1
紹興統元乾道淳熙會元曆
The Calendars of Shaoxing, Tongyuan, Qiandao, Chunxi, and Huiyuan.
2
演紀上元甲子，距紹興五年乙卯，歲積九千四百二十五萬一千五百九十一。 《乾道》上元甲子，距乾道三年丁亥，歲積九千一百六十四萬五千八百二十三。 《淳熙》上元甲子，距淳熙三年丙申，歲積五千二百四十二萬一千九百七十二。 《會元》上元甲子，距紹熙三年辛亥，歲積二千五百四十九萬四千七百六十七。
By the era-cycle reckoning from the Superior Origin in the year jiazi, the elapsed count to the fifth year of Shaoxing (1135), year yimao, stands at 94,251,591 years. For the Qiandao calendar, from the Superior Origin in the year jiazi to the third year of Qiandao (1167), year dinghai, the accumulated count is 91,645,823 years. For the Chunxi calendar, from the Superior Origin in the year jiazi to the third year of Chunxi (1176), year bingshen, the accumulated count is 52,421,972 years. For the Huiyuan calendar, from the Superior Origin in the year jiazi to the third year of Shaoxi (1190), year xinhai, the accumulated count is 25,494,767 years.
3
步氣朔
Procedure for Calculating Qi and New Moons.
4
元法：六千九百三十。 《乾道》三萬。 《淳熙》五千六百四十。 《會元》統率三萬八千七百。
Origin divisor: 6,930. Qiandao: 30,000. Chunxi: 5,640. Huiyuan: unified rate 38,700.
5
歲周：二百五十三萬一千一百三十八； 歲周日：三百六十五、餘一千六百八十八。 《乾道》期實一千九十五萬七千三百八，歲週三百六十五、餘七千三百八。 《淳熙》歲實二百五萬九千九百七十四，歲周日三百六十五、餘一千三百七十四。 《會元》氣率一千四百一十三萬四千九百二十二。
Tropical year cycle: 2,531,138; tropical year in days: 365, remainder 1,688. Qiandao: period constant 10,957,308; year cycle 365 days, remainder 7,308. Chunxi: year constant 2,059,974; year-day 365, remainder 1,374. Huiyuan: qi rate 14,134,922.
6
氣策：一十五日、餘一千五百一十四、秒十五。 《乾道》餘六千五百五十四半。 《淳熙》餘一千二百二十二、秒二十五。 《會元》餘八千四百五十五半。
Qi interval: 15 days, remainder 1,514, 15 seconds. Qiandao: remainder 6,554½. Chunxi: remainder 1,222, 25 seconds. Huiyuan: remainder 8,455½.
7
朔實：二十萬四千六百四十七。 《乾道》八十八萬五千九百一十七、秒七十六。 《淳熙》一十六萬六千五百五十二、秒五十六。 《會元》朔率一百一十四萬二千八百三十四。
New-moon constant: 204,647. Qiandao: 885,917, 76 seconds. Chunxi: 166,552, 56 seconds. Huiyuan: new-moon rate 1,142,834.
8
歲閏：七萬五千三百七十四。 《乾道》三十二萬六千二百九十四、秒八十八，又有閏限八十五萬八千七百二十六、秒五十二，月閏二萬七千一百九十一、秒二十四。 《會元》四十二萬九百二十四，又有閏限七十二萬一千九百一十。 《乾道》又有沒限二萬二千四百四十五半。 《淳熙》四千四百七、秒七十五。 《會元》三萬二百四十四半。
Year intercalation: 75,374. Qiandao: 326,294, 88 seconds; also an intercalation limit of 858,726, 52 seconds; month intercalation 27,191, 24 seconds. Huiyuan: 420,924; also an intercalation limit of 721,910. Qiandao also has a disappearance limit of 22,445½. Chunxi: 4,407, 75 seconds. Huiyuan: 30,244½.
9
朔策：二十九日、余三萬六千七十七。 《乾道》余一萬五千九百一十七、秒七十六。 《淳熙》餘三千九百九十二、秒五十六。 《會元》余二萬五百三十四，約分五十三、秒五。
New-moon interval: 29 days, remainder 36,077. Qiandao: remainder 15,917, 76 seconds. Chunxi: remainder 3,992, 56 seconds. Huiyuan: remainder 20,534; reduced fraction 53, 5 seconds.
10
望策：十四日、餘五千三百三半。 《乾道》余一萬二千九百五十八、秒八十八。 《淳熙》餘四千三百一十六、秒二十八。 《會元》余二萬九千六百一十七。
Full-moon interval: 14 days, remainder 5,303½. Qiandao: remainder 12,958, 88 seconds. Chunxi: remainder 4,316, 28 seconds. Huiyuan: remainder 29,617.
11
弦策：七日、餘二千六百五十一太。 《乾道》余一萬一千四百七十九，秒四十四。 《淳熙》餘二千一百五十八、秒一十四。 《會元》余一萬四千八百八半。
Quarter-moon interval: 7 days, remainder 2,651¾. Qiandao: remainder 11,479, 44 seconds. Chunxi: remainder 2,158, 14 seconds. Huiyuan: remainder 14,808½.
12
中盈分：三千三百二十八、秒三十。 《乾道》一萬三千一百九。 《淳熙》二千四百六十四、秒五十。 《會元》一萬六千九百一十一。
Mid-climate excess fraction: 3,328, 30 seconds. Qiandao: 13,109. Chunxi: 2,464, 50 seconds. Huiyuan: 16,911.
13
朔虛分：三千二百五十三。 《乾道》一萬四千八十二、秒二十四。 《淳熙》二千六百四十七、秒四十四。 《會元》一萬八千一百六十六。
New-moon void fraction: 3,253. Qiandao: 14,082, 24 seconds. Chunxi: 2,647, 44 seconds. Huiyuan: 18,166.
14
旬周：四十一萬五千八百。 漙《乾道》一百八十萬。 《淳熙》三十三萬八千四百、秒一。 《會元》二百三十二萬二千。
Ten-day cycle: 415,800. Qiandao: 1,800,000. Chunxi: 338,400, 1 second. Huiyuan: 2,322,200.
15
紀法：六十。 三曆同。
Era-cycle divisor: 60. The same for all three calendars.
16
推天正冬至：置距所求積年，以歲周乘之，為氣積分； 以旬周去之，不盡，總法約之為大餘，不滿為小餘。 大餘命甲子，算外，即得所求年天正冬至日辰及餘。 其小餘總法退除為約分，即百為母。
To calculate the winter solstice of the civil new year: take the accumulated years from the epoch sought, multiply by the tropical year cycle to obtain the qi accumulated parts; divide out the ten-day cycle; the remainder, reduced by the general divisor, yields the major remainder; what is left below that is the minor remainder. Count the major remainder from jiazi, one step outside the count, and one obtains the day, double-hour, and remainder of the winter solstice of the civil new year for the year sought. For the minor remainder, divide back through the general divisor to obtain reduced fractions, with one hundred as the denominator.
17
求次氣：置冬至大、小餘，以氣策及餘秒加之，秒盈秒法從一小余，小余滿總法從一大余，滿紀法去，命甲子，算外，合得次氣日辰及餘秒。
To find the next qi: set the winter solstice major and minor remainders, add the qi interval together with its remainder and seconds; when seconds fill the second divisor carry one into the minor remainder, when the minor remainder fills the general divisor carry one into the major remainder, when full remove the era-cycle divisor, count from jiazi one step outside the count, and thereby obtain the day, double-hour, remainder, and seconds of the next qi.
18
求天正經朔：置天正冬至氣積分，以朔實去之，不盡為閏餘； 以減冬至氣積分，餘為天正十一月經朔加時朔積分； 以旬周去之，不滿，總法約之為大餘，不滿為小餘。 命甲子，算外，即得所求年天正十一月經朔加時朔積分。 以旬周去之，不滿，總法約之為大餘，不滿為小餘。 大餘命甲子，算外，即得所求天正十一月經朔日辰及餘。
To find the canonical new moon of the civil new year: set the qi accumulated parts of the civil winter solstice, divide out the new-moon constant; the remainder is the intercalary remainder; subtract this from the winter solstice qi accumulated parts; the remainder is the accumulated new-moon parts at the hour of the canonical new moon of the eleventh month of the civil new year; divide out the ten-day cycle; what remains, reduced by the general divisor, yields the major remainder; what is left below that is the minor remainder. Count from jiazi one step outside the count, and one obtains the accumulated new-moon parts at the hour of the canonical new moon of the eleventh month of the civil new year for the year sought. Divide out the ten-day cycle; what remains, reduced by the general divisor, yields the major remainder; what is left below that is the minor remainder. Count the major remainder from jiazi one step outside the count, and one obtains the day, double-hour, and remainder of the canonical new moon of the eleventh month of the civil new year for the year sought.
19
求弦望及次朔經日：置天正十一月經朔大、小餘，以弦策加之，為上弦； 累加之，去命如前，各得弦、望及次月朔經日及餘也。
To find the quarter-moon, full-moon, and subsequent canonical new-moon days: set the major and minor remainders of the civil eleventh-month canonical new moon, add the quarter-moon interval to obtain the first quarter; add cumulatively, and by removing and counting as before one obtains in each case the canonical days and remainders for the quarter-moon, full moon, and new moon of the following month.
20
求沒日：置有沒之氣小餘，以一百八十乘之，秒從之，用減一百二十六萬五千五百六十九，余以一萬八千一百六十九除之為日，不滿為餘。 命其氣初日，算外，即得其氣辰。 凡二十四氣，小餘五千四百一十五、秒一百六十五。
To find the disappearance day: set the minor remainder of the qi subject to disappearance, multiply it by 180 with seconds carried along, subtract this from 1,265,569; divide the remainder by 18,169 to obtain days, and what is left is the remainder. Count from the first day of that qi one step outside the count, and one obtains its qi double-hour. For all twenty-four qi, the minor remainder is 5,415, 165 seconds.
21
求滅日：置有滅經朔小餘，三十乘之，滿朔虛分除為日，不滿為餘。 命經朔初日，算外，即得其月滅日辰。 經朔小餘不滿朔虛分者，為有滅之朔。
To find the extinction day: set the minor remainder of the canonical new moon subject to extinction, multiply by 30, and when full divide by the new-moon void fraction to obtain days; what is left is the remainder. Count from the first day of the canonical new moon one step outside the count, and one obtains the extinction day and double-hour of that month. When the minor remainder of the canonical new moon falls short of the new-moon void fraction, that new moon is subject to extinction.
22
步發斂
Procedure for Issuing and Collecting.
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候策：五日、餘五百四、秒一百二十五。 《乾道》餘二千一百八十四、秒二十五。 《淳熙》餘四百一十、秒七十五。 《會元》餘二千八百一十二、秒五十。
Hou interval: 5 days, remainder 504, 125 seconds. Qiandao: remainder 2,184, 25 seconds. Chunxi: remainder 410, 75 seconds. Huiyuan: remainder 2,812, 50 seconds.
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卦策：六日、餘六百五、秒一百一十四。 《乾道》餘二千六百二十一、秒二十四。 《淳熙》餘四百九十二、秒九十。 《會元》餘三千三百八十二、秒二十。
Gua interval: 6 days, remainder 605, 114 seconds. Qiandao: remainder 2,621, 24 seconds. Chunxi: remainder 492, 90 seconds. Huiyuan: remainder 3,382, 20 seconds.
25
土王策：三日、餘三百二、秒一百四十七。 《乾道》餘二千三百一十、秒二十七。 《淳熙》餘二百四十六、秒四十五。 《會元》一千六百九十一、秒一十。
Earth-sovereign interval: 3 days, remainder 302, 147 seconds. Qiandao: remainder 2,310, 27 seconds. Chunxi: remainder 246, 45 seconds. Huiyuan: 1,691, 10 seconds.
26
辰法：五百七十七半。 《乾道》二千五百。 《淳熙》四百七十。 《會元》三千二百二十五。
Double-hour divisor: 577½. Qiandao: 2,500. Chunxi: 470. Huiyuan: 3,225.
27
半辰法：二百八十八太。 《乾道》一千二百五十。 《淳熙》二百三十五。 《會元》一千六百一十二半。
Half double-hour divisor: 288¾. Qiandao: 1,250. Chunxi: 235. Huiyuan: 1,612½.
28
刻法：六百九十三。 《乾道》三百。 《淳熙》五百六十四。 《會元》三百八十七。
Quarter-divisor: 693. Qiandao: 300. Chunxi: 564. Huiyuan: 387.
29
秒法：一百八十。 《乾道》三十。 《淳熙》、《會元》同一百。 《淳熙》又有月閏五千一百一十一、秒九十四。
Second divisor: 180. Qiandao: 30. Chunxi and Huiyuan: 100 each. Chunxi also has month intercalation 5,111, 94 seconds.
30
求六十四卦、五行用事日、二十四氣、七十二候。 四曆俱與前曆同，此不載。
To find the sixty-four hexagrams, the days when the Five Phases take charge, the twenty-four qi, and the seventy-two hou. All four calendars follow the previous calendar in this respect; the methods are not recorded here.
31
求發斂去經朔日：置天正閏餘，以中盈及朔虛分累益之，即每月閏餘； 滿總法除之為閏日，不盡為小餘，即各得其月中氣去經朔日辰。 因求卦候者，各以卦、候、土王策依次累加減之，中氣前減，中氣後加。
To find the issuing-and-collecting day as an offset from the canonical new-moon day: set the civil new-year intercalary remainder, cumulatively add the mid-climate excess and new-moon void fractions, and thereby obtain each month's intercalary remainder; when full, divide by the general divisor to obtain intercalary days; what remains is the minor remainder; and in each case one obtains the day and double-hour by which that month's mid-climate stands apart from the canonical new moon. To find the hexagram and hou thereafter, add or subtract cumulatively in sequence using the gua, hou, and earth-sovereign intervals; subtract before the mid-climate and add after it.
32
各得其月卦、候去經朔日辰。
In each case one obtains the day and double-hour by which that month's hexagram and hou stand apart from the canonical new moon.
33
求發斂加時：置所求小餘，以辰法除之為辰數，不滿，進一位，以刻法而一為刻，不盡為刻分。 其辰數命子正，算外，各得加時所在辰、刻及分。 加辰刻即命起子初。
To find the hour of issuing and collecting: set the minor remainder sought, divide by the double-hour divisor to obtain the double-hour count; for what remains, advance one place and divide by the quarter-divisor to obtain quarters; what is left below that is quarter-fractions. Count its double-hour number from midnight of zi one step outside the count, and in each case obtain the double-hour, quarter, and fraction of the hour sought. With the double-hours and quarters added, count from the beginning of zi.
34
步日躔
Procedure for the Sun's Motion.
35
周天分：二百五十三萬一千二百二十六、秒八十七。 《乾道》分一千九十五萬七千七百一十七、秒五。
Circumference of heaven in parts: 2,531,226, 87 seconds. Qiandao: 10,957,717 parts, 5 seconds.
36
歲差：八十八、秒八十七。 《乾道》四百九、秒五。 《淳熙》一萬一千五百一十三。 《會元》軌差五百二十五、秒一十三。
Annual difference: 88, 87 seconds. Qiandao: 409, 5 seconds. Chunxi: 11,513. Huiyuan: orbital difference 525, 13 seconds.
37
周天度：三百六十五、約分二十五、秒六十四。 三曆同。
Circumference of heaven in degrees: 365, reduced fraction 25, 64 seconds. The same for all three calendars.
38
乘法：五十五。 《乾道》八十七。 《淳熙》一百一十九。 《會元》一百一十九。
Multiplication factor: 55. Qiandao: 87. Chunxi: 119. Huiyuan: 119.
39
除法：八百三十七。 《乾道》一千三百二十四。 《淳熙》一千八百一十二。 《會元》一千八百一十一。
Division factor: 837. Qiandao: 1,324. Chunxi: 1,812. Huiyuan: 1,811.
40
秒法：一百。 三曆同。
Second divisor: 100. The same for all three calendars.
41
《乾道》又有象限九十一度、分三十一、秒九，《淳熙》、《會元》同。 《淳熙》又有乾實三億九百萬七千六百一十三，半周天一百八十二度、分二十五、秒七十二。 《會元》半周天度同、分六十二、秒八十六。
Qiandao also has a quadrant limit of 91 degrees, 31 minutes, 9 seconds; Chunxi and Huiyuan are the same. Chunxi also has the qian constant 309,007,613; half the circumference of heaven 182 degrees, 25 minutes, 72 seconds. Huiyuan: half the circumference of heaven in degrees is the same, 62 minutes, 86 seconds.
42
求每月盈縮分，朔、弦、望入氣朏朒定數，赤道宿度，冬至赤道日度，赤道宿積度入初末限，二十八宿黃道度，天正冬至加時黃道日度，二十四氣加時黃道日度，二十四氣初日晨前夜半黃道日躔宿次，晨前夜半黃道日躔宿次，太陽入宮日時刻及分。 法同前曆，此不載。
To find each month's excess-and-deficit parts; the fixed numbers for new moon, quarter-moon, and full moon entering qi and for waxing and waning; equatorial lodge degrees; the sun's equatorial degree at the winter solstice; equatorial lodge accumulated degrees entering the initial and final limits; the twenty-eight lodges' ecliptic degrees; the ecliptic solar degree at the hour of the civil winter solstice; the ecliptic solar degree at the hour of each of the twenty-four qi; the ecliptic solar motion and lodge position at dawn before midnight on the first day of each qi; the ecliptic solar motion and lodge position at dawn before midnight; and the day, double-hour, quarter, and fraction when the sun enters a palace. The methods follow the previous calendar; they are not recorded here.
43
步月離
Procedure for Lunar Motion.
44
轉周分：一十九萬九百五十三、秒二千五百六十三。 《乾道》八十二萬六千六百三十七、秒七千三百九十五。 《淳熙》一十五萬五千四百七、秒九千七百四十。 《會元》轉率一百六萬六千三百六十一、秒七千三百一十。
Rotation cycle in parts: 190,953, 2,563 seconds. Qiandao: 826,637, 7,395 seconds. Chunxi: 155,407, 9,740 seconds. Huiyuan: rotation rate 1,066,361, 7,310 seconds.
45
轉周日：二十七、餘三千八百四十三、秒二千五百六十三。 《乾道》余一萬六千六百三十七、秒七千三百九十五。 《淳熙》餘三千一百二十七、秒九千七百四。 《會元》余三萬一千四百六十一、秒七千三百一十。
Rotation cycle in days: 27, remainder 3,843, 2,563 seconds. Qiandao: remainder 16,637, 7,395 seconds. Chunxi: remainder 3,127, 9,704 seconds. Huiyuan: remainder 31,461, 7,310 seconds.
46
朔差日：一、餘六千七百六十三、秒七千四百三十七。 《乾道》余二萬九千二百八十、秒二百五。 《淳熙》餘五千五百四十、秒五千八百六十。 《會元》余三萬七千七百七十二、秒二千六百一十。
New-moon difference in days: 1, remainder 6,763, 7,437 seconds. Qiandao: remainder 29,280, 205 seconds. Chunxi: remainder 5,540, 5,860 seconds. Huiyuan: remainder 37,772, 2,610 seconds.
47
望策：一十四、餘五千三百三、秒五千。
Full-moon interval: 14, remainder 5,303, 5,000 seconds.
48
弦策：七、餘二千六百五十一、秒七千五百。 《乾道》余一萬一千四百七十九、秒四千四百。 《淳熙》餘二千一百五十八、秒一千四百。 《會元》一萬四千八百八、秒五十。
Quarter-moon interval: 7, remainder 2,651, 7,500 seconds. Qiandao: remainder 11,479, 4,400 seconds. Chunxi: remainder 2,158, 1,400 seconds. Huiyuan: 14,808, 50 seconds.
49
七日：初數六千一百五十八，約分八十九； 末數七百七十二，約分一十一。
Seventh day: initial number 6,158, reduced fraction 89; final number 772, reduced fraction 11.
50
十四日：初數五千三百八十七，約分七十八； 末數一千五百四十三，約分二十二。
Fourteenth day: initial number 5,387, reduced fraction 78; final number 1,543, reduced fraction 22.
51
二十一日：初數四千六百一十五，約分六十七； 末數二千三百一十五，約分三十三。
Twenty-first day: initial number 4,615, reduced fraction 67; final number 2,315, reduced fraction 33.
52
二十八日：初數三千八百四十三，約分五十五； 末數空。
Twenty-eighth day: initial number 3,843, reduced fraction 55; final number: none.
53
以上秒母一萬。
Above, the second denominator is 10,000.
54
以下秒母一百。
Below, the second denominator is 100.
55
上弦：九十一度三十一分、秒四十一。 三曆同。
First quarter: 91 degrees 31 minutes, 41 seconds. The same for all three calendars.
56
望：一百八十二度六十二分、秒八十二。 三曆秒八十六。
Full moon: 182 degrees 62 minutes, 82 seconds. For all three calendars: 86 seconds.
57
下弦：二百七十三度九十四分、秒二十三。 三曆秒二十九。
Last quarter: 273 degrees 94 minutes, 23 seconds. For all three calendars: 29 seconds.
58
平行分：一十三度三十六分、秒八十七半。
Mean motion in parts: 13 degrees 36 minutes, 87½ seconds.
59
推天正十一月經朔入轉，經弦、望及次朔入轉。 法同前曆，此不載。
To calculate entry into rotation for the civil eleventh-month canonical new moon, and for the canonical quarter-moon, full moon, and subsequent new moon. The methods follow the previous calendar; they are not recorded here.
60
《乾道》又有七日初數二萬六千六百五十九、初約八十九，末數三千三百四十一、末約一千一； 十四日初數二萬二千三百一十九、初約七十八，末數六千六百八十一，末約二十三； 二十一日初數一萬九千九百九十八、初約六十七，末數一萬二十二、末約三十三； 二十八日初數一萬六千六百三十七、初約五十五，末數空、末約空。 《淳熙》七日初數五千一十一，末數六百二十，初約八十九、末約一千二； 十四日初數四千三百八十三、末數一千二百五十七，初約七十八、末約二十二； 二十一日初數三千七百五十五、末數一千八百八十五，初約六十七、末約三十三； 二十八日初數三千一百二十七、初約五十五。 《會元》七日初數三萬四千三百九十、安約八十九，末數四千三百一十、末約一十一； 十四日初數三萬八千、初約七十八，末數八千六百二十、末約二十一； 二十一日初數二萬五千七百七十二、初約六十七，末數一萬二千九百二十九、末約三十三； 二十八日初數二萬一千四、初約五十五，末數一百六十一。
Qiandao also has for the seventh day initial number 26,659, initial reduced fraction 89, final number 3,341, final reduced fraction 1,001; fourteenth day initial number 22,319, initial reduced fraction 78, final number 6,681, final reduced fraction 23; twenty-first day initial number 19,998, initial reduced fraction 67, final number 10,022, final reduced fraction 33; twenty-eighth day initial number 16,637, initial reduced fraction 55, final number none, final reduced fraction none. Chunxi: seventh day initial number 5,011, final number 620, initial reduced fraction 89, final reduced fraction 1,002; fourteenth day initial number 4,383, final number 1,257, initial reduced fraction 78, final reduced fraction 22; twenty-first day initial number 3,755, final number 1,885, initial reduced fraction 67, final reduced fraction 33; twenty-eighth day initial number 3,127, initial reduced fraction 55. Huiyuan: seventh day initial number 34,390, initial reduced fraction 89, final number 4,310, final reduced fraction 11; fourteenth day initial number 38,000, initial reduced fraction 78, final number 8,620, final reduced fraction 21; twenty-first day initial number 25,772, initial reduced fraction 67, final number 12,929, final reduced fraction 33; twenty-eighth day initial number 21,004, initial reduced fraction 55, final number 161.
61
求朔、弦、望入轉朏朒定數：朔、弦、望定日朔、弦、望加時日所在度； 推月行九道； 求九道宿度，月行九道平交入氣，平定入轉，朏朒定數，正交入氣，正交加時黃道日度，正交加時月離九道宿度，定朔、弦、望月所在宿度，定朔夜半入轉，次朔夜半入轉，月晨昏度，朔、弦、望晨昏定程，轉定度，晨昏月，天正十一月經朔加時平行月，天正十一月定朔日晨前夜半平行月，次朔夜半平行月，定弦、望夜半平行月，天正定朔夜半入轉，弦、望及後朔定日夜半入轉，定朔、弦、望夜半月度。 法同前曆，此不載。
To find the fixed numbers for new moon, quarter-moon, and full moon entering rotation and for waxing and waning: the fixed days of new moon, quarter-moon, and full moon; the degrees where the sun stands at the hour of new moon, quarter-moon, and full moon; calculate the moon's motion along the nine paths; find the lodge degrees of the nine paths; the moon's motion along the nine paths at mean crossing entering qi; fixed entry into rotation; fixed numbers for waxing and waning; true crossing entering qi; the ecliptic solar degree at the hour of true crossing; the moon's departure from the nine-path lodge degree at the hour of true crossing; the lodge degrees where the fixed new moon, quarter-moon, and full moon stand; fixed new moon at midnight entering rotation; the next new moon at midnight entering rotation; the moon's degrees at dawn and dusk; the fixed schedules for dawn and dusk at new moon, quarter-moon, and full moon; the fixed rotation degree; the moon at dawn and dusk; the mean moon at the hour of the civil eleventh-month canonical new moon; the mean moon at dawn before midnight on the fixed civil eleventh-month new-moon day; the mean moon at midnight of the next new moon; the mean moon at midnight of the fixed quarter-moon and full moon; the civil fixed new moon at midnight entering rotation; fixed day at midnight entering rotation for quarter-moon, full moon, and subsequent new moon; the lunar degree at midnight for the fixed new moon, quarter-moon, and full moon. The methods follow the previous calendar; they are not recorded here.
62
步晷漏
Procedure for Gnomon Shadow and Clepsydra.
63
二至限：一百八十二、六十三分。 《乾道》分同，秒一十八。 《淳熙》、《會元》同。
Solstice limit: 182, 63 minutes. Qiandao: minutes the same, 18 seconds. Chunxi and Huiyuan: the same.
64
象限：九十一、三十一分。 三曆同，秒九。
Quadrant limit: 91, 31 minutes. The same for all three calendars: 9 seconds.
65
消息法：一萬二千二百十一。
Message method: 12,211.
66
辰法：五百七十七半，計八刻二百三十一分。 《乾道》餘一百。 《淳熙》餘一百八十八。 《會元》餘一百二十九。
Double-hour divisor: 577½, totaling 8 quarters 231 parts. Qiandao: remainder 100. Chunxi: remainder 188. Huiyuan: remainder 129.
67
昏明刻：三百四十六半。 《乾道》餘一百五十。 《淳熙》餘二百八十二。
Dusk-and-dawn quarters: 346½. Qiandao: remainder 150. Chunxi: remainder 282.
68
昏明餘數：一百七十三少。 《乾道》昏明分七百五十。 《淳熙》昏明分一百四十一。 《會元》九百六十七半。
Dusk-and-dawn remainder count: 173 and seven-eighths. Qiandao: dusk-and-dawn parts 750. Chunxi: dusk-and-dawn parts 141. Huiyuan: 967½.
69
冬至岳台晷景：一丈二尺八寸三分。
Winter solstice gnomon shadow at Yue Terrace: 12 chi 8 cun 3 fen.
70
夏至岳台晷景：一尺五寸六分。
Summer solstice gnomon shadow at Yue Terrace: 1 chi 5 cun 6 fen.
71
冬至後初限夏至後末限：六十二日。 分空。
Initial limit after the winter solstice and final limit after the summer solstice: 62 days. Minutes: none.
72
夏至後初限冬至後末限：一百二十日六十二分。
Initial limit after the summer solstice and final limit after the winter solstice: 120 days 62 minutes.
73
求每日消息定數：黃道去極度及赤道內、外度，晨昏日出、入分及半晝分，每日距中度，夜半定漏，晝、夜刻及日出、入辰刻，更籌辰刻，昏、明度，五更攢點中星，九服距差日，九服晷景，九服所在晝、夜漏刻。 法與前曆同，此不載。
To find the fixed message numbers for each day: ecliptic departure from the pole and equatorial inner and outer degrees; dawn-and-dusk sunrise and sunset parts and half-day parts; daily distance from the mid-degree; fixed clepsydra at midnight; day and night quarters and sunrise and sunset double-hour quarters; watch and tally double-hour quarters; dusk and dawn degrees; the culminating star at the fifth watch tally-point; the nine domains' distance-difference days; the nine domains' gnomon shadows; and the day-and-night clepsydra quarters for each of the nine domains. The methods follow the previous calendar; they are not recorded here.
74
步交會
Procedure for Conjunction.
75
交終分：一十八萬八千五百八十、秒六千四百五十七。 《乾道》八十一萬六千三百六十六、秒六千三十四。 《淳熙》交實一十五萬三千四百七十六、秒九千五百四十六。 《會元》交率一百五萬三千一百一十二、秒二千一百四十。
Conjunction cycle in parts: 188,580, 6,457 seconds. Qiandao: 816,366, 6,034 seconds. Chunxi: conjunction constant 153,476, 9,546 seconds. Huiyuan: conjunction rate 1,053,112, 2,140 seconds.
76
交終日：二十七、餘一千四百七十、秒六千四百五十七。 《乾道》餘六千三百六十六、秒六千三十四。 《淳熙》餘一千一百九十六、秒九千五百四十二。 《會元》餘八千二百一十三、秒二千一百四十。
Conjunction cycle in days: 27, remainder 1,470, 6,457 seconds. Qiandao: remainder 6,366, 6,034 seconds. Chunxi: remainder 1,196, 9,542 seconds. Huiyuan: remainder 8,213, 2,140 seconds.
77
交中日：一十三、餘四千二百、秒三千二百二十八半。 《乾道》余一萬八千一百八十三、秒三十七。 《淳熙》餘三千四百一十八、秒四千七百七十一半。 《會元》余二萬三千四百五十六、秒六千七十。
Mid-conjunction day: 13, remainder 4,200, 3,228½ seconds. Qiandao: remainder 18,183, 37 seconds. Chunxi: remainder 3,418, 4,771½ seconds. Huiyuan: remainder 23,456, 6,070 seconds.
78
朔差：二日、餘二千二百六、秒三千五百四十三。 《乾道》餘九千五百五十一、秒一千五百六十六。 《淳熙》餘一千七百九十五、秒六千五十七。 《會元》余一萬二千三百二十、秒七千八百六十。
New-moon difference: 2 days, remainder 2,206, 3,543 seconds. Qiandao: remainder 9,551, 1,566 seconds. Chunxi: remainder 1,795, 6,057 seconds. Huiyuan: remainder 12,320, 7,860 seconds.
79
後限：一日、餘一千一百十三、秒一千七百七十一半。 《乾道》餘四千七百七十五、秒五千七百八十三。
Later limit: 1 day, remainder 1,113, 1,771½ seconds. Qiandao: remainder 4,775, 5,783 seconds.
80
前限：十二日、餘三千九十七、秒一千四百五十。 《乾道》余一萬三千四百九、秒七千二百三十四。
Earlier limit: 12 days, remainder 3,097, 1,450 seconds. Qiandao: remainder 13,409, 7,234 seconds.
81
望策：十四日、餘五千三百三、秒五十。 《乾道》余二萬二千九百五十八、秒八千八百。 《淳熙》餘四千三百一十六、秒二千八百。 《會元》余二萬九千六百三十七。
Full-moon interval: 14 days, remainder 5,303, 50 seconds. Qiandao: remainder 22,958, 8,800 seconds. Chunxi: remainder 4,316, 2,800 seconds. Huiyuan: remainder 29,637.
82
交率：四十二。 《乾道》八十。 《淳熙》六十一。 《會元》五百七。
Conjunction rate: 42. Qiandao: 80. Chunxi: 61. Huiyuan: 507.
83
交數：五百三十五。 《乾道》一千一十九。 《淳熙》七百七十七。
Conjunction number: 535. Qiandao: 1,019. Chunxi: 777.
84
交終度：三百六十三度七十六分。 《乾道》分七十九、秒四十。 《淳熙》同。 《會元》分同、秒四十四。
Conjunction cycle in degrees: 363 degrees 76 minutes. Qiandao: 79 minutes, 40 seconds. Chunxi: the same. Huiyuan: minutes the same, 44 seconds.
85
交象度：九十度九十四分。 《乾道》分同、秒八十五。 《淳熙》同。 《會元》分同、秒八十六。
Conjunction arc in degrees: 90 degrees 94 minutes. Qiandao: minutes the same, 85 seconds. Chunxi: the same. Huiyuan: minutes the same, 86 seconds.
86
半交象度：一百八十一度八十八分。 《乾道》度四十五、分四十七、秒四十二半。 《淳熙》同。 《會元》秒四十二。
Half-conjunction arc in degrees: 181 degrees 88 minutes. Qiandao: 45 degrees, 47 minutes, 42½ seconds. Chunxi: the same. Huiyuan: 42 seconds.
87
陽曆食限：二千七百四十五。 《乾道》一萬四千四百。 《淳熙》二千六百三十。 《會元》一萬八千。
Solar eclipse limit: 2,745. Qiandao: 14,400. Chunxi: 2,630. Huiyuan: 18,000.
88
陽曆定法：二百七十四半。 《乾道》一千四百四十。 《淳熙》二百六十三。
Solar eclipse fixed method: 274½. Qiandao: 1,440. Chunxi: 263.
89
陰曆食限：四千五百八十五。 《乾道》一萬八千。 《淳熙》三千二百四十。 《會元》二萬二千五百。
Lunar eclipse limit: 4,585. Qiandao: 18,000. Chunxi: 3,240. Huiyuan: 22,500.
90
陰曆定法：四百五十八半。 《乾道》三百二十四。
Lunar eclipse fixed method: 458½. Qiandao: 324.
91
《乾道》又有月食限二萬九千一百，《淳熙》五千四百六十，《會元》三萬六千。 《乾道》月食定法一千八百，《淳熙》三百五十六。 《乾道》月食既限一萬一千一百。 《淳熙》月食既限一千九百。
Qiandao also has a lunar eclipse limit of 29,100; Chunxi 5,460; Huiyuan 36,000. Qiandao lunar eclipse fixed method 1,800; Chunxi 356. Qiandao lunar eclipse totality limit 11,100. Chunxi lunar eclipse totality limit 1,900.
92
推天正十一月加時入交泛日：求次朔及望入交泛日，定朔、望夜半交泛，次朔夜半入交泛日，朔、望加時入交常日，朔望加時入交定日，月行陰陽曆，朔、望加時入陰陽曆積度，朔、望加時月去黃道度，食甚定餘，日月食甚入氣，日月食甚中積、氣差、刻差，日入食限，日入食分，日食泛用分，月入食限，月入食分，月食泛用分，日月食定用分，日月食虧初、復滿小餘，月食既內、外分，日月食所起，月食更、點定法，月食入更點，日月帶食出入所見分數，日月食甚宿次。 法同前曆，此不載。
To calculate the generalized conjunction day at the hour of the civil eleventh-month new moon: find the generalized conjunction days for the next new moon and full moon; fixed new-moon and full-moon generalized conjunction at midnight; the next new moon at midnight entering the generalized conjunction day; canonical new-moon and full-moon entry into the regular conjunction day at the hour; canonical new-moon and full-moon entry into the fixed conjunction day at the hour; the moon's motion along the solar and lunar paths; canonical new-moon and full-moon entry into the accumulated degrees of the solar and lunar paths at the hour; the moon's departure from the ecliptic at the hour of canonical new moon and full moon; fixed remainder at greatest eclipse; the qi entered at greatest solar and lunar eclipse; central accumulation, qi difference, and quarter difference at greatest solar and lunar eclipse; the sun entering the eclipse limit; the sun entering the eclipse fraction; generalized applicable fraction for solar eclipse; the moon entering the eclipse limit; the moon entering the eclipse fraction; generalized applicable fraction for lunar eclipse; fixed applicable fraction for solar and lunar eclipse; minor remainders for the beginning of eclipse loss and restoration to fullness; inner and outer fractions for lunar eclipse totality; where solar and lunar eclipses begin; fixed method for lunar eclipse watches and tally-points; lunar eclipse entry into watches and tally-points; visible fractions when solar and lunar eclipses accompany rising and setting; and the lodge position at greatest solar and lunar eclipse. The methods follow the previous calendar; they are not recorded here.
93
步五星
Procedure for the Five Planets.
94
五星會策：一十五度、二十一分、秒九十。
Five-planet conjunction interval: 15 degrees, 21 minutes, 90 seconds.
95
木星終率：二百七十六萬四千二百三十八、秒三十二。 《乾道》一千一百九十六萬六千五百八十一、秒五十五。 《淳熙》周實二百二十四萬九千七百一十五、秒六十五。 《會元》周率一千五百四十三萬六千八百三十四、秒九十八。
Jupiter cycle rate: 2,764,238, 32 seconds. Qiandao: 11,966,581, 55 seconds. Chunxi: cycle constant 2,249,715, 65 seconds. Huiyuan: cycle rate 15,436,834, 98 seconds.
96
終日：三百九十八、約分八十八、秒七十九。 《乾道》分八十八、秒六十。 《淳熙》約分八十八、秒五十七。 《會元》分八十八、秒四十六。
Cycle in days: 398, reduced fraction 88, 79 seconds. Qiandao: 88 minutes, 60 seconds. Chunxi: reduced fraction 88, 57 seconds. Huiyuan: 88 minutes, 46 seconds.
97
歲差：六十七、秒九十八。 《乾道》周差一百萬八千八百六十四、秒五十。 《淳熙》一十八萬九千七百四十一、秒六十五。
Annual difference: 67, 98 seconds. Qiandao: cycle difference 1,008,864, 50 seconds. Chunxi: 189,741, 65 seconds.
98
伏見度：一十三。
Stationary-visibility degrees: 13.
99
《乾道》曆率一千九十五萬七千二百四十九、秒九。 《淳熙》二百五萬九千九百八十一、秒一十。 《會元》一千四百一十三萬五千四百五十六、秒九。 《乾道》曆中度一百八十三、分六十三、秒二十四。 《淳熙》同。 《會元》秒八十六。 《乾道》曆策度一十五、分二十一、秒八十五。 《淳熙》同。 《會元》秒九十。
Qiandao: calendar rate 10,957,249, 9 seconds. Chunxi: 2,059,981, 10 seconds. Huiyuan: 14,135,456, 9 seconds. Qiandao: calendar mid-degree 183 degrees, 63 minutes, 24 seconds. Chunxi: the same. Huiyuan: 86 seconds. Qiandao: calendar interval degree 15 degrees, 21 minutes, 85 seconds. Chunxi: the same. Huiyuan: 90 seconds.