole conjunction months: twenty-six.
169
月餘，二萬三。
Month remainder: 23,000.
170
朔大餘，四十七。
Conjunction day index forty-seven.
171
朔小餘，三千六百二十七。
Fractional conjunction part 3,627.
172
入月日，一十三。
Conjunction occurs on the thirteenth day of the lunation.
173
日餘，三百五十八萬五千二百三十。
Intramonth fractional remainder: 3,585,230.
174
朔虛分，九百三十二。
Void fraction to complete the day divisor: 932.
175
斗分，一百八萬六千五百四十。
Dipper arc fine structure: 1,086,540.
176
行星度，五十。
Mean heliocentric advance: fifty degrees in this segment.
177
度餘，一百四十一萬二千一百五十。
Arc remainder after the whole degrees: 1,412,150.
178
土：合終歲數，三千九百四十三。
Saturn synodic cycle constant: 3,943 years.
179
合終合數，三千八百九。
Saturn completes 3,809 conjunctions in that span.
180
合月法，七萬二千三百七十一。
Saturn conjunction-month divisor: 72,371.
181
日度法，七百一萬九千九百八十七。
Saturn day-degree divisor: 7,019,987.
182
合月數，一十二。
Whole conjunction months: twelve.
183
月餘，五萬八千一百五十三。
Fractional month remainder: 58,153.
184
朔大餘，五十四。
Conjunction sexagenary index: fifty-four.
185
朔小餘，一千六百七十四。
Fractional conjunction parts: 1,674.
186
入月日，二十四。
The planet enters on the twenty-fourth civil day.
187
日余，六十七萬五千三百六十四。
Day remainder within the month: 675,364.
188
朔虛分，二千八百八十五。
Void fraction: 2,885.
189
斗分，一百七十三萬三千九十五。
Dipper fine arc: 1,733,095.
190
行星度，一十二。
Whole-degree advance: twelve.
191
度餘，五百九十六萬二千二百五十六。
Arc remainder: 5,962,256.
192
金：合終歲數，一千九百七。
Venus synodic year constant: 1,907.
193
合終合數，二千三百八十五。
Venus completes 2,385 conjunctions in the grand cycle.
194
合月法，四萬五千三百一十五。
Venus conjunction-month divisor: 45,315.
195
日度法，四百三十九萬五千五百五十五。
Venus day-degree divisor: 4,395,555.
196
合月數，九。
Whole conjunction months: nine.
197
月餘，四萬三百一十。
Month remainder: 40,310.
198
朔大餘，二十五。
Conjunction day index twenty-five.
199
朔小餘，三千五百三十五。
Fractional conjunction: 3,535 parts.
200
入月日，二十七。
Conjunction falls on the twenty-seventh day.
201
日余，十九萬四千九百九十。
Intramonth remainder: 194,990.
202
朔虛分，一千二十四。
Void fraction: 1,024.
203
斗分，一百八萬五千一百七十五。
Dipper fine arc: 1,085,175.
204
行星度，二百九十二。
Heliocentric arc for this line: 292 degrees.
205
度余，十九萬四千九百九十。
Degree remainder matches the day remainder line.
206
水：合終歲數，一千八百七十。
Mercury synodic year constant: 1,870.
207
合終合數，一萬一千七百八十九。
Mercury completes 11,789 conjunctions in the cycle.
208
合月法，二十二萬三千九百九十一。
Mercury conjunction-month divisor: 223,991.
209
日度法，二千一百七十二萬七千一百二十七。
Mercury day-degree divisor: 21,727,127.
210
合月數，一。
Whole months entered: one.
211
月餘，二十一萬五千四百五十九。
Fractional month: 215,459.
212
朔大餘，二十九。
Conjunction day index twenty-nine.
213
朔小餘，二千四百一十九。
Fractional conjunction: 2,419.
214
入月日，二十八。
Mercury appears on the twenty-eighth day.
215
日餘，二千三十四萬四千二百六十一。
Day remainder: 20,344,261.
216
朔虛分，二千一百四十。
Void fraction: 2,140.
217
斗分，五百三十六萬三千九百九十五。
Dipper fine arc: 5,363,995.
218
行星度，五十七。
Arc count: fifty-seven degrees.
219
度餘，二千三十四萬千三百六十一。
Degree remainder aligns with the parallel day line at 20,344,261 parts.
220
推五星術曰：置壬辰元以來盡所求年，以合終合數乘之，滿合終歲數得一，名積合，不盡名為合餘。 以合終合數減合餘，得一者星合往年，得二者合前往年，無所得，合其年。 餘以減合終合數，為度分。 金、水積合，偶為晨，奇為夕。
To count synodic events since the epoch: multiply years by conjunctions per cycle and divide by conjunctions per year for the quotient and remainder. Compare the remainder to the cycle length to tell whether the latest conjunction fell this year or one or two years earlier. The complement of that remainder yields the arc still to complete the synodic template. Inferior planets alternate dawn appearances on even counts and dusk appearances on odd counts.
221
推五星合月：以月數、月餘各乘積合，余滿合月法從月，為積月，不盡為月餘。 以紀月除積月，所得算外，所入紀也，余為入紀月。 副以章閏乘之，滿章月得一為閏，以減入紀月，餘以歲中去之，餘為入歲月，命以天正起，算外，星合月也。 其在閏交際，以朔禦之。
Scale the tabulated month numerator and denominator by the accumulated conjunction index. Divide by 22,795 to learn the cycle index and intra-cycle month offset. Apply the seven-in-nineteen leap rule, strip full civil years, and read the month from the winter count. At leap junctions anchor the count to the nearest conjunction rule.
222
推合月朔：以通數乘入紀月，滿日法得一，為積日，不盡為小餘。 以六十去積日，余為大餘，命以所入紀，算外，星合朔日也。
Convert intra-cycle months to days with factor 134,630 ÷ 4,559. Modulo sixty names the conjunction’s sexagenary day.
223
推入月日：以通數乘月餘，合月法乘朔小餘，並之，通法約之，所得滿日度法得一，則星合入月日也，不滿日餘。 命日以朔，算外，入月日也。
Blend fractional month and day terms, reduce by 47, then divide by the planet’s day-degree divisor. Index the result from conjunction to mark the day within the lunation.
224
推星合度：以周天乘度分，滿日度法得一為度，不盡為餘。 命以牛前五度起，算外，星所合度也。
Multiply the arc template by 673,150 and divide by the day-degree divisor. Step the result through the lodge system beginning five degrees before Ox.
225
求後全月：以月數加入歲月，以余加月餘，余滿合月法得一月。 月不滿歲中，即在其年； 滿去之，有閏計焉，餘為後年； 再滿，在後二年。 金、水加晨得夕，加夕得晨也。
Advance month tallies and carry at the conjunction-month divisor. A sub-year month count keeps the event inside the current year. Overflow past twelve civil months pushes the conjunction into the next year, with leap adjustments noted. Two full overflows place the event two years ahead. Inferior planets flip between dawn and dusk apparitions each synodic step.
226
求後合朔：以朔大、小餘數加合朔月大、小餘，其月餘上成月者，又加大餘二十九，小餘二千四百一十九，小餘滿日法從大餘，命如前法。
Propagate lunation fractions with the standard 29-day, 2,419-part increment.
227
求後入月日：以入月日、日餘加入月日及余，余滿日度法得一。 其前合朔小餘滿其虛分者，去一日； 後小餘滿二千四百一十九以上，去二十九日； 不滿，去三十日，其餘則後合入月日，命以朔。 求後合度，以度數及分，如前合宿次命之。
Sum the intramonth lines and carry at the day-degree divisor. Borrow a day when the prior void fraction is saturated. Large fractional tails trigger a 29-day rollback. Otherwise subtract thirty civil days and read from conjunction. Advance heliocentric longitude with the same lodge-walking rule as before.
228
木：晨與日合，伏，順，十六日九十九萬七千八百三十二分行星二度百七十九萬五千二百三十八分，而晨見東方，在日後。 順，疾，日行五十七分之十一，五十七日行十一度。 順，遲，日行九分，五十七日行九度而留。 不行二十七日而旋。 逆，日行七分之一，八十四日退十二度而復留。 二十七日復遲，日行九分，五十七日行九度而復順。 疾，日行十一分，五十七日行十一度，在日前，夕伏西方。 順，十六日九十九萬七千八百三十二分行星二度百七十九萬五千二百三十八分，而與日合。 凡一終，三百九十八日百九十九萬五千六百六十四分，行星三十三度百四十七萬二千八百六十九分。
Jupiter’s dawn apparition opens after sixteen days plus the tabulated fraction while moving about two degrees direct while still invisible. Fast direct motion is 11/57° per day, covering 11° in fifty-seven days. Slow direct motion is 9 fen daily for fifty-seven days until station. After a twenty-seven-day halt it turns retrograde. Retrograde speed is 1/7° per day, yielding 12° of retreat in eighty-four days before a second station. Another twenty-seven-day pause leads back to slow direct motion at nine fen per day. Accelerating direct motion brings it ahead of the sun for an evening disappearance. The closing hidden arc mirrors the opening sixteen-day segment before inferior conjunction. One complete synodic cycle spans 398 days plus the listed fraction and advances 33° with remainder.
229
火：晨與日合，伏，七十二日百七十九萬二千六百一十五分行星五十六度百二十四萬九千三百四十五分，而晨見東方，在日後。 順，日行二十三分之十四，百八十四日行百一十二度。 更順，遲，日行十二分，九十二日行四十八度而留。 不行十一日而旋。 逆，日行六十二分之十七，六十二日退十七度而復留。 十一日復順，遲，日行十二分，九十二日行四十八度而復疾。 日行十四分，百八十四日行百一十二度，在日前，夕伏西方。 順，七十二日百七十九萬二千六百一十五分行星五十六度百二十四萬九千三百四十五分，而與日合。 凡一終，七百八十日三百五十八萬五千二百三十分，行星四百一十五度二百四十九萬八千六百九十分。
Mars emerges after seventy-two days of direct invisibility while advancing fifty-six degrees. Fast Mars averages 14/23° per day for 184 days and 112°. Slow direct motion is twelve fen daily for ninety-two days until station. An eleven-day standstill precedes retrograde motion. Retrograde Mars loses seventeen degrees in sixty-two days at 17/62° per day. Eleven days later slow direct motion resumes, returning to fast motion after another ninety-two days. Fast direct motion carries it 112° in 184 days until evening invisibility. The evening arc closes with the same seventy-two-day segment as the morning opening. Mars’s full synodic cycle lasts 780 days with the tabulated fractional tail and 415° of travel.
230
土：晨與日合，伏，十九日三百八十四萬七千六百七十五分半行星二度六百四十九萬一千一百二十一分半，而晨見東方，在日後。 順，行百七十二分之十三，八十六日行六度半而留。 不行三十二日半而旋。 逆，日行十七分之一，百二日退六度而復留。 不行三十二日半復順，日行十三分，八十六日行六度半，在日前，夕伏西方。 順，十九日三百八十四萬七千六百七十五分半行星二度六百四十九萬一千一百二十一分半，而與日合。 凡一終，三百七十八日六十七萬五千三百六十四分，行星十二度五百九十六萬二千二百五十六分。
Saturn’s dawn visibility follows nineteen days of tiny direct motion while still hidden. Direct Saturn creeps 13/172° per day for 86 days to a station near 6.5°. A thirty-two-and-a-half-day pause initiates retrograde motion. Retrograde motion at 1/17° per day removes six degrees in 102 days. The outbound arc mirrors the inbound slow direct leg before evening disappearance. The cycle closes with the same nineteen-day micro-step as at morning conjunction. Saturn completes one synodic period in 378 days plus fraction, advancing only twelve degrees of sky.
231
金：晨與日合，伏，六日退四度，而晨見東方，在日後而逆。 遲，日行五分之三，十日退六度。 留，不行七日而旋。 順，遲，日行四十五分之三十三，四十五日行三十三度而順。 疾，日行一度九十一分之十四，九十一日行百五度而順。 益疾，日行一度九十一分之二十一，九十一日行百一十二度，在日後，而晨伏東方。 順，四十二日十九萬四千九百九十分行星五十二度十九萬四千九百九十分，而與日合。 一合，二百九十二日十九萬四千九百九十分，行星如之。
Venus first drops four degrees in six hidden retrograde days before dawn apparition. Slow retrograde motion removes six degrees over ten days at three fifths of a fen daily. After a seven-day halt Venus turns from station. Slow direct motion covers 33° in forty-five days at 33/45° per day. Fast direct leg advances 105° in ninety-one days. Peak direct speed reaches 112° before morning disappearance. The closing direct arc lasts forty-two days plus the tabulated fraction for 52°. A single synodic period spans 292 days with the listed fractional tail.
232
金：夕與日合，伏，順，四十二日十九萬四千九百九十分行星五十二度十九萬四千九百九十分，而夕見西方，在日前。 順，疾，日行一度九十一分之二十一，九十一日行百一十二度而更順。 遲，日行一度十四分，九十一日行百五度而順。 益遲，日行四十五分之三十三，四十五日行三十三度而留。 不行七日而旋。 逆，日行五分之三，十日退六度，在日前，夕伏西方。 逆，六日退四度，而與日合。 凡再合一終，五百八十四日三十八萬九千九百八十分，行星如之。
Evening apparition begins after the same forty-two-day direct segment while west of the sun. Evening fast direct motion mirrors the morning leg. Slowing to one degree fourteen fen daily for ninety-one days. Final deceleration leads to station after forty-five days. Another week-long standstill precedes retrograde motion. Evening retrograde removes six degrees in ten days before invisibility. Six retrograde days erase four degrees down to inferior conjunction. Two synodic cycles total 584 days with the stated remainder.
233
水：晨與日合，伏，十一日退七度，而晨見東方，在日後。 逆，疾，一日退一度而留。 不行一日而旋。 順，遲，日行八分之七，八日行七度而順。 疾，日行一度十八分之四，十八日行二十二度，在日後，晨伏東方。 順，十八日二千三十四萬四千二百六十一分行星三十六度二千三十四萬四千二百六十一分，而與日合。 凡一合，五十七日二千三十四萬四千二百六十一分，行星如之。
Mercury’s dawn arc opens with eleven retrograde degrees. Rapid retrograde cancels a full degree in a single day before halting. A one-day station separates retrograde from direct motion. Slow direct motion adds seven degrees in eight days. Accelerating direct motion reaches twenty-two degrees before morning disappearance. The hidden direct arc closes in eighteen days with the tabulated fractional degrees. Mercury’s synodic period is fifty-seven days plus the listed fraction.
234
水：夕與日合，伏，十八日二千三十四萬四千二百六十一分行星三十六度二千三十四萬四千二百六十一分，而夕見西方，在日前。 順，疾，日行一度十八分之四，十八日行二十二度而更順。 遲，日行八分之七，八日行七度而留。 不行一日而旋。 逆，一日退一度，在日前，夕伏西方。 逆，十一日退七度，而與日合。 凡再合一終，百一十五日千八百九十六萬一千三百九十五分，行星如之。
Evening apparition follows the same eighteen-day segment while east of the sun. Evening fast direct motion repeats the eighteen-day pattern. Deceleration ends in an eight-day station. A single idle day marks the retrograde pivot. Fast retrograde erases a degree daily until evening invisibility. Eleven retrograde days remove seven degrees to inferior conjunction. Mercury’s double synodic cycle totals 115 days with remainder.
235
五星曆步術
Heading: stepwise planetary computation.
236
以法伏日度餘加星合日度余，余滿日度法得一從全，命之如前，得星見日及度餘也。 以星行分母乘見度分，如日度法得一，分不盡，半法以上亦得一，而日加所行分，分滿其母得一度。 逆順母不同，以當行之母乘故分，如故母而一，當行分也。 留者承前，逆則減之，伏不盡度，除斗分，以行母為率。 分有損益，前後相禦。
Sum the tabulated hidden arc to the conjunction line, carrying at the day-degree divisor. Convert fractional motion using each planet’s denominator with rounding at half divisor. Rescale fractional steps whenever retrograde and direct legs use different divisors. Station lines continue prior totals; retrograde subtracts; hidden legs drop dipper fractions. Sequential segments borrow or lend fractional parts across the synodic table.
237
武帝侍中平原劉智，以鬥曆改憲，推《四分法》，三百年而減一日，以百五十為度法，三十七為斗分。 推甲子為上元，至泰始十年，歲在甲午，九萬七千四百一十一歲，上元天正甲子朔夜半冬至，日月五星始于星紀，得元首之端。 飾以浮說，名為《正曆》。
Liu Zhi revised the calendrical constants, tightening the quarter-day system with divisor 150 and dipper fraction 37. His epoch placed a grand conjunction of luminaries at winter solstice 274 CE minus 97,411 years. He packaged the scheme rhetorically as the Zhengli calendar.
238
當陽侯杜預著《春秋長曆》，說云：
Du Yu’s Chunqiu Changli prefaces a methodological note.
239
日行一度，月行十三度十九分之七有奇，日官當會集此之遲疾，以考成晦朔，以設閏月。 閏月無中氣，而北斗邪指兩辰之間，所以異於他月。 積此以相通，四時八節無違，乃得成歲，其微密至矣。 得其精微，以合天道，則事敘而不愆。 故《傳》曰：「閏以正時，時以作事。」 然陰陽之運，隨動而差，差而不已，遂與曆錯。 故仲尼、丘明每于朔閏發文，蓋矯正得失，因以宣明歷數也。
Du Yu urges averaging solar and lunar speeds to regulate conjunctions and intercalations. Leap months sit without major qi while the Dipper handle straddles two celestial positions. Stacking such checks preserves the agrarian year with maximal precision. Matching heaven’s subtle ticks keeps human schedules aligned. Hence the classic line that leap months fix seasons and seasons shape labor. Natural drift means any fixed table eventually diverges from the sky. Confucius and Zuo Qiuming foregrounded calendrical notes to expose systematic error.
240
劉子駿造《三正曆》以修《春秋》，日蝕有甲乙者三十四，而《三正曆》惟得一蝕，比諸家既最疏。 又六千餘歲輒益一日，凡歲當累日為次，而故益之，此不可行之甚者。
Liu Xin’s scheme failed thirty-three of thirty-four dated eclipses in the annals. Its long-cycle leap stole whole days without astronomical warrant.
241
自古已來，諸論《春秋》者多違謬，或造家術，或用黃帝已來諸曆，以推經傳朔日，皆不諧合。 日蝕於朔，此乃天驗，《經傳》又書其朔蝕，可謂得天，而劉賈諸儒說，皆以為月二日或三日，公違聖人明文，其弊在於守一元，不與天消息也。
Later scholars forced mismatched epochal calendars onto the Chunqiu text. Du Yu faults commentators who moved eclipses off conjunction to save flawed calendars.
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餘感《春秋》之事，嘗著《曆論》，極言曆之通理。 其大指曰：「天行不息，日月星辰各運其舍，皆動物也。 物動則不一，雖行度有大量可得而限，累日為月，累月為歲，以新故相涉，不得不有毫末之差，此自然之理也。 故春秋日有頻月而蝕者，有曠年不蝕者，理不得一，而算守恆數，故曆無不有先後也。 始失於毫毛，而尚未可覺，積而成多，以失弦望晦朔，則不得不改憲以從之。 《書》所謂「欽若昊天，曆象日月星辰」，《易》所謂「治曆明時」，言當順天以求合，非為合以驗天者也。 推此論之，春秋二百餘年，其治曆變通多矣。 雖數術絕滅，遠尋《經傳》微旨，大量可知，時之違謬，則《經傳》有驗。 學者固當曲循《經傳》月日、日蝕，以考晦朔，以推時驗； 而皆不然，各據其學，以推春秋，此無異於度己之跡，而欲削他人之足也。
Du Yu authored a calendar treatise to defend conjunction-based interpretation. Heaven’s bodies are in constant motion, each wheeling through its mansions. Discrete month-year stacking inevitably accumulates tiny drift. Eclipse frequency varies, so rigid constants must eventually fail. Neglected fractions snowball until only calendar reform recovers conjunctions. Classical dicta demand fitting the calendar to the sky, not the reverse. Lu state’s methods evolved across the Chunqiu centuries. Internal textual evidence exposes which calendars fit which reigns. Scholars should privilege manuscript dates over private algorithms. Du Yu likens partisan calendar claims to self-referential violence.
243
餘為《曆論》之後，至咸寧中，善算者李修、卜顯，依論體為術，名《乾度曆》，表上朝廷。 其術合日行四分數而微增月行，用三百歲改憲之意，二元相推，七十餘歲，承以強弱，強弱之差蓋少，而適足以遠通盈縮。 時尚書及史官，以《乾度》與《泰始曆》參校古今記注，《乾度曆》殊勝《泰始曆》，上勝官曆四十五事。 今其術具存。 又並考古今十曆以驗《春秋》，知《三統》之最疏也。
Li Xiu and Bu Xian codified Du Yu’s ideas as the Qiandu system. The Qiandu scheme layered gentle solar-lunar tweaks across linked epochs. Court validation favored Qiandu in forty-five head-to-head tests. The Qiandu algorithm remains fully documented. Comparative testing ranked Liu Xin’s San tong worst.
244
《春秋》大凡七百七十九日，三百九十三《經》，三百八十六《傳》。 其三十七日食。 三無甲乙。
Du Yu counts 779 dated lines split between text and Zuo. Thirty-seven lines note solar eclipses. Three eclipse entries omit the sexagenary day.
245
《黃帝》曆得四百六十六日，一蝕。
The Huangdi table aligns only one eclipse in 466 test days.
246
《顓頊曆》得五百九日，八蝕。
The Zhuanxu scheme scores eight hits in 509 days.
247
《夏曆》得五百三十六日，十四蝕。
The Xia calendar nets fourteen hits over 536 days.
248
《真夏曆》得四百六十六日，一蝕。
The revised Xia table matches one eclipse.
249
《殷曆》得五百三日，十三蝕。
The Yin calendar scores thirteen hits in 503 days.
250
《周曆》得五百六日，十三蝕。
The Zhou calendar yields thirteen hits in 506 days.
251
《真周曆》得四百八十五日，一蝕。
The refined Zhou line scores one hit.
252
《魯曆》得五百二十九日，十三蝕。
Lu’s official calendar scores thirteen hits in 529 days.
253
《三統曆》得四百八十四日，一蝕。
Liu Xin’s system manages one hit in 484 days.
254
《乾象曆》得四百九十五日，七蝕。
Liu Hong’s Qianxiang scores seven hits in 495 days.
255
《泰始曆》得五百一十日，十九蝕。
The Jin Taishi calendar reaches nineteen hits in 510 days.
256
《乾度曆》得五百三十八日，十九蝕。
Qiandu edges slightly ahead with nineteen hits in 538 days.
257
今《長曆》得七百四十六日，三十三蝕。 失三十三日，《經傳》誤； 四日蝕，三無甲乙。
Du’s own reconstruction scores thirty-three hits across 746 days. Thirty-three day mismatches stem from textual corruption, not the model. Four eclipses are misdated; three entries lack sexagenary tags.
258
漢末，宋仲子集七曆以考《春秋》，案其夏、周二曆術數，皆與《藝文志》所記不同，故更名為《真夏》、《真周曆》也。
Song Zhongzi distinguished authentic Xia and Zhou reconstructions from Han catalogue versions.
259
穆帝永和八年，著作郎琅邪王朔之造《通曆》，以甲子為上元，積九萬七千年，四千八百八十三為紀法，千二百五為斗分，因其上元為開闢之始。
Wang Shuozhi’s Tongli used a mythic deep epoch with custom divisors.
260
後秦姚興時，當孝武太元九年，歲在甲申，天水姜岌造《三紀甲子元曆》，其略曰：「治曆之道，必審日月之行，然後可以上考天時，下察地化。 一失其本，則四時變移。 故仲尼之作《春秋》，日以繼月，月以繼時，時以繼年，年以首事，明天時者人事之本，是以王者重之。 自皇羲以降，暨于漢魏，各自製曆，以求厥中。 考其疏密，惟交會薄蝕可以驗之。 然書契所記，惟《春秋》著日蝕之變，自隱公訖於哀公，凡二百四十二年之間，日蝕三十有六，考其晦朔，不知用何曆也。 班固以為《春秋》因《魯曆》，《魯曆》不正，故置閏失其序。 魯以閏餘一之歲為蔀首，檢《春秋》置閏不與此蔀相符也。 《命曆序》曰：孔子為治《春秋》之故，退修殷之故曆，使其數可傳於後。 如是，《春秋》宜用《殷曆》正之。 今考其交會，不與《殷曆》相應，以《殷曆》考《春秋》，月朔多不及其日，又以檢《經》，率多一日，《傳》率少一日。 但《公羊》、《經》、《傳》異朔，於理可從，而《經》有蝕朔之驗，《傳》為失之也。 服虔解《傳》用太極上元，太極上元乃《三統曆》劉歆所造元也，何緣施於《春秋》？ 於《春秋》而用《漢曆》，於義無乃遠乎？ 《傳》之違失多矣，不惟斯事而已。 襄公二十七年冬十有一月乙亥朔，日有蝕之。 《傳》曰：'辰在申，司曆過，再失閏也。 '考其去交分，交會應在此月，而不為再失閏也。 案歆曆於《春秋》日蝕一朔，其餘多在二日。 因附《五行傳》，著朓與側匿之說云：春秋時諸侯多失其政，故月行恆遲。 歆不以曆失天，而為之差說。 日之食朔，此乃天驗也，而歆反以曆非此，冤天而負時曆也。 杜預又以為周衰世亂，學者莫得其真，今之所傳七曆，皆未必是時王之術也。 今誠以七家之曆，以考古今交會，信無其驗也，皆由斗分疏之所致也。 《殷曆》以四分一為斗分，《三統》以一千五百三十九分之三百八十五為斗分，《乾象》以五百八十九分之一百四十五為斗分，今《景初》以一千八百四十三分之四百五十五為斗分，疏密不同，法數各異。 《殷曆》斗分粗，故不施於今。 《乾象》斗分細，故不得通于古。 《景初》斗分雖在粗細之中，而日之所在乃差四度，日月虧已，皆不及其次，假使日在東井而蝕，以月驗之，乃在參六度，差違乃爾，安可以考天時人事乎？ 今治新曆，以二千四百五十一分之六百五為斗分，日在鬥十七度，天正之首，上可以考合於《春秋》，下可以取驗於今世。 以之考《春秋》三十六蝕，正朔者二十有五，蝕二日者二，蝕晦者二，誤者五，凡三十三蝕，其餘蝕經無日諱之名，無以考其得失。 圖緯皆云「三百歲鬥曆改憲」。 以今新施於春秋之世，日蝕多在朔，春秋之世，下至於今，凡一千餘歲，交會弦望故進退於三蝕之間，此法乃可永載用之，豈三百歲鬥曆改憲者乎？
Jiang Ji’s preface stresses observing sun and moon before tuning the civil year. If solar and lunar roots err, seasonal markers wander. Confucius chained day-month-season-year to show royal governance depends on heaven’s timetable. Every dynasty revised constants to chase the celestial middle. Only conjunctions and grazing eclipses falsify or confirm a scheme. The Chunqiu eclipse set spans 242 years with thirty-six events of uncertain calendrical backing. Ban Gu blamed Lu’s irregular leap placement. Lu’s supposed leap rule fails against the annals’ actual leap months. The Ming li xu credits Confucius with an Yin-based editorial calendar. That tradition would privilege the Yin table for Chunqiu dates. Empirical checks show Yin’s conjunctions misalign with both text layers. Eclipse evidence vindicates the Classic’s conjunctions over the Zuo variants. Fu Qian’s borrowing of Liu Xin’s Han-era origin is anachronistic. Applying Eastern Han mathematics to Spring and Autumn chronology is historically strained. The Zuo text is flawed on more than calendrics alone. The record cites the eleventh month’s yihai day, year 546 BCE, for a solar eclipse (opening a longer gloss). The Zuo gloss: the branch mismatch shows the official calendar missed two leap insertions. Checking the nodal residue shows the month placement is wrong, not the leap count. Liu Xin’s table hits only one true conjunction eclipse; most land a day late. Liu Xin blamed politics for systematic lunar lag instead of adjusting his fractions. He invented moralized excuses rather than fixing constants. Conjunction eclipses prove his model wrong, yet he impugned the sky. Du Yu doubted any extant system truly matched Zhou court practice. Comparative eclipse work exposes loose dipper fractions across all seven schools. Historical dipper fractions range from 1/4 to the Jingchu ratio, each implying different mean motions. Yin’s crude quarter cannot serve modern reckoning. Qianxiang’s tight fraction fails deep retrotests. Even Jingchu mislocates solar longitude by several degrees, vitiating eclipse predictions. The proposed 605/2451 ratio anchors the sun at 17° in Dipper at year head for dual classical and modern fit. The new table scores twenty-five exact conjunctions among thirty-six, with several edge cases and five clear misses. Apocrypha prescribe a three-century calendrical overhaul. The author argues his scheme tracks eclipses across millennia, not just per apocryphal 300-year cycle.
261
元法，七千三百五十三。
The new yuanfa is 7,353.
262
紀法，二千四百五十一。
Cycle divisor jifa: 2,451.
263
通數，十七萬九千四十四。
Common multiple: 179,044.
264
日法，六千六十二。
Day divisor: 6,062.
265
月周，三萬二千七百六十六。
Lunar circuit constant: 32,766.
266
氣分，萬二千八百六十。
Solar-term fraction numerator: 12,860.
267
元月，九萬九百四十五。
Origin-month constant: 90,945.
268
紀月，三萬三百一十五。
Cycle-month total: 30,315.
269
沒分，四萬四千七百六十一。
Hidden-month numerator: 44,761.
270
沒法，六百四十三。
Oblivion divisor: 643.
271
斗分，六百五。
Dipper fractional part: 605.
272
周天，八十九萬五千二百二十。 一名紀日。
Full revolution modulus: 895,220. That constant doubles as the cycle-day line.
273
章月，二百三十五。
Months per cycle: 235.
274
章歲，十九。
Intercalation cycle years: nineteen.
275
章閏，七。
Seven leap months each cycle.
276
歲中，十二。
Twelve months in a standard year.
277
會數，四十七。 日月八百九十三歲，凡四十七會，分盡。
Lunisolar meeting constant: forty-seven. 893 solar years contain exactly forty-seven lunisolar beat cycles.
278
氣中，十二。
Twelve qi per civil year.
279
甲子紀交差，九千一百五十七。
Jiazi-cycle nodal increment: 9,157.
280
甲申紀交差，六千三百三十七。
Jiashen-cycle nodal increment: 6,337.
281
甲辰紀交差，三千五百一十七。
Jiachen-cycle nodal increment: 3,517.
282
周半，一百二十七。
Half-revolution constant: 127.
283
朔望合數，九百四十一。
Conjunction–opposition sum: 941.
284
會歲，八百九十三。
Beat-cycle length in years: 893.
285
會月，萬一千四十五。
Months per meeting cycle: 11,045.
286
小分，二千一百九十六。
Micro-fraction parameter: 2,196.
287
章數，一百二十九。
Chapter count: 129.
288
小分，二千一百八十三。
Alternate micro-fraction: 2,183.
289
周閏大分，七萬六千二百六十九。
Half-orbit leap numerator: 76,269.
290
曆周，四十四萬七千六百一十。 半周天
Calendrical week modulus: 447,610. Label: half-celestial arc.
291
會分，三萬八千一百三十四。
Meeting-cycle fraction: 38,134.
292
差分，一萬一千九百八十六。
Difference numerator: 11,986.
293
會率，一千八百八十二。
Meeting divisor rate: 1,882.
294
小分法，二千二百九。
Micro-fraction modulus: 2,209.
295
入交限，一萬一百四。
Lunar-node limit: 10,104.
296
小周，二百五十四。
Minor cycle: 254.
297
甲子紀差率，四萬九千一百七十八。
Jiazi anomaly increment: 49,178.
298
甲申紀差率，五萬八千二百三十一。
Jiashen anomaly increment: 58,231.
299
甲辰紀差率，六萬七千二百八十四。
Jiachen anomaly increment: 67,284.
300
通周，十六萬七千六十三。
Passage-cycle modulus: 167,063.
301
周日日餘，三千三百六十二。
Weekly day remainder: 3,362.
302
周虛，二千七百一。
Weekly complement: 2,701.
303
五星約法，據出見以為正，不系於元本。 然則算步究于元初，約法施於今用，曲求其趣，則各有宜，故作者兩設其法也。 岌以月食檢日宿度所在，為曆術者宗焉。 又著《渾天論》，以步日于黃道，駁前儒之失，並得其中矣。
Planetary short tables anchor on observed elongations rather than deep epoch. Full and abbreviated planetary algorithms serve different practical needs. Jiang Ji’s eclipse triangulation of solar longitude became authoritative for later specialists. His Huntian lun corrected prior armillary theory by tracking the ecliptic accurately.