1
志第三曆下
Treatise 3: Calendar, Part 2.
2
《元嘉曆法》：
The Yuanjia Calendar Method:
3
上元庚辰甲子紀首至太甲元年癸亥，三千五百二十三年，至元嘉二十年癸未，五千七百三年，算外。
From the gengchen-year superior-origin era-head on a jiazi day to the first year of King Taiai (guihai) is 3,523 years; from there to the twentieth year of Yuanjia (guiwei) is 5,703 years in all, exclusive in the count.
4
元法，三千六百四十八。
The origin divisor (yuanfa): 3,648.
5
章歲，十九。
Rule years (zhang sui): nineteen.
6
紀法，六百八。
The era divisor (jifa): 608.
7
章月，二百三十五。
Rule months (zhang yue): 235.
8
紀月，七千五百二十。
Months per full era cycle (jiyue): 7,520.
9
章閏，七。
Rule intercalations (zhang run): seven.
10
紀日，二十二萬二千七十。
Days per era cycle (jiri): 222,070.
11
度分，七十五。
Degree parts (du fen): seventy-five.
12
度法，三百四。
The degree divisor (dufa): 304.
13
氣法，二十四。
The qi divisor (qifa): twenty-four.
14
餘數，一千五百九十五。
The remainder number (yu shu): 1,595.
15
歲中，十二。
Major solar terms per year (sui zhong): twelve.
16
日法，七百五十二。
The day divisor (rifa): 752.
17
沒餘，三十六。
The extinction remainder (mo yu): thirty-six.
18
通數，二萬二千二百七。
The communication number (tong shu): 22,207.
19
通法，四十七。
The communication divisor (tongfa): forty-seven.
20
沒法，三百一十九。
The extinction divisor (mofa): 319.
21
月周，四千六十四。
The lunar circuit (yue zhou): 4,064.
22
周天，十一萬一千二十五。
The celestial circumference (zhoutian): 111,025.
23
通周，二萬七百二十一。
The communication circuit (tong zhou): 20,721.
24
周日日餘，四百一十七。
The circuit-day remainder (zhouri ri yu): 417.
25
周虛，三百三十五。
The circuit void (zhou xu): 335.
26
會數，一百六十。
The coincidence number (hui shu): 160.
27
交限數，八百五十九。
The crossing-limit number (jiao xian shu): 859.
28
會月，九百二十九。
Coincidence months (hui yue): 929.
29
朔望合數，八十。
The new- and full-moon conjunction number (shuowang he shu): 80.
30
甲子紀第一遲疾差一萬七千六百六十三，交會差八百七十七
Jiazi Era 1: lunar anomaly offset 17,663; nodal coincidence offset 877.
31
甲戌紀第二遲疾差三千四十三，交會差二百七十九
Jiaxu Era 2: lunar anomaly offset 3,043; nodal coincidence offset 279.
32
甲申紀第三遲疾差九千一百四十四，交會差六百二十
Jiashen Era 3: lunar anomaly offset 9,144; nodal coincidence offset 620.
33
甲午紀第四遲疾差一萬五千二百四十五，交會差二十二
Jiawu Era 4: lunar anomaly offset 15,245; nodal coincidence offset 22.
34
甲辰紀第五遲疾差六百二十五，交會差三百六十三
Jiachen Era 5: lunar anomaly offset 625; nodal coincidence offset 363.
35
甲寅紀第六遲疾差六千七百二十六，交會差七百四
Jiayin Era 6: lunar anomaly offset 6,726; nodal coincidence offset 704.
36
推入紀法：置上元庚辰盡所求年，以元法除之，不滿元法，以紀法除之，余不滿紀法，入紀年也。 滿法去之，得後紀。 入甲午紀壬辰歲來，至今元嘉二十年歲在癸末，二百三十一年，算外。
To find which era a year falls in, count from the gengchen superior origin through the target year, divide by 3,648, then divide the remainder by 608; the final remainder is the year within that era. When a divisor is fully exhausted, drop it and advance to the following era. From entry into the Jiawu Era (renchen year) down to the present the twentieth year of Yuanjia (guiwei) is 231 years, exclusive in the count.
37
推積月術：置入紀年數算外，以章月乘之，如章歲為積月，不盡為閏餘。 閏餘十二以上，其年閏。
For accumulated months, take the years within the current era (exclusive count), multiply by 235, divide by 19; the quotient is total months, the remainder the leap fraction. A leap year occurs when the intercalary remainder reaches twelve or higher.
38
推朔術：以通數乘積分，為朔積分，滿日法為積日，不盡為小餘。 以六旬去積日，不盡為大餘，命以紀，算外，所求年正月朔日也。
To compute conjunction instants, multiply accumulated months by 22,207; divide by 752 for whole days; the residue is the fractional day. Reduce the day count modulo sixty for the sexagenary index, then name the day against the era cycle: that is the first-month conjunction of the target year.
39
求次月，加大餘二十九，小餘三百九十九，小余滿日法從大餘，即次月朔也。 小餘三百五十三以上，其月大也。
For the following month add 29 to the day count and 399 to the fraction, carrying overflows by 752 into the day count. If the fractional part is 353 or greater, the month is a long thirty-day month.
40
推弦望法：加朔大餘七，小餘二百八十七，小分三，小分滿四從小余，小余滿日法從大餘，命如前，上弦日也。 又加之得望，又加之得下弦。
For the first quarter add 7 days, 287 parts, and 3 small parts to the new-moon remainder, carrying at 4 small parts and at 752 parts per day, then read off the date as before. Repeat the same addition to reach full moon, and once more for last quarter.
41
推二十四氣術：置入紀年算外，以餘數乘之，滿度法三百四為積沒，不盡為小餘。 以六旬去積沒，不盡為大餘，命以紀，算外，所求年雨水日也。 求次氣，加大餘十五，小餘六十六，小分十一，小分滿氣法從小余，小余滿度法從大餘，次氣日也。 雨水在十六日以後者，如法減之，得立春。
For the twenty-four solar terms, take years within the era (exclusive), multiply by 1,595, divide by 304 for whole cycles; the remainder is the fractional part. Reduce modulo sixty for the stem-branch index and name against the era: that gives Rain Water for the target year. For the following term add 15 days, 66 parts, and 11 small parts, carrying at 24 small parts and at 304 parts per day. When Rain Water would fall after the sixteenth, apply the prescribed reduction to obtain Beginning of Spring instead.
42
推閏月法：以閏余減章歲，餘以歲中乘之，滿章閏得一，數從正月起，閏所在也。 閏有進退，以無中氣禦之。
For the leap month, subtract the leap fraction from 19, multiply the result by 12, divide by 7; count forward from the first month by the quotient to locate the intercalary month. Leap placement may need adjustment; the month without a major solar term governs the decision.
43
立春正月節限數一百九十四，間數一百九十
Beginning of Spring (first-month solar term): limit number 194; interval number 190.
44
雨水正月中限數一百八十六，間數一百八十二
Rain Water (first-month mid-term): limit number 186; interval number 182.
45
驚蟄二月節限數一百七十七，間數一百七十二
Awakening of Insects (second-month solar term): limit number 177; interval number 172.
46
春分二月中限數一百六十七，間數一百六十二
Spring Equinox (second-month mid-term): limit number 167; interval number 162.
47
清明三月節限數一百五十八，間數一百五十四
Pure Brightness (third-month solar term): limit number 158; interval number 154.
48
穀雨三月中限數一百四十九，間數一百四十五
Grain Rain (third-month mid-term): limit number 149; interval number 145.
49
立夏四月節限數一百四十二，間數一百三十九
Beginning of Summer (fourth-month solar term): limit number 142; interval number 139.
50
小滿四月中限數一百三十六，間數一百三十四
Small Fullness (fourth-month mid-term): limit number 136; interval number 134.
51
芒種五月節限數一百三十三，間數一百三十二
Bearded Grain (fifth-month solar term): limit number 133; interval number 132.
52
夏至五月中限數一百三十一，間數一百三十二
Summer Solstice (fifth-month mid-term): limit number 131; interval number 132.
53
小暑六月節限數一百三十三，間數一百三十四
Small Heat (sixth-month solar term): limit number 133; interval number 134.
54
大暑六月中限數一百三十六，間數一百三十九
Great Heat (sixth-month mid-term): limit number 136; interval number 139.
55
立秋七月節限數一百四十二，間數一百四十五
Beginning of Autumn (seventh-month solar term): limit number 142; interval number 145.
56
處暑七月中限數一百四十九，間數一百五十三
End of Heat (seventh-month mid-term): limit number 149; interval number 153.
57
白露八月節限數一百五十七，間數一百六十二
White Dew (eighth-month solar term): limit number 157; interval number 162.
58
秋分八月中限數一百六十七，間數一百七十二
Autumn Equinox (eighth-month mid-term): limit number 167; interval number 172.
59
寒露九月節限數一百七十七，間數一百八十二
Cold Dew (ninth-month solar term): limit number 177; interval number 182.
60
霜降九月中限數一百八十六，間數一百九十
Frost Descent (ninth-month mid-term): limit number 186; interval number 190.
61
立冬十月節限數一百九十四，間數一百九十七
Beginning of Winter (tenth-month solar term): limit number 194; interval number 197.
62
小雪十月中限數二百，間數二百三
Small Snow (tenth-month mid-term): limit number 200; interval number 203.
63
大雪十一月節限數二百五，間數二百六
Great Snow (eleventh-month solar term): limit number 205; interval number 206.
64
冬至十一月中限數二百七，間數二百六
Winter Solstice (eleventh-month mid-term): limit number 207; interval number 206.
65
小寒十二月節限數二百五，間數二百三
Small Cold (twelfth-month solar term): limit number 205; interval number 203.
66
大寒十二月中限數二百，間數一百九十七
Great Cold (twelfth-month mid-term): limit number 200; interval number 197.
67
推沒滅術：因雨水積，以沒餘乘之，滿沒法為大餘，不盡為小餘，如前，所求年為雨水前沒日也。 求次沒，加大餘六十九，小餘一百九十六，滿沒法從大餘，命如前，雨水後沒日也。 雨水前沒多在故歲，常有五沒，官以沒正之，一年常有五沒或六沒。 小餘盡為滅日也。 雨水小餘三十九以還，雨水六旬後乃有。
For extinction days before Rain Water, multiply the Rain Water accumulation by 36 and divide by 319 for the day index and remainder, naming the date as before. For the next extinction add 69 days and 196 parts, carrying at 319 parts per day, then read off the post-Rain Water extinction. Pre-Rain Water extinctions usually belong to the prior civil year; five such days are normal, and the bureau aligns the calendar accordingly—five or six per year is standard. When the fractional part reaches zero, that day is a disappearance (mie) day. When Rain Water's fraction is 39 or below, the next such event does not occur until sixty days after Rain Water.
68
推土用事法：置立春大小餘小分之數，減大餘十八，小餘七十九，小分十八，命以紀，算外，立春前土用事日也。 大餘不足加六十，小餘不足減，減大餘一，加度法而後減之。 立夏、立冬求土用事皆如上法。
For Earth taking reign before Beginning of Spring, take the spring-term remainders, subtract 18 days, 79 parts, and 18 small parts, and name the date against the era cycle. Borrow 60 days when the day count runs short; borrow one degree-divisor (304 parts) per day when fractions underflow. For Beginning of Summer and Beginning of Winter, compute Earth's reign by the same procedure.
69
推日所在度法：以度法乘朔積度，不盡為分。 命度起室二，次宿除之，算外，正月朔夜半日在度及分也。 求次日，日加一度，經室去度分。
To find the sun's ecliptic longitude at conjunction, multiply the accumulated conjunction degrees by 304; the remainder is fractional parts. Index from Room 2 through the lodges; the result is the sun's position at midnight on the first-month new moon. For each following day add one du; when crossing the Room lodge, apply the prescribed part reduction.
70
推月所在度法：以月周乘朔積日，周天去之，余滿度法為積度，不盡為分，命度如前，正月朔夜半月所在度及分。 求次月，小月加度二十二，分一百三十三，大月加度三十五，分二百四十五，分滿度法成一度，命如前，次月朔月所在度及分也。 曆先月法：以十六除月行分為大分，如所入遲疾加之，經室去度分。
For the moon's position, multiply conjunction days by 4,064, reduce modulo 111,025, divide by 304 for du and parts, and index from Room 2 as before. For the next month add 22 du 133 parts (short month) or 35 du 245 parts (long month), carrying at 304 parts per du. In the month-ahead procedure, divide lunar motion parts by 16, add the entered anomaly correction, and reduce when crossing Room.
71
推合朔月食術：置所求年積月，以會數一百六十乘之，以所入交會紀差二十二加之，滿會月去之，餘則其年正月朔去交分也。 求次月，以會數加之，滿會月去之。 求望，加合數。 朔望去交分如合數以下，交限數以上，朔則交會，望則月食。
For syzygy and eclipse limits, multiply year lunations by 160, add the nodal offset for the current era (22 for this cycle), and reduce modulo 929 for the first month's nodal distance. For each following month add 160 and reduce modulo 929. For full moon add the conjunction number (80). When nodal distance is between 80 and 859, new moon yields a solar eclipse and full moon a lunar eclipse.
72
推入遲疾曆法：置所求年朔積分，所入遲疾差一萬五千二百四十五加之，滿通周去之，余滿日法得一日，不盡為日餘，命日算外，所求年正月朔入曆。 求次月，加一日，日餘七百三十四。 求望，加十四日，日餘五百七十五半。 余滿日法成一日，日滿二十七去之，除日余如周日日餘，不足減，減一日，加周虛。 日滿二十七而日余不滿周日日餘，為損。 周日滿去之，為入曆一日。
For lunar anomaly entry at the year's first conjunction, add the era's anomaly offset (15,245 for Jiawu), reduce modulo 20,721, then divide by 752 for days and remainder. For the following month add 1 day and 734 parts. For full moon add 14 days and 575½ parts. Carry at 752 parts per day; reduce days modulo 27; at circuit day apply the 417-part remainder rule, borrowing a day and adding 335 when needed. When the day count reaches 27 but the remainder has not reached 417, apply the decrease rule. When the circuit-day condition is satisfied, drop it and count one day of calendar entry.
73
推合朔月食定大小餘法：以入曆日餘乘入曆下損益率，入一日，益二十五是也。 以損益盈縮積分，值損則損之，值益則益之。 為定積分。 以入曆日餘乘列差，滿日法盈減縮加差法，為定差法。 以除定積分，所得減加本朔望小餘，值盈則減，縮則加之。 為定小餘。 加之滿日法，合朔月食進一日； 減之不足減者，加日法而後減之，則退一日。 值周日者，用周日定數。
To fix new- and full-moon remainders for syzygy and eclipse, multiply the entered day-fraction by the anomaly table's excess-deficit rate; on day 1 the rate is +25. Apply that rate to the surplus-shrinkage accumulator, subtracting on deficit rows and adding on excess rows. The result is the corrected accumulated fraction. Multiply the entered day-fraction by the column difference, divide by 752, and adjust the difference divisor by the surplus or shrinkage sign. Divide the corrected accumulator by that divisor and apply the quotient to the base conjunction fraction, subtracting for surplus and adding for shrinkage. That yields the corrected fractional day. If the correction carries past 752 parts, advance the syzygy one day. If subtraction underflows, borrow 752 parts and push the date back one day. On a circuit day use the special circuit-day correction.
74
推加時：以十二乘定小余，滿日法得一辰，數從子起，算外，則朔望加時所在辰也。 有餘者四之，滿日法得一為少，二為半，三為太半。 又有餘者三之，滿日法得一為強，半法以上排成一，不滿半法棄之。 以強並少為少強，並半為半強，並太為太強。 得二者為小弱，以並少為半弱，以並半為太弱，以並太為一辰弱。 以所在辰名之。
Multiply the corrected fraction by twelve and divide by 752 to name the twelve double-hours from midnight. Split the leftover into quarters of the divisor for the shao, ban, and tai 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.
75
推合朔月食加時滿刻法：各以百刻乘定小餘，如日法而一； 不盡什之，求分。 先除夜漏之半，即晝漏加時刻及分也。 晝漏盡，又入夜漏。 在中節前後四日以還者，視限數。 在中節前後五日以上者，視間限數。 月食加時定小餘不滿限數、間數者，皆以算上為日。
For syzygy and eclipse times, scale the corrected fraction to 100 day-parts by dividing by 752. Express the remainder in tenths for the fractional mark. Subtract half the night-water total to obtain the daytime hour and parts. After the day portion is used up, count into the night clepsydra. When a syzygy falls within four days of a major solar term, consult the tight limit number. Beyond five days from the major term, apply the wider gap-limit rule. When the eclipse fraction falls under both margin thresholds, promote the count to the next day.
76
月行遲疾度損益率盈縮積分列差差法一日十四度十三分益二十五盈二二百六十二日十四度十一分益二十三盈萬八千八百三二百五十八三日十四度八分益二十盈三萬六千九十六四二百五十五四日十四度四分益十六盈五萬一千一百三十六五二百五十一五日十三度十八分益十一盈六萬三千一百六十八五二百四十六六日十三度十三分益六盈七萬一千四百四十六二百四十一七日十三度七分益盈七萬五千九百五十二五二百三十五八日十三度二分損五盈七萬五千九百五十二四二百三十九日十二度十七分損九盈七萬二千一百九十二三二百二十六十日十二度十四分損十二盈六萬五千四百二十四三二百二十三十一日十二度十一分損十五盈五萬六千四百三二百二十十二日十二度八分損十八盈四萬五千一百二十二二百一十七十三日十二度六分損二十盈三萬一千五百八十四二二百一十五十四日十二度四分損二十二盈一萬六千五百四十四二二百一十三十五日十二度二分益二十四縮二二百一十一十六日十二度四分益二十二縮一萬八千四十八二二百一十三十七日十二度六分益二十縮三萬四千五百九十二三二百一十五十八日十二度九分益十七縮四萬九千六百三十二五二百一十八十九日十二度十四分益十二縮六萬二千四百一十六六二百二十三二十日十三度一分益六縮七萬一千四百四十六二百二十九廿一日十三度七分益縮七萬五千九百五十二五二百三十五廿二日十三度十二分損五縮七萬五千九百五十二四二百四十廿三日十三度十六分損九縮七萬二千一百九十二四二百四十四廿四日十四度一分損十三縮六萬五千四百二十四四二百四十八廿五日十四度五分損十七縮五萬五千六百四十八三二百五十二廿六日十四度八分損二十縮四萬二千八百六十四三二百五十五廿七日十四度十一分損二十三縮二萬七千八百二十四二二百五十八周日十四度十三分損二十五定縮一萬五百二十八定備二百六十定
Lunar anomaly table: for each of 28 days the columns give slow-fast degrees, excess-deficit rate, surplus-deficit accumulation, column difference, and difference method, from Day 1 at 14°13′ with excess 25 through the circuit day at deficit 25 with fixed shrinkage 10,528.
77
小分一百三損二百二十四九萬三千四百八意差法二千
Table note: small parts 103; deficit 224; accumulation 93,408; difference method 2,309.
78
三百九
309.
79
推合朔度：以章歲乘朔小余，滿通法為大分，不盡為小分。 以大分從朔夜半日日分，滿度命如前，正月朔日月合朔所在共合度也。
For syzygy longitude, multiply the conjunction fraction by 19 and divide by 47 for large and small parts. Add those parts to the midnight sun position and index through the lodges to obtain the shared solar-lunar longitude at the first-month conjunction.
80
求次月，加度二十九，大分一百六十一，小分十四，小分滿通法從大分，大分滿度法從度。 經室除度分。 求望，加十四度，大分二百三十二，小分三十半。 求望月所在度，加日度一百八十二，分一百八十九，小分二十三半。
Each following month add 29° plus 161 large and 14 small parts, carrying at 47 and 304, with the Room-lodge reduction. When crossing the Room lodge, apply the prescribed part reduction. For full moon add 14° plus 232 large and 30½ small parts. For the moon's position at full moon add 182° plus 189 parts and 23½ small parts to the day's longitude.
81
推五星法：二十四氣日所在度日中晷影晝漏刻夜漏刻雨水室太強八尺二寸二分五十五分四十九五分驚蟄壁一強六尺七寸二分五十二九分四十七一分春分奎七少強五尺三寸九分五十五五分四十四五分清明婁六半四尺二寸五分五十八四十二穀雨胃九太弱三盡二寸五分六十三分三十九七分立夏昴十一弱二尺五寸六十二三分三十七七分小滿畢十五少弱一尺九寸七分六十三九分三十六一分芒種井三半弱一尺六寸九分六十四八分三十五二分夏至井十八一尺五寸六十五三十五小暑鬼一弱一尺六寸九分六十四八分三十五二分大暑柳十二弱一尺九寸七分六十三九分三十六一分立秋張五半強二尺五寸六十二三分三十七七分處暑翼二半三尺二寸五分六十三分三十九七分白露翼十七太弱四尺二寸五分五十八四十二秋分軫十五五尺三寸九分五十五五分四十四五分寒露亢一少六尺七寸二分五十二九分四十七一分霜降氐七半八尺二寸八分五十五分四十九五分立冬心二半弱九尺九寸一分四十八四分五十一六分小雪尾十二太強一丈一尺三寸四分四十六七分五十三三分大雪箕十一丈二尺四寸八分四十五六分五十四四分冬至鬥十四強一丈三尺四十五五十五小寒牛三半強一丈二尺四寸八分四十五六分五十四四分大寒女十半強一丈一尺三寸四分四十六七分五十三三分立春危四九尺九寸一分四十八四分五十一六分二十四氣昏中星明中星雨水觜一少強尾十一強驚蟄井九半箕四少弱春分井二十九半強鬥四弱清明柳十二太鬥十四半穀雨張十鬥二十五半立夏翼十太弱女三少小滿軫十弱虛二弱芒種角十太弱危七弱夏至氐五少弱室五少強小暑房四太弱壁六太弱大暑尾八太弱奎十二太弱立秋箕三胃二太弱處暑鬥三半昴七太弱白露鬥十四半弱畢十六半弱秋分鬥二十五少強井九少強寒露牛八半強井二十九弱霜降女十一半弱柳十一半強立冬危二弱張八太弱小雪危十三半強翼八太強大雪室九半強軫八少強冬至壁八太強角七少強小寒奎十五少亢九大寒胃四半強氐十三太強立春昴九少心四強
Five-planet method: a combined table of the twenty-four qi listing solar lodge, ecliptic degree, midday shadow grade and length, day and night clepsydra marks, and dusk and dawn culminating stars for each term from Rain Water through Great Cold.
82
推五星法：
Method for computing the five planets:
83
合歲合數日度法室分木三百四十四三百一十五九萬五千七百六十二萬三千六百二十五火四百五十九二百一十五六萬五千三百六十一萬六千一百二十五土三百八十三三百七十一十一萬二千四百八十二萬七千七百五十金二百六十七一百六十七五萬七百六十八一萬二千五百二十五水七十九二百四十九七萬五千六百九十六一萬八千六百七十五
Synodic constants by planet: Wood—344-year cycle, 315 conjunctions, divisors 95,762 and 3,625; Fire—459/215 with 65,361 and 6,125; Earth—383/371 with 112,482 and 7,750; Metal—267/167 with 57,068 and 12,525; Water—79/249 with 75,696 and 18,675.
84
火後元乙亥，元嘉十二年，至元嘉二十年癸未，九年算上。
Fire's sub-epoch begins yihai; from the twelfth year of Yuanjia to the twentieth year of Yuanjia is nine years on the count.
85
土後元甲戌，元嘉十一年，至元嘉二十年癸未，十年算上。
Earth's sub-epoch begins jiaxu; from the eleventh year of Yuanjia to the twentieth year of Yuanjia is ten years on the count.
86
金後元甲申，晉太元九年，至元嘉二十年癸未，六十年算上。
Metal's sub-epoch begins jiashen; from Jin Taiyuan 9 to the twentieth year of Yuanjia is sixty years on the count.
87
水後元乙丑，元嘉二年，至元嘉二十年癸未，十九年算上。
Water's sub-epoch begins yichou; from the second year of Yuanjia to the twentieth year of Yuanjia is nineteen years on the count.
88
推五星法：各設其元至所求年算上，以合數乘之，滿合歲為積合，不盡曰合餘，多者以合數除之，得一，星合往年，得二，合前往年，不滿合數，其年。 木、土、金則有往年合，火有前往年合，水一年三合或四合也。 以合餘減合數為度分，水度分滿合歲則去之也。 以周天十一萬一千三十五乘度分，滿日度法為積度，不盡曰度餘。 命度以室二，算外，星合所在度也。 以合數乘其年，內雨水小餘，並度余為日余，滿日度法從積度為日，命以雨水，算外，星合日也。 求星見日法，以法伏日及餘木則十六日及金是也。 加星合日及余，滿日度法成一日，命如前，星見日也。 求星見度法，以法伏度及餘木則二度及餘是也。 加星合度及余，滿日度法成一度，命如前，所見度也。 以星行分母木則二十三見也。 乘見度余，滿日度法得一，分乃日加所行分。 木順日行四分。 分滿其母成一度，逆順母不同，木逆分母七也。 當各乘度餘，留者承前，逆則減之，伏不書度，經室去分，不足減者，破全度。 五星室分各異，若在行分，各依室分去之。
For each planet, count from its sub-epoch to the target year, multiply by its conjunction number, and divide by its cycle length; the quotient gives prior conjunctions, the remainder locates the current conjunction year. Wood, Earth, and Metal may conjoin in the prior year; Fire may do so two years back; Water yields three or four conjunctions per year. Subtract the conjunction remainder from the conjunction number for ecliptic parts; for Water, reduce when parts exceed the synodic year. Multiply those parts by 111,035 and divide by the day-degree divisor for accumulated degrees; the remainder is degree residue. Index from Room 2 outward to name the conjunction longitude. Combine the synodic count with the Rain Water fraction and degree residue, carry into days, and name the date from Rain Water outward. For first visibility add the hiding interval—16 days for Wood and the same for Metal. Add that interval to the conjunction date, carry at the day-degree divisor, and read off the appearance day. For appearance longitude add the hiding arc—2° plus residue for Wood. Add to the conjunction longitude and carry at the day-degree divisor to obtain the visibility degree. Wood's motion divisor is based on 23 apparitions per cycle. Scale the appearance-degree remainder by the day-degree divisor and add daily motion parts thereafter. Wood in direct motion advances 4 parts per day. Parts carry into whole degrees at different denominators; Wood's retrograde divisor is 7. Apply the appropriate divisor to degree remainders; hold during stations, subtract in retrograde, omit degrees while hidden, reduce at Room, and borrow a full degree when needed. Each planet has its own Room-lodge reduction when computing motion parts.
89
木：初與日合，伏，十六日，日余四萬一千七百八十，行二度，余七萬七千八百四十七半，晨見東方。 去日十三度半強。 順，日行二十三分之四，一百一十五日行二十度。 留，不行，二十六日而逆。 日行七分之一，八十四日退十二度。 又留二十六日。 順，一百一十五日行二十度，夕伏西方，日度余如初，與日合。 一終三百九十八日，日余八萬三千五百六十，行星三十三度，余五萬九千九百三十五。
Wood: at conjunction it hides 16 days with remainder 41,780, advancing 2° with residue 77,847½, then appears in the east at dawn. It stands 13½° strong east of the sun. Direct: 4/23 of a degree per day, 20° in 115 days. It stations for 26 days, then turns retrograde. Retrograde at 1/7° per day, retreating 12° in 84 days. It stations again for 26 days. Direct motion 20° in 115 days, then hides in the west at dusk with the same remainders as at first conjunction. One synodic cycle: 398 days with remainder 83,560, planetary travel 33° with residue 59,935.
90
火：初與日合，伏，七十一日，日余二萬四千八百一十二半，行五十四度，度余四萬九千四百三十，晨見東方。 去日十七度半強。 順，疾，日行七分之五，一百八日半行七十七度半。 小遲，日行七分之四，一百二十六日行七十二度而大遲。 日行七分之二，四十二日行十二度。 留，不行，十二日而遲。 日行十分之三，六十日退十八度。 又留十二日。 順，遲，四十二日行十二度。 小疾，一百二十六日，行七十二度。 一百八日半行七十七度半，夕伏西方，日度余如初，與日合。 一終七百七十九日，日余四萬九千六百二十五，行星四百一十四，度余三萬三千五百。 除一周，定四十九度，度余一萬七千三百七十五。
Fire: hides 71 days with remainder 24,812½, advancing 54° with residue 49,430, then appears in the east at dawn. It stands 17½° strong east of the sun. Direct fast: 5/7° per day, 77½° in 108½ days. Slightly slow: 4/7° per day, 72° in 126 days, then very slow. Very slow: 2/7° per day, 12° in 42 days. It stations for 12 days, then turns retrograde. Retrograde at 3/10° per day, retreating 18° in 60 days. It stations again for 12 days. Direct slow: 12° in 42 days. Slightly fast: 72° in 126 days. Fast direct motion 77½° in 108½ days, then hides in the west at dusk and reconjoins the sun. One synodic cycle: 779 days with remainder 49,625, planetary travel 414° with residue 33,500. Reduce by one full circuit to 49° with residue 17,375.
91
土：初與日合，伏，十八日，日餘四千四百八十二半，行二度，度余四萬六千八百四十七半，晨見東方。 去日十五度半強，順，日行十二分之一，八十四日，行七度。 留，不行，三十六日而逆。 日行十七分之一，一百二日退六度。 又留三十六日。 順，八十四日行七度，夕伏西方，日度余如初，與日合。 一終三百七十八日，日餘八千九百六十五，行星十二度，度余九萬三千六百九十五。
Earth: hides 18 days with remainder 4,482½, advancing 2° with residue 46,847½, then appears in the east at dawn. It stands 15½° strong east of the sun; direct at 1/12° per day, 7° in 84 days. It stations for 36 days, then turns retrograde. Retrograde at 1/17° per day, retreating 6° in 102 days. It stations again for 36 days. Direct motion 7° in 84 days, then hides in the west at dusk and reconjoins the sun. One synodic cycle: 378 days with remainder 8,965, planetary travel 12° with residue 93,695.
92
金：初與日合，伏，四十一日，日余四萬九千六百八十四半，行五十一度，度余四萬九千六百八十四半，見西方。 去日十度。 順，疾，日行一度十三分之三，九十一日行一百十二度而小遲。 日行一度十三分之二，九十一日行一百五度。 又大遲。 日行十五之十一，四十五日行三十三度。 留，不行，八日而遲。 日行三分之二，九日退六度，伏西方。 伏六日，退四度而與日合。 又六日退四度，晨見東方。 逆，九日退六度。 又留八日。 順，四十五日行三十三度。 小疾，九十一日行一百五度。 大疾，九十一日行百一十二度，晨伏東方，日度余如初，與日合。 一終五百八十三日，日余四萬八千六百一。 除一周，行星定二百一十八度，度余三萬六千七十六。 一合二百九十一日，余四萬九千六百八十四半，行星如之。
Metal: hides 41 days with remainder 49,684½, advancing 51° with the same residue, then appears in the west. It stands 10° from the sun. Direct fast: 1° 3/13 per day, 112° in 91 days, then slightly slow. Slightly slow: 1° 2/13 per day, 105° in 91 days. Then very slow. Very slow: 11/15° per day, 33° in 45 days. It stations for 8 days, then turns retrograde. Retrograde at 2/3° per day, retreating 6° in 9 days, then hides in the west. Hidden 6 days while retreating 4°, then reconjoins the sun. After another 6 days retreating 4°, it appears in the east at dawn. Retrograde 6° in 9 days. It stations again for 8 days. Direct motion 33° in 45 days. Slightly fast: 105° in 91 days. Fast direct 112° in 91 days, then hides in the east at dawn and reconjoins the sun. One synodic cycle: 583 days with remainder 48,601. Reduce by one circuit to net travel of 218° with residue 36,076. One inferior conjunction: 291 days with remainder 49,684½ and the same planetary arc.
93
水：初與日合，伏，十七日，日余七萬一千二百一十半，行三十四度，度余七萬一千二百一十半，見西方。 去日十七度。 順，疾，日行一度三分之一，十八日行二十四度而遲。 日行七分之五，七日行五度。 留，不行，四日，夕伏西方。 伏十一日，退六度，而與日合。 又十一日退六度，而晨見東方。 留四日。 順，遲，七日行五度。 疾，十八日行二十四度，晨伏東方，日度余如初，與日合。 一終一百一十五日，日余六萬六千七百二十五，行星如之。 一合五十七日，日余七萬一千二百一十半，行星亦如之。 盈加縮減，十六除月行分，日法除盈縮分，以減度分，盈加縮減。
Mercury: at initial solar conjunction it disappears for 17 days with remainder 71,210½, advancing 34° with the same residue, then appears in the west. It stands 17° from the sun. Direct fast: 1° ⅓ per day, 24° in 18 days, then slower. Slow direct: 5/7° per day, 5° in 7 days. It stations for 4 days, then hides in the west at dusk. Hidden 11 days while retreating 6°, then reconjoins the sun. After another 11 days retreating 6°, it appears in the east at dawn. It stations for 4 days. Direct slow: 5° in 7 days. Fast direct 24° in 18 days, then hides in the east at dawn and reconjoins the sun with the initial day and degree residues. One synodic cycle: 115 days with remainder 66,725 and the same planetary arc. One inferior conjunction: 57 days with remainder 71,210½ and the same planetary travel. Apply surplus by addition and shrinkage by subtraction: scale lunar motion parts by 1/16, convert excess-deficit parts through the day divisor to adjust degree parts, then add or subtract accordingly.
94
推卦：因雨水大小余，加大餘六，小餘三百一十九，小余滿三千六百四十八成日。 日滿二十七日餘不足加減不加周虛。
For hexagram rulership, take the Rain Water sexagenary line and add 6 to the day count and 319 to the fractional part; carry at 3,648 parts to the day. Once the count reaches 27 days with a remainder too small to adjust, omit the circuit-void correction.
95
元嘉二十年，承天奏上尚書：「今既改用《元嘉曆》，漏刻與先不同，宜應改革。 按《景初曆》春分日長，秋分日短，相承所用漏刻，冬至後晝漏率長於冬至前。 且長短增減，進退無漸，非唯先法不精，亦各傳寫謬誤。 今二至二分，各據其正。 則至之前後，無複差異。 更增損舊刻，參以晷影，刪定為經，改用二十五箭。 請台勒漏郎將考驗施用。」 從之。
In the twentieth year of Yuanjia, He Chengtian memorialized the Minister of Works: "Having adopted the Yuanjia Calendar, the water-clock graduations no longer match the old system and should be revised. The Jingchu Calendar fixes a long day at the spring equinox and a short one at the autumn equinox, yet the inherited clepsydra tables make daytime water longer after midwinter than before it. Day lengths rise and fall without smooth gradation, and the discrepancies are not due to the old formula alone—copyists have compounded the fault. Let the two solstices and two equinoxes each serve as their own fixed anchors. Then the intervals before and after each solstice will no longer disagree. Revise the old graduations against sundial shadows, establish the corrected norm, and adopt a twenty-five-tube clepsydra. He asked that the palace clepsydra officer test and put the new scale into service. The court assented.
96
前世諸儒依圖緯雲，月行有九道。 故畫作九規，更相交錯，檢其行次，遲疾換易，不得順度。 劉向論九道雲：「青道二出黃道東，白道二出黃道西，黑道二出北，赤道二出南。」 又雲：「立春、春分，東從青道； 立夏、夏至，南從赤道。 秋白冬黑，各隨其方。」 按日行黃道，陽路也，月者陰精，不由陽路，故或出其外，或入其內，出入去黃道不得過六度。 入十三日有奇而出，出亦十三日有奇而入，凡二十七日而一入一出矣。 交于黃道之上，與日相掩，則蝕焉。 漢世劉洪推檢月行，作陰陽曆法。 元嘉二十年，太祖使著作令史吳癸依洪法，制新術，令太史施用之。
Earlier scholars, relying on cosmological charts and apocrypha, held that the moon travels nine paths. They mapped nine interlocking circles, but the moon's observed pace refused to march evenly through such a scheme. Liu Xiang on the nine paths wrote: "Two green paths lie east of the yellow path, two white paths west of it, two black paths to the north, and two red paths to the south. He added: "From Establishment of Spring through the spring equinox the moon runs the green path in the east; from Establishment of Summer through the summer solstice it follows the red path in the south. In autumn it takes the white path, in winter the black, each aligned with its season." In fact the sun keeps to the yellow path, the highway of yang; the moon, as yin essence, cannot stay on that road and so swings above or below it, never more than six degrees from the ecliptic. It takes a little over thirteen days to cross inward and the same to cross outward—one complete in-out cycle in twenty-seven days. When the crossing coincides with the sun on the ecliptic, an eclipse occurs. In the Han, Liu Hong verified the moon's motion and devised the yin-yang calendar method. In the twentieth year of Yuanjia Emperor Wen had archive clerk Wu Gui adapt Liu Hong's method into a new procedure and put it in use at the Astronomical Bureau.
97
《元嘉曆》月行陰陽法：
Yuanjia Calendar: method for the moon's yin-yang motion:
98
陰陽曆損益率兼數
Yin-yang calendar: excess-deficit rates and combined numbers
99
一日益十七初
Day 1: excess rate +17, opening entry.
100
二日前限餘六百六十五微分一千七百三十八益十六十七
Day 2: front-limit remainder 665, minute fraction 1,738, excess rate 16, combined number 17.
101
三日益十五三十三
Day 3: excess rate +15, combined number 33.
102
四日益十二四十八
Day 4: excess rate +12, combined number 48.
103
五日益八六十
Day 5: excess rate +8, combined number 60.
104
六日益四六十八
Day 6: excess rate +4, combined number 68.
105
七日益一七十二
Day 7: excess rate +1, combined number 72.
106
八日損二七十三
Day 8: deficit rate −2, combined number 73.
107
九日損六七十一
Day 9: deficit rate −6, combined number 71.
108
十日損十六十五
Day 10: deficit rate −10, combined number 65.
109
十一日損十三五十五
Day 11: deficit rate −13, combined number 55.
110
十二日損十五四十二
Day 12: deficit rate −15, combined number 42.
111
十三日後限餘二千一十九微分一千七十九損十六二十七
Day 13: rear-limit remainder 2,019, minute fraction 1,079, deficit rate 16, combined number 27.
112
分日二千六百八十五半損十六大大者五千三百
Fraction-day row: 2,685½ parts, deficit rate 16, double-full combined number 5,300.
113
七十一分之三千四百七十二十一
Fraction 3,470/71 with combined number 21.
114
曆周，五萬五千五百一十七半。
Calendar cycle (li zhou): 55,517½.
115
差率，一萬一百九十。
Difference rate (cha lü): 10,190.
116
微分法，一千八百七十八。
Minute-fraction divisor (weifen fa): 1,878.
117
推入陰陽曆術曰：以會月去入紀積月，餘以會數乘之，以所入紀交會差加之，周天乘之，滿微分法為大分，不盡為微分。 大分滿周天去之，余不滿曆周者為入陽曆。 余皆如月周得一日，算外，所求年正月合朔入曆也。 不盡為日餘。
To find yin-yang calendar entry: reduce accumulated months since era entry by synodic months, multiply the remainder by the synodic constant, add that era's conjunction offset, scale by the circuit of Heaven, and split the product at 1,878 into large and minute fractions. Remove full circuits of Heaven from the large parts; any remainder under the calendar cycle counts as yang-calendar entry. Divide the residue by the lunar circuit for whole days; the result, exclusive in the count, is the yin-yang entry at the year's first-month new moon. The leftover is the day-fraction.
118
求次月，加二日，日餘一千三百三十一，微分一千五百九十八，如法成日，日滿十三去之，除日餘如分日。 陰陽曆竟平入端，入曆在前限餘前，後限餘後者，月行中道。
For each following month add 2 days plus remainder 1,331 and minute fraction 1,598, carrying at the divisor; drop full cycles of 13 days and apply the fraction-day reduction. At cycle turnover, if entry falls before the front-limit remainder or after the rear-limit remainder the moon is off the middle path; between those bounds it runs the central track.
119
求朔弦望定數：各置入遲疾曆盈縮定積分，以章歲乘之，差法除之，所得滿通法為大分。 不盡，以微分法乘之，如法為微分。 盈減縮加陰陽日餘，盈不足，以月周進退日而定，以定日餘乘損益兼數，為加時定數。
To fix syzygy and quarter-moon times, take each slow-fast surplus-shrinkage corrected accumulator, multiply by rule years, divide by the difference divisor, and convert the quotient at the communication divisor into large parts. For the remainder, scale by the minute-fraction divisor to obtain minute parts. Subtract surplus and add shrinkage to the yin-yang day-fraction, carrying days as needed, then multiply the fixed remainder by the table's combined number for the corrected hour mark.
120
推夜半入曆：以差率朔小餘，如微分法得一，以減入曆餘，不足，加月周而減之，卻一日，卻得分日，加其分，半微分為小分，即朔日夜半入歷歷餘小分也。
For midnight entry, convert the conjunction fraction through the difference rate and minute-fraction divisor, subtract from the calendar remainder (borrowing a lunar circuit and one day if needed), add the fraction-day parts, and halve the minute fraction for the small-fraction term at new-moon midnight.
121
求次日，加一日，日餘十六，小分三百二十，小分如會從余，余滿月周去之，又加一日。 曆竟，下日余滿分日去之，于入曆初也。 不滿分日者，值之，加餘一千二百九十四，小分七百八十九半，為入次曆。
For the following day add 1 day, remainder 16, and small fraction 320, carrying at the divisor and lunar circuit as required. At cycle end, drop a full fraction-day from the day-remainder to reset calendar entry. When the remainder falls short of a fraction-day at turnover, add 1,294 plus small fraction 789½ to enter the next cycle.
122
求夜半定日：以朔小余減入遲疾曆日餘，不足一日，卻得周日，加餘四百一十七，即月夜半入曆日及餘也。 以日餘乘損益率，盈縮積分，為定積分。 滿通法為大分，不盡以會月乘之，如法為小分，以盈加縮減入陰陽日餘，盈不足進退日而定也。 以定日餘乘損益率，如月周，以損益兼數，為夜半定數。
For midnight correction, subtract the conjunction fraction from the entered slow-fast day-remainder; if that underflows a day, borrow the circuit day and add remainder 417 for the moon's midnight calendar entry. Multiply the day-fraction by the excess-deficit rate and surplus-shrinkage accumulator to get the corrected accumulated fraction. Convert at the communication divisor, finish the tail with synodic months for small parts, then add surplus or subtract shrinkage from the yin-yang day-fraction, carrying days to fix the date. Multiply the fixed day-fraction by the excess-deficit rate, scale by the lunar circuit and combined number, for the corrected midnight mark.
123
求昏明數：以損益率乘所近節氣夜漏，二百而一為明，以減損益率為昏，而以損益夜半數為昏明定數也。
For dawn and dusk corrections, take the nearest term's night water, allocate the excess-deficit rate in a 1:200 ratio for dawn and the balance for dusk, and shift the midnight value accordingly.
124
求月去黃道度：置加時若昏明定數，以十二除之為度，其餘三而一為少，不盡為強，二少弱也。 所得為月去黃道度。
For lunar latitude, take the corrected hour, quarter, or dusk/dawn number, divide by 12 for whole degrees, then split the remainder into strong and weak parts in thirds. The result is the moon's ecliptic latitude in degrees.
125
大明六年，南徐州從事史祖沖之上表曰：
In the sixth year of Daming, Zu Chongzhi, aide to the governor of South Xu, submitted a memorial stating:
126
古曆疏舛，頗不精密，群氏糾紛，莫審其要。 何承天所奏，意存改革，而置法簡略，今已乖遠。 以臣校之，三睹厥謬：日月所在，差覺三度； 二至晷影，幾失一日； 五星見伏，至差四旬，留逆進退，或移兩宿。 分至乖失，則節閏非正； 宿度違天，則伺察無准。 臣生屬聖辰，逮在昌運，敢率愚瞽，更創新曆。 謹立改易之意有二，設法之情有三。
The old calendars were loose and inaccurate, and rival schools disputed them without settling the core principles. He Chengtian's proposal aimed at reform, yet the rules he set were too spare and have already drifted badly from the heavens. In three separate checks I found the same faults: the sun and moon are off by almost three degrees; solstice shadows err by nearly a whole day; planetary phases can be wrong by forty days, and stations, retrogrades, and reversals may wander two lodges off. If the quarter and solstice points slip, intercalation and the qi nodes fall out of true; if lodge longitudes part from the sky, observation loses its measure. Born under a sage reign and a prosperous age, I venture despite my limited sight to draft a new calendar. I respectfully propose two grounds for reform and three principles for the new system.
127
改易者，其一，以舊法一章十九歲有七閏，閏數為多，經二百年，輒差一日。 節閏既移，則應改法，曆紀屢遷，實由此條。 今改章法，三百九十一年有一百四十四閏。 令卻合周、漢，則將來永用，無複差動。 其二，以《堯典》雲：「日短星昴，以正仲冬。」 以此推之，唐代冬至，日在今宿之左五十許度。 漢代之初，即用秦曆，冬至日在牽牛六度。 漢武改立《太初曆》，冬至日在牛初。 後漢《四分法》，冬至日在鬥二十二。 晉時薑岌以月蝕檢日，知冬至在鬥十七。 今參以中星，課以蝕望，冬至之日，在鬥十一。 通而計之，未盈百載，所差二度。 舊法並令冬至日有定處，天數既差，則七曜宿度漸與曆舛。 乖謬既著，輒應改制，僅合一時，莫能通遠，遷革不已，又由此條。 今令冬至所在，歲歲微差，卻檢漢注，並皆審密，將來久用，無煩屢改。
First change: the old rule of seven intercalations in nineteen years inserts too many leap months; in two centuries it slips a full day. Once seasons and intercalation drift, the calendar must be revised—and that, I argue, is why systems have been rewritten again and again. My new intercalation rule yields 144 leap months in 391 years. Aligned with Zhou and Han practice, it should serve indefinitely without further drift. Second: the Canon of Yao says, "The days are short and the star is Mao—thereby fix mid-winter." From that line, at the Tang dynasty midwinter the sun stood some fifty degrees west of its present lodge. At the Han founding the Qin calendar placed midwinter at 6° in Ox. Emperor Wu's Taichu Calendar moved it to the start of Ox. The Later Han Quarter-Remainder system put it at Dipper 22. In Jin, Jiang Ji checked eclipses and found midwinter at Dipper 17. Comparing culminating stars with eclipse and full-moon tests, midwinter now falls at Dipper 11. Taken together, the shift is two degrees in less than a century. Older systems pinned midwinter to a fixed longitude; as the sky moves, the seven luminaries steadily part from the calendar. Once the mismatch shows, reform is due; a fixed midwinter served one age but not the long term—and that too drove endless revisions. I would let midwinter advance slightly year by year; checked against Han observations it holds closely, and long use should not require repeated overhauls.
128
又設法者，其一，以子為辰首，位在正北，爻應初九，鬥氣之端，虛為北方，列宿之中，元氣肇初，宜在此次。 前儒虞喜，備論其義。 今曆上元日度，發自虛一。 其二，以日辰之號，甲子為先，曆法設元，應在此歲。 而黃帝以來，世代所用，凡十一曆，上元之歲，莫值此名。 今曆上元，歲在甲子。 其三，以上元之歲，曆中眾條，並應以此為始，而《景初曆》交會遲疾，亦置紀差，裁合朔氣而已。 條序紛互，不及古意。 今設法，日月五緯，交會遲疾，悉以上元歲首為始。 則合璧之曜，信而有征，連珠之暉，於是乎在，群流共源，實精古法。
First principle of the new methods: take zi as the head of the earthly branches, due north, matching the first nine of the Changes—the Dipper's qi begins there, Emptiness marks the northern pivot among the lodges, and the primal breath rightly starts at this point. The earlier scholar Yu Xi set out that doctrine in full. The present calendar's superior-origin solar longitude begins at Emptiness 1. Second, jiazi leads the cycle of days and double-hours, so the calendar's epoch ought to fall in that year. Yet from the Yellow Emperor onward, eleven calendars have been used across the ages, and not one placed the superior-origin year on jiazi. The present calendar's superior origin is the jiazi year. Third, every calendar rule ought to begin from the superior-origin year; the Jingchu Calendar, however, offsets conjunction and lunar anomaly by era and trims only syzygy and solar-term alignment. Its rules are jumbled and fall short of the classical intent. The present method starts sun, moon, five planets, syzygy, and lunar anomaly all from the superior-origin year-head. Then the joined disc of sun and moon rests on solid evidence, the strung-pearl gathering of the five planets finds its place, and all streams share one source—truly refining the classical method.
129
若夫測以定形，據以實效，縣象著明，尺表之驗可推，動氣幽微，寸管之候不忒。 今臣所立，易以取信。 但深練始終，大存整密，革新變舊，有約有繁。 用約之條，理不自懼，用繁之意，顧非謬然。 何者？ 夫紀閏參差，數各有分，分之為體，非細不密。 臣是用深惜毫釐，以全求妙之准，不辭積累，以成永定之制。 非為思而莫悟，知而不改也，竊恐贊有然否，每崇遠而隨近； 論有是非，或貴耳而遺目。 所以竭其管穴，俯洗同異之嫌，披心日月，仰希葵藿之照。 若臣所上，萬一可采，伏願頒宣群司，賜垂詳究，庶陳錙銖，少增盛典。
Measured against form and verified in practice, the suspended heavens stand clear; gnomon tests can be extrapolated, and even subtle qi and clepsydra graduations hold true. What I have established is easy to verify. It works through every stage with full rigor, renews the old system, and mixes concise rules with detailed ones. The simplified rules need not be feared on principle, and the detailed provisions are not arbitrary. Why so? Era cycles and intercalation are uneven, each constant has its proper fraction, and precision requires fine parts. I therefore guard every smallest fraction to reach an exact standard and do not shrink from cumulative detail to build a permanently fixed system. This is not from ignorance or obstinacy; I fear that in debate yea and nay alike exalt distant precedent while clinging to what is near at hand; and that in judging right and wrong some trust hearsay over what the eye can see. I therefore lay out all I know, bowing to clear away suspicion of partiality, lay my heart open before sun and moon, and look up hoping for the sun's impartial light. If even a trace of what I submit may be adopted, I humbly ask that it be sent to all offices for full review, so that this tiny offering may add something to the grand canonical tradition.
130
○曆法
○ The Calendar Method
131
上元甲子至宋大明七年癸卯，五萬一千九百三十九年算外。
From the jiazi superior origin to Song the seventh year of Daming (guimao) is 51,939 years, exclusive in the count.
132
元法，五十九萬二千三百六十五。
The origin divisor (yuanfa): 592,365.
133
紀法，三萬九千四百九十一。
The era divisor (jifa): 39,491.
134
章歲，三百九十一。
Rule years (zhang sui): 391.
135
章月，四千八百三十六。
Rule months (zhang yue): 4,836.
136
章閏，一百四十四。
Rule intercalations (zhang run): 144.
137
閏法，十二。
The intercalation divisor (runfa): twelve.
138
月法，十一萬六千三百二十一。
The month divisor (yuefa): 116,321.
139
日法，三千九百三十九。
The day divisor (rifa): 3,939.
140
餘數，二十萬七千四十四。
The remainder number (yu shu): 207,044.
141
歲餘，九千五百八十九。
The year remainder (sui yu): 9,589.
142
沒分，三百六十萬五千九百五十一。
The extinction parts (mo fen): 3,605,951.
143
沒法，五萬一千七百六十一。
The extinction divisor (mofa): 51,761.
144
周天，一千四百四十二萬四千六百六十四。
The celestial circumference (zhoutian): 14,424,664.
145
虛分，萬四百四十九。
The void parts (xu fen): 10,449.
146
行分法，二十三。
The motion-parts divisor (xingfen fa): twenty-three.
147
小分法，一千七百一十七。
The small-parts divisor (xiaofen fa): 1,717.
148
通周，七十二萬六千八百一十。
The communication circuit (tong zhou): 726,810.
149
會周，七十一萬七千七百七十七。
The conjunction circuit (hui zhou): 717,777.
150
通法，二萬六千三百七十七。
The communication divisor (tongfa): 26,377.
151
差率，三十九。
The difference rate (cha lü): thirty-nine.
152
推朔術：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。 求次月，加大餘二十九，小餘二千九十，小余滿日法從大余，大余滿六旬去之，命如前，次月朔也。 求弦望：加朔大餘七，小餘千五百七，小分一，小分滿四從小余，小余滿日法從大餘，命如前，上弦日也。 又加得望，又加得下弦，又加得後月朔也。
To compute conjunction: from the superior-origin year count, multiply by rule months and divide by 391 for accumulated months; the remainder is the intercalation residue. If the intercalation remainder is 247 or higher, the year receives a leap month. Multiply accumulated months by 116,321 and divide by 3,939 for accumulated days; the remainder is the fractional day. Reduce accumulated days modulo sixty for the day index. Index the day from jiazi outward to obtain the civil eleventh-month new moon of the target year. If the fractional day is 1,849 parts or more, the month is long (30 days). For the following month add 29 days and 2,090 parts, carrying at 3,939 and reducing modulo sixty, then name the next new moon as before. For first quarter add 7 days, 1,507 parts, and 1 small part, carrying at 4 small parts, 3,939 parts, and sixty days, to name the upper-quarter date. Repeat the step for full moon, last quarter, and the next month's new moon.
153
推閏術：以閏余減章歲，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。 推二十四氣術：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，加大餘十五，小餘八千六百二十六，小分五，小分滿六從小余，小余滿紀法從大餘，命如前，次氣日也。 求土用事：加冬至大餘二十七，小余萬五千五百二十八，季冬土用事日也。 又加大餘九十一，小余萬二千二百七十，次土用事日也。 推沒術：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。
For the leap month subtract the intercalation remainder from 391, divide by 12, and index from the civil first month outward. Leap placement may shift, but a month without a major solar term is the governing rule. For the twenty-four qi, multiply years since the superior origin by 207,044 and divide by 39,491 for accumulated days; the remainder is the fractional day. Reduce accumulated days modulo sixty for the day index. Index from jiazi outward to obtain the civil eleventh-month Winter Solstice. For each following qi add 15 days, 8,626 parts, and 5 small parts, carrying at 6 small parts and 39,491 parts. For Earth's reign in late winter add 27 days and 15,528 parts to the Winter Solstice index. Add another 91 days and 12,270 parts for the second Earth-reign day. For extinction days subtract ninety times the Winter Solstice fraction from 3,605,951 and divide by 51,761, naming the date from Winter Solstice outward.
154
求次沒，加日六十九，日余三萬四千四百四十二，余滿沒法從日，次沒日也。 日餘盡為滅。
For the next extinction add 69 days and 34,442 parts, carrying at 51,761 parts per day. When the fractional part reaches zero, that day is a disappearance (mie).
155
推日所在度術：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，大月加度三十，小月加度二十九，入虛去度分。 求行分，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。
For solar longitude at the civil eleventh-month new moon, multiply accumulated days by 39,491, reduce modulo 14,424,664, index from Emptiness 1 through the lodges. Each month add 30° for a long month or 29° for a short month, with the Emptiness-lodge reduction. Divide degree remainder by 1,717 for motion parts; the tail is small parts. Carry small parts into motion parts at 1,717 and motion parts into whole degrees at 23. Add 1° per day thereafter. When crossing Emptiness subtract 6 motion parts and 147 small parts.
156
推月所在度術：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。 微分滿月法從度余，度余滿紀法為度，以減朔夜半日所在，則月所在度。
For lunar longitude multiply the conjunction fraction by 124 for degree remainder. Multiply the same fraction by 860 for minute parts. Carry minute parts at 116,321 and degree parts at 39,491, then subtract from midnight solar longitude to obtain the moon's position.
157
求次月，大月加度三十五，度余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，度余萬七千二百六十一，微分六萬三千七百三十六，入虛去度分也。
Each long month add 35° plus 31,834 degree parts and 77,967 minute parts; each short month add 22° plus 17,261 and 63,736, with the Emptiness reduction.
158
遲疾曆：
Lunar anomaly table:
159
月行度損益率盈縮積分差法一日十四行分十三益七十盈初5304二日十四十一益六十五盈百八十四萬二千三百一十六5270三日十四八益五十七盈三百五十五萬七百六5219四日十四四益四十七盈五百五萬八千二百八5151五日十三二十二益三十四盈六百二十九萬七千八百五十七5066六日十三十七益二十二盈七百二十萬二千六百九十一4981七日十三十一益六盈七百七十七萬二千七百一十4879八日十三五損九盈七百九十四萬九百五十二4777九日十二二十二損二十四盈七百七十萬七千四百一十五4675十日十二十六損三十九盈七百七萬二千一百4573十一日十二十一損五十二盈六百三萬五千七4488十二日十二八損六十盈四百六十六萬三千一百4437十三日十二六損六十五盈三百九萬三百二4403十四日十二四損七十盈百三十八萬三千五百八十4369十五日十二五益六十七縮四十五萬七千六十九4368十六日十二七益六十二縮二百二十三萬七百五十五4420十七日十二十益五十五縮三百八十七萬五百一十四4471十八日十二十四益四十四縮五百三十萬九千三百八十五4539十九日十二十九益三十二縮六百四十八萬四百四4624二十日十三一益十九縮七百三十一萬六千六百八4709二十一日十三七益四縮七百八十一萬七千九百九十六4811二十二日十三十三損十一縮七百九十一萬七千六百七4913二十三日十三十九損二十七縮七百六十一萬五千四百四十5015二十四日十四一損三十九縮六百九十萬一千四百九十五5100二十五日十四六損五十二縮五百八十七萬二千七百三十五5185二十六日十四十損六十二縮四百四十九萬九千一百五十九5253二十七日十四十二損六十七縮二百八十五萬七千七百三十二5287二十八日十四十四損七十四縮百八萬二千三百七十九5321
Lunar anomaly table: for each of 28 days the columns give motion degrees, motion parts, excess-deficit rate, surplus-shrinkage accumulation, and column difference—from Day 1 at 14°13′ with excess 70 and initial surplus 5,304 through Day 28 at 14°14′ with deficit 74 and shrinkage 1,082,379.
160
推入遲疾曆術：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曆日也。
To enter the anomaly table multiply accumulated days by 26,377, reduce modulo 726,810, and split into days and day remainder. Index the day outward to obtain midnight calendar entry on the civil eleventh-month new moon.
161
求次月，大月加二日，小月加一日，日余皆萬一千七百四十六。 曆滿二十七日，日余萬四千六百三十一，則去之。
Each long month add 2 days and each short month 1 day, always with remainder 11,746. After 27 days with remainder 14,631, drop one full anomaly cycle.
162
求次日，加一日。 求日所在定度：以夜半入曆日餘乘損益率，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餘為定度。 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。
Add one day per day thereafter. For corrected solar longitude multiply the entered day-fraction by the table rate, adjust the surplus-shrinkage accumulator by the difference rate of 39, and add or subtract from mean longitude. Borrow or carry at 39,491 parts when addition overflows or subtraction underflows. Compute motion parts by the same carry rules. Advance daily by the entered anomaly increment and apply the Emptiness reduction as above.
163
陰陽曆
The Yin-Yang (nodal) calendar:
164
損益率
Excess-deficit rates and combined numbers:
165
益十六
Day 1: excess 16
166
益十五
Day 2: excess 15
167
益十四
Day 3: excess 14
168
三十一
Combined number 31
169
益十二
Day 4: excess 12
170
四十五
Combined number 45
171
五十七
Combined number 57
172
六十六
Combined number 66
173
七十一
Combined number 71
174
七十二
Combined number 72
175
六十四
Combined number 64
176
十一日
Day 11
177
損十三
Deficit 13
178
五十四
Combined number 54
179
十二日
Day 12
180
損十五
Deficit rate 15.
181
四十一
Combined number 41.
182
十三日
Day 13.
183
損十六
Deficit rate 16.
184
二十六
Combined number 26.
185
十四日
Day 14.
186
損十六
Deficit rate 16.
187
推入陰陽曆術：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曆分，滿去之，為朔入陰曆分。 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曆日也。
To enter the yin-yang calendar, take the communication dividend, subtract the conjunction circuit (717,777), and read the remainder under crossing number 358,888½ as the yang-calendar fraction at new moon; subtract that amount again for the yin-calendar fraction. Divide each fraction by the communication divisor (26,377) for whole days and a day-remainder, then name the date outside the count: midnight calendar entry on the first-month new moon of the civil year.
188
求次月，大月加二日，小月加一日，日余皆二萬七百七十九。 曆滿十三日，日余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。 求朔望差，以二千二十九乘朔小余，滿三百三為日餘，不盡倍之為小分，則朔差數也。 加一十四日，日余二萬一百八十六，小分百二十五，小分滿六百六從日余，日余滿通法為日，即望差數也。 又加之，後月朔也。
For the following month add 2 days (long month) or 1 day (short month), always adding day-remainder 20,779. When the sequence reaches 13 days with day-remainder 15,987½, drop that full yin-yang cycle. At the end of the yang half-cycle enter yin; at the end of the yin half-cycle enter yang. For the next day add one day. For the syzygy correction, multiply the conjunction fraction by 2,029, divide by 303 for the day-remainder, and double the tail for small parts to obtain the new-moon difference. Add 14 days, day-remainder 21,186, and small parts 125; carry at 606 small parts and 26,377 day-parts to obtain the full-moon difference. Repeat the addition for the following month's conjunction.
189
求合朔月食：置朔望夜半入陰陽曆日及餘，有半者去之，置小分三百三，以差數加之，小分滿六百六從日余，日余滿通法從日，日滿一曆去之。 命日算外，則朔望加時入曆也。 朔望加時入曆一日，日餘四千一百九十八，小分四百二十八以下，十二日，日余萬一千七百八十八，小分四百八十一以上，朔則交會，望則月食。
For syzygy and eclipse timing, take midnight yin-yang calendar entry and remainder at new or full moon, drop halves, add difference number 303 in small parts, and carry through 606, 26,377, and the 13-day cycle as needed. Name the day outside the count to obtain the yin-yang calendar entry at the syzygy instant. If hour-entry is ≤1 day 4,198 parts 428, or ≥12 days 11,788 parts 481, new moon yields a solar-lunar conjunction and full moon yields a lunar eclipse.
190
求合朔月食定大小餘：令差數日餘加夜半入遲疾曆余，日余滿通法從日，則朔望加時入曆也。 以入曆餘乘損益率，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。
To fix syzygy remainders, add the difference day-remainder to the midnight anomalistic entry, carrying at 26,377 parts, to obtain hour-of-syzygy calendar entry. Multiply the entered day-fraction by the table's excess-deficit rate, apply it to the surplus-shrinkage accumulator divided by the difference method (39), and adjust the base syzygy fraction—subtracting surplus and adding shrinkage. On overflow or underflow in the correction, carry or borrow a day using the day divisor (3,939).
191
求合朔月食加時：以十二乘定小余，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，以並太為一辰弱，以前辰名之。
For syzygy or eclipse time, multiply the corrected fraction by 12 and divide by 3,939 to name the double-hour from midnight. Split any leftover into quarters of the day divisor for the shao, ban, and tai subdivisions. Further triple the tail for strong (qiang) units and combine them with weak, half, or full subdivisions per the clepsydra notation. Two strongs collapse to a weak grade; adding to tai yields a weak double-hour mark named from the prior hour.
192
求月去日道度：置入陰陽曆餘乘損益率，如通法而一，以損益兼數為定，定數十二而一為度，不盡三而一，為少、半、太。 又不盡者，一為強，二為少弱，則月去日道數也。 陽曆在表，陰曆在裏。
For lunar latitude, multiply the yin-yang remainder by the table rate, divide by 26,377, adjust the combined number, then convert to degrees at 12 parts per degree with third-degree shao, ban, and tai subdivisions. Any further remainder yields one strong unit or two for a weak grade—the final lunar distance from the ecliptic. Yang-calendar motion lies outside the solar path; yin-calendar motion inside it.
193
求昏明中星：各以度數加夜半日所在，則中星度也。
For dusk and dawn culminating stars, add each star's offset to the midnight solar longitude.
194
推五星術：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。 水率：四百五十七萬六千二百四。 推五星術：置度實各以率去之，餘以減率，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。
Five-planet method: 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. Mercury rate (shui lu): 4,576,204. For planetary conjunctions, divide each planet's degree dividend by its rate, subtract the remainder from the rate, and divide by the era divisor (39,491) for days and day-remainder entered in the year. Name the date against the first-month new moon outside the count to obtain the conjunction day.
195
求星合度：以入歲日及餘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。 求星見日術：以伏日及餘，加星合日及余，余滿紀法從日，命如前，見日也。 求星見度術：以伏度及餘，加星合度及余，余滿紀法從度，入虛去度分，命如前，星見度也。 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； 逆行出虛，則加之。
For conjunction longitude, add the year's entered days and parts to the new-moon accumulated longitude, carry at 39,491 and 360°, index from Emptiness 1, and read off the conjunction degree. For first visibility, add the planet's hiding interval to the conjunction date and carry at 39,491 parts to name the appearance day. For appearance longitude, add the hiding arc to the conjunction degree, carry at 39,491, apply the Emptiness-lodge reduction, and index as before. For daily planetary motion, convert degree-remainders to motion parts via the small-parts divisor (1,717), add them day by day with carry, hold stations unchanged, subtract in retrograde, and omit degrees while hidden. When direct motion crosses Emptiness, subtract 6 motion parts and 147 small parts. When retrograde motion exits Emptiness, add them back.
196
木：初與日合，伏，十六日，余萬七千八百三十二，行二度，度余三萬七千五百四，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。 日度余如初。 一終，三百九十八日，日余三萬五千六百六十四，行三十三度，度余二萬五千二百一十五。
Jupiter: at conjunction it hides 16 days with remainder 17,832, advancing 2° with residue 37,504, then appears in the east at dawn. Direct motion: 4 parts per day, 19° 11 parts in 112 days. It stations for 28 days. Retrograde: 3 parts per day, retreating 11° 5 parts in 86 days. It stations again for 28 days. Direct motion at 4 parts per day for 112 days, then hides in the west at dusk. Day and degree remainders return to their initial values. One synodic cycle: 398 days with remainder 35,664, planetary travel 33° with residue 25,215.
197
火：初與日合，伏，七十二日，日餘六百八，行五十五度，度余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，日度余如初。 一終，七百八十日，日餘千二百一十六，行四百一十四度，度余三萬二百五十八。 除一周，定行四十九度，度余萬九千八百九。
Mars: hides 72 days with remainder 608, advancing 55° with residue 28,865, then appears in the east at dawn. Direct fast: 17 parts per day, 68° in 92 days. Slightly slow: 14 parts per day, 56° in 92 days. Very slow: 9 parts per day, 36° in 92 days. It stations for 10 days. Retrograde: 6 parts per day, retreating 16° 16 parts in 64 days. It stations again for 10 days. Direct slow: 9 parts per day for 92 days. Slightly fast: 14 parts per day for 92 days. Fast direct: 17 parts per day for 92 days, then hides in the west at dusk with the same remainders as at first conjunction. One synodic cycle: 780 days with remainder 1,216, planetary travel 414° with residue 30,258. Reduce by one full circuit to net travel of 49° with residue 19,809.
198
土：初與日合，伏，十七日，日餘千三百七十八，行一度，度余萬九千三百三十三，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，日度余如初。 一終，三百七十八日，日餘二千七百五十六，行十二度，度余三萬一千七百九十八。
Saturn: hides 17 days with remainder 1,378, advancing 1° with residue 19,333, then appears in the east at dawn. Direct: 2 parts per day, 7° 7 parts in 84 days. It stations for 33 days. Retrograde: 1 part per day, retreating 4° 18 parts in 110 days. It stations again for 33 days. Direct motion at 2 parts per day for 84 days, then hides in the west at dusk with the same remainders as at first conjunction. One synodic cycle: 378 days with remainder 2,756, planetary travel 12° with residue 31,798.
199
金：初與日合，伏，三十九日，余三萬八千一百二十六，行四十九度，度余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，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，日度余如初。 一終，五百八十三日，日余三萬六千七百六十一，行星如之。 除一周，定行二百十八度，度余二萬六千三百一十二。 一合，二百九十一日，日余三萬八千一百二十六，行星亦如之。
Venus: hides 39 days with remainder 38,126, advancing 49° with the same residue, then appears in the west at dusk. Direct fast: 1° 5 parts per day, 112° in 92 days. Slightly slow: 1° 4 parts per day, 108° in 92 days. Very slow: 17 parts per day, 33° 6 parts in 45 days. It stations for 9 days. Slow retrograde: 16 parts per day, retreating 6° 6 parts. It hides in the west at dusk. Hidden 5 days while retreating 5°, then reconjoins the sun. After another 5 days retreating 5°, it appears in the east at dawn. Retrograde: 16 parts per day for 9 days. It stations for 9 days. Direct slow: 17 parts per day for 45 days. Slightly fast: 1° 4 parts per day for 92 days. Fast direct: 1° 5 parts per day for 92 days, then hides in the east at dawn with the same remainders as at first conjunction. One synodic cycle: 583 days with remainder 36,761 and the same planetary arc. Reduce by one circuit to net travel of 218° with residue 26,312. One inferior conjunction: 291 days with remainder 38,126 and the same planetary arc.
200
水：初與日合，伏，十四日，日余三萬七千一百十五，行三十度，度余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，日度余如初。 一終，百一十五日，日余三萬四千七百三十九，行星如之。 一合，五十七日，日余三萬七千一百一十五，行星亦如之。
Mercury: hides 14 days with remainder 37,115, advancing 30° with the same residue, then appears in the west at dusk. Direct fast: 1° 6 parts per day, 29° in 23 days. Slow: 20 parts per day, 6° 22 parts in 8 days. It stations for 2 days. Slow retrograde: 11 parts per day, retreating 22 parts in 2 days. It hides in the west at dusk. Hidden 8 days while retreating 8°, then reconjoins the sun. After another 8 days retreating 8°, it appears in the east at dawn. Retrograde: 11 parts per day for 2 days. It stations for 2 days. Direct slow: 20 parts per day for 8 days. Fast direct: 1° 6 parts per day for 23 days, then hides in the east at dawn with the same remainders as at first conjunction. One synodic cycle: 115 days with remainder 34,739 and the same planetary arc. One inferior conjunction: 57 days with remainder 37,115; planetary motion matches.
201
上元之歲，歲在甲子，天正甲子朔夜半冬至，日月五星，聚于虛度之初，陰陽遲疾，並自此始。
At the superior origin, in the jiazi year, the first-month jiazi new moon fell at midnight on the winter solstice; sun, moon, and five planets stood at the start of the Void lodge, and yin-yang slow-fast reckoning all begin from that moment.
202
世祖下之有司，使內外博議，時人少解歷數，竟無異同之辨。 唯太子旅賁中郎將戴法興議，以為：
Emperor Xiaowu ordered the ministries to open debate inside and outside the court, but few then understood calendrical computation, and no clear consensus emerged. Only Dai Faxing, Palace Attendant-in-Ordinary to the Crown Prince, offered a formal objection, saying:
203
三精數微，五緯會始，自非深推測，窮識晷變，豈能刊古革今，轉正圭宿。 案沖之所議，每有違舛，竊以愚見，隨事辨問。 案沖之新推曆術，「今冬至所在，歲歲微差」。 臣法興議：夫二至發斂，南北之極，日有恆度，而宿無改位。 古曆冬至，皆在建星。 戰國橫騖，史官喪紀，爰及漢初，格候莫審，後雜覘知在南斗二十二度，元和所用，即與古曆相符也。 逮至景初，而終無毫忒。 《書》雲：「日短星昴，以正仲冬。」 直以月維四仲，則中宿常在衛陽，羲、和所以正時，取其萬世不易也。 沖之以為唐代冬至日在今宿之左五十許度，遂虛加度分，空撤天路。 其置法所在，近違半次，則四十五年九月，率移一度。 在《詩》「七月流火」，此夏正建申之時也。 「定之方中」，又小雪之節也。 若冬至審差，則豳公火流，晷長一尺五寸，楚宮之作，晝漏五十三刻，此詭之甚也。 仲尼曰：「丘聞之，火伏而後蟄者畢。 今火猶西流，司曆過也。」 就如沖之所誤，則星無定次，封有差方。 名號之正，古今必殊，典誥之音，代不通軌，堯之開、閉，今成建、除。 今之壽星，乃周之鶉尾，即時東壁，已非玄武，軫星頓屬蒼龍，誣天痛經，乃至於此。
The three luminaries' parameters are fine-grained and the five planets' conjunction epoch is delicate; without deep calculation and full mastery of gnomon and seasonal change, how could one revise antiquity, reform the present, and reset the jade-standard lodges? Reviewing Zu Chongzhi's proposals, I find repeated errors; I shall therefore question each point in turn from my own limited understanding. Zu Chongzhi's new calendar holds that "the winter solstice moves slightly from year to year." Dai Faxing replies: The solstices mark the turning points of seasonal expansion and contraction; the sun has fixed ecliptic measure, but the lodges do not shift their places. Ancient calendars always placed the winter solstice in the Jian lodge. Warring States calendrics were chaotic and court historiography broke down; through early Han, gnomon readings were unreliable, until later observation fixed the solstice at Southern Dipper 22°—the Han Yuanhe calendar adopted that and matched the old reckoning. Down to the Wei Jingchu calendar there was not the slightest discrepancy. The Documents say: "The days are short and the Pleiades culminate at dusk—thus mid-winter is fixed." If the four seasonal lodges are read by the moon's axis, their central stars always fall in the Weiyang quarter—the very reason Xi and He fixed time by what would never change for ten thousand generations. Zu Chongzhi claims that at the Tang epoch the winter solstice stood some fifty degrees west of today's position, and on that basis he has added empty degrees and parts, dismantling the celestial road. Where his method is anchored, it departs by nearly half a lodge; on his rate the solstice would shift one degree every forty-five years and nine months. The Odes' "In the seventh month the Fire Star sets" marks the month when the Xia calendar makes shen the first month. "Ding stands centered in the square" likewise belongs to the Lesser Snow node. If the solstice were truly off by his amount, the Fire Star would set in Duke Bin's day, noon shadows would measure one chi five cun, and the Chu palace would have been built with a fifty-three-mark day clepsydra—an extreme absurdity. Confucius said: "I have heard that after the Fire Star sets, every hibernating creature has finished its dormancy. Yet in Confucius' day the Fire Star was still moving westward—the calendar officer had fallen behind." If Zu Chongzhi were wrong in that way, stars would have no fixed order and regional field-allotments would be displaced. The proper names of antiquity and the present would necessarily differ; the sounds of canonical edicts could not match across ages; Yao's Opening and Closing would become today's Establishment and Removal. Today's Longevity Star was Zhou's Chimei Tail; Eastern Wall is no longer Dark Warrior, and Chariot suddenly belongs to Azure Dragon—distorting Heaven and violating the classics to this degree.
204
沖之又改章法三百九十一年有一百四十四閏。 臣法興議：夫日有緩急，故鬥有闊狹，古人制章，立為中格，年積十九，常有七閏，晷或虛盈，此不可革。 沖之削閏壞章，倍減餘數，則一百三十九年二月，於四分之科，頓少一日； 七千四百二十九年，輒失一閏。 夫日少則先時，閏失則事悖。 竊聞時以作事，事以厚生，以此乃生人之大本，歷數之所先，愚恐非沖之淺慮妄可穿鑿。
Zu Chongzhi also changed the intercalation rule to 391 years with 144 intercalations. Dai Faxing replies: Because the sun's motion varies, the Dipper's opening varies in width; the ancients fixed the rule at a middle standard—over nineteen years there are always seven intercalations, and gnomon readings may run slightly long or short; this cannot be changed. Zu Chongzhi cuts intercalations and breaks the rule, halving the remainder numbers; then in 139 years and two months, on the Quarter-Remainder standard, one day is suddenly lost; in 7,429 years one intercalation is lost. Too few days make the seasons run early; lost intercalations make affairs fall out of step. Seasons govern work, and work sustains life—that is the foundation of human existence and the first concern of calendrical science; I doubt Zu Chongzhi's shallow thinking can arbitrarily rework it.
205
沖之又命上元日度發自虛一，雲虛為北方列宿之中。 臣法興議：沖之既雲冬至歲差，又謂虛為北中，舍形責影，未足為迷。 何者？ 凡在天非日不明，居地以鬥而辨。 借令冬至在虛，則黃道彌遠，東北當為黃鐘之宮，室壁應屬玄枵之位，虛宿豈得複為北中乎？ 曲使分至屢遷，而星次不改，招搖易繩，而律呂仍往，則七政不以璣衡致齊，建時亦非攝提所紀，不知五行何居，六屬安托？
Zu Chongzhi also set the superior-origin solar degree to begin at Void 1, claiming Void as the center of the northern lodges. Dai Faxing replies: Zu Chongzhi both claims solstice precession and calls Void the northern center; abandoning form for shadow alone is not yet proof of error. Why? In the sky nothing is clear without the sun; on earth one orients by the Dipper. Suppose the solstice stood in Void: the ecliptic would lie farther out, northeast ought to be Yellow Bell's palace, and Room-Wall ought to fall in Dark Establishment—how could Void still be the northern center? Forcing the seasonal nodes to keep shifting while lodge order stays fixed, moving the Dipper cord while pitch-pipes remain unchanged—the seven regulators would not align by the armillary, Establishment time would not be marked by Sagittarius, and one would not know where the Five Phases or six domains belong.
206
沖之又令上元年在甲子。 臣法興議：夫置元設紀，各有所尚，或據文於圖讖，或取效於當時。 沖之雲，「群氏糾紛，莫審其會」。 昔《黃帝》辛卯，日月不過； 《顓頊》乙卯，四時不忒； 《景初》壬辰，晦無差光； 《元嘉》庚辰，朔無錯景，豈非承天者乎！ 沖之苟存甲子，可謂為合以求天也。
Zu Chongzhi also set the superior-origin year in jiazi. Dai Faxing replies: Setting the origin and establishing the era each follow different preferences—some rely on apocryphal texts, some take proof from their own time. Zu Chongzhi says, "Schools dispute endlessly and none can verify the conjunction." The Yellow Emperor calendar used xinmao; sun and moon never exceeded their bounds; Zhuanxu used yimao; the four seasons were never off; Jingchu used renchen; last-day moons had no light error; Yuanjia used gengchen; new moons had no shadow error—were these not Heaven-conforming calendars! If Zu Chongzhi arbitrarily keeps jiazi, one may say he forced a fit to seek Heaven's approval.
207
沖之又令日月五緯，交會遲疾，悉以上元為始。 臣法興議：夫交會之元，則食既可求，遲疾之際，非凡夫所測。 昔賈逵略見其差，劉洪觕著其術。 至於疏密之數，莫究其極。 且五緯所居，有時盈縮，即如歲星在軫，見超七辰，術家既追算以會今，則往之與來，斷可知矣。 《景初》所以紀首置差，《元嘉》兼又各設後元者，其並省功於實用，不虛推以為煩也。 沖之既違天于改易，又設法以遂情，愚謂此治曆之大過也。
Zu Chongzhi also set sun, moon, and five-planet conjunctions, slow-fast motion, and all nodes to begin from the superior origin. Dai Faxing replies: From the conjunction origin eclipses can already be computed; slow-fast junctures are not what ordinary men can measure. Jia Kui roughly saw their discrepancy; Liu Hong crudely set out their method. As for the fine and coarse numbers, none reached the ultimate. Moreover the five planets' positions sometimes expand and contract; when Jupiter is in Chariot it appears seven lodges ahead, and calendrists already trace calculation to match the present—past and future can surely be known. Jingchu placed an initial discrepancy at the era-head and Yuanjia set separate later origins to save labor in practical use, not to push emptily and make trouble. Zu Chongzhi both violates Heaven in reform and sets methods to suit his wishes; I deem this a great fault in governing the calendar.
208
臣法興議：日有八行，各成一道，月有一道，離為九行，左交右疾，倍半相違，其一終之理，日數宜同。 沖之通周與會周相覺九千四十，其陰陽七十九周有奇，遲疾不及一匝。 此則當縮反盈，應損更益。
Dai Faxing replies: The sun has eight motions, each forming one path; the moon has one path divided into nine; crossing left and fast right, double and half mutually diverge—for one complete cycle the day-count ought to be the same. Zu Chongzhi's communication cycle and conjunction cycle differ by 9,040; his yin-yang seventy-nine cycles have a fraction, slow-fast falling short of one circuit. This means where there should be shrinkage there is surplus, where there should be decrease there is increase.
209
沖之隨法興所難辯折之曰：
Zu Chongzhi answered Dai Faxing's objections point by point, saying:
210
臣少銳愚尚，專功數術，搜練古今，博采沈奧，唐篇夏典，莫不揆量，周正漢朔，鹹加該驗。 罄策籌之思，究疏密之辨。 至若立圓舊誤，張衡述而弗改； 漢時斛銘，劉歆詭謬其數，此則算氏之劇疵也。 《乾象》之弦望定數，《景初》之交度周日，匪謂測候不精，遂乃乘除翻謬，斯又曆家之甚失也。 及鄭玄、闞澤、王蕃、劉徽，並綜數藝，而每多疏舛。 臣昔以暇日，撰正眾謬，理據炳然，易可詳密，此臣以俯信偏識，不虛推古人者也。 按何承天曆，二至先天，閏移一月，五星見伏，或違四旬，列差妄設，當益反損，皆前術之乖遠，臣曆所改定也。 既沿波以討其源，刪滯以暢其要，能使躔次上通，晷管下合，反以譏詆，不其惜乎！ 尋法興所議六條，並不造理難之關楗。 謹陳其目。
From youth I was dull but eager, specializing in numerical methods, searching ancient and modern sources and collecting hidden depths; Tang documents and Xia canons I have all measured; Zhou first month and Han new moon I have all tested. I exhausted counting-rod analysis and pursued the distinction of coarse and fine. As for the old error in establishing the circle, Zhang Heng described it and did not change it; the Han-era hu inscription, Liu Xin falsified its numbers—this is a gross flaw of the calculators. Qianxiang's fixed numbers for quarter and full moon, Jingchu's crossing-degree circuit day—this was not because observation was imprecise, but because multiplication and division were reversed in error; this again is a grave lapse of calendrists. Zheng Xuan, Kan Ze, Wang Fan, and Liu Hui all combined numerical arts, yet each had many coarse errors. In leisure days I corrected common errors; the reasoning is clear and can be examined in detail—this is how I rely on partial knowledge and do not push the ancients emptily. Examining He Chengtian's calendar: the two solstices preceded Heaven, intercalation shifted a month, five-planet visibility sometimes missed by four ten-day periods, listed discrepancies were falsely set, where increase was needed decrease was applied—all distant errors of earlier methods that my calendar has corrected. Having traced the stream to its source and removed stagnation to clarify the essentials, it can make lodge order connect above and gnomon and clepsydra match below—yet you turn to ridicule; is this not regrettable! Examining Dai Faxing's six objections, none reaches the key of rational difficulty. I respectfully set out the items.
211
其一，日度歲差，前法所略，臣據經史辨正此數，而法興設難，徵引《詩》《書》，三事皆謬。 其二，臣校晷景，改舊章法，法興立難，不能有詰，直雲「恐非淺慮，所可穿鑿」。 其三，次改方移，臣無此法，求術意誤，橫生嫌貶。 其四，曆上元年甲子，術體明整，則苟合可疑。 其五，臣其曆七曜，鹹始上元，無隙可乘，複雲「非凡夫所測」。 其六，遲疾陰陽，法興所未解，誤謂兩率日數宜同。 凡此眾條，或援謬目譏，或空加抑絕，未聞折正之談，厭心之論也。 謹隨詰洗釋，依源征對。 仰照天暉，敢罄管穴。
First, solar-degree annual precession: earlier methods omitted this; I corrected the number according to classics and history, yet Dai Faxing objects, citing the Odes and Documents—on all three points he is wrong. Second, I corrected gnomon shadows and changed the old rule; Dai Faxing objects but cannot rebut, saying only "I fear this is not shallow thinking that can be reworked." Third, changing lodge position—this method I do not have; he misread the intent of the method and groundlessly slandered it. Fourth, the superior-origin year jiazi—the method's form is clear and whole, yet he calls a forced fit suspicious. Fifth, my calendar's seven luminaries all begin from the superior origin with no gap to exploit, yet he again says "not what ordinary men can measure." Sixth, slow-fast yin-yang—what Dai Faxing has not understood he wrongly says the two rates' day-counts ought to be the same. Of all these items, some cite false points to ridicule, some suppress emptily; I have not heard correcting discussion, only arguments that weary the heart. I respectfully follow each question to clarify, tracing the source and matching evidence. Looking up to Heaven's brightness, I dare exhaust my narrow understanding.
212
法興議曰：「夫二至發斂，南北之極，日有恆度，而宿無改位。 故古曆冬至，皆在建星」。 沖之曰：周漢之際，疇人喪業，曲技競設，圖緯實繁，或借號帝王以崇其大，或假名聖賢以神其說。 是以讖記多虛，桓譚知其矯妄； 古曆舛雜，杜預疑其非直。 按《五紀論》黃帝曆有四法，顓頊、夏、周並有二術，詭異紛然，則孰識其正，此古曆可疑之據一也。 夏曆七曜西行，特違眾法，劉向以為後人所造，此可疑之據二也。 殷曆日法九百四十，而《乾鑿度》雲殷曆以八十一為日法。 若《易緯》非差，殷曆必妄，此可疑之據三也。 《顓頊》曆元，歲在乙卯，而《命曆序》雲：「此術設元，歲在甲寅。」 此可疑之據四也。 《春秋》書食有日朔者凡二十六，其所據曆，非周則魯。 以周曆考之，檢其朔日，失二十五，魯曆校之，又失十三。 二曆並乖，則必有一偽，此可疑之據五也。 古之六術，並同《四分》，《四分》之法，久則後天。 以食檢之，經三百年，輒差一日。 古曆課今，其甚疏者，朔後天過二日有餘。 以此推之，古術之作，皆在漢初週末，理不得遠。 且卻校《春秋》，朔並先天，此則非三代以前之明征矣，此可疑之據六也。 尋《律曆志》，前漢冬至日在鬥牛之際，度在建星，其勢相鄰，自非帝者有造，則儀漏或闕，豈能窮密盡微，纖毫不失。 建星之說，未足證矣。
Dai Faxing submitted: "The solstices mark expansion and contraction, the poles of north and south; the sun has fixed measure, but lodges do not change position. Therefore ancient calendars always placed the winter solstice in the Jian lodge." Zu Chongzhi said: Between Zhou and Han, calendrists lost their profession; crooked arts competed and apocryphal charts flourished; some borrowed imperial titles to magnify themselves, some borrowed sage names to make their doctrines divine. Therefore apocryphal records are mostly empty; Huan Tan knew their falsity; ancient calendars were mixed and wrong; Du Yu doubted they were not straight. Examining the Five-Era Treatise, the Yellow Emperor calendar had four methods; Zhuanxu, Xia, and Zhou each had two—strange and conflicting; then who knows which is correct? This is the first ground for doubting ancient calendars. The Xia calendar made the seven luminators move westward, specially violating common methods; Liu Xiang held it was made by later men—this is the second ground for doubt. The Yin calendar's day divisor was 940, yet the Qianzao Du says the Yin calendar used 81 as day divisor. If the Changes Apocrypha is not wrong, the Yin calendar must be false—this is the third ground for doubt. The Zhuanxu calendar's origin year was in yimao, yet the Mingli Xu says: "This method sets the origin year in jiayin." This is the fourth ground for doubt. The Spring and Autumn Annals record twenty-six eclipses with day and new moon; the calendar used was either Zhou or Lu. Testing by the Zhou calendar, checking new-moon days, twenty-five fail; testing by the Lu calendar, thirteen fail again. Both calendars are wrong, so one must be false—this is the fifth ground for doubt. The six ancient methods all match the Quarter-Remainder; the Quarter-Remainder method, over long time, lags Heaven. Testing by eclipses, over three hundred years it suddenly misses one day. Testing ancient calendars against the present, where most coarse, new moons lag Heaven by more than two days. Pushing from this, ancient methods were all made in early Han and late Zhou—it cannot logically be far earlier. Moreover testing against the Spring and Autumn Annals, new moons all precede Heaven—this is not clear proof from before the Three Dynasties; this is the sixth ground for doubt. Examining the Treatise on Pitch-pipes and Calendars, in Former Han the winter solstice stood at the juncture of Dipper and Ox, the degree in Jian—the positions are adjacent; unless the emperor made it, instruments and clepsydra were perhaps lacking—how could it be exhaustively fine without the slightest loss. The Jianxing theory is not sufficient proof.
213
法興議曰：「戰國橫騖，史官喪紀，爰及漢初，格候莫審，後雜覘知在南斗二十二度，元和所用，即與古曆相符也。 逮至景初，終無毫忒。」 沖之曰：古術訛雜，其詳闕聞，乙卯之曆，秦代所用，必有效於當時，故其言可征也。 漢武改創，檢課詳備，正儀審漏，事在前史，測星辨度，理無乖遠。 今議者所是不實見，所非徒為虛妄，辨彼駭此，既非通談，運今背古，所誣誠多，偏據一說，未若兼今之為長也。 《景初》之法，實錯五緯，今則在沖口，至曩已移日。 蓋略治朔望，無事檢候，是以晷漏昏明，並即《元和》，二分異景，尚不知革，日度微差，宜其謬矣。
Dai Faxing submitted: "In the Warring States calendrical practice ran wild, historiographers lost proper reckoning; down to early Han, gnomon observations were uncertain; later mixed observation placed it at Southern Dipper 22°; the Yuanhe calendar used this, matching the ancient calendar. Down to the Jingchu era there was never the slightest error." Zu Chongzhi said: Ancient methods were wrong and mixed; details are missing from record; the yimao calendar was used in the Qin era and must have been effective in its time, so its words can be verified. Emperor Wu of Han reformed and created anew; testing and checking were detailed and complete, correcting instruments and examining clepsydra—the matter is in earlier histories; measuring stars and distinguishing degrees, the principle is not far wrong. What the debaters approve is not real observation; what they reject is merely empty falsity; distinguishing that and rejecting this is not general discussion; applying the present while turning against antiquity, the distortions are indeed many—relying on one theory alone is not as good as combining the present. The Jingchu method in fact mis-set the five planets; Dai Faxing now cites it, yet long ago its solar day had already shifted. It roughly handled new and full moons without testing observation; therefore gnomon, clepsydra, dusk and dawn all followed Yuanhe; at the two seasonal nodes the shadows differed, yet they still did not know to reform—a slight solar discrepancy was bound to go wrong.
214
法興議曰：「《書》雲'日短星昴，以正仲冬'。 直以月推四仲，則中宿常在衛陽，羲、和所以正時，取其萬代不易也。 沖之以為唐代冬至，日在今宿之左五十許度，遂虛加度分，空撤天路。」 沖之曰：《書》以上四星昏中審分至者，據人君南面而言也。 且南北之正，其詳易准，流見之勢，中天為極。 先儒注述，其義僉同，而法興以為《書》說四星，皆在衛陽之位，自在巳地，進失向方，退非始見，迂回經文，以就所執，違訓詭情，此則甚矣。 舍午稱巳，午上非無星也。 必據中宿，餘宿豈複不足以正時。 若謂舉中語兼七列者，觜參尚隱，則不得言，昴星雖見，當雲伏矣，奎婁已見，複不得言伏見囗囗不得以為辭，則名將何附。 若中宿之通非允，當實謹檢經旨，直雲星昴，不自衛陽，衛陽無自顯之義，此談何因而立。 苟理無所依，則可愚辭成說，曾泉、桑野，皆為明證，分至之辨，竟在何日，循複再三，竊深歎息。
Dai Faxing submitted: "The Documents say 'The days are short and the Pleiades culminate at dusk—thus mid-winter is fixed. Taking the moon to push the four seasonal lodges, the central lodges always stand in Weiyang; this is why Xi and He fixed the seasons, taking what would never change for ten thousand generations. Zu Chongzhi holds that at the Tang epoch the winter solstice stood some fifty degrees west of today's position, and on that basis falsely added degrees and parts, emptying the celestial road." Zu Chongzhi said: The Documents, using the four stars' dusk culmination to verify the seasonal nodes, speak from the ruler facing south. Moreover true north and south are easy to standardize; the trend of setting stars takes mid-heaven as the pole. Earlier Ru commentaries all agree in meaning, yet Dai Faxing holds that the Documents' four stars all stand in Weiyang, naturally in the si quarter—advancing, he loses direction; retreating, it is not first appearance; twisting the text to fit his hold, violating instruction and distorting meaning—this is extreme. Abandoning wu and calling it si—above wu there are certainly stars. If one must rely on central lodges, are the other lodges not also sufficient to fix time? If mentioning central stars includes all seven lodges, then Zi and Shen would still be hidden and could not be named; though the Pleiades are visible, one would have to call them hidden; when Kui and Lou are already visible, one again could not speak of hidden or visible [text defective]—this cannot serve as an argument; then to what would the names attach? If the general rule of central lodges is not allowed, one should strictly examine the classic's intent and simply say "the stars are the Pleiades," not Weiyang; Weiyang has no meaning of self-manifestation—on what basis does this talk stand? If principle has no basis, foolish words can become doctrine; Zeng Spring and Sang Wild are all clear proofs—the distinction of the seasonal nodes, in the end on what day? Going back and forth again and again, I deeply sigh.
215
法興議曰：「其置法所在，近違半次，則四十五年九月率移一度。」 沖之曰：《元和》日度，法興所是，唯征古曆在建星，以今考之，臣法冬至亦在此宿，鬥二十二了無顯證，而虛貶臣曆乖差半次，此愚情之所駭也。 又年數之餘有十一月，而議雲九月，涉數每乖，皆此類也。 月盈則食，必在日沖，以檢日則宿度可辨，請據效以課疏密。 按太史注記，元嘉十三年十二月十六日中夜月蝕盡，在鬼四度，以沖計之，日當在牛六。 依法興議：「在女七」。 又十四年五月十五日丁夜月蝕盡，在鬥二十六度，以沖計之，日當在井三十，依法興議曰：「日在柳二。」 又二十八年八月十五日丁夜月蝕，在奎十一度，以沖計之，日當在角二； 依法興議曰：「日在角十二。」 又大明三年九月十五日乙夜月蝕盡，在胃宿之末，以沖計之，日當在氐十二； 依法興議曰：「日在心二。」 凡此四蝕，皆與臣法符同，纖毫不爽，而法興所據，頓差十度，違沖移宿，顯然易睹。 故知天數漸差，則當式遵以為典，事驗昭晰，豈得信古而疑今。
Dai Faxing submitted: "Where his method is set, it nearly violates half a lodge; then every forty-five years and nine months the position shifts one degree." Zu Chongzhi said: The Yuanhe solar degree—Dai Faxing approves it; only citing the ancient calendar in Jian—testing by the present, my winter solstice is also in this lodge; Dipper 22 has no clear proof, yet he emptily slanders my calendar for half a lodge's error—this is what my humble mind finds shocking. Moreover in the remainder of years there is an eleventh month, yet the objection says the ninth month—when numbers are involved it is often wrong; all are of this kind. When the moon is full it eclipses, necessarily at solar opposition; testing by the day, the lodge degree can be distinguished—please test by results to examine coarse and fine. Examining the Grand Clerk's annotated record: in the thirteenth year of Yuanjia, on the sixteenth day of the twelfth month, at midnight the moon was fully eclipsed at Ghost 4°; by my calculation the sun ought to have been at Ox 6. By Dai Faxing's objection: "at Woman 7." Again in the fourteenth year, on the fifteenth day of the fifth month, during the ding night, the moon was fully eclipsed at Dipper 26°; by my calculation the sun ought to have been at Well 30; Dai Faxing objected: "the sun was at Willow 2." Again in the twenty-eighth year, on the fifteenth day of the eighth month, during the ding night, there was a lunar eclipse at Kui 11°; by my calculation the sun ought to have been at Horn 2; By Dai Faxing's objection: "the sun at Horn 12." Again in the third year of Daming, on the fifteenth day of the ninth month, during the yi night, the moon was fully eclipsed at the end of Stomach lodge; by my calculation the sun ought to have been at Base 12; By Dai Faxing's objection: "the sun at Heart 2." All four eclipses match my method exactly, not the slightest discrepancy, while Dai Faxing's basis suddenly differs by ten degrees, violating the shifted lodge—clear and easy to see. Therefore one knows celestial numbers gradually shift; one ought then to follow them as standard; the test is clear and bright—how can one trust antiquity and doubt the present?
216
法興議曰：「在《詩》'七月流火'，此夏正建申之時也。 '定之方中'，又小雪之節也。 若冬至審差，則豳公火流，晷長一尺五寸，楚宮之作，晝漏五十三刻，此詭之甚也。」 沖之曰：臣按此議三條皆謬。 《詩》稱流火，蓋略舉西移之中，以為驚寒之候。 流之為言，非始動之辭也。 就如始說，冬至日度在鬥二十二，則火星之中，當在大暑之前，豈鄰建申之限。 此專自攻糾，非謂矯失。 《夏小正》：「五月昏，大火中。」 此複在衛陽之地乎？ 又謂臣所立法，楚宮之作，在九月初。 按《詩》傳箋皆謂定之方中者，室辟昏中，形四方也。 然則中天之正，當在室之八度。 臣曆推之，元年立冬後四日，此度昏中，乃處十月之初，又非寒露之日也。 議者之意，蓋誤以周世為堯時，度差五十，故致此謬。 小雪之節，自信之談，非有明文可據也。
Dai Faxing submitted: "In the Odes, 'In the seventh month the Fire Star sets,' marks the time when the Xia calendar takes shen as first month. 'Ding stands centered in the square' is again the Lesser Snow node. If the solstice were truly off by his amount, the Fire Star would set in Duke Bin's day, noon shadows would measure one chi five cun, and the Chu palace would have been built with a fifty-three-mark day clepsydra—an extreme absurdity." Zu Chongzhi said: I hold that on all three points of this objection he is wrong. The Odes speak of the setting Fire Star, broadly taking the middle of westward movement as the sign of approaching cold. "Setting" as a word is not the language of first motion. Just as in his first claim, if the solstice's solar degree were at Dipper 22, then the Fire Star's central passage would fall before Great Heat—how could it border the limit of jian-shen? This is solely self-contradictory attack—not saying he falsifies error. The Xia Small Calendar: "In the fifth month at dusk, the Great Fire is central." Is this again in Weiyang's place? He also says the method I established places the Chu palace's building in early ninth month. Examining Odes commentary and notes, all say "Ding stands centered in the square" means Room's wall is centrally dark, forming a square. Then mid-heaven's correct point ought to be at Room 8°. My calendar pushes it: four days after Beginning of Winter in year 1, this degree is centrally dark, and it falls at the start of the tenth month—not again the Cold Dew day. The objector's intent probably mistakes the Zhou age for Yao's time; the fifty-degree difference therefore causes this error. The Lesser Snow node is self-confident talk, with no clear text for proof.
217
法興議曰：「仲尼曰：'丘聞之，火伏而後蟄者畢。 今火猶西流，司曆過也。 '就如沖之所誤，則星無定次，卦有差方，名號之正，古今必殊，典誥之音，時不通軌。 堯之開、閉，今成建、除，今之壽星，乃周之鶉尾也。 即時東壁，已非玄武，軫星頓屬蒼龍，誣天背經，乃至於此。」 沖之曰：臣以為辰極居中，而列曜貞觀，群像殊體，而陰陽區別，故羽介咸陳，則水火有位，蒼素齊設，則東西可准，非以日之所在，定其名號也。 何以明之？ 夫陽爻初九，氣始正北，玄武七列，虛當子位。 若圓儀辨方，以日為主，冬至所舍，當在玄枵； 而今之南極，乃處東維，違體失中，其義何附。 若南北以冬夏稟稱，則卯酉以生殺定號，豈得春躔義方，秋麗仁域，名舛理乖，若此之反哉！ 因茲以言，因知天以列宿分方，而不在於四時，景緯環序，日不獨守故轍矣。 至於中星見伏，記籍每以審時者，蓋以歷數難詳，而天驗易顯，各據一代所合，以為簡易之政也。 亦猶夏禮未通商典，《濩》容豈襲《韶》節，誠天人之道同差，則藝之興，因代而推移矣。 月位稱建，諒以氣之所本，名隨實著，非謂鬥杓所指。 近校漢時，已差半次，審鬥節時，其效安在。 或義非經訓，依以成說，將緯候多詭，偽辭間設乎？ 次隨方名，義合宿體。 分至雖遷，而厥位不改，豈謂龍火貿處，金水亂列，名號乖殊之譏，抑未詳究。 至如壁非玄武，軫屬蒼龍，瞻度察晷，實效鹹然。 《元嘉曆法》，壽星之初，亦在翼限，參校晉注，顯驗甚眾。 天數差移，百有餘載，議者誠能馳辭騁辯，令南極非冬至，望不在沖，則此談乃可守耳。 若使日遷次留，則無事屢嫌，乃臣曆之良證，非難者所宜列也。 尋臣所執，必據經史，遠考唐典，近征漢籍，讖記碎言，不敢依述，竊謂循經之論也。 月蝕檢日度，事驗昭著，史注詳論，文存禁閣，斯又稽天之說也。 《堯典》四星，並在衛陽，今之日度，遠淮元和，誣背之誚，實此之謂。
Dai Faxing submitted: "Confucius said: 'I have heard that after the Fire Star sets, every hibernating creature has finished its dormancy. Yet in Confucius' day the Fire Star was still moving westward—the calendar officer had fallen behind. If it were as Zu Chongzhi errs, then stars would have no fixed order and the hexagrams would have displaced directions; correct names of old and present must differ; the sounds of canonical edicts cannot match across ages. Yao's Opening and Closing become today's Establishment and Removal; today's Longevity Star was Zhou's Chimei Tail. At present Eastern Wall is no longer Dark Warrior, and Chariot suddenly belongs to Azure Dragon—distorting Heaven and violating the classics to this degree." Zu Chongzhi said: I hold that the pole star stands at center while the arrayed luminaries keep fixed observation; the images differ in form while yin and yang are distinguished; therefore feathered and shelled are all arrayed, then water and fire have their places; azure and white are equally set, then east and west can be standardized—not by where the sun stands to fix names. How is this clear? At the first nine of the yang line, qi first stands due north; the seven ranks of Dark Warrior, Void occupies the zi position. If the circular instrument distinguishes direction with the sun as master, where the winter solstice lodges ought to be in Dark Establishment; yet today's south pole is in the eastern cord—violating form and losing center; to what principle does this attach? If north and south are named by winter and summer's endowment, then mao and you ought to be fixed by birth and killing—how could spring lodge in the domain of righteousness and autumn stand in the domain of benevolence—names wrong and principle twisted, so contrary! From this one speaks, and knows Heaven divides direction by arrayed lodges, not by the four seasons; the bright cords circle in order, and the sun does not alone keep its old track. As for central stars' visibility and hiding, records often use them to verify seasons because calendrical numbers are hard to detail while celestial tests are easy to show—each relies on what matched one age as a simple policy. It is also like Xia rites not matching Shang canons—how could Hu's music inherit Shao's measure; truly if Heaven and man's ways differ alike, then the rise of arts shifts with the age. Calling the month's position Establishment plainly takes the qi's root; the name follows the reality—it does not mean the Dipper handle's pointing. Testing near Han times, it already differed by half a lodge; verifying Dipper-node seasons, where is the effect? Or if meaning is not from classic instruction, relying on it to make doctrine—will apocryphal observations be mostly strange and false words occasionally set? Next, following direction names, meaning matches lodge form. Though the seasonal nodes shift, their positions do not change—does one mean Dragon and Fire trade places, Metal and Water disorder the array—the reproach of mismatched names? I have not yet examined it in detail. As for Wall not being Dark Warrior and Chariot belonging to Azure Dragon—observing degrees and examining gnomon, the real effect is indeed so. In the Yuanjia calendar method, at the start of the Longevity Star it was also in Wings' limit; comparing Jin commentary, clear proofs are many. Celestial numbers shift over more than a hundred years; if the debater can truly gallop words and race argument, making the south pole not the winter solstice and full moon not at opposition—then this talk could be kept. If the sun keeps shifting lodge and pausing, then there is no need for repeated suspicion—this is good proof of my calendar, not what objectors ought to list. Examining what I hold, I must rely on classics and history, examining Tang canons in the distance and Han records near at hand; apocryphal fragments I dare not follow—I deem this discussion that follows the classics. Testing solar degree by lunar eclipse, the proof is clear and bright; history's annotated discussion survives in the forbidden archive—this again is investigation of Heaven's sayings. The Canons of Yao's four stars all stand in Weiyang; today's solar degree is far from Yuanhe—the reproach of distortion and violation is exactly this.
218
法興議曰：「夫日有緩急，故鬥有闊狹，古人制章，立為中格，年積十九，常有七閏，晷或盈虛，此不可革。 沖之削閏壞章，倍減餘數，則一百三十九年二月，於四分之科，頓少一日； 七千四百二十九年，輒失一閏。 夫日少則先時，閏失則事悖。 竊聞時以作事，事以厚生，此乃生民之所本，歷數之所先。 愚恐非沖之淺慮，妄可穿鑿。」 沖之曰：按《後漢書》及《乾象說》，《四分曆法》，雖分章設篰創自元和，而晷儀眾數定于嘉平三年。 《四分志》，立冬中影長一丈，立春中影九尺六寸。 尋冬至南極，日晷最長，二氣去至，日數既同，則中影應等，而前長後短，頓差四寸，此曆景冬至後天之驗也。 二氣中影，日差九分半弱，進退均調，略無盈縮。 以率計之，二氣各退二日十二刻，則晷影之數，立冬更短，立春更長，並差二寸，二氣中影俱長九尺八寸矣。 即立冬、立春之正日也。 以此推之，曆置冬至，後天亦二日十二刻也。 嘉平三年，時曆丁醜冬至，加時正在日中。 以二日十二刻減之，天定以乙亥冬至，加時在夜半後三十八刻。 又臣測景曆紀，躬辨分寸，銅表堅剛，暴潤不動，光晷明潔，纖毫盡然。 據大明五年十月十日，影一丈七寸七分半，十一月二十五日，一丈八寸一分太，二十六日，一丈七寸五分強，折取其中，則中天冬至，應在十一月三日。 求其蚤晚，令後二日影相減，則一日差率也。 倍之為法，前二日減，以百刻乘之為實，以法除實，得冬至加時在夜半後三十一刻，在《元嘉曆》後一日，天數之正也。 量檢竟年，則數減均同，異歲相課，則遠近應率。 臣因此驗，考正章法。 今以臣曆推之，刻如前，竊謂至密，永為定式。 尋古曆法並同《四分》，《四分》之數久則後天，經三百年，朔差一日。 是以漢載四百，食率在晦。 魏代已來，遂革斯法，世莫之非者，誠有效於天也。 章歲十九，其疏尤甚，同出前術，非見經典。 而議雲此法自古，數不可移。 若古法雖疏，永當循用，謬論誠立，則法興複欲施《四分》於當今矣，理容然乎？ 臣所未譬也。 若謂今所革創違舛失衷者，未聞顯據有以矯奪臣法也。 《元嘉曆》術，減閏餘二，直以襲舊分粗，故進退未合。 至於棄盈求正，非為乖理。 就如議意，率不可易，則分無增損，承天置法，複為違謬。 節氣蚤晚，當循《景初》，二至差三日，曾不覺其非，橫謂臣曆為失，知日少之先時，未悟增月甚惑也。 誠未睹天驗，豈測歷數之要，生民之本，諒非率意所斷矣。 又法興始雲窮識晷變，可以刊舊革今，複謂晷數盈虛，不可為准，互自違伐，罔識所依。 若推步不得准，天功絕於心目，未詳曆紀何因而立。 案《春秋》以來千有餘載，以食檢朔，曾無差失，此則日行有恆之明征也。 且臣考影彌年，窮察毫微，課驗以前，合若符契，孟子以為千歲之日至，可坐而知，斯言實矣。 日有緩急，未見其證，浮辭虛貶，竊非所懼。
Dai Faxing submitted: "Because the sun's motion varies, the Dipper's opening varies in width; the ancients fixed the rule at a middle standard—over nineteen years there are always seven intercalations, and gnomon readings may run slightly long or short—this cannot be changed. Zu Chongzhi cuts intercalations and breaks the rule, halving the remainder numbers; then in 139 years and two months, on the Quarter-Remainder standard, one day is suddenly lost; in 7,429 years one intercalation is lost. Too few days make the seasons run early; lost intercalations make affairs fall out of step. Seasons govern work, and work sustains life—this is the foundation of the people and the first of calendrical numbers. I fear this is not something Zu Chongzhi's shallow thinking can arbitrarily rework." Zu Chongzhi said: Examining the Book of Later Han and the Qianxiang Treatise, the Quarter-Remainder calendar method, though dividing the rule and setting bamboo markers was created from Yuanhe, gnomon instruments and many numbers were fixed in Jiaoping year 3. The Quarter-Remainder Treatise: at Beginning of Winter the central shadow is one zhang long; at Beginning of Spring the central shadow is nine chi six cun. Examining the winter solstice south pole, the sun's gnomon is longest; the two qi depart from the solstice with equal day-count, so the central shadow ought to be equal—yet the former is long and the latter short, suddenly differing four cun—this is the calendar shadow's proof that the winter solstice lags Heaven. The two qi's central shadows differ by nine fen and a half weak in day-count; advance and retreat are evenly adjusted, roughly without surplus or shrinkage. Calculating by rate, each qi retreats two days twelve marks; then the gnomon shadow numbers—Beginning of Winter shorter, Beginning of Spring longer—both differ two cun; the two qi's central shadows are both nine chi eight cun long. That is the correct day of Beginning of Winter and Beginning of Spring. Pushing from this, the calendar's set winter solstice also lags Heaven by two days twelve marks. In Jiaoping year 3, the calendar's dingchou winter solstice, the added hour was exactly at midday. Subtracting two days twelve marks, Heaven fixes yihai winter solstice, the added hour at thirty-eight marks after midnight. Moreover I measured shadows over calendar eras, personally distinguishing fen and cun; the bronze table was firm and rigid, swelling and wetting did not move it; the light gnomon was bright and clean, the slightest hair was clear. According to the measurements for the fifth year of Daming: on the tenth day of the tenth month, the shadow was one zhang, seven cun, seven and a half fen; on the twenty-fifth day of the eleventh month, one zhang, eight cun, and one large fen; on the twenty-sixth, one zhang, seven cun, and a little over five fen. Taking the mean, the mid-heaven winter solstice should fall on the third day of the eleventh month. Seeking early and late, let the shadows of the next two days be subtracted—then the one-day difference rate. Doubling it makes the divisor; subtract the previous two days, multiply by one hundred marks for the dividend, divide the dividend by the divisor—the winter solstice added hour is thirty-one marks after midnight, one day after the Yuanjia calendar—the correction of celestial numbers. Measuring and checking the whole year, the numbers decrease equally; testing different years, near and far match the rate. From this test I corrected the rule. Now pushing by my calendar, the marks as before—I deem it extremely precise and forever the fixed pattern. Examining ancient calendar methods, all match the Quarter-Remainder; the Quarter-Remainder numbers over long time lag Heaven; over three hundred years new moons miss one day. Therefore in Han's four hundred years, eclipses often fell on last days. From Wei onward they reformed this method; no one faulted it—truly it was effective with Heaven. Rule years nineteen—its coarseness is especially great; it comes from earlier methods alike, not seen in classics. Yet the objection says this method is ancient and the number cannot be moved. If ancient methods though coarse must forever be followed, the false argument is truly established—then Dai Faxing again wants to apply the Quarter-Remainder to the present day; is that reasonable? This is what I have not understood. If one says what I now reform violates error and misses the mean, I have not heard clear proof sufficient to overturn my method. The Yuanjia calendar method reduced intercalation remainder by two, directly inheriting the old coarse fraction, so advance and retreat did not match. As for abandoning surplus to seek correctness, it is not violating principle. Just as in the objection's intent, if the rate cannot be changed, then the fraction has no increase or decrease—setting the method to match Heaven again becomes wrong. Whether solar nodes come early or late ought to follow Jingchu; the two solstices differ by three days, yet he never notices the error, and groundlessly says my calendar is wrong—knowing days are too few makes seasons early, yet not understanding the added month is very confused. Truly not seeing celestial tests, how can one measure the essentials of calendrical numbers—the foundation of the people is surely not what arbitrary intent can decide. Moreover Dai Faxing first said that fully knowing gnomon change could edit the old and reform the present, yet again says gnomon numbers empty and full cannot be standard—mutually contradicting himself, not knowing what he relies on. If computation cannot be standard, Heaven's work is cut off from eye and mind—I do not know on what basis the calendar was established. Examining from the Spring and Autumn Annals for more than a thousand years, testing new moons by eclipses, there was never error—this is clear proof that the sun's motion is constant. Moreover I tested shadows over many years, examining to the slightest hair; testing against earlier methods, the match is like seal and tally; Mencius held that a thousand years' solstice could be known sitting—this saying is true. The sun's slow and fast motion—I have not seen its proof; floating words and empty slander I am not afraid of.
219
法興議曰：「沖之既雲冬至歲差，又謂虛為北中，舍形責影，未足為迷。 何者？ 凡在天非日不明，居地以鬥而辨，借令冬至在虛，則黃道彌遠，東北當為黃鐘之宮，室壁應屬玄枵之位，虛宿豈得複為北中乎？ 曲使分至屢遷，而星次不改，招搖易繩，而律呂仍往，則七政不以璣衡致齊，建時亦非攝提所紀，不知五行何居，六屬安托。」 沖之曰：此條所嫌，前牒已詳。 次改方移，虛非中位，繁辭廣證，自構紛惑，皆議者所謬誤，非臣法之違設也。 七政致齊，實謂天儀，鄭、王唱述，厥訓明允，雖有異說，蓋非實義。
Dai Faxing submitted: "Zu Chongzhi both claims solstice precession and calls Void the northern center; abandoning form for shadow alone is not yet proof of error. Why? In the sky nothing is clear without the sun; on earth one orients by the Dipper; suppose the solstice stood in Void, the ecliptic would lie farther out, northeast ought to be Yellow Bell's palace, and Room-Wall ought to fall in Dark Establishment—how could Void still be the northern center? Forcing the seasonal nodes to keep shifting while lodge order stays fixed, moving the Dipper cord while pitch-pipes remain unchanged—the seven regulators would not align by the armillary, Establishment time would not be marked by Sagittarius, and one would not know where the Five Phases or six domains belong." Zu Chongzhi said: The objection on this item was already detailed in the earlier memorial. Next, changing lodge position—Void is not the central position; verbose words and broad proofs self-construct confusion—all are the debater's errors, not violations of my method's setting. The seven regulators' alignment truly means the celestial instrument; Zheng and Wang proclaimed and set forth—their instruction is clear and fair; though there are differing sayings, they are mostly not real meaning.
220
法興議曰：「夫置元設紀，各有所尚，或據文於圖讖，或取效於當時。 沖之雲'群氏糾紛，莫審其會。 '昔《黃帝》辛卯，日月不過，《顓頊》乙卯，四時不忒，《景初》壬辰，晦無差光，《元嘉》庚辰，朔無錯景，豈非承天者乎？ 沖之苟存甲子，可謂為合以求天也。」 沖之曰：夫曆存效密，不容殊尚，合讖乖說，訓義非所取，雖驗當時，不能通遠，又臣所未安也。 元值始名，體明理正。 未詳辛卯之說何依，古術詭謬，事在前牒，溺名喪實，殆非索隱之謂也。 若以曆合一時，理無久用，元在所會，非有定歲者，今以效明之。 夏、殷以前，載籍淪逸，《春秋》漢史，鹹書日蝕，正朔詳審，顯然可徵。 以臣曆檢之，數皆協同，誠無虛設，循密而至，千載無殊，則雖遠知矣。 備閱曩法，疏越實多，或朔差三日，氣移七晨，未聞可以下通於今者也。 元在乙丑，前說以為非正，今值甲子，議者複疑其苟合，無名之歲，自昔無之，則推先者，將何從乎？ 曆紀之作，幾於息矣。 夫為合必有不合，願聞顯據，以核理實。
Dai Faxing submitted: "Setting the origin and establishing the era each follow different preferences—some rely on apocryphal texts, some take proof from their own time. Zu Chongzhi says 'Schools dispute endlessly and none can verify the conjunction. The Yellow Emperor calendar used xinmao, sun and moon never exceeded; Zhuanxu used yimao, the four seasons were never off; Jingchu used renchen, last-day moons had no light error; Yuanjia used gengchen, new moons had no shadow error—were these not Heaven-conforming calendars? If Zu Chongzhi arbitrarily keeps jiazi, one may say he forced a fit to seek Heaven's approval." Zu Chongzhi said: A calendar preserves effectiveness and fineness—it cannot tolerate different preferences; matching apocrypha and violating doctrine, interpretive meaning is not taken; though tested in its time, it cannot extend far—this again is what I am not settled about. The origin value begins the name; the form is clear and principle correct. I do not know on what the xinmao saying relies; ancient methods are strange and wrong—the matter is in the earlier memorial; drowning the name and losing the reality is hardly what seeking the hidden means. If because the calendar matches one time, principle has no long use—the origin lies where things meet, not in a fixed year—now I make it clear by effect. Before Xia and Yin, records were lost; the Spring and Autumn Annals and Han histories all record solar eclipses; correct new moon is examined in detail, clearly verifiable. Testing by my calendar, the numbers all agree; truly no empty setting; following fineness to the end, a thousand years without difference—then though distant, it is known. Fully reading former methods, coarseness and distance are really many; some new moons miss three days, some qi shift seven lodges—I have not heard that one can extend downward to the present. The origin in yichou—the earlier saying held it not correct; now the value is jiazi, and the debater again doubts a forced fit; a year without name never existed from old—then for pushing earlier, from what will one proceed? The making of calendar eras is nearly at an end. For a fit there must be misfits—I wish to hear clear proof to examine principle and reality.
221
法興曰：「夫交會之元，則蝕既可求，遲疾之際，非凡夫所測。 昔賈逵略見其差，劉洪粗著其術，至於疏密之數，莫究其極。 且五緯所居，有時盈縮，即如歲星在軫，見超七辰，術家既追算以會今，則往之與來，斷可知矣。 《景初》所以紀首置差，《元嘉》兼又各設後元者，其並省功於實用，不虛推以為煩也。 沖之既違天于改易，又設法以遂情，愚謂此治曆之大過也。」 沖之曰：遲疾之率，非出神怪，有形可檢，有數可推，劉、賈能述，則可累功以求密矣。 議又雲「五緯所居，有時盈縮」。 「歲星在軫，見超七辰」。 謂應年移一辰也。 案歲星之運，年恆過次，行天七匝，輒超一位。 代以求之，曆凡十法，併合一時，此數咸同，史注所記，天驗又符。 此則盈次之行，自其定準，非為衍度濫徙，頓過其沖也。 若審由盈縮，豈得常疾無遲。 夫甄耀測象者，必料分析度，考往驗來，准以實見，據以經史。 曲辯碎說，類多浮詭，甘、石之書，互為矛盾。 今以一句之經，誣一字之謬，堅執偏論，以罔正理，此愚情之所未厭也。 算自近始，眾法可同，但《景初》之二差，承天之後元，實以奇偶不協，故數無盡同，為遺前設後，以從省易。 夫建言倡論，豈尚矯異，蓋令實以文顯，言勢可極也。 稽元曩歲，群數鹹始，斯誠術體，理不可容譏； 而譏者以為過，謬之大者。 然則《元嘉》置元，雖七率舛陳，而猶紀協甲子，氣朔俱終，此又過謬之小者也。 必當虛立上元，假稱曆始，歲違名初，日避辰首，閏餘朔分，月緯七率，並不得有盡，乃為允衷之制乎？ 設法情實，謂意之所安； 改易違天，未睹理之譏者也。
Dai Faxing said: "From the conjunction origin eclipses can already be computed; slow-fast junctures are not what ordinary men can measure. Jia Kui roughly saw their discrepancy; Liu Hong crudely set out their method; as for the fine and coarse numbers, none reached the ultimate. Moreover the five planets' positions sometimes expand and contract; when Jupiter is in Chariot it appears seven lodges ahead, and calendrists already trace calculation to match the present—past and future can surely be known. Jingchu placed an initial discrepancy at the era-head and Yuanjia set separate later origins to save labor in practical use, not to push emptily and make trouble. Zu Chongzhi both violates Heaven in reform and sets methods to suit his wishes; I deem this a great fault in governing the calendar." Zu Chongzhi said: Slow-fast rates are not born from spirits and monsters; there is form that can be tested, numbers that can be pushed; if Liu and Jia could set them forth, one can accumulate work to seek fineness. The objection again says "the five planets' positions sometimes expand and contract." "When Jupiter is in Chariot it appears seven lodges ahead." Meaning it ought to shift one lodge each year. Examining Jupiter's motion, each year it always passes its lodge; circling Heaven seven circuits, it suddenly advances one position. Replacing generations to seek it, calendars in all ten methods combined at one time—this number is everywhere the same; history's annotated record and celestial tests again match. This is surplus-lodge motion from its fixed standard—not expanding degrees and recklessly shifting, suddenly passing its opposition. If it truly came from expansion and contraction, how could it always be fast without slow? One who selects brilliance and measures images must calculate and analyze degrees, examine past and test future, standardize by real observation, rely on classics and history. Crooked argument and fragmentary talk are mostly floating and strange; the books of Gan and Shi mutually contradict. Now with one line from a classic to slander one character's error, stubbornly holding partial argument to obscure correct principle—this is what my humble mind has not yet tired of. Calculation from the near start—many methods can agree; only Jingchu's two discrepancies, sub-epoch after matching Heaven, in fact because odd and even do not harmonize, therefore numbers cannot all match—setting former and later to follow simplification. Proposing words and initiating discussion—how could one esteem perverse difference; it is simply to make reality show through text, and speech reach its limit. Examining the former origin year, all numbers begin—this is truly the method's form; principle cannot tolerate ridicule; yet the ridiculer holds it excessive—the greater error. Then Yuanjia set the origin—though seven rates were wrongly arrayed, it still matched jiazi in the era, qi and new moon both ending—this again is the lesser excessive error. Must one emptily set a superior origin, falsely call it calendar beginning, year violating name's start, day avoiding lodge head, intercalation remainder and new-moon fraction, moon's seven rates—all cannot be exhausted—then it is the balanced-center system? Setting method's reality and intent means where the mind finds rest; reforming and violating Heaven—I have not seen ridicule of principle.
222
法興曰：「日有八行，合成一道，月有一道，離為九行，左交右疾，倍半相違，其一終之理，日數宜同。 沖之通同與會周相覺九千四十，其陰陽七十九周有奇，遲疾不及一匝，此則當縮反盈，應損更益。」 沖之曰：此議雖遊漫無據，然言跡可檢。 按以日八行譬月九道，此為月行之軌，當循一轍，環匝於天，理無差動也。 然則交會之際，當有定所，豈容或鬥或牛，同麗一度。 去極應等，安得南北無常。 若日月非例，則八行之說是衍文邪？ 左交右疾，語甚未分，為交與疾對？ 為舍交即疾？ 若舍交即疾，即交在平率入曆七日及二十一日是也。 值交蝕既當在盈縮之極，豈得損益，或多或少。 若交與疾對，則在交之沖，當為遲疾之始，豈得入曆或深或淺，倍半相違，新故所同，複摽此句，欲以何明。 臣覽曆書，古今略備，至如此說，所未前聞，遠乖舊准，近背天數，求之愚情，竊所深惑。 尋遲疾陰陽不相生，故交會加時，進退無常，昔術著之久矣，前儒言之詳矣。 而法興雲日數同。 竊謂議者未曉此意，乖謬自著，無假驟辯，既雲盈縮失衷，複不備記其數，或自嫌所執，故泛略其說乎？ 又以全為率，當互因其分，法興所列二數皆誤，或以八十為七十九，當縮反盈，應損更益，此條之謂矣。 總檢其議，豈但臣曆不密，又謂何承天法乖謬彌甚。 若臣曆宜棄，則承天術益不可用。 法興所見既審，則應革創。 至非景極，望非日沖，凡諸新說，必有妙辯乎？
Dai Faxing said: "The sun has eight motions, forming one path together; the moon has one path, divided into nine motions; crossing left and fast right, double and half mutually diverge—for one complete cycle the day-count ought to be the same. Zu Chongzhi's communication cycle and conjunction cycle differ by 9,040; his yin-yang seventy-nine cycles have a fraction, slow-fast falling short of one circuit—this means where there should be shrinkage there is surplus, where there should be decrease there is increase." Zu Chongzhi said: This objection though wandering without basis, yet its words can be tested. Examining comparing the sun's eight motions to the moon's nine paths—this is the moon's track; it ought to follow one rut, circling Heaven in circuits, principle allowing no discrepancy. Then at conjunction junctures there ought to be a fixed place—how could it be Dipper or Ox, both standing at one degree? Distance from pole ought to be equal—how can north and south be without constant rule? If sun and moon are not the pattern, then is the eight-motion saying derivative text? Left crossing right fast—the words are very unclear; does crossing pair with fast? Or abandoning crossing is immediately fast? If abandoning crossing is immediately fast, then crossing at mean-rate calendar entry on day seven and day twenty-one is it. When crossing meets eclipse it ought to be at expansion-contraction extremes—how can there be decrease or increase, more or less? If crossing pairs with fast, then at crossing's opposition ought to be slow-fast's beginning—how can calendar entry be deep or shallow, double and half diverging, new and old the same—again marking this line, intending to clarify what? I read calendar books ancient and modern, roughly complete—to such a saying I have never heard before; far violating old standard, near betraying celestial numbers—seeking my humble mind, I am deeply confused. Examining slow-fast yin-yang—they do not generate each other; therefore conjunction added hour advance and retreat have no constant rule; former methods long set it forth, earlier Ru spoke in detail. Yet Dai Faxing says day-counts are the same. I deem the debater has not understood this meaning—error is self-evident, needing no sudden argument; since he says expansion-contraction misses the mean, again he does not fully record the numbers—or he suspects his own hold, therefore broadly abbreviates the saying? Again taking whole as rate ought mutually to follow their fractions; the two numbers Dai Faxing listed are both wrong—or taking eighty for seventy-nine: where there should be shrinkage there is surplus, where there should be decrease there is increase—this is what this item means. Summarily examining his objections—not only is my calendar imprecise, he also says He Chengtian's method is even more wrong. If my calendar ought to be abandoned, then Chengtian's art is even less usable. If Dai Faxing's view is already examined, then reform ought to be undertaken. Until non-Jing extreme and full moon non-solar opposition—all new sayings must have clever argument?
223
時法興為世祖所寵，天下畏其權，既立異議，論者皆附之。 唯中書舍人巢尚之是沖之之術，執據宜用。 上愛奇慕古，欲用沖之新法，時大明八年也。 故須明年改元，因此改曆。 未及施用，而宮車晏駕也。
At the time Dai Faxing was favored by Emperor Xiaowu; the realm feared his power; having set different objections, debaters all attached to him. Only Palace Secretary Chao Shangzhi approved Zu Chongzhi's art, holding that evidence showed it should be used. The emperor loved the strange and admired antiquity, wishing to use Zu Chongzhi's new method—it was then the eighth year of Daming. Therefore he had to change the era name the next year, and thereby reform the calendar. Before it could be applied, the emperor died.