1
志第十三曆二
Treatise 13: Calendars, Part Two
2
○麟德甲子元曆
○ The Origin Calendar of the Linde jiazi Era
3
上元甲子，距今大唐甲子，歲積二十六萬九千八百八十算。 推法：一千三百四十。 期實：四十八萬九千四百二十八。 旬周：六十。
From the Superior Origin in the jiazi year to the present jiazi year of Great Tang, the elapsed years total 269,880 counts. Computation divisor: 1,340. True period value: 489,428. Ten-day cycle: 60.
4
○推氣序術
○ Procedure for calculating the order of the solar terms
5
置入甲子元積，算距今所求年，以期乘之，為期總。 滿法得一為積日，不滿為小餘。 旬去積日，不盡為大餘。 命大餘起甲子算外，即所求年天正中氣冬至恆日及大小餘。 天正建子，律氣所由，故陰陽發斂，皆從其時為自。
Enter the jiazi-era accumulated count; compute the years from the present back to the year sought; multiply by the period constant to obtain the period total. When it fills the divisor, that counts as one accumulated day; the remainder is the small remainder. Strip out full ten-day cycles from the accumulated days; what remains is the large remainder. Starting from the large remainder, count from jiazi outside the reckoning—that gives the mean day of the winter solstice at the celestial first month for the year sought, with its large and small remainders. The celestial first month is established in the zi month; the pitch-pipes and solar terms take their origin from it—so the waxing and waning of yin and yang all begin from this point.
6
○求恆次氣術
○ Procedure for finding the mean successive solar terms
7
因冬至大小餘，加大餘十五、小餘二百九十二、小分六之五。 小分滿，從小餘。 小餘滿總法之，從大餘一。 大餘滿旬周之。 以次轉加，而命各得其所求。 他皆放此。 凡氣餘朔大餘為日，小餘為辰也。
Take the winter solstice large and small remainders; add 15 to the large remainder, 292 to the small remainder, and five-sixths of a minor fraction. When minor fractions fill their unit, carry into the small remainder. When the small remainder fills the common divisor, carry one to the large remainder. When the large remainder fills the ten-day cycle, discard the cycle. Continue adding in sequence and assigning stem-branch names in turn—each step yields the term sought. All other cases follow this pattern. In general, the large remainder in a qi or new-moon remainder is the day, and the small remainder is the double-hour.
8
○求土王
○ Finding the intercalary Earth days
9
置清明、小暑、寒露、小寒、大寒小餘，各加大餘十二、小餘二百四十四、小分八。 互乘氣小分通之，加八。 若滿三十，去，從小餘一。 凡分餘相並不同者，互乘而並之。 母相乘為法。 其並滿法一為全，此即齊同之術。 小餘滿總法，從命如前，即各其氣從土王日。
Take the small remainders for Pure Brightness, Minor Heat, Cold Dew, Minor Cold, and Great Cold; to each add 12 to the large remainder, 244 to the small remainder, and eight minor fractions. Cross-multiply the solar-term minor fractions to a common denominator and add eight. If the total reaches thirty, discard thirty and carry one to the small remainder. Whenever fractional remainders to be combined have different denominators, cross-multiply and combine them. Multiply the denominators together to form the divisor. When the combined numerator fills the divisor, that counts as one whole—this is the method of reduction to a common denominator. When the small remainder fills the common divisor, carry into the stem-branch name as before—each yields the intercalary Earth day following that solar term.
10
沒日法：一千七百五十七。
Buried-day divisor: 1,757.
11
沒分：十二萬二千三百五十七。
Buried-day fraction: 122,357.
12
求沒日術
Procedure for finding buried days
13
以九十乘有沒氣小餘，十五乘小分，從之，以減沒分，餘，法得一，為日。 不盡，餘，以日數加其氣大餘。 去命如前，即其氣內沒日也。 小氣餘一千四十已上，其氣有沒者，勿推也。 沒餘皆盡者為減。 求次沒：因前沒加日六十九，餘一千一百四，餘滿從沒日一，因而命之，以氣別日。
Multiply the small remainder of the qi that has a buried day by 90; multiply its minor fraction by 15 and add; subtract the sum from the buried-day fraction; divide the remainder by the divisor to obtain one day. The remainder is what is left; add the day-count to that qi's large remainder. Discard cycles and assign names as before—the buried day falls within that solar term. When a minor solar term's remainder is 1,040 or greater, do not compute a buried day for that term. When the buried remainders are fully exhausted, subtract. To find the next buried day: from the previous buried day add 69 days with remainder 1,104; when the remainder fills the divisor, carry one buried day and assign names in turn, distinguishing days by solar term.
14
盈朔實：三萬九千九百三十三。
Excess new-moon constant: 39,933.
15
朒朔實：三萬九千二百二十。
Deficiency new-moon constant: 39,220.
16
恆朔實：三萬九千五百七十一。
Mean new-moon constant: 39,571.
17
推朔端
Deriving the new-moon epoch
18
列期總，以恆朔實除之為積月，不滿為閏餘。 滿總法為閏日，不滿為閏辰。 以閏日減冬至大餘，辰減小餘，即所求年天正月恆朔大小餘。 命大餘以甲子算外，即其日也。 天正者，日南至之月也。 恆朔者，不朒不盈之常數也。 凡減者，小餘不足減，退大餘一，如總法而減之。 大餘不足減者，加旬周，乃減之。 其須減分奇者，退分餘一，如其法而減，以其在宿度遊實不足減者，加在宿過周連餘及奇，乃減之。 以天正恆朔小餘加閏餘，以減期總，餘為總實。
Set out the period total; divide by the mean new-moon constant to obtain accumulated months; the remainder is the intercalary remainder. When it fills the common divisor, that is one intercalary day; the remainder is the intercalary double-hour. Subtract the intercalary day from the winter solstice large remainder and the intercalary double-hour from the small remainder—that gives the large and small remainders for the mean new moon of the celestial first month for the year sought. From the large remainder, count from jiazi outside the reckoning—that is the day. The celestial first month is the month in which the sun reaches its southern limit. The mean new moon is the standard interval neither shortened nor lengthened. In general when subtracting, if the small remainder is insufficient, borrow one from the large remainder and subtract a full common divisor. When the large remainder is insufficient, add a full ten-day cycle and then subtract. When fractional subtraction is required, borrow one from the fractional remainder and subtract according to its divisor; when the wandering constant in lodge-degrees is insufficient, add the full lodge-wandering cycle with remainder and odd part, then subtract. Add the intercalary remainder to the celestial first-month mean new-moon small remainder and subtract from the period total—the remainder is the total constant.
19
求恆弦望術
Procedure for finding the mean first quarter, full moon, and last quarter
20
因天正恆朔大小餘，加大餘十，小餘五百一十二太，凡四分一為少，二為半，三為太。 滿法者，去命如前，即天正上弦恆日及大小餘。 以次轉加，得望下弦及來月朔。 以次轉加，去命如前，合得所求。 他皆放此。 因朔徑求望，加大餘十四，小餘一百二十五分半。 因朔徑求下弦，加大餘二十二，小餘一百九十八少。 因朔徑次朔，加大餘二十九，小餘七百十一。 半總：六百七十。 辰率：三百三十五。
From the celestial first-month mean new-moon large and small remainders, add 10 to the large remainder and 512 plus a major fraction to the small remainder—of four parts, one is minor, two is half, three is major. When it fills the divisor, discard the cycle and assign names as before—that gives the mean day of the first quarter for the celestial first month, with large and small remainders. Continue adding in sequence to obtain the full moon, last quarter, and the next month's new moon. Continue adding, discarding cycles and assigning names as before—together these yield the date sought. All other cases follow this pattern. To derive the full moon directly from the new moon: add 14 to the large remainder and 125 plus a half fraction to the small remainder. To derive the last quarter directly from the new moon: add 22 to the large remainder and 198 plus a minor fraction to the small remainder. To derive the next new moon directly from the new moon: add 29 to the large remainder and 711 to the small remainder. Half common divisor: 670. Double-hour rate: 335.
21
檢律候氣日術
Procedure for verifying pitch-pipe response days and the added double-hour
22
求恆氣初日影泛差術
Procedure for finding the provisional difference in gnomon shadow on the first day of a mean solar term
23
見所求氣陟降率，並後氣率，半之，十五而一，為泛末率。 又二率相減，餘，十五而一，為總差。 前少，以總差減泛末率； 前多，以總差加泛末率。 加減泛末率訖，即為泛初率。 其後氣無同率，因前末率即為泛初率。 以總差減初率，餘為泛末率。
Take the ascending or descending rate of the term sought and of the following term; average them and divide by fifteen—this is the provisional terminal rate. Also subtract the two rates; divide the difference by fifteen—this is the total difference. If the earlier rate is smaller, subtract the total difference from the provisional terminal rate; if the earlier rate is larger, add the total difference to the provisional terminal rate. When adjustment of the provisional terminal rate is complete, the result is the provisional initial rate. When the following solar term has no corresponding rate, the preceding terminal rate serves as the provisional initial rate. Subtract the total difference from the initial rate; the remainder is the provisional terminal rate.
24
求恆氣初日影定差術
Procedure for finding the fixed difference in gnomon shadow on the first day of a mean solar term
25
十五除總差，為別差為限。 前少者，以限差加泛初末率； 前多者，以限差減泛初末率。 加減泛初末率訖，即為定初末率，即恆氣初日影定差。
Divide the total difference by fifteen—the quotient is the interval difference used as the limit. If the earlier rate is smaller, add the interval difference to the provisional initial and terminal rates; if the earlier rate is larger, subtract the interval difference from the provisional initial and terminal rates. When adjustment is complete, these are the fixed initial and terminal rates—the fixed shadow difference for the first day of the mean solar term.
26
求次日影差術
Procedure for finding the next day's shadow difference
27
以別定差，前少者加初日影定差，前多者減初日影定差。 加減初日影定差訖，即為次日影定差。 以次積累歲，即各得所求。 每氣皆十五日為限。 其有皆以十六除取泛末率及總差別差。
Using the separate fixed difference: if the earlier rate is smaller, add it to the first-day fixed shadow difference; if larger, subtract it. When this adjustment is complete, the result is the next day's fixed shadow difference. Accumulate day by day in sequence—each step yields the value sought. Each solar term spans a limit of fifteen days. Cases that use division by sixteen instead to obtain the provisional terminal rate and the total and separate differences.
28
求恆氣日中影定數術
Procedure for finding the fixed midday gnomon shadow for a mean solar term
29
置其恆氣小餘，以半總減之，餘為中後分。 不足減者反減半總，餘為中前分。 置前後分，影定差乘之，總法而一，為變差。 冬至後，午前以變差減氣影，午後以變差加氣影。 夏至後，午前以變差加氣影，午後以變差減氣影。 冬至一日，有減無加。 夏至一日，有加無減。 加減訖，各其恆氣日中定影。
Set the small remainder of that mean solar term; subtract half the common divisor—the remainder is the post-midday fraction. When it cannot be subtracted, subtract from half the common divisor instead—the remainder is the pre-midday fraction. Set the pre- and post-midday fractions; multiply by the fixed shadow difference and divide by the common divisor—the quotient is the variable difference. After the winter solstice, before noon subtract the variable difference from the term's shadow; after noon add it. After the summer solstice, before noon add the variable difference; after noon subtract it. On the winter solstice day there is only subtraction, never addition. On the summer solstice day there is only addition, never subtraction. When adjustment is complete, each term has its fixed midday gnomon shadow.
30
求次日中影術
Procedure for finding the next day's midday shadow
31
迭以定差陟減降加恆氣日中定影，各得次日中影。 後漢及魏宋曆，冬至日中影一丈二尺，夏至一尺五寸，於今並短。 各須隨時影校其陟降，及氣日中影應二至率。 他皆仿此。 前求每日中影術，古曆並無，臣等創立斯法也。
Step by step, add or subtract the fixed difference according to whether the rate ascends or descends from the mean term's fixed midday shadow—each step yields the next day's midday shadow. In the calendars of Later Han and of Wei and Song, the winter solstice midday shadow was twelve feet and the summer solstice shadow one foot five inches—both are shorter than present-day measurements. Each calendar must adjust its ascending and descending rates against shadows measured at the proper season, so that the midday shadow at each term matches the solstice rates. All other cases follow this model. The earlier procedure for finding each day's midday shadow—no ancient calendar had it; we, your ministers, devised this method anew.
32
求律呂應日及加時術
Procedure for finding the days when pitch-pipes respond and the added double-hour
33
十二律各以其月恆中氣日加時，應列其氣小餘，六乘之，辰率而一，為半總之數，不盡，為辰餘。 命時起子算半，為加時所在辰。 六乘辰餘，如法得一為初，二為少弱，三為少，四為少強，五為半弱。 若在辰半後者，得一為半強，二為太弱，三為太，四為太強，五為辰末。
Each of the twelve pitch-pipes corresponds to the added double-hour of its month's mean central solar term; set out that term's small remainder, multiply by six, and divide by the double-hour rate—the quotient counts half common divisors; the remainder is the double-hour remainder. From the double-hour, begin counting from zi at the half-interval—that is the double-hour of the added moment. Multiply the double-hour remainder by six; when it fills the divisor, one counts as initial, two as minor-weak, three as minor, four as minor-strong, five as half-weak. If the moment falls after the middle of the double-hour, one counts as half-strong, two as major-weak, three as major, four as major-strong, five as the end of the double-hour.
34
求七十二候術
Procedure for finding the seventy-two pentads
35
恆氣日，即初候日也。 加其大餘五，小餘九十七，小分十一。 三乘氣小分加十一，滿十八從小餘一。 滿法，去命如前，即次候日。 以次轉加，得末候日。
The mean solar-term day is the day of the first pentad. Add 5 to its large remainder, 97 to the small remainder, and 11 to the minor fraction. Multiply the solar term's minor fraction by three and add 11; when the total reaches 18, carry one to the small remainder. When it fills the divisor, discard the cycle and assign names as before—that gives the next pentad day. Continue adding in sequence to obtain the day of the final pentad.
36
求次氣日檢盈虛術
Procedure for finding the next solar-term day and verifying excess and deficiency
37
進綱一十六退紀一十七
Advance guideline: 16; retreat cycle: 17
38
泛差一十一總辰一十二六十並平闕
General difference: 11; total double-hours: 12; sixty combined with mean shortfall
39
秋分後春分前日行速，春分後秋分前日行遲。 速為進綱，遲為退紀。 若取其數，綱為名； 用其時，春分為至。 進日分前，退日分後。 凡用綱紀，皆准此例。
After the autumn equinox and before the spring equinox the sun's daily motion is fast; after the spring equinox and before the autumn equinox it is slow. Fast motion is advance guideline; slow motion is retreat cycle. When taking their numbers, the guideline serves as the name; when applying their seasons, the spring equinox marks the turning point. Advance before the day-fraction; retreat after the day-fraction. In general, all uses of guidelines and cycles follow this pattern.
40
見所在氣躔差率，並後氣率，半之，總辰乘之，綱紀而一，得氣末率。 各以泛差通其綱紀，以同差辰也。 又二率相減，餘以總辰乘而紀除之，為總差。 辰之綱紀除之，為別差率。 前少者，以總差減末率； 前多者，以總差加末率。 加減訖，皆為其氣初日損益率。 前多者，以別差率減； 前少者，以別差率加。 加減氣初日損益率訖，即次日損益率。 亦名每日躔差率。 以次加減，得每日所求。 各累所損益，隨曆定氣損益消息總，各為其日消息數。 其後氣無同率，及有數同者，皆因前少，以前末率為初率，加總差為末率，別差漸加初率，為每日率。 前多者，總差減初率為末率，別差漸減為日率。 其有氣初末計會及綱紀所校多少不葉者，隨其增損調而禦之，使際會相准。
Take the lodge-difference rate of the term in question and of the following term; average them, multiply by the total double-hours, and divide by guideline and cycle—this yields the term's terminal rate. Each applies the general difference through its guideline and cycle to align the double-hour differences. Also subtract the two rates; multiply the difference by the total double-hours and divide by the cycle—this is the total difference. Divide by the double-hour guideline and cycle—this is the separate difference rate. If the earlier rate is smaller, subtract the total difference from the terminal rate; if the earlier rate is larger, add the total difference to the terminal rate. When adjustment is complete, each is the gain-loss rate for the first day of that solar term. If the earlier rate is larger, subtract the separate difference rate; if the earlier rate is smaller, add the separate difference rate. When the solar term's first-day gain-loss rate has been adjusted, the result is the next day's gain-loss rate. This is also called the daily lodge-difference rate. Continue adding and subtracting in sequence to obtain the value for each day. Accumulate the gains and losses; following the mean solar terms, the running total of gain-loss messages for each day is that day's message number. When the following term has no matching rate, or when rates coincide, if the earlier rate is smaller take the preceding terminal rate as the initial rate, add the total difference for the terminal rate, and gradually add the separate difference to the initial rate to obtain the daily rate. If the earlier rate is larger, subtract the total difference from the initial rate to obtain the terminal rate; gradually subtract the separate difference to obtain the daily rate. When a term's initial and terminal reckonings do not match what the guideline and cycle require, adjust gains and losses until the junctions align.
41
求氣盈朒所入日辰術
Procedure for finding the day and double-hour entered by solar-term excess and deficiency
42
冬夏二至，即以恆氣為定。 自外，各以氣下消息數，息減消加其恆氣小餘，滿若不足，進退其日。 即其氣朒日辰。 亦因別其日，命以甲子，得所求。 加之為盈氣，減之為朒氣，定其盈朒所在，故日定。 凡推日月度及推發斂，皆依定氣推之。 若注曆，依恆氣日。
For the winter and summer solstices, take the mean solar term as definitive. For all other terms, take the message number below the term; where it indicates rest, subtract, and where it indicates motion, add to the mean solar term's small remainder; when full or insufficient, advance or retreat the day. That gives the true day and double-hour of the solar term. Also determine its day and assign stem-branch names from jiazi to obtain the date sought. Addition yields an excess term; subtraction yields a deficient term—fix where the excess or deficiency falls, and the day is thereby determined. In general, when computing solar and lunar positions and when computing waxing and waning, all calculations rely on true solar terms. If annotating the calendar, use mean solar-term days.
43
求定氣恆朔弦望夜半後辰數術
Procedure for finding double-hours after midnight for true solar terms, true new moons, and true quarters and full moons
44
各置其小餘，三乘，如辰率而一，為夜半後辰數。
Set each small remainder; multiply by three; divide by the double-hour rate—the quotient is the number of double-hours after midnight.
45
求每日盈朒積術
Procedure for finding the daily accumulated excess and deficiency
46
各置其氣先後率與盈朒積，乃以先率後率加躔差率，盈朒積加消息總，亦如求消息法，即得每日所入盈朒及先後之數。
Set each term's early and late rates with its excess-deficiency accumulation; then add the lodge-difference rate to the early and late rates and add the message total to the excess-deficiency accumulation, using the same method as for messages—this yields the daily entered excess-deficiency and early-late numbers.
47
求朔弦望恆日恆所入盈縮數術
Procedure for finding the mean entered excess and contraction numbers for new moon, first quarter, full moon, and last quarter on mean days
48
各以總辰乘其所入定氣日，算朒朔弦望夜半後辰數，乃以所入定氣夜半後辰數減之，餘為辰總。 其恆朔弦望與定氣同日而辰多者，其朔弦望即在前氣氣末，而辰總時有多於進綱紀通數者，疑入後氣之初也。 以乘其氣前多之末率、前少之初率，總辰而一，為總率。 凡須相乘有分餘者，母必通全子乘訖報母，異者齊同也。 其前多者，辰總減紀乘總差，綱紀而一，為差。 並於總率差，辰總乘之，倍總辰除之，以加總率。 前少者，辰總再乘別差，總辰自辰乘，倍而除之，以加總率，皆為總數。 乃以先加後減其氣盈朒為定積，凡分餘不成全而更不復須者，過半更不後夜無氣也。 以盈朒定積，盈加朒減其日小餘，滿若不足，進退之，各其入盈朒日及小餘。 若非朔望有交從者速粗舉者，以所入定氣日算乘先後率，加十五而一，先加減盈朒為定積。 入氣日十五算者，加十六而一。
For each, multiply the entered true solar-term day by the total double-hours and compute the double-hours after midnight for new moon, first quarter, full moon, and last quarter; then subtract the entered true solar term's double-hours after midnight—the remainder is the double-hour total. When a mean new moon or quarter falls on the same day as a true solar term but with more double-hours, the new moon or quarter lies at the end of the preceding term; when the double-hour total exceeds the advance guideline and cycle common number, it may enter the beginning of the following term. Multiply the terminal rate if the earlier rate is larger, or the initial rate if the earlier rate is smaller; divide by the total double-hours—the quotient is the total rate. Whenever multiplication involves fractional remainders, reduce to a common denominator, multiply the numerators, then report to the denominator; when denominators differ, reduce them to a common measure. If the earlier rate is larger, subtract the cycle-multiplied total difference from the double-hour total; divide by guideline and cycle—the quotient is the difference. Add to the total-rate difference; multiply by the double-hour total; divide by twice the total double-hours; add to the total rate. If the earlier rate is smaller, multiply the double-hour total again by the separate difference, multiply by the total double-hours from the double-hour itself, double and divide; add to the total rate—these are all total numbers. Then apply later-addition and earlier-subtraction to the term's excess-deficiency to obtain the fixed accumulation; whenever fractional remainders do not form wholes and need not be continued, if more than half, do not carry to the following night without a term. With the excess-deficiency fixed accumulation, add for excess and subtract for deficiency from the day's small remainder; when full or insufficient, advance or retreat—each yields the entered excess-deficiency day and small remainder. If not a new moon or full moon with an eclipse and one wants a rough quick reckoning, multiply the entered true solar-term day by the early-late rate, add, and divide by fifteen—first add or subtract excess-deficiency for the fixed accumulation. For days entering a term counted as fifteen, add and divide by sixteen.
49
曆變周：四十四萬三千七十七。
Calendar change cycle: 443,077.
50
變奇率：十二。
Change odd rate: 12.
51
曆變日：二十七； 變餘，七百四十三； 變奇，一。
Calendar change day: 27; change remainder, 743; change odd, 1.
52
月程法：六十三。
Lunar circuit divisor: 63.
53
推曆變術
Procedure for deriving calendar change
54
以曆變周去總實，餘，以變奇率乘之，滿變周又去之。 不滿者，變奇率約之，為變分。 不盡，為變奇。 分滿總法為日，不滿為餘。 命日算外，即所求年天正恆朔夜半入變日及餘，以天正恆朔小餘加之，即經辰所入。
Remove the calendar change cycle from the total constant; multiply the remainder by the change odd rate; when it fills the change cycle, remove again. What does not fill is reduced by the change odd rate to a change fraction. What is not exhausted is the change odd. When the fraction fills the common divisor, that counts as one day; the remainder is what does not fill. Count days outside the reckoning—that gives the change day and remainder entered at midnight of the celestial first-month mean new moon for the year sought; add the celestial first-month mean new-moon small remainder—that is the lodge entered.
55
求朔弦望經辰所入
Finding the lodge entered at new moon, first quarter, full moon, and last quarter
56
因天正經辰所入日餘奇，加日七、餘五百一十二、奇九。 奇滿率成餘。 餘，如總法為日，得上弦經辰所入。 以次轉加，得望、下弦及來月朔。 所入滿變日及餘奇，則去之。 凡相連去者，皆仿於此。 徑求望者，加朔所入日十四、餘一千二十五、奇六。 徑求次朔，加一日、餘一千三百七、奇十一。
From the celestial first month's lodge day, remainder, and odd entered at midnight, add 7 days, remainder 512, and odd 9. When the odd fills the rate, it becomes remainder. For the remainder: when it matches the common divisor, that counts as one day—this yields the lodge entered at the first quarter. Continue adding in sequence to obtain the full moon, last quarter, and the next month's new moon. When the entered value fills the change day, remainder, and odd, discard them. All successive removals follow this pattern. To derive the full moon directly, add to the new moon's entered day 14, remainder 1,025, and odd 6. To derive the next new moon directly, add one day, remainder 1,307, and odd 11.
57
求朔望弦盈朒減辰所入術
Procedure for finding the entered day and double-hour of new moon, full moon, and quarters with excess-deficiency adjustment
58
各以其日所入盈朒定積，盈加朒減其恆經辰所入，餘即各所求。
Each takes that day's entered excess-deficiency fixed accumulation; add for excess and subtract for deficiency from the mean lodge entered—the remainder is the value sought for each.
59
求朔弦望盈朒日辰入變遲速定數術
Procedure for finding the fixed change slow-fast number for the entered day and double-hour of new moon, quarters, and full moon with excess-deficiency
60
各列其所入日增減率，並後率而半之，為通率。 又二率相減，餘為率差。 增者，以入餘減總法，餘乘率差，總法而一，並率差而半之。 減者，半入餘乘率差，亦總法而一，並以加於通率，入餘乘之，總法而一，所得為經辰變轉半經辰變。 速減遲加盈朒經辰所入餘，為轉餘。 應增者，減法。 應減者，因餘。 皆以乘率差，總法而一，加於通率。 變率乘之，總法而一，以速減遲加變率為定率。 乃以定率增減遲速積為定。 此法微密至當，以示算理通途。 若非朔望有交及欲考校速要者，但以入餘乘增減率，總法而一，增減速為要耳。 其後無同率者，亦因前率，應增者以通率為初數，半率差而減之； 應減入餘進退日者分為二日，隨餘初末，如法求之。 所得並以加減變率為定。
List each entered day's increase-decrease rate; add it to the following rate and halve—the result is the common rate. Also subtract the two rates—the difference is the rate difference. For increase, subtract the entered remainder from the common divisor; multiply the remainder by the rate difference; divide by the common divisor and halve together with the rate difference. For decrease, halve the entered remainder multiplied by the rate difference, also divide by the common divisor and add to the common rate; multiply the entered remainder and divide by the common divisor—the result is lodge change at the turn and half lodge change at the turn. Fast subtracts and slow adds to the excess-deficiency lodge entered remainder—this is the turn remainder. Where increase is required, subtract from the divisor. Where decrease is required, use the remainder. In all cases multiply by the rate difference, divide by the common divisor, and add to the common rate. Multiply by the change rate and divide by the common divisor; subtract fast and add slow to the change rate to obtain the fixed rate. Then use the fixed rate to increase or decrease the slow-fast accumulation to obtain the fixed value. This method is minute and exact throughout, showing the thoroughfare of computational principle. If not a new moon or full moon with an eclipse and one wishes to examine or verify quickly, simply multiply the entered remainder by the increase-decrease rate, divide by the common divisor, and adjust slow—that suffices. When the following term has no matching rate, also follow the preceding rate; where increase is required, take the common rate as the initial number and subtract half the rate difference; where decrease requires advancing or retreating the day from the entered remainder, split into two days and compute according to whether the remainder is initial or terminal. Add or subtract the results to the change rate to obtain the fixed value.
61
其入前件日餘，如初數已下者為初，已上者以初數減總法，餘為末之數。 增減相反，約以九分為限。 初雖少弱，而末微強，餘差不多，理況兼舉，皆今有雜差，各隨其數。 若恆算所求，七日與二十一日得初率，而末之所減，隱而不顯。 且數與平行正算，亦初末有數，而恆算所無。 其十四日、二十八日既初末數存，而虛差亦減其數，數當去恆法不見。
For the entered remainder on the preceding item's day, if below the initial number it is initial; if above, subtract the initial number from the common divisor—the remainder is the terminal number. Increase and decrease are opposite; approximate with nine-ninths as the limit. Though the initial value is slightly weak and the terminal slightly strong, the remainders are nearly equal—reason suggests taking both, yet now there are mixed differences, each following its number. If by mean calculation one seeks, on the seventh and twenty-first days one obtains the initial rate, while what the terminal subtracts remains hidden and not displayed. Moreover, the numbers used in parallel regular calculation also have initial and terminal values, which mean calculation does not provide. On the fourteenth and twenty-eighth days, since initial and terminal numbers remain, the spurious difference also reduces those numbers—the numbers should be dropped from the mean divisor and not appear.
62
求朔弦望盈朒所入日名及小餘術
Procedure for finding the entered day name and small remainder of excess-deficiency for new moon, quarters, and full moon
63
各以其所入變曆速定數速減遲加其盈朒小餘。 滿若不足，進退其日。 命以甲子算外，各其盈朒日反餘。 加其恆日，餘者為盈； 減其恆日，餘者為朒。 其日不動者，依恆朔日而定其小餘，推擬日月行度。 其定小餘二十四已下，一千三百一十六已上者，其入氣盈朒、入曆遲速，皆須覆依本術推算，不得從粗舉速要之限。 乃前朔後朔，迭相推校。 盈朒之課，據實為准。 損不侵朒，益不過盈。
Each takes the fixed change slow-fast number of the entered change cycle—fast subtracts and slow adds to its excess-deficiency small remainder. When full or insufficient, advance or retreat the day. Assign from jiazi beyond the count; for each, take the reversed remainder of the excess-deficiency day. Add to its mean day—the remainder is excess; Subtract its mean day—the remainder is deficiency. Where the day does not move, fix the small remainder according to the mean new-moon day and derive the solar and lunar motions in degrees. When the fixed small remainder is twenty-four or below, or thirteen hundred sixteen or above, the entered-qi excess-deficiency and entered-cycle slow-fast all require recalculation by the original procedure—they must not follow the rough shortcut for speed. Then compare the preceding new moon with the following new moon in turn. The reckoning of excess and deficiency takes actuality as the standard. Reduction must not encroach on deficiency; increase must not exceed excess.
64
求定朔月大小術
Procedure for finding whether the month of the fixed new moon is long or short
65
凡朔盈朒日名，即為定朔日名。 其定朔日名，十幹與來月同者大，不同者小。 其月無中氣者為閏月。 其正月朔有定加時正月者，消息前後各一兩月，以定月之大小。 合虧在晦二者，弦望亦隨事消息。 凡置月朔，盈朒之極，不過頻三。 其或過者，觀定小餘近夜半者量之。
In general the day name of the new-moon excess-deficiency is the fixed new-moon day name. For the fixed new-moon day name, if the ten stem matches the coming month the month is long; if not, it is short. A month without mid-qi is an intercalary month. When the first-month new moon has a fixed addition falling in first month, adjust one or two months before and after to determine whether the month is long or short. When conjunction and eclipse both fall on the last day of the month, the quarters and full moon are also adjusted as the case requires. In general when setting monthly new moons, the extreme of excess and deficiency does not exceed three in succession. If it exceeds this, observe whether the fixed small remainder is near midnight and measure accordingly.
66
檢宿度術
Procedure for checking lodge degrees
67
前件周天二十八宿，相距三百六十五度，前漢唐都以渾儀赤道所量。 其數常定，紘帶天中，儀圖所准。 日月往來，隨交損益。 所入宿度，進退不同。
The above twenty-eight lodges of the circuit of heaven, spaced at three hundred sixty-five degrees, were measured on the equatorial armillary sphere in Former Han and the Tang capitals. Their numbers are constantly fixed, girdling the center of heaven—the standard of armillary diagrams. Sun and Moon come and go, gaining and losing with the crossing. The lodge degrees entered differ in advance and retreat.
68
黃道宿度左中郎將賈達檢日月所去赤道不同，更鑄黃道渾儀所檢者。
The ecliptic lodge degrees—Left Palace Attendant Jia Da checked that the Sun and Moon's departure from the equator differs—were verified with a newly cast ecliptic armillary sphere.
69
臣等今所修撰討論，更造木渾圖交絡調賦黃赤二道三百六十五度有奇，校量大率，與此符會。 今曆以步日行月及五星出入循此。 其月行交絡黃道，進退亦宜有別。 每交輒差，不可詳盡。 今亦依黃道推步。
We who compile and discuss here have newly made a wooden armillary diagram interlacing and balancing the yellow and red two paths at three hundred sixty-five odd degrees; measured at large rate, it agrees with this. The present calendar derives solar daily motion, lunar motion, and the ingress and egress of the five stars according to this. The Moon's motion crossing and interlacing the ecliptic should also differ in advance and retreat. At each crossing there is always difference—it cannot be detailed exhaustively. Now we also derive according to the ecliptic.
70
推日躔術
Procedure for deriving solar lodges
71
置冬至初日躔差率，加總法，乘冬至小餘，如總法而一，以減天宿度分。 其餘命起黃道鬥十二度，宿次去之，經鬥去宿分度，不滿宿算外，即所求年冬至夜半所在宿度算及分。
Set the winter-solstice first-day lodge-difference rate; add the common divisor; multiply by the winter-solstice small remainder; divide by the common divisor; subtract from the celestial lodge degree and fraction. With the remainder, assign beginning from Yellow Way Dipper twelve degrees; remove by lodge sequence; pass Dipper and remove lodge fraction and degree; outside the count when insufficient to fill a lodge—that is the lodge degree count and fraction where the winter solstice midnight of the year sought lies.
72
求每定氣初日夜半日所在定度術
Procedure for finding the fixed degree where the Sun lies at midnight on the first day of each fixed qi
73
各以其定氣初日躔差率，乘氣定餘，總法而一，進加退減餘為分，以減定氣日度及分，命以宿次如前，即其夜半度及春秋二分定氣初日為進退之始，當平行一度。 自餘依進加退減度之。
Each takes that fixed qi's first-day lodge-difference rate, multiplies by the qi's fixed remainder, and divides by the common divisor; advance-add and retreat-subtract the remainder as parts; subtract from the fixed qi's solar degree and fraction; assign by lodge sequence as before—that is its midnight degree, and the fixed qi's first days of spring and autumn equinoxes are the start of advance and retreat, corresponding to one degree of uniform motion. For the rest, add degrees by advance and subtract by retreat.
74
求次日夜半日所在定度術
Procedure for finding the fixed degree where the Sun lies at midnight on the next day
75
各因定氣夜半所在為本，加度一。 又以其日躔差率，進加退減度分。 滿若不足，並依前例。 去命如上，即得所求。 其定朔弦望夜半日度，各隨定氣，以其日月名亦直而分別之。 勘右依恆有餘，從定恆行度，不用躔差。
Each takes the fixed-qi midnight position as base and adds one degree. Also with that day's lodge-difference rate, advance-add and retreat-subtract degree and fraction. When full or insufficient, all follow the preceding pattern. Remove and assign as above—then obtain what is sought. The midnight solar degrees for fixed new moon, quarters, and full moon each follow fixed qi; assign their day and month names directly and distinguish them. Examined at right: following the mean there is remainder; proceed from fixed-mean motion in degrees—do not use lodge difference.
76
求朔弦望定日夜辰所加日度術
Procedure for finding the solar degrees added at the fixed day and double-hour of new moon, quarters, and full moon
77
各以其定小餘為平分。 又定小餘乘其日所躔差率，總法而一，乃進加退減其平分，以加其夜半日度，即各定辰所加。 其與五星加減者，半其分，消息月朔者，應推月度所須，皆依本朔大小。 若注曆，依甲子乙丑各擬入。
Each takes its fixed small remainder as the equal fraction. Also multiply the fixed small remainder by that day's lodge-difference rate and divide by the common divisor; then advance-add and retreat-subtract its equal fraction; add to its midnight solar degree—that is what each fixed double-hour adds. Where it adds and subtracts with the five stars, halve the fraction; where the monthly new moon is adjusted, what lunar motion requires—all follow the original new moon's long or short month. If annotating the calendar, assign entry according to jiazi and yichou respectively.
78
推月離術
Procedure for deriving lunar separation
79
求朔望定日辰月所在度術
Procedure for finding the degree where the Moon lies at the fixed day and double-hour of new and full moon
80
各置朔弦望定辰所加日度及分。
Set the solar degree and fraction added at the fixed double-hour of each new moon, quarter, and full moon.
81
凡朔定辰所加為合朔，日月同度。 上弦加度九十一、分四百一十七。
In general what the fixed new-moon double-hour adds is conjunction new moon—Sun and Moon at the same degree. First quarter: add ninety-one degrees, four hundred seventeen parts.
82
望加度一百八十三、分八百三十四。
Full moon: add one hundred eighty-three degrees, eight hundred thirty-four parts.
83
下弦加度二百七十三、分一千二百五十一。 訖，各半而十退之，為程度分。
Last quarter: add two hundred seventy-three degrees, one thousand two hundred fifty-one parts. When complete, halve each and retreat by ten—the result is cycle degree and parts.
84
求次月定朔夜半入變曆術
Procedure for finding entry into the change cycle at midnight of the next month's fixed new moon
85
置天正恆朔夜半所入變日及餘。 定朔有進退一日者，進退一日，為定朔夜半所入。
Set the change day and remainder entered at midnight of the celestial first month's mean new moon. If the fixed new moon advances or retreats one day, advance or retreat one day for what is entered at fixed-new-moon midnight.
86
月大加二日，月小加一日。 餘皆五百九十六、奇十六。
Long month adds two days; short month adds one day. Remainders are all five hundred ninety-six, with fraction sixteen.
87
求次日夜半所入變曆術
Procedure for finding entry into the change cycle at the next day's midnight
88
因定朔夜半所入日算，加日一，滿皆如前。 其弦皆依前定日所在求之。
From the day count entered at fixed-new-moon midnight, add one day; when full, all as before. The quarters are all sought according to where the preceding fixed day lies.
89
求變日定離程術
Procedure for finding the fixed separation course for change days
90
各以其日夜半入變餘，乘離差，總法而一，為見差。 以進加退減其日離程，為月每日所離定程。
Each multiplies that day's midnight entered change remainder by separation difference and divides by the common divisor—that is the visible difference. Advance-add and retreat-subtract to its daily separation course—that is the Moon's fixed daily separation course.
91
求朔弦望之定日夜半月所在度術
Procedure for finding the degree where the Moon lies at midnight on the fixed day of new moon, quarters, and full moon
92
各以其日定小餘，乘所入變日離定程，總法而一，為夜半後分。 滿程法為度，餘為度分。 以減其日加辰所在度及分，命以黃道宿度，即其所求。 次日夜半，各以離定程加朔弦望夜半所在分，滿程法從度，去命以黃道宿度算外，則次日夜半月度。 求晨昏度，以其日離定程乘其日夜刻，二百而一，為昏分，滿程法為度。 望前以昏，後以晨，加夜半度，得所求。 其弦望以五乘定小餘，程法一，為刻，即各其辰所入刻數。 皆減其晨前刻，不盡為晨後刻。 不滿晨前刻者，從前日注曆，伺候推。
Each multiplies that day's fixed small remainder by the entered change day's fixed separation course and divides by the common divisor—that is the fraction after midnight. When it fills the course divisor it becomes degrees; the remainder is degree-parts. Subtract from the degree and fraction where that day's added double-hour lies; assign by ecliptic lodge degrees—that is what is sought. At the next day's midnight, add the fixed separation course to the fraction where new moon, quarters, and full moon lie at midnight; when it fills the course divisor carry to degrees; remove and assign by ecliptic lodge degree count outside—that is the Moon's degree at the next midnight. To find dawn and dusk degrees, multiply that day's fixed separation course by that day's day-and-night clepsydra marks and divide by two hundred—that is the dusk fraction; when it fills the course divisor it becomes degrees. Before full moon use dusk; after use dawn; add to midnight degrees—to obtain what is sought. For quarters and full moon, multiply the fixed small remainder by five and divide by course divisor one—that is clepsydra marks; that is the clepsydra-mark number entered for each double-hour. All subtract the marks before dawn; what remains is the marks after dawn. If insufficient for marks before dawn, follow the previous day's calendar annotation and wait to derive.
93
總刻：一百。 辰刻：分十一。 刻分法：七十二。
Total marks: one hundred. Double-hour marks: divide eleven. Clepsydra-mark fraction divisor: seventy-two.
94
求定氣日晝夜漏刻及日出沒術
Procedure for finding day-and-night clepsydra marks of fixed-qi days and sunrise and sunset
95
倍其氣晨前刻及分，滿法從刻，為日不見漏。 以減百刻，餘為日見漏。 五刻晝漏刻。 以晝漏刻減百刻，餘夜漏刻。 以四刻十二分加晨前漏刻，命起子初刻算外，即日出辰刻。 以日見漏加日出刻辰，以次如前，即日沒所在辰刻。 以二十五除從夜漏，得每更一籌之數。 以二刻三十六分加日沒辰刻，即甲辰刻，又以更籌數加之，得甲夜一籌數。 以次累加，滿辰去命之，即五更夜籌所以當辰刻及也，以配二十一箭漏之法也。
Double its qi's marks before dawn and fraction; when full carry from marks—that is the clepsydra of invisible sun. Subtract from one hundred marks; the remainder is the clepsydra of visible sun. Day clepsydra marks: five marks. Subtract day clepsydra marks from one hundred marks; the remainder is night clepsydra marks. Add four marks twelve parts to clepsydra marks before dawn; assign beginning from zi initial mark beyond the count—that is the double-hour mark of sunrise. Add visible-sun clepsydra to sunrise mark and double-hour; in order as before—that is the double-hour mark where the sun sets. Divide night clepsydra by twenty-five to obtain the count for each watch tally. Add two marks thirty-six parts to sunset double-hour mark—that is first-yin mark; also add watch-tally count to obtain the first watch tally of the night. Accumulate in order; when full remove by double-hour and assign—that is how the five watches' night tallies correspond to double-hour marks, matching the method of twenty-one-arrow clepsydra.
96
求每日並屈申數術
Procedure for finding daily combined flexion and extension numbers
97
每氣准為一十五日，各置其氣屈申率。 每以發斂差損益之，差滿十從分，分滿十從率一，即各每日屈申率。 各累計屈申率為刻分，乃以一百八十乘刻分，泛差十一乘綱紀而除之，得為刻差，滿法為刻。 隨氣所在，以申減屈加不見漏而半之，為晨前定刻。 每求次日，各如前法。 時加其如始，隨加辰日晚，以率課之。
Each qi is taken as fifteen days; set each qi's flexion-extension rate. Each uses waxing-waning difference to gain and lose; when difference fills ten carry to parts, when parts fill ten carry one to rate—that is each day's flexion-extension rate. Accumulate flexion-extension rates as clepsydra parts; then multiply clepsydra parts by one hundred eighty; multiply general difference by guideline and cycle and divide—to obtain clepsydra difference; when full of divisor it becomes marks. Following where the qi lies, subtract extension and add flexion to invisible clepsydra and halve—that is the fixed marks before dawn. Each time the next day is sought, all as the preceding method. Add time as at the beginning; as added double-hours make the day late, assess by rate.
98
求黃道去極每日差術
Procedure for finding the daily difference of the ecliptic's departure from the pole
99
置刻差，三十而一為度。 不滿三約為分。 申減屈加其氣初黃道度，即每日所求。
Set the clepsydra difference; divide by thirty to obtain degrees. What does not fill thirty counts as fractional parts. Subtract extension and add flexion to that qi's initial ecliptic degree—that gives what is sought for each day.
100
求昏旦去中星度術
Procedure for finding dusk and dawn departure from culmination-star degrees
101
每日求其晝漏刻數，以乘期實，二百乘總法而除之，得昏去中星度。 以減周天度，餘為晨去中星度。 以昏旦去中星度，加其辰日所在，即各其日中宿度。 其梗概粗舉者，加其夜半日度，各其日中星宿度。
Each day find its day clepsydra mark count; multiply by the period constant; divide by two hundred times the common divisor—to obtain the dusk departure from culmination-star degree. Subtract from the circuit-of-heaven degrees; the remainder is the dawn departure from culmination-star degree. Add the dusk and dawn departure from culmination-star degrees to where the double-hour day lies—that gives each noon lodge degree. For the rough outline method, add the midnight solar degree to obtain each noon star lodge degree.
102
因求次日者，各置其四刻差，七十二乘之，二百八十八而一度。 冬至後加，夏至後減。 隨日加，各得每日去中度。 晨昏所距日在黃道中星准度，以赤道計之。 其赤道同太初星距。
To seek the next day, set each day's four-mark difference; multiply by seventy-two; two hundred eighty-eight make one degree. After the winter solstice add; after the summer solstice subtract. Add day by day; each step yields the daily departure from center degree. The dawn and dusk distance from the Sun at the ecliptic median-star standard degree is calculated by the equator. Its equatorial distances match those of the Taichu calendar.
103
推遊交術
Procedure for calculating wandering-node crossing
104
終率：一千九十三萬九千三百一十三。 奇率：三百。
Terminal rate: 10,939,313. Odd rate: 300.
105
約終：三萬六千四百六十四奇一百一十三。
Reduced terminal: 36,464 odd 113.
106
交中：一萬八千二百三十二奇五十六半。
Crossing center: 18,232 odd 56½.
107
交中日：二十七餘二百八十四奇一百一十三。
Crossing middle days: 27 remainder 284 odd 113.
108
中日：十三餘八百一十二奇五十六半。
Middle day: 13 remainder 812 odd 56½.
109
虧朔：三千一百六奇一百八十七。
Waning new moon: 316 odd 187.
110
實望：一萬九千七百八十五奇一百五十。
True full moon: 19,785 odd 150.
111
後准：一百五十二奇九百三半。
Rear criterion: 152 odd 903½.
112
前准：一萬六千六百七十八奇二百六十三。
Front criterion: 16,678 odd 263.
113
求月行入交表裏術
Procedure for finding the Moon's entry into crossing outside and inside
114
置總實，以終率去之。 不足去者，奇率乘之。 滿終率，又去之。 不滿者，奇率約之，為天正恆朔夜半入交分。 不盡，為奇。 以總法約入交分，為日。 不盡，為餘。 命日算外，即天正恆朔夜半入交日算及餘、奇。 天正定朔有進退日者，依所進退一日，為朔所入。 日不滿中日及餘、奇者，為月在外； 滿，去之，餘皆一為月在內。 大月加二日，小月加一日，餘皆一千五十五、奇一百八十七。 求次日，加一日，滿中日者，皆去之，餘為入次。 一表一里，迭互入之。
Set the total constant; remove by the terminal rate. What does not suffice to remove, multiply by the odd rate. When it fills the terminal rate, remove again. What does not fill, reduce by the odd rate—that is the crossing-entry fraction at midnight of the celestial first-month mean new moon. What is not exhausted is the odd fraction. Reduce the crossing-entry fraction by the common divisor—that gives days. What is not exhausted is the remainder. Count days outside the reckoning—that gives the crossing-entry day count, remainder, and odd at midnight of the celestial first-month mean new moon. When the celestial first-month fixed new moon advances or retreats one day, advance or retreat one day for what the conjunction enters. When the day does not fill the middle day and remainder and odd, the Moon is outside; When full, remove; when the remainder is all one, the Moon is inside. Long month add two days; short month add one day; remainders are all 1,055 odd 187. To seek the next day, add one day; when full of the middle day remove all; the remainder is the next entry. One outside and one inside, entering alternately.
115
求月入交去日道遠近術
Procedure for finding the Moon's entry into crossing and its distance from the solar path
116
置所入日差，並後差半之，為通率。 進，以入日餘減總法，以乘差，總法而一，並差以半之。 退者，半入餘，以乘差，總法而一。 皆加通率，為交定率。 乃以入餘乘定總法。 乃進退差積，滿十為度，不滿為分，即各其日月去日道度數。 每求日道宿度去極數，其入七日，餘一千七十六、奇二十八少已下者，進，已上，盡全； 餘二百六十三、奇二百七十一大者，退入十四日，如交餘奇已下者，退； 其入已上，盡全； 餘五百二十七、奇二百四十二半者，進。 而終其要為五分。 初則七日四分，十四日三分； 末則七日後一分，十四日後二分。 雖初強末弱，差率有檢，月道一度半強已下者，為沾黃道。 當朔望，則有虧。 遇五星在黃道者，則相侵掩。
Set the entered day difference; combine with the rear difference and halve—that is the common rate. For advance: subtract the entered day remainder from the common divisor; multiply by the difference and divide by the common divisor; combine with the difference and halve. For retreat: halve the entered remainder; multiply by the difference and divide by the common divisor. In all cases add to the common rate—that is the crossing fixed rate. Then multiply the entered remainder by the fixed total divisor. Then apply the advance-and-retreat difference accumulation; when full of ten counts as degrees, what does not fill counts as parts—that gives each day's Sun and Moon departure from the solar path in degrees. Each time the solar-path lodge degree departure-from-pole number is sought: on entered day seven, when the remainder is 1,076 odd 28 minor or below, advance; when above, take entirely; When the remainder is 263 odd 271 major or greater, retreat and enter day fourteen; when like the crossing remainder odd or below, retreat; When the entered value is above, take entirely; When the remainder is 527 odd 242½, advance. In the end the essential value is five parts. Initially, seven days yield four parts and fourteen days yield three parts; Finally, after seven days one part and after fourteen days two parts. Though initially strong and finally weak, the difference rate has a check; when the lunar path is one and a half degrees strong or below, it is touching the ecliptic. At new and full moon, there is then eclipse waning. When the five planets lie on the ecliptic, mutual invasion and occultation then occur.
117
求所在宿術
Procedure for finding the lodge entered
118
求夜半入交日十三算者及餘，以減中日及餘，不盡者，以乘其日離定程，總法而一，為離分，滿程為度，以加其日夜半月所在宿度算及分，求次交准此，各得其定交所在度。 置前後定交所宿度算及分，半之，即各表裏極所在宿度及分。
Find the midnight crossing-entry day count of thirteen and its remainder; subtract from the middle day and remainder; multiply what is not exhausted by that day's fixed separation course and divide by the common divisor—that is the separation fraction; when full of the course counts as a degree; add to that day-and-night half-Moon lodge degree count and fraction; seek the next crossing likewise—each yields the fixed crossing lodge degree entered. Set the front and rear fixed crossing lodge degree count and fraction; halve—that gives each outside-and-inside pole lodge degree and fraction.
119
求恆朔望泛交分野
Finding the general crossing fraction field for mean new and full moon
120
因天正恆朔夜半入交分，以天正恆朔泛交分求望泛交，以實望加之。 又加，得次月恆朔泛交分。 滿約終及奇，去之。 次求次朔，以虧望加之。
From the crossing-entry fraction at midnight of the celestial first-month mean new moon, use the celestial first-month mean new moon general crossing fraction to find the full-moon general crossing; add the true full moon. Add again to obtain the next month's mean new moon general crossing fraction. When full of the reduced terminal and odd, remove. Next seek the following new moon; add the waning full moon.
121
求朔望入常交分術
Procedure for finding new and full moon entry into regular crossing fraction
122
以入氣盈朒定積，盈加朒減其恆泛交分，滿若不足，進退約終。 即其常分交。
Use the entered-qi excess-deficiency fixed accumulation; excess adds and deficiency subtracts from the mean general crossing fraction; when full or insufficient, advance and retreat the reduced terminal. That is its regular crossing fraction.
123
求朔望定交分術
Procedure for finding the fixed crossing fraction at new and full moon
124
以六十乘定遲速，以七百七十七降除之，所得為限數。 速減遲加如常。 其數朔入交月在日道里者，以所入限數減定遲速，餘以速減遲加其定交分。 而出日道表者，為變交分。 加減不出日道表，即依定交分求蝕分。 其變交分出日道表三時半內者，檢其前後月望入交分數多少，依月虧初復末定蝕術，注消息，以定蝕不。
Multiply the fixed slow-fast by sixty; reduce and divide by 777—the result is the limit number. Fast subtracts and slow adds as usual. When at new moon the Moon entering the crossing is inside the solar path, subtract the limit number entered from the fixed slow-fast; with the remainder, fast subtracts and slow adds to its fixed crossing fraction. When it exits outside the solar path, that is the changed crossing fraction. When addition and subtraction do not exit outside the solar path, follow the fixed crossing fraction to find the eclipse fraction. When the changed crossing fraction exits outside the solar path within three and a half double-hours, check the preceding and following month's full-moon crossing-entry fraction amounts; follow the lunar waning initial, restoration, and final fixed eclipse procedure; note waxing and waning—to determine whether there is an eclipse.
125
求入蝕限術
Procedure for finding entry into eclipse limit
126
其入交定分，如交中已下者，為月在外道； 交中已上者，以交中減之，餘為月在內。 其分如後准已下、前准已上者，為入蝕限。 望則月蝕，朔入限，月在裏者，日蝕。 入限如後准已下者，為交後分； 前准已上者，反減交中，餘為交前分。 以一百一十二約之，為交時。
When the crossing-entry fixed fraction is at the crossing center or below, the Moon is on the outer path; When above the crossing center, subtract the crossing center; the remainder is the Moon inside. When the fraction is at the rear criterion or below and the front criterion or above, that is entry into the eclipse limit. At full moon there is lunar eclipse; at new moon entering the limit, when the Moon is inside there is solar eclipse. When the entry into the limit is at the rear criterion or below, that is the post-crossing fraction; When at the front criterion or above, subtract inversely from the crossing center; the remainder is the pre-crossing fraction. Divide by 112—that gives the crossing double-hour.
127
求月蝕所在辰術
Procedure for finding the double-hour where lunar eclipse lies
128
置望日不見刻，六十七乘之，十而一，所得，若蝕望定小餘與之等已下，又以此得減總法餘與之等已為蝕正見數定小餘。 如求律氣應加時法，得加時所在辰月在沖辰蝕，若非正見者，於日出後日沒前十二刻半內，求其初末以候之。 又以半總減蝕定小餘，不足減者半總加減訖，以六乘之，如辰率而一，命起子半算外，即月蝕所在辰。
Set the full-moon day's invisible marks; multiply by sixty-seven and divide by ten—the result: if the eclipse full-moon fixed small remainder equals it or is below, also use this result to subtract from the common-divisor remainder and when equal or below, that is the eclipse true-visibility fixed small remainder. As in the procedure for finding the pitch-pipe qi response added double-hour, obtain the added double-hour where it lies; the Moon in the opposing double-hour at eclipse; if not true visibility, within twelve and a half marks after sunrise or before sunset seek the initial and final moments to await it. Also subtract the eclipse fixed small remainder by half the common divisor; when insufficient to subtract, add the half common divisor and subtract when done; multiply by six; divide by the double-hour rate; assign beginning from zi at the half-interval beyond the count—that is the double-hour where lunar eclipse lies.
129
求日蝕所在辰術
Procedure for finding the double-hour where solar eclipse lies
130
置有蝕朔定小餘副之，以辰率除之，所得以艮、坤、巽、乾為次，命退算外。 不滿法者，半法減之。 無可減者，為初； 所減之餘，為末。 初則減法，各為差率。 月在內道者，乃以十加去交時數而三除之，以乘差率，十四而一，為差。 其朔在二分前後一氣內，即以差為定。 近冬至以去寒露雨水、近夏至以去清明白露氣數倍之，又三除去交時數增之。 近冬至，艮巽以加，坤乾以減； 近夏至，艮巽以減、坤乾以加其差，為定差。 艮坤加副，巽乾減副。 月在外道者，三除去交時數，以乘差率，十四而一，為之差。 艮坤以減副，巽乾以加副，各加減副訖，為定副小餘。 如求律氣應加時術，即日蝕所在辰及少太。 其求入辰刻，以半辰刻乘朔，辰率而一，得刻及分。 若蝕近朝夕者，以朔所入氣日出沒刻校蝕所在，知蝕見不之多少，所在辰為正見日月蝕既，在起復初末，亦或變常退於見前後十二刻半候之。
Set the eclipsing new-moon fixed small remainder aside; divide by the double-hour rate; take gen, kun, xun, qian in sequence; assign retreat beyond the count. When not full of the divisor, subtract half the divisor. When nothing can be subtracted, that is initial; The remainder after subtraction is final. If initial, subtract the divisor; each gives the difference rate. When the Moon is on the inner path, add ten to the distance-from-crossing double-hour count and divide by three; multiply by the difference rate and divide by fourteen—that is the difference. When the new moon falls within one qi before or after either equinox, use the difference as fixed. Near the winter solstice use the distance from Cold Dew and Rain Water; near the summer solstice use the distance from Clear and Bright and White Dew; multiply the qi count; also increase by three-tenths of the distance-from-crossing double-hour count. Near the winter solstice gen and xun add and kun and qian subtract; Near the summer solstice gen and xun subtract and kun and qian add to the difference—that is the fixed difference. Gen and kun add to the aside; xun and qian subtract from the aside. When the Moon is on the outer path, reduce the distance-from-crossing double-hour count by three-tenths; multiply by the difference rate and divide by fourteen—that is the difference. Gen and kun subtract from the aside and xun and qian add to the aside; when all additions and subtractions to the aside are done, that is the fixed aside small remainder. As in the procedure for finding the pitch-pipe qi response added double-hour, obtain the double-hour where the solar eclipse lies and its minor and major degree. To find the entry double-hour marks: multiply the new moon by half the double-hour marks, divide by the double-hour rate, and obtain marks and fractions. If the eclipse falls near dawn or dusk, use the new moon's entering qi sunrise and sunset marks to verify the eclipse position and determine how visible it will be; the double-hour where it lies is true visibility for a total solar or lunar eclipse; at emergence and restoration, initial and final moments may also deviate from the norm—retreat twelve and a half marks before or after visibility to await it.
131
求月起復依蝕分後術
Procedure for finding lunar emergence and restoration according to eclipse fraction, following the procedure above
132
求月在日道表朔不應蝕准。 朔在夏至初日，准去交前後二百四十八分為初准； 已下，加時在午正前後七刻內者，食。 朔去夏至前後，每一日損初准二分，畢於前後九十四日，各為每日變准。 其朔去交如變准已下，加時如前者，蝕。
Procedure for the criterion when a new moon with the Moon on the outer side of the sun's path should not eclipse When the new moon falls on the first day of summer solstice, the criterion is: a distance from the crossing before or after of 248 fractions is the initial criterion; below that, if the added double-hour falls within seven marks before or after noon culmination, there is an eclipse. For the new moon's distance from summer solstice before or after, each day reduce the initial criterion by two fractions; this completes at ninety-four days before or after, each becoming a daily variable criterion. When the new moon's distance from the crossing is at the variable criterion or below, and the added double-hour is as above, there is an eclipse.
133
又以末准六十減初准及變准，餘以十八約之，為刻准。 以並午正前後七刻數為時准。 加時准內去交分，如末准已下，並蝕。 又置末准，每一刻加十八，為差准。 每加時刻，去午前後如差准刻已下，去交分如差已下者，並蝕。 自秋分至春分，去交如末准已下，加時南方三辰者，亦蝕。 凡定交分在辰前後半時外者，雖入蝕准前為蝕。 求月在日道里朔應蝕而不蝕准。 朔在夏至日，去交一千三百七十三，為初准； 已上，加時在午正前後十八刻內者，或不蝕。 朔去夏至前後，每一日益初准一分半，畢於前後九十四日，各為每日變准。 以初減變，餘十而一，為刻准。 以刻減午正前後十八刻，餘，十而一為時准。 其去交在變准已上，加時在准內者，或不蝕。
Also subtract the initial and variable criterion by the final criterion of 60; divide the remainder by 18—that is the mark criterion. Take the combined count of seven marks before or after noon culmination as the time criterion. When the distance-from-crossing fraction is within the added double-hour criterion and at the final criterion or below, both conditions indicate eclipse. Also set the final criterion: each mark adds 18—that is the difference criterion. For each added double-hour mark: if the distance from noon before or after is at the difference-criterion marks or below, and the distance-from-crossing fraction is at the difference or below, both indicate eclipse. From autumn equinox to spring equinox: if the distance from the crossing is at the final criterion or below and the added double-hour falls in the three southern double-hours, there is also an eclipse. Whenever the fixed crossing fraction lies outside half a double-hour before or after the double-hour, even if it enters the eclipse criterion beforehand, that counts as eclipse. Procedure for the criterion when a new moon with the Moon on the inner side of the sun's path should eclipse but does not On summer solstice day at new moon: a distance from the crossing of 1,373 is the initial criterion; above that, if the added double-hour falls within eighteen marks before or after noon culmination, there may be no eclipse. For the new moon's distance from summer solstice before or after, each day increase the initial criterion by one and a half fractions; this completes at ninety-four days before or after, each becoming a daily variable criterion. Subtract the variable from the initial; divide the remainder by ten—that is the mark criterion. Subtract the marks from eighteen marks before or after noon culmination; divide the remainder by ten—that is the time criterion. When the distance from the crossing is above the variable criterion and the added double-hour falls within the criterion, there may be no eclipse.
134
求月蝕分術
Procedure for finding lunar eclipse fraction
135
置去交前後定分，冬交前後，皆去二百二十四。 春交後去一百，交前去二百。 夏不問前後，去五十。 秋交後去二百，交前去一百。 不足去者，蝕既。 有餘者，以減後准，一百四而一。 餘半已下，為半弱； 半已上，為半強。 命以十五為限，得月蝕之大分。
Set the fixed distance-from-crossing before or after fraction; for winter crossing before or after, each subtract 224. In spring: after the crossing subtract 100; before the crossing subtract 200. In summer, regardless of before or after: subtract 50. In autumn: after the crossing subtract 200; before the crossing subtract 100. When insufficient to subtract, it is a total eclipse. When there is a remainder, subtract from the rear criterion and divide by 104. If the remainder is half the divisor or below, it is half-weak; if half the divisor or above, it is half-strong. Assign with fifteen as the limit to obtain the major lunar eclipse fraction.
136
求月蝕所起術
Procedure for finding where lunar eclipse begins
137
月在內道：蝕東方三辰，虧自月下斜南上，月從西而漸北，自東而漸南。 蝕南方三辰，虧起左下，甚于正南，復於右下。 蝕西方三辰。 虧自南而漸東，月從北而漸西，起於月上，斜南而下。 月在外道：蝕東方三辰，虧起自月下，斜北而上，虧起東而漸北，月從西漸南。 蝕南方三辰，虧起左上，甚於正北，復於右上。 蝕西方三辰。 虧自北而漸東，月從南而漸西，起於月上，斜北而上。 凡蝕十二分已上，皆隨黃道所在起復，於正傍逆順上下每過其分。 又道有升降，每各不同，各隨時取正。
When the Moon is on the inner path: in an eclipse of the eastern three double-hours, obscuration begins from below the moon slanting southward and upward; the moon moves from west gradually northward, from east gradually southward. In an eclipse of the southern three double-hours, obscuration begins at the lower left, reaches greatest at due south, and again at the lower right. In an eclipse of the western three double-hours: obscuration proceeds from south gradually eastward; the moon from north gradually westward; it begins above the moon, slanting southward and downward. When the Moon is on the outer path: in an eclipse of the eastern three double-hours, obscuration begins from below the moon slanting northward and upward; obscuration begins east gradually northward, the moon from west gradually southward. In an eclipse of the southern three double-hours, obscuration begins at the upper left, reaches greatest at due north, and again at the upper right. In an eclipse of the western three double-hours: obscuration proceeds from north gradually eastward; the moon from south gradually westward; it begins above the moon, slanting northward and upward. Whenever the eclipse is twelve parts or above, emergence and restoration all follow the ecliptic position; at due flank, reverse and direct, above and below—each passing its fraction. The path also has ascent and descent, each differing; each follows the time to take the due position.
138
求日蝕分術
Procedure for finding solar eclipse fraction
139
月在內道者，朔入冬至，畢朒雨水，及盈秋分，畢大雪，皆以五百五十八為蝕差。 自入朒春分已後，日損六分，畢于白露。 置蝕去交前後定分，皆以蝕差減之。 但去交分不足減者，皆反以減蝕差為不蝕餘。 自入朒小滿，畢盈小暑，加時在午正前後七刻外者，皆去不蝕餘一時； 三刻內，加不蝕餘一時。 朒大寒畢朒立春，交前五時外，大暑畢盈立冬，交後五時外，皆去不蝕餘一時，五時內加一時。 諸加時蝕差應減者，交後減之，交前加之。 應加者，交後加之，交前減之。 但不足減去者，蝕既。 加減入不蝕限者，或不蝕。 其月在外道者，冬至初日無蝕差。 自後日益六分，累計以為蝕差，畢于朒雨水。 自入朒春分，畢于盈白露，皆以五百二十二為蝕差。 自入盈秋分已後，日損六分，畢于大雪。 所損之餘，為蝕差。 以蝕差加去交定分，為蝕分。 以減後准，餘為不蝕分。 各置其朔蝕差，十五約之，以減一百四，餘為定法。 不蝕分餘，各如定法得一分。 餘半法已上，為半強； 已下，為半弱。 減十五，餘為蝕之大分。
When the Moon is on the inner path: from new moon entering winter solstice through waning Rain Water, and from waxing autumn equinox through Great Snow—all use 558 as the eclipse difference. From entering waning spring equinox onward, each day reduce six fractions until completed at White Dew. Set the eclipse distance-from-crossing before or after fixed fraction; subtract the eclipse difference from all. But when the distance-from-crossing fraction is insufficient to subtract, inversely subtract the eclipse difference as the non-eclipse remainder. From entering waning Minor Fullness through waxing Minor Heat, if the added double-hour falls outside seven marks before or after noon culmination, subtract one time from the non-eclipse remainder; within three marks, add one time to the non-eclipse remainder. From waning Great Cold through waning Start of Spring, outside five times before the crossing; from Great Heat through waxing Start of Winter, outside five times after the crossing—all subtract one time from the non-eclipse remainder; within five times add one time. For all added double-hour eclipse differences that should be subtracted: after the crossing subtract; before the crossing add. Those that should be added: after the crossing add; before the crossing subtract. But when insufficient to subtract, it is a total eclipse. When addition and subtraction enter the non-eclipse limit, there may be no eclipse. When the Moon is on the outer path: the winter solstice first day has no eclipse difference. From then on each day increase six fractions, accumulating as the eclipse difference until completed at waning Rain Water. From entering waning spring equinox through waxing White Dew—all use 522 as the eclipse difference. From entering waxing autumn equinox onward, each day reduce six fractions until completed at Great Snow. The remainder after reduction is the eclipse difference. Add the eclipse difference to the distance-from-crossing fixed fraction—that is the eclipse fraction. Subtract from the rear criterion; the remainder is the non-eclipse fraction. Set each new-moon eclipse difference; divide by fifteen; subtract from 104; the remainder is the fixed divisor. For the non-eclipse fraction remainder: each division by the fixed divisor obtains one part. If the remainder is half the divisor or above, it is half-strong; below, it is half-weak. Subtract fifteen; the remainder is the major eclipse fraction.
140
求日蝕所起術
Procedure for finding where solar eclipse begins
141
日在內道：日蝕東方三辰，虧自日上近北而斜下，月漸西北，日漸東南。 日蝕南方三辰，虧起右下，甚正北，復左下。 月在南而漸東，日在北而漸西。 日蝕西方三辰。 月漸東北，日漸西南，虧自日下近西而斜上。 日在外道：日蝕東方三辰，虧自日上近南而斜下，月漸東南，日漸西北。 日蝕南方三辰，虧起右下，甚正北，復左下。 月在南而漸東，日在北而漸西。 日蝕西方三辰。 月漸西南，日漸東北，虧自日下近南而斜上。 凡蝕十二分已上，起於正旁。 各據黃道升降，以准其體。 隨其所處，每各不同。 蝕有初末，動涉其時，隨便益損，以定虧復所在之方也。
When the Sun is on the inner path: in a solar eclipse of the eastern three double-hours, obscuration begins from above the sun near north slanting downward; the moon moves gradually northwest, the sun gradually southeast. In a solar eclipse of the southern three double-hours, obscuration begins at the lower right, reaches greatest at due north, and again at the lower left. The moon is in the south gradually eastward; the sun in the north gradually westward. In a solar eclipse of the western three double-hours: the moon moves gradually northeast, the sun gradually southwest; obscuration begins from below the sun near west slanting upward. When the Sun is on the outer path: in a solar eclipse of the eastern three double-hours, obscuration begins from above the sun near south slanting downward; the moon moves gradually southeast, the sun gradually northwest. In a solar eclipse of the southern three double-hours, obscuration begins at the lower right, reaches greatest at due north, and again at the lower left. The moon is in the south gradually eastward; the sun in the north gradually westward. In a solar eclipse of the western three double-hours: the moon moves gradually southwest, the sun gradually northeast; obscuration begins from below the sun near south slanting upward. Whenever the eclipse is twelve parts or above, it begins at the due flank. Each follows the ecliptic's ascent and descent to gauge its disk. Following where each lies, all differ. Eclipses have initial and final moments; their movement spans its time; increase and decrease as needed to fix the direction of obscuration and restoration.
142
求日月蝕虧初及復末時刻術
Procedure for finding the moments of eclipse waning initial and restoration final
143
置朔望所蝕大分數為率。 四分已上，因增二。 五分已上，因增三。 九分已上，因增四。 十三分已上，因增五。 各為泛用刻率，副之。 以乘所入率，副之。 以乘所入變增減率，總法而一，應速增損、減加，應遲依其增減副，訖，為蝕定用刻數。 乃四乘之，十而一，以減蝕甚辰刻，為虧初。 又六乘之，十而一，加蝕甚辰刻，為復末。 依其定加時所在辰刻加減命之，各其辰、其月蝕甚初末更籌。 因其日月所入辰刻及分，依前定氣所遇夜刻更籌術，求其初末及甚時更籌。
Set the new and full moon eclipse major fraction count as the rate. Four parts or above: accordingly increase by two. Five parts or above: accordingly increase by three. Nine parts or above: accordingly increase by four. Thirteen parts or above: accordingly increase by five. Each becomes a general-use mark rate; set it aside. Multiply by the entered rate; set aside. Multiply by the entered change increase-decrease rate; divide by the common divisor; when fast, increase or decrease and subtract or add; when slow, follow its increase-decrease to the aside; when done, that is the fixed eclipse-use mark count. Then multiply by four and divide by ten; subtract from the maximum eclipse double-hour marks—that is waning initial. Also multiply by six and divide by ten; add to the maximum eclipse double-hour marks—that is restoration final. According to its fixed added double-hour position in double-hour marks, add and subtract to assign; each to its double-hour; the lunar eclipse maximum initial and final night-watch tallies. Following the sun and moon's entered double-hour marks and fractions, according to the prior fixed qi night-mark night-watch tally procedure, find the initial, final, and maximum night-watch tallies.
144
迦葉孝威等天竺法，先依日月行遲疾度，以推入交遠近日月蝕分加時，日月蝕亦為十五分。 去交十五度、十四度、十三度，影虧不蝕法，自此已下，乃依驗蝕。 十二度十五分，蝕二分少強，以漸差降，自五度半已上，蝕既，十四分強。 若五度無餘分已下，皆蝕盡。 又用前蝕多少，以定後蝕分餘。 若既，其後蝕度及分，即加七度以為蝕度。 若望月蝕既，來月朔日雖入而不注蝕。 若蝕半已下，五分取一分； 若半已上，三分取一分，以加來月朔蝕度及分。 若今歲日餘度及分，然後可驗蝕度分數多少。 又云：六月依節一蝕。 是月十五日是月蝕節，黑月儘是月蝕節，亦以吉凶之象，警告王者奉順正法，蒼生福盛，雖時應蝕，由福故也，其蝕即退。 更經六月，欲蝕之前，皆有先兆。 月欲有蝕，先月形搖振，狀若驚懼，月兔及側月色黃如有憂狀。 自常暈，月初生時，光不顯盛，或極細微。 日欲有蝕，先日形搖振，極如驚懼狀。 或光色微昧，不赫盛，或黎慘。 日月蝕先同候，光隕墜，或旦暮際有赤色起，如火燒，金銀珠玉諸寶失光。 或有闕盡如雲入日，或有黑盡入月，鳥聲細隱，烏不顯亮，雲交擾擾，光景渾亂，忽極令諸乳卒竭，月濕如汗狀，日形段裂無光，犬嗥貓叫，虹見有聲，三辰失闕，月時有缺，水赤色有膩。 十四日、十五日，辟鳥圓集者，亦是蝕之先候。 此等與中國法數稍殊，自外梗概相似也。
The Indian method of Kāśyapa Xiaowei and others: first according to the sun and moon's slow and fast degrees, derive the crossing-entry distance, eclipse fraction, and added double-hour; solar and lunar eclipses also use fifteen parts. At distances from the crossing of fifteen, fourteen, and thirteen degrees—the shadow-waning non-eclipse method applies; from here below, one then relies on verified eclipse. At twelve degrees fifteen parts—an eclipse of two parts minor-strong, gradually decreasing by difference; from five and a half degrees above—a total eclipse of fourteen parts strong. If five degrees with no remainder below, all is entirely eclipsed. Also use the magnitude of the prior eclipse to fix the fraction remainder of the next. If the eclipse was total, add seven degrees to its later eclipse degree and parts to obtain the eclipse degree. If a full-moon eclipse is total, the next month's new moon may enter the criterion yet is not registered as an eclipse. If the eclipse is half or less, take one part in five; if half or more, take one part in three and add it to the next month's new-moon eclipse degree and parts. Once the current year's solar remainder in degrees and parts is set, one may then verify the eclipse degree and fraction. It also states that, by the nodes, there is one eclipse every six months. The fifteenth of that month is the full-moon eclipse node; the dark of the moon is wholly the full-moon eclipse node. These omens of fortune and misfortune warn the king to uphold the righteous law: when the people are deeply blessed, though an eclipse is seasonally due, it retreats for that very blessing. Six months later, before an eclipse is due, there are always foretokens. When the moon is about to be eclipsed, its disk first trembles and shakes as if in alarm; the hare in the moon and the moon's limb turn yellow, as though stricken with grief. Apart from its usual halo, at first waxing the light does not blaze forth, or is exceedingly faint. When the sun is about to be eclipsed, its disk first trembles and shakes, as if in extreme alarm. Its light and color may grow faint and dim, no longer blazing bright, or turn dusky and grim. Solar and lunar eclipses share the same foretokens: light falls away; at dawn or dusk a red glow rises as if afire; gold, silver, pearls, jade, and every treasure lose their luster. Gaps may be eaten away as though clouds swallowed the sun, or blackness swallowed the moon; birds cry thinly and hidden, crows lose their brightness; clouds tangle in turmoil and light grows wholly confused; suddenly, to the extreme, the milk of all nursing creatures fails; the moon grows damp as with sweat; the sun's disk splits segment by segment and goes dark; dogs howl and cats wail; a rainbow appears with sound; the three luminaries lose their wholeness; the moon at times shows gaps; water turns red and greasy. On the fourteenth and fifteenth days, warblers gathering in circles are also a foretoken of eclipse. These differ slightly from China's numerical methods, yet in broad outline they are much the same.
145
步五星術
Procedure for the Five Planets
146
見伏五十二日，晨見伏六十三日，餘、奇同終分奇。
Appearance-invisibility: 52 days; morning appearance-invisibility: 63 days; remainder and odd parts match the terminal fraction odd.
147
求五星平見術
Procedure for Finding the Five Planets' Mean Appearances
148
各以伏分減總實，餘以其星總率去之。 不足去者，反減其餘總率。 餘以總法約之，為日，不盡為餘奇，即所求年天正恆朔夜半後星晨夕平見日算及餘奇。 天正定朔進退日者，進減退加一日為定朔夜半後星平見日及餘奇。 其金水二星，先得夕平見，其滿見伏日及餘者去之，餘為晨平見日及餘奇。 命見日天正曆月大小，以次去之，不滿月者為入其月，命日算外，即晨夕平見所在月日及餘奇。
For each planet, subtract the invisibility fraction from the accumulated remainder and divide the remainder by that star's total cycle rate. If the division cannot be completed, subtract the remainder inversely from the total rate. Reduce the remainder by the total factor to obtain days; what remains undivided is the remainder odd—that gives the morning or evening mean appearance day count and remainder odd after midnight of the celestial first month's mean new moon for the year sought. When the celestial first month's fixed new moon advances or retreats a day, subtract a day for advance and add a day for retreat to obtain the mean appearance day and remainder odd after midnight of the fixed new moon. For Venus and Mercury, first obtain the evening mean appearance; cast out the full appearance-invisibility days and remainder; the remainder is the morning mean appearance day and remainder odd. From the appearance day, cast out full months according to the celestial first month calendar's long and short months; what does not fill a month is the month entered; count the day outside the reckoning—that gives the month, day, and remainder odd of the morning or evening mean appearance.
149
求後平見在月日術
Procedure for Finding Later Mean Appearance Month and Day
150
各以其星終日算及餘奇，如前平見所在月日算及餘奇。 奇滿奇率，從餘。 餘滿總法，為日。 去命如前，即後平見所在月日及餘奇，其金水二星，加夕得晨，加晨得夕。 各半見餘，以同半總。
For each planet, add that star's terminal day count and remainder odd to the prior mean appearance's month, day count, and remainder odd. When the odd parts fill the odd rate, carry into the remainder. When the remainder fills the total factor, convert it to days. Cast out and assign as before to obtain the later mean appearance's month, day, and remainder odd; for Venus and Mercury, adding the evening date yields the morning, and adding the morning yields the evening. Halve each planet's appearance remainder to match the half-total.
151
求五星常見術
Procedure for Finding the Five Planets' Regular Appearances
152
各依其星平見所入恆氣，計日損益。 分滿半總為日，不滿為分，以損益所加減。 訖，餘以加減訖平見日及分，即其常見日及分。 星日初見去日度，平見入氣曆。 加減日。 損益率。
For each planet, according to the mean qi entered at its mean appearance, calculate the daily diminish-increase. When the parts fill the half-total, convert them to days; what does not fill remains as parts; apply the diminish-increase to add or subtract. When this is done, apply the remainder to add or subtract from the completed mean appearance day and parts—that yields the regular appearance day and parts. At first appearance, distance from the sun in degrees; mean appearance entering-qi calendar. Add-subtract days. Diminish-increase rates.
153
歲星初見，去日十四度。 見入冬至，畢小寒，均減六日。 自入大寒已後，日損六十七分。 見入春分初日，依平。 自後日加八十九分。 入立夏，畢小滿，均加六日。 自入芒種已後，日損八十九分。 入夏至，畢立秋，均加四日。 自入處暑已後，日損一百七十八分。 入白露，初日依平均，自後日減五十二分。 入小雪，畢大雪，均減六日。
Jupiter: at first appearance it stands fourteen degrees from the sun. At mean appearance entering the Winter Solstice through Minor Cold, uniformly subtract six days. From Major Cold onward, subtract 67 parts per day. At mean appearance entering the Spring Equinox on the first day, follow the mean rate. From then on, add 89 parts per day. From the Start of Summer through Minor Fullness, uniformly add six days. From Grain in Ear onward, subtract 89 parts per day. From the Summer Solstice through the Start of Autumn, uniformly add four days. From End of Heat onward, subtract 178 parts per day. At White Dew, follow the mean rate on the first day; from then on, subtract 52 parts per day. From Minor Snow through Major Snow, uniformly subtract six days.
154
熒惑初見，去日十七度。 見入冬至，初日減二十七日。 自後日損六百三分。 入大寒，初日依平。 自後日加四百二分。 入雨水，畢穀雨，均加二十七日。 入自立夏已後，日損一百九十八分。 入立秋，依平。 自入處暑已後，日減一百九十分。 入小雪，畢大寒，均減二十七日。
Mars: at first appearance it stands seventeen degrees from the sun. At mean appearance entering the Winter Solstice, subtract 27 days on the first day. From then on, subtract 603 parts per day. At Major Cold, follow the mean rate on the first day. From then on, add 402 parts per day. From Rain Water through Grain Rain, uniformly add 27 days. From the Start of Summer onward, subtract 198 parts per day. At the Start of Autumn, follow the mean rate. From End of Heat onward, subtract 190 parts per day. From Minor Snow through Major Cold, uniformly subtract 27 days.
155
鎮星初見，去日十七度。 見入冬至，初日減四日。 自後日益八十九分。 入大寒，畢春分，均減八日。 自入清明已後，日損五十九分。 入小暑，初日依平。 自後日加八十九分。 入白露，初日加八日。 自後日損一百七十八分。 入秋分，均加四日。 自入寒露已後，日損五十九分。 入小雪，初日依平。 自平後日減八十九分。
Saturn: at first appearance it stands seventeen degrees from the sun. At mean appearance entering the Winter Solstice, subtract four days on the first day. From then on, add 89 parts per day. From Major Cold through the Spring Equinox, uniformly subtract eight days. From Clear Brightness onward, subtract 59 parts per day. At Minor Heat, follow the mean rate on the first day. From then on, add 89 parts per day. At White Dew, add eight days on the first day. From then on, subtract 178 parts per day. At the Autumn Equinox, uniformly add four days. From Cold Dew onward, subtract 59 parts per day. At Minor Snow, follow the mean rate on the first day. From the mean rate onward, subtract 89 parts per day.
156
太白初見，去日十一度。 夕見：入冬至，初日依平。 自後日減一百分。 入啟蟄，畢春分，均減九日。 自入清明已後，日損一百分。 入芒種，依平。 自入夏至已後，日加一百分。 入處暑，畢秋分，均加九日。 自入寒露已後，日損一百分。 入大雪，依平。 晨見：入冬至，依平。 自入小寒已後，日加六十七分。 入立春，畢立夏，均加三日。 自入小滿已後，日損六十七分。 入夏至，依平。 自入小暑已後，日減六十七分。 入立秋，畢立冬，均減三日。 自入小雪已後，日損六十七分。
Venus: at first appearance it stands eleven degrees from the sun. Evening appearance: from the Winter Solstice, follow the mean rate on the first day. From then on, subtract 100 parts per day. From Awakening of Insects through the Spring Equinox, uniformly subtract nine days. From Clear Brightness onward, subtract 100 parts per day. At Grain in Ear, follow the mean rate. From the Summer Solstice onward, add 100 parts per day. From End of Heat through the Autumn Equinox, uniformly add nine days. From Cold Dew onward, subtract 100 parts per day. At Major Snow, follow the mean rate. Morning appearance: from the Winter Solstice, follow the mean rate. From Minor Cold onward, add 67 parts per day. From the Start of Spring through the Start of Summer, uniformly add three days. From Minor Fullness onward, subtract 67 parts per day. At the Summer Solstice, follow the mean rate. From Minor Heat onward, subtract 67 parts per day. From the Start of Autumn through the Start of Winter, uniformly subtract three days. From Minor Snow onward, subtract 67 parts per day.
157
辰星初見，去日十七度。 夕見：入冬至，畢清明，依平。 入穀雨，畢芒種，均減二日。 入夏至，畢大暑，依平。 入立秋，畢霜降，應見不見。 其在立秋及霜降二氣之內，夕去日十八度外，三十六度內，有木火土金一星已上者亦見。 入立冬，畢大雪，依平。 晨見：入冬至，均減四日。 入小寒，畢大寒，依平。 入立春，畢啟蟄，均減三日。 其在啟蟄氣內，去日度如前，晨無木火土金，一星已上者不見。 入雨水，畢立夏，應見不見。 其在立夏氣內，去日度如前，晨有木火土金一星已上者，亦見。 入小滿，畢寒露，依平。 入露降，畢立冬，均加一日。 入小雪，畢大雪，依平。
Mercury: at first appearance it stands seventeen degrees from the sun. Evening appearance: from the Winter Solstice through Clear Brightness, follow the mean rate. From Grain Rain through Grain in Ear, uniformly subtract two days. From the Summer Solstice through Major Heat, follow the mean rate. From the Start of Autumn through Frost's Descent, it should appear but does not. If within the two qi of Start of Autumn and Frost's Descent, in the evening it stands more than eighteen degrees but less than thirty-six degrees from the sun, and Jupiter, Mars, Saturn, or Venus is also visible, it is seen as well. From the Start of Winter through Major Snow, follow the mean rate. Morning appearance: from the Winter Solstice, uniformly subtract four days. From Minor Cold through Major Cold, follow the mean rate. From the Start of Spring through Awakening of Insects, uniformly subtract three days. If within the Awakening of Insects qi, at the same solar distance as before, with no Jupiter, Mars, Saturn, or Venus visible in the morning, it is not seen. From Rain Water through the Start of Summer, it should appear but does not. If within the Start of Summer qi, at the same solar distance as before, with one or more of Jupiter, Mars, Saturn, or Venus visible in the morning, it is seen as well. From Minor Fullness through Cold Dew, follow the mean rate. From White Dew and Frost's Descent through the Start of Winter, uniformly add one day. From Minor Snow through Major Snow, follow the mean rate.
158
求五星定見術
Procedure for Finding the Five Planets' Fixed Appearances
159
各置其星常見日消息定數半之，息減消加常見日，即為定見日及分。 五星休王光不同，喜怒盛衰大小尤異。 苟變於常見或先後，今依日躔遲速考其行，度其格，以去日為之定準。
For each planet, take half of that star's daily correction constant at its regular appearance; apply diminish-increase at rest to the regular appearance day to obtain the fixed appearance day and parts. The five planets differ in brightness at rest and at exaltation; their phases of joy, anger, flourishing, and waning, and their apparent sizes, differ especially. When appearance varies from the regular date, whether early or late, examine its motion according to the sun's varying speed, measure its position, and take its distance from the sun as the fixed standard.
160
求星見所在度術
Procedure for Finding the Degree Where the Star Appears
161
置星定見日夜半日所在宿度算及分，半其日躔差，乘定見餘，半總而一，進加退減定見餘，以加夜半度分，乃以其星初見去日度數，晨減夕加之，即星初見辰所在。
Set the lodge-degree count and parts of the sun at midnight on the star's fixed appearance day; halve the solar motion difference, multiply by the fixed appearance remainder, and divide by the half-total; add for advance and subtract for retreat to the fixed appearance remainder, and add this to the midnight degree and parts; then apply that star's first-appearance solar distance—in the morning subtract and in the evening add—to obtain the lodge where the star first appears.
162
宿度等及分行星術
Procedure for Lodge Degrees and Stepping the Planets
163
各置其星初見日消息定數，半之，息加消減，其星初見行留日率。 其土木二星不須加減，即依本術。 其加減不滿日者，與見通之。 過半從一日，無半不從論。 乃依行星日度之率，求日之行分。
For each planet, set its first-appearance day's correction constant, halve it, and at rest add or diminish to adjust its first-appearance motion-and-station day rate. For Jupiter and Saturn, no add-subtract adjustment is needed; follow the base procedure. If the add-subtract adjustment does not amount to a full day, combine it with the appearance date. If the remainder exceeds half, carry one day; if it does not reach half, do not carry. Then, according to the planet's day-and-degree motion rate, obtain the day's motion parts.
164
求初見日後夜半星所在術
Procedure for Finding Where the Star Stands at Midnight After the First Appearance Day
165
置其星定見餘，以減半總，以其星初見行分乘之，半總而一，以順加逆減星初見定辰所在度分。 加之滿法，減之不足，進退一度。 依前命之算外，即星見後夜半所在宿度及分。 自此已後，每依其星計日行度，所至日度及益疾，皆從夜半為始。 辰有少，隨所近也。
Set the star's fixed appearance remainder, subtract it from the half-total, multiply by that star's first-appearance motion parts, and divide by the half-total; add for direct motion and subtract for retrograde to the star's first-appearance fixed chen location in degrees and parts. If addition fills the divisor, or subtraction falls short, advance or retreat one degree. Count outside the prior assignation to obtain the star's lodge degree and parts at midnight after appearance. From here onward, for each star compute the daily motion in degrees; the day-degree reached and any increase in speed are all reckoned from midnight. If the chen has a fractional remainder, follow whichever is nearest.
166
轉求次日夜半星行所至術
Procedure for Stepping to Where the Star Reaches at the Next Day's Midnight
167
各以其星一日所行度及分，順逆加減之。 其行有小分者，以日率為母。 小分滿母，去之，從行分一。 行分滿半總，去之，從度一。 其行有益疾益遲者，副置一日行分。 各以其差遲損疾加之，留者因前，逆則依減。 順行出鬥去其分，逆行入鬥先加分。 訖，皆以程法約行分為度分，各得每日所至。 其五星後順留退所終日度，各依伏度，求其去日遠近，消息日度之所在，以定伏日所在。 若注曆，其日度及金水等星，皆棄其分也。
For each star, add or subtract its one-day motion in degrees and parts according to direct or retrograde motion. If the motion has fractional small parts, use the day rate as denominator. When the small parts fill the denominator, remove them and carry one into the motion parts. When the motion parts fill the half-total, remove them and carry one into the degrees. If the motion has increasing speed or increasing slowness, set aside the one-day motion parts separately. For each, add the slowness-or-speed difference as diminish or increase; if stationary, continue from the prior value; if retrograde, subtract accordingly. In direct motion exiting the Dipper, remove its parts; in retrograde motion entering the Dipper, first add the parts. When this is done, reduce the motion parts to degree parts by the rule factor for each day's arrival. For the five planets' later direct, stationary, and retrograde terminal day-degrees, use each star's invisibility degree to find its solar distance, near or far, and the day-degree reached by diminish-increase, thereby fixing the invisibility day. When annotating the calendar, discard the fractional parts for day-degrees and for Venus, Mercury, and the other stars.
168
求平行度及分術
Procedure for Finding Mean Motion Degree and Parts
169
置定度率，以半總乘之，以有分者從之，以日率除之，所得，為一日行分。 不盡小分滿其行分。 滿半總為度。 即是一日所行度及行分、小分。 置定日率，減一日，以所差分乘之，二而一，為差率。 益疾者以差率減平行分，益遲者以差率加平行分，即是初日所行度及分。
Set the fixed degree rate, multiply by the half-total, carry any parts forward, divide by the day rate, and the quotient is the one-day motion parts. Any undivided small parts are carried into the motion parts. When the parts fill the half-total, convert them to a degree. This gives the one-day motion in degrees, motion parts, and small parts. Set the fixed day rate, subtract one day, multiply by the applied difference in parts, and divide by two to obtain the difference rate. For increasing speed, subtract the difference rate from the mean motion parts; for increasing slowness, add it; this yields the first day's motion in degrees and parts.
170
星名星行變日初行入氣曆行日率行度及度分率：損益率。
Star name, star motion, change day, first motion entering-qi calendar, motion day rate, motion degree and degree-parts rate: diminish-increase rate.
171
歲星：初順，差行一百一十四日，行十八度五百九遲一分先疾，日益十四日。 前留，二十六日。 旋退西行，差行三十日，退六度十二分。 先遲，日益疾二分。 又退西行，差行四十二日，退六度十二分。 先疾，日益遲二分。 後留，二十五日。 後順，差行一百一十四日，行十八度五百九。 先進遲，日益疾分日盡而夕伏十四日。
Jupiter: first direct motion, differential 114 days, travels 18°509 parts—one part slow at first, then fast—with a daily increase of 14 parts. Prior station: 26 days. Revolving retreat, moving westward: differential 30 days, retreats 6°12 parts. Slow at first, with speed increasing by 2 parts per day. Again retreats westward: differential 42 days, retreats 6°12 parts. Fast at first, with slowness increasing by 2 parts per day. Later station: 25 days. Later direct motion: differential 114 days, travels 18°509 parts. Slow at first while advancing, with speed increasing daily in parts until the days are exhausted; evening invisibility: 14 days.
172
熒惑：初順，入冬至初日，率二百四十三日行一百六十五度。 自後三日損日及度各三。 小寒初日，二百三十五日行一百五十四度。 自後二日損日及度各三。 穀雨四日，平，畢小滿九日。 一百七十八日行一百度。 自入小滿九日已後，二日益日及度各一。 夏至初日，平，畢六日。 一百七十一日行九十三度。 自入夏至六日已後，三日益日及度各一。 立秋初日，一百八十四日行一百六度。 自後一日益日及度各一。 白露初日，二百一十四日行一百三十六度。 自後五日益日及度各一。 秋分初日，二百三十二日行一百五十四度。 自後一日益日及度各一。 寒露初日，二百四十七日行一百六十九度。 自後五日益日及度各二。 霜降五日，平，畢立冬十三日。 二百五十九日行一百八十一度。 自入立冬十三日已後，二日損日及度各一。 復冬至初日，二百四十二日行一百六十五度。
Mars: first direct motion, from the Winter Solstice on the first day, rate 243 days for 165 degrees. From then on, every three days diminish the day count and degree count each by three. At Minor Cold on the first day: 235 days for 154 degrees. From then on, every two days diminish the day count and degree count each by three. From Grain Rain for four days at the mean rate through Minor Fullness for nine days. 178 days for 100 degrees. From nine days after entering Minor Fullness, every two days increase the day count and degree count each by one. From the Summer Solstice on the first day at the mean rate through six days. 171 days for 93 degrees. From six days after entering the Summer Solstice, every three days increase the day count and degree count each by one. At the Start of Autumn on the first day: 184 days for 106 degrees. From then on, every day increase the day count and degree count each by one. At White Dew on the first day: 214 days for 136 degrees. From then on, every five days increase the day count and degree count each by one. At the Autumn Equinox on the first day: 232 days for 154 degrees. From then on, every day increase the day count and degree count each by one. At Cold Dew on the first day: 247 days for 169 degrees. From then on, every five days increase the day count and degree count each by two. From Frost's Descent for five days at the mean rate through the Start of Winter for thirteen days. 259 days for 181 degrees. From thirteen days after entering the Start of Winter, every two days diminish the day count and degree count each by one. Again at the Winter Solstice on the first day: 242 days for 165 degrees.
173
各依所入恆氣，平者依率，自餘計日損益，名為前疾日度定率。 其前遲及留退入氣有損益日度者，計日損益，皆同此疾之法，以為遲留旋退定日度之率也。
For each mean qi entered, where the rate is mean follow the base rate; from the remainder compute daily diminish-increase—this is called the prior-fast fixed day-and-degree rate. For prior slow motion and station-retreat entering qi where day-and-degree diminish-increase applies, compute daily diminish-increase by the same method as for fast motion, to obtain the fixed day-and-degree rates for slow, stationary, and revolving-retreat phases.
174
求變日率術：此疾，入大寒六日，損日率一，畢雨水。 入春分，畢立夏，減日率十。 入小滿初，減日率十。 後三日損所減一。 畢芒種，依平。 若入立秋，三日益日率一，畢處暑。 入白露，畢秋分，均加率十。 入寒露初，加率十。 後一日半損所加一。 畢氣盡，依平。
Procedure for finding the change day rate: in this fast phase, from six days into Major Cold, diminish the day rate by one through Rain Water. From the Spring Equinox through the Start of Summer, reduce the day rate by ten. At Minor Fullness on the first day, reduce the day rate by ten. Afterward, every three days diminish the reduction by one. Through Grain in Ear, follow the mean rate. From the Start of Autumn, increase the day rate by one every three days through End of Heat. From White Dew through the Autumn Equinox, uniformly add ten to the rate. At Cold Dew on the first day, add ten to the rate. Afterward, every day and a half diminish the addition by one. Once the qi period is exhausted, follow the mean rate.
175
求變度率術：此疾，若入大寒，畢於啟蟄，立夏至大暑氣盡，霜降畢小雪，皆加度率四。 清明畢穀雨，加率度十二。 初行入處暑，減日率六十，度率三十。 別為初遲半度之行，行盡此日度，及來所減之餘日度之率續為疾。 入白露，畢秋分，四十四日行二十二度。 皆為初遲半度之率。 初行入大寒，畢大暑，差行，先疾，日益遲一分。 各如上法，求其行分。 其前遲後日率，既有增損，而益遲益疾若分，皆檢括前疾末日行分，為前遲初日行分。 以前遲平行分減之，餘為前遲總差。 後疾日分，為後遲末日行分。 為後遲日行分減之，餘為後總差。 減為後別日差分。 其不滿者，皆調為小分。 遲疾之際，行分衰殺不論。 所差多者，依此推算。 若所差不多者，各依本法。
Procedure for finding the change degree rate: in this fast phase, from Major Cold through Awakening of Insects, from the Summer Solstice through Major Heat until the qi is exhausted, and from Frost's Descent through Minor Snow, add four to the degree rate in each case. From Clear Brightness through Grain Rain, add twelve to the rate in degrees. At first motion entering End of Heat, reduce the day rate by sixty and the degree rate by thirty. Set apart a prior-slow half-degree motion; when this day-degree is exhausted, the remaining day-and-degree rate from the reduction continues as fast motion. From White Dew through the Autumn Equinox: 44 days for 22 degrees. All of these use the prior-slow half-degree rate. First motion from Major Cold through Major Heat: differential motion, fast at first, with slowness increasing by one part per day. For each, obtain the motion parts by the method above. When the prior slow phase's latter day rate has been increased or diminished, and increasing slowness or increasing speed applies to the parts, take the prior fast phase's last-day motion parts as the prior slow phase's first-day motion parts. Subtract this from the prior slow mean motion parts; the remainder is the prior slow total difference. The latter fast phase's day parts serve as the latter slow phase's last-day motion parts. Subtract this from the latter slow phase's daily motion parts; the remainder is the latter total difference. The result of the subtraction is the latter separate day-difference parts. Any remainder that does not amount to a full unit is adjusted into minor parts. At transitions between slow and fast motion, decay and diminish of motion parts are not counted. Where the discrepancy is large, compute by this procedure. If the discrepancy is slight, each case follows the base method.
176
前遲：順，差行，入冬至，六十日行二十五度。 先疾，日益。 自入小寒已後，二遲二分，日損日及度各一。 大寒初日，五十五日行二十度。 自後三日益日及度各一。 立春初日平。 畢清明，六十日行二十五度。 自穀雨氣別減一氣。 立夏初日平。 畢小滿，六十日行二十二度。 自入芒種，別益一度。 夏至初日平。 畢處暑，六十日行二十五度。 自入白露已後，三日損一度。 秋分初日，六十日行二十王度。 自後一日益一，日半益一度。 寒露初日，六十日行二十五度。 自後二日損一度。 立冬一日平。 畢氣，六十日行十七度。 自大雪已後，五日益一度。 大雪初日，六十日行二十度。 自後三日益一度。
Prior slow motion: direct differential motion from the Winter Solstice—60 days for 25 degrees. Fast at first, with speed increasing daily. From entering Minor Cold onward, every two days slow by two parts; diminish the day count and degree count each by one per day. At Major Cold on the first day: 55 days for 20 degrees. From then on, every three days increase the day count and degree count each by one. At the Start of Spring on the first day, follow the mean rate. Through Clear Brightness: 60 days for 25 degrees. From the Grain Rain qi onward, subtract one degree per solar term. At the Start of Summer on the first day, follow the mean rate. Through Minor Fullness: 60 days for 22 degrees. From entering Grain in Ear, add one degree per solar term. At the Summer Solstice on the first day, follow the mean rate. Through End of Heat: 60 days for 25 degrees. From entering White Dew onward, diminish one degree every three days. At the Autumn Equinox on the first day: 60 days for 25 degrees. From then on, add one to the day count each day and one to the degree count every day and a half. At Cold Dew on the first day: 60 days for 25 degrees. From then on, diminish one degree every two days. One day into the Start of Winter, follow the mean rate. Through the end of the qi period: 60 days for 17 degrees. From Major Snow onward, add one degree every five days. At Major Snow on the first day: 60 days for 20 degrees. From then on, add one degree every three days.
177
前留：十三日。 前疾減日率一度，以其數分益此留及後遲日率。 前疾加日率者，以其數分遲日率。 旋退，西行。 入冬至安裝日，六十三日退二十一度。 自自後四日益一度。 小寒一日，六十三日退二十六度。 自入小寒已後，三日半損一度。 立春三日平。 畢啟蟄，六十二日退十七度。 自入雨水已後，二日益日及度各一。 雨水八日平。 畢氣盡，六十七日退二十一度。 自入春分已後，一日損日及度各一。 春分四日平。 畢芒種，六十三日退七十度。 自入夏至已後，六日損日及度各一。 大暑初日平。 畢氣盡，五十八日退十二度。 立秋初日平。 畢氣盡，五十七日退十一度。 自入白露已後，二日益日及度各一。 白露十二日平。 畢秋分，六十三日退七十度。 自入寒露已後，三日益日及度各一。 寒露九日平。 畢氣盡，六十六日退二十度。 自入霜降已後，三日損日及度各一。 霜降六日平。 畢氣盡，六十三日退十七度。 自立冬已後，三日益日及度各一。 立冬十一日平。 畢氣盡，六十七日退二十一度。 自入小雪已後，二日損日及度各一。 小雪八日平。 畢氣盡，六十三日退十七度。 自入大雪已後，三日益一度。
Prior station: 13 days. If the prior fast phase diminished the day-rate by one, distribute that amount in parts to increase this station and the later slow day-rate. If the prior fast phase added to the day-rate, distribute that amount in parts to diminish the slow day-rate. Then revolving retreat, moving westward. On the first day of the Winter Solstice: 63 days retreating 21 degrees. From then on, add one degree every four days. One day into Minor Cold: 63 days retreating 26 degrees. From entering Minor Cold onward, diminish one degree every three and a half days. Three days into the Start of Spring, follow the mean rate. Through Awakening of Insects: 62 days retreating 17 degrees. From entering Rain Water onward, every two days increase the day count and degree count each by one. Eight days into Rain Water, follow the mean rate. Through the end of the qi period: 67 days retreating 21 degrees. From entering the Spring Equinox onward, every day diminish the day count and degree count each by one. Four days into the Spring Equinox, follow the mean rate. Through Grain in Ear: 63 days retreating 70 degrees. From entering the Summer Solstice onward, every six days diminish the day count and degree count each by one. At Major Heat on the first day, follow the mean rate. Through the end of the qi period: 58 days retreating 12 degrees. At the Start of Autumn on the first day, follow the mean rate. Through the end of the qi period: 57 days retreating 11 degrees. From entering White Dew onward, every two days increase the day count and degree count each by one. Twelve days into White Dew, follow the mean rate. Through the Autumn Equinox: 63 days retreating 70 degrees. From entering Cold Dew onward, every three days increase the day count and degree count each by one. Nine days into Cold Dew, follow the mean rate. Through the end of the qi period: 66 days retreating 20 degrees. From entering Frost's Descent onward, every three days diminish the day count and degree count each by one. Six days into Frost's Descent, follow the mean rate. Through the end of the qi period: 63 days retreating 17 degrees. From the Start of Winter onward, every three days increase the day count and degree count each by one. Eleven days into the Start of Winter, follow the mean rate. Through the end of the qi period: 67 days retreating 21 degrees. From entering Minor Snow onward, every two days diminish the day count and degree count each by one. Eight days into Minor Snow, follow the mean rate. Through the end of the qi period: 63 days retreating 17 degrees. From entering Major Snow onward, add one degree every three days.
178
後留：冬至留十三日。 自後二日半益一日。 大寒初平，畢氣盡，留二十五日。 自入立春已後，二日半日損一。 雨水初，留十三日。 自後三日益一日。 清明初，留二十三日。 自後一日損一日。 清明十日平，畢氣盡，留十五日。 自入白露已後，二日損一日益一日。 秋分十一日，無留。 自入秋分十一日已後，一日益一日。 霜降初日，留十九日。 自後三日損一日。 立冬三日平，畢大雪，留十三日。
Later station: stationary for 13 days at the Winter Solstice. From then on, add one day every two and a half days. From Major Cold on the first day at the mean rate through the end of the qi, stationary for 25 days. From entering the Start of Spring onward, diminish one day every two and a half days. At Rain Water on the first day, stationary for 13 days. From then on, add one day every three days. At Clear Brightness on the first day, stationary for 23 days. From then on, diminish one day each day. Ten days into Clear Brightness at the mean rate through the end of the qi, stationary for 15 days. From entering White Dew onward, every two days diminish one day and increase one day. Eleven days into the Autumn Equinox: no stationary period. From eleven days after the Autumn Equinox onward, add one day each day. At Frost's Descent on the first day, stationary for 19 days. From then on, diminish one day every three days. Three days into the Start of Winter at the mean rate through Major Snow, stationary for 13 days.
179
後遲：順，差行六十日行二十五度。 先疾，日益疾二日。 前後疾加度者，此遲依數減之為定度； 前疾無加度者，此遲入秋分至立冬，減三度，入冬至減五度，後留定日朒十三日者，以所朒日數，加此遲日率也。
Later slow motion: direct differential motion—60 days for 25 degrees. Fast at first, with speed increasing by two parts per day. If the prior or later fast phase added degrees, subtract that amount in this slow phase to obtain the fixed degrees; If the prior fast phase added no degrees, subtract 3 degrees from this slow phase between the Autumn Equinox and the Start of Winter, and 5 degrees entering the Winter Solstice; if the later stationary fixed days fall short by 13, add the shortfall in days to this slow day-rate.
180
後疾：冬至初日，率二百一一日行一百三十一度。 自後一日損日及度各一。 大寒八日，一百七十二日行九十四度。 自入大寒八日已後，一日損日及度各一。 啟蟄，平。 畢氣盡，一百六十一日行八十三度。 自入雨水已後，三日益日及度各一。 穀雨三日，一百七十七日行九十九度。 自入穀雨後，三日益日及度各一。 芒種十四日平。 畢夏至，二百三十三日行一百五十度。 自入夏至已後，十日益日及度各一。 小暑五日，二百五十三日行一百七十五度。 自入小暑已後，五日益日及度各一。 大暑初日平，畢處暑，二百六十三日行一百八十五度。 自入白露已後，二日損日及度各一。 秋分一日，二百五十五日行一百七十七度。 自入秋分一日已後，一日半復日及度各一。 大雪初日，二百五十日行一百二十度。 自入秋分，三日益日及度各一。 冬至初日，復二百一十日行一百二十七度。 其入恆氣日度之率有損益者，，計日損益，並同前疾之法，以為後疾定度之率。
Later fast motion: at the Winter Solstice on the first day, rate 211 days for 131 degrees. From then on, every day diminish the day count and degree count each by one. Eight days into Major Cold: 172 days for 94 degrees. From eight days after entering Major Cold onward, every day diminish the day count and degree count each by one. At Awakening of Insects, follow the mean rate. Through the end of the qi period: 161 days for 83 degrees. From entering Rain Water onward, every three days increase the day count and degree count each by one. Three days into Grain Rain: 177 days for 99 degrees. From entering Grain Rain onward, every three days increase the day count and degree count each by one. Fourteen days into Grain in Ear, follow the mean rate. Through the Summer Solstice: 233 days for 150 degrees. From entering the Summer Solstice onward, every ten days increase the day count and degree count each by one. Five days into Minor Heat: 253 days for 175 degrees. From entering Minor Heat onward, every five days increase the day count and degree count each by one. From Major Heat on the first day at the mean rate through End of Heat: 263 days for 185 degrees. From entering White Dew onward, every two days diminish the day count and degree count each by one. One day into the Autumn Equinox: 255 days for 177 degrees. From one day after the Autumn Equinox onward, every day and a half restore the day count and degree count each by one. At Major Snow on the first day: 250 days for 120 degrees. From the Autumn Equinox onward, every three days increase the day count and degree count each by one. At the Winter Solstice on the first day, again: 210 days for 127 degrees. Where the fixed-qi day and degree rates require adjustment, tally the daily diminution and addition by the same method as the prior fast phase, and use the result as the later-fast fixed-degree rate.
181
求變日率術：其前遲定日朒六十，及退行定日朒六十三者，皆以所朒日數加此疾定日率，前遲定日盈六十三，後留定日盈十三者，皆以所盈日數減此疾定日率。 加減訖，即變日率。
Procedure for finding the transformed day-rate: if the prior slow fixed days fall short by 60, or the retrograde fixed days fall short by 63, add the shortfall in days to this fast fixed day-rate; if the prior slow fixed days exceed by 63, or the later stationary fixed days exceed by 13, subtract the surplus in days from this fast fixed day-rate. Once addition and subtraction are complete, that is the transformed day-rate.
182
求變度率術：其前遲定度朒二十五，退行定度盈十七，後遲入秋分至冬至減度者，皆以所盈朒度數，加此疾定度率。 前遲定度盈二十五，及退行定度朒十七者，皆以所盈朒度數，減此疾定度率。 加減訖，即變度率。
Procedure for finding the transformed degree-rate: if the prior slow fixed degrees fall short by 25, the retrograde fixed degrees exceed by 17, or the later slow phase diminishes degrees between the Autumn Equinox and the Winter Solstice, add the surplus or shortfall in degrees to this fast fixed degree-rate. If the prior slow fixed degrees exceed by 25, or the retrograde fixed degrees fall short by 17, subtract the surplus or shortfall in degrees from this fast fixed degree-rate. Once addition and subtraction are complete, that is the transformed degree-rate.
183
初行，入春分，畢穀雨，差行。 先遲，日益疾一分。 初行，入立夏，畢夏至，日行半度。 六十六日行二十二度。 小暑，五十日行二十五度。 立秋畢氣盡，二十日行十度，減率續行，並同前疾初遲法。 損益依前，求其行分。 各盡度而夕伏。
First motion from the Spring Equinox through Grain Rain: differential motion. Slow at first, with speed increasing by one part per day. First motion from the Start of Summer through the Summer Solstice: half a degree per day. 66 days for 22 degrees. At Minor Heat: 50 days for 25 degrees. From the Start of Autumn through the end of the qi: 20 days for 10 degrees; diminish the rate and continue motion by the same initial-slow method as the prior fast phase. Apply diminish-increase as before to obtain the motion parts. Each leg completes its degrees, then evening invisibility.
184
鎮星：初順，差行，八十三日行七度二百九十分。 先疾，日益遲半分。 前留，三十七日。 旋退，西行，差行，五十一日退三十分。 先遲，日益疾少半。
Saturn: initial direct motion, differential 83 days, travels 7°290 parts. Fast at first, with slowness increasing by half a part per day. Prior station: 37 days. Then retrograde, moving westward: differential 51 days, retreats 30 parts. Slow at first, with speed increasing by a small half part per day.
185
太白：夕見，順，入冬至畢立夏，入立秋畢大雪。 一百七十二日行二百六度。 自入小滿後，十日益一度，為定疾。 初入白露，畢春分，差行。 疾，日益遲二分。 自餘平行。 夏至畢小暑，一百七十二日行二百九度。 自入大暑已後，五日損一度，畢氣盡。 平行：入冬至初日及大暑，各畢氣盡。 一十三日行一十三度。 自入冬至後，十日損一，畢已後立春，入立秋，日益一，畢秋分。 啟蟄畢芒種，七日行七度。 自入夏至後，五日益一，畢于小雪。 寒露初日，三十三日行二十二度。 自後六日損一，畢于小雪。 順遲：差行，三十二日行三十度。 先疾，日益遲八分。 前疾加度過二百六度者，准數損此度。 夕留，七日。 夕退，西行，一十日退五度。 日盡而夕伏。 晨初退，西行，十日退五度。 日退半度。 晨留，七日。 順遲，差行，冬至畢立夏，大雪畢氣盡。 三十二日，先遲，日益疾八分。 自入小滿已後，率十日損一度，畢芒種。 平行，冬至畢氣盡，立夏畢氣盡。 一十三日行一十三度。 日行一度。 自入小寒已後，六日益日及度各一，畢於啟蟄。 入小滿後，七日損日度各一，畢立秋。 雨水初日，二十三日行二十三度。 自後六日損日及度各一，畢于穀雨。 處暑畢寒露，無此平行。 自入霜降後，五日益日及度各一，畢大雪。 前遲行損度不滿三十度者，此疾依數益之。 疾行，一百七十二日行二百六度。 處暑畢寒露，差行，先遲，日益疾一分。 餘平行，行日盡而晨伏。
Venus: evening appearance, direct motion from the Winter Solstice through the Start of Summer, and from the Start of Autumn through Major Snow. 172 days for 206 degrees. From entering Minor Fullness onward, add one degree every ten days to obtain the fixed fast rate. First entering White Dew through the Spring Equinox: differential motion. Fast motion, with slowness increasing by two parts per day. The remainder follows uniform motion. From the Summer Solstice through Minor Heat: 172 days for 209 degrees. From entering Major Heat onward, diminish one degree every five days through the end of the qi. Uniform motion: from the Winter Solstice on the first day and from Major Heat, each through the end of the qi. 13 days for 13 degrees. From entering the Winter Solstice onward, diminish one every ten days through the Start of Spring; from entering the Start of Autumn, add one each day through the Autumn Equinox. From Awakening of Insects through Grain in Ear: 7 days for 7 degrees. From entering the Summer Solstice onward, add one every five days through Minor Snow. At Cold Dew on the first day: 33 days for 22 degrees. From then on, diminish one every six days through Minor Snow. Direct slow motion: differential 32 days for 30 degrees. Fast at first, with slowness increasing by eight parts per day. If the prior fast phase added degrees beyond 206, subtract that excess from this degree count. Evening station: 7 days. Evening retrograde, moving westward: 10 days retreating 5 degrees. When the days are exhausted, evening invisibility. Morning initial retrograde, moving westward: 10 days retreating 5 degrees. Half a degree of retrograde motion per day. Morning station: 7 days. Direct slow motion, differential from the Winter Solstice through the Start of Summer, and from Major Snow through the end of the qi. 32 days: slow at first, with speed increasing by eight parts per day. From entering Minor Fullness onward, diminish one degree every ten days through Grain in Ear. Uniform motion: from the Winter Solstice through the end of the qi, and from the Start of Summer through the end of the qi. 13 days for 13 degrees. One degree per day. From entering Minor Cold onward, every six days increase the day count and degree count each by one through Awakening of Insects. From entering Minor Fullness onward, every seven days diminish the day count and degree count each by one through the Start of Autumn. At Rain Water on the first day: 23 days for 23 degrees. From then on, every six days diminish the day count and degree count each by one through Grain Rain. From End of Heat through Cold Dew: no such uniform motion. From entering Frost's Descent onward, every five days increase the day count and degree count each by one through Major Snow. If the prior slow phase diminished degrees by less than 30, add that amount in this fast phase. Fast motion: 172 days for 206 degrees. From End of Heat through Cold Dew: differential motion, slow at first, with speed increasing by one part per day. The remainder follows uniform motion until the motion days are exhausted, then morning invisibility.
186
辰星：夕見，順疾，一十二日行二十一度六分。 日行一度五百三分。 大暑畢處暑，一十二日行一十七度二分。 日行一度二百八十分。 平行，七日行七度。 自入大暑後，二日損日及度各一。 入立秋，無此平行。 順遲行，六日行二度四分。 日行二百二十四分，前疾行十一度者，無此遲行。 日盡而夕伏。 夕留，五日。 晨見，留五日。 順遲行，六日行二度四分。 日行二百二十四分。 自入大寒，畢於啟蟄，無此遲行。 平行，七日行七度。 日行一度。 大寒已後，二日損日及度各一。 入立春，無此平行。 順疾行，一十二日行二十一度六分。 日行一度五百三分。 前無遲行者，一十三日行十七度十分。 日行一度二百八十分。 各日盡而晨伏。
Mercury: evening appearance, direct fast motion, 12 days for 21°6 parts. One degree 503 parts per day. From Major Heat through End of Heat: 12 days for 17°2 parts. One degree 280 parts per day. Uniform motion: 7 days for 7 degrees. From entering Major Heat onward, every two days diminish the day count and degree count each by one. From entering the Start of Autumn: no such uniform motion. Direct slow motion: 6 days for 2°4 parts. 224 parts per day; if the prior fast phase traveled eleven degrees, there is no such slow motion. When the days are exhausted, evening invisibility. Evening station: 5 days. Morning appearance: stationary for 5 days. Direct slow motion: 6 days for 2°4 parts. 224 parts per day. From Major Cold through Awakening of Insects: no such slow motion. Uniform motion: 7 days for 7 degrees. One degree per day. From Major Cold onward, every two days diminish the day count and degree count each by one. From entering the Start of Spring: no such uniform motion. Direct fast motion: 12 days for 21°6 parts. One degree 503 parts per day. If there was no prior slow phase: 13 days for 17°10 parts. One degree 280 parts per day. Each leg completes its days, then morning invisibility.
187
凡五星終日分奇，皆於伏分消遁，故於行星更不別見。
Generally, the five planets' leftover fractional day-parts are all absorbed into invisibility; they therefore need not appear separately in the planetary tables.
188
武太后稱制，詔曰：「頃者所司造曆，以臘月為閏。 稽考史籍，便紊舊章，遂令去歲之中，晦仍月見。 重更尋討，果差一日。 履端舉正，屬在於茲。 宜改曆於惟新，革前非於既往。 可以今月為閏十月，來月為正月。」 是歲得甲子合朔冬至。 於是改元聖曆，以建子月為正，建醜為臘，建寅為一月。 命太史瞿曇羅造新曆。 至三年，復用夏時，《光宅曆》亦不行用。 中宗反正，太史丞南宮說奏：「《麟德曆》加時浸疏。 又上元甲子之首，五星有入氣加時，非合璧連珠之正也。」 乃詔說與司曆徐保乂、南宮季友，更治《乙巳元曆》。 至景龍中，曆成，詔令施用。 俄而睿宗即位，《景龍曆》寢廢不行。 《麟德歷經》，今略載其法大端。
While Empress Wu held regency, an edict said: "Recently the office charged with making the calendar took the twelfth month as intercalary. Examination of the historical records showed that this violated longstanding precedent, with the result that within the previous year the moon was still visible on the last day of the month. On further investigation, it was indeed off by one day. Establishing the year's beginning and setting the calendar right belong to this matter. The calendar should be revised afresh, and past errors corrected. This month may be taken as intercalary tenth month, and the coming month as first month." That year obtained a jiazi-day conjunction at new moon on the Winter Solstice. Thereupon the era was changed to Sacred Calendar, with the zi-established month as the first month, the chou-established month as the twelfth month, and the yin-established month as the first month of the year. The Grand Astrologer Gautama Siddhartha was ordered to compose a new calendar. By the third year, the Xia seasonal system was restored, and the Guangzhai Calendar was likewise not used. When Emperor Zhongzong restored the dynasty, Vice Director of the Grand Astrologer's Office Nangong Shuo memorialized: "The Linde Calendar's added times have grown increasingly loose. Moreover, at the head of the upper origin in the jiazi year, when the five stars entered qi and added times, this did not match the correct conjunction of jade disk and linked pearls." Thereupon an edict ordered Shuo, together with Calendar Commissioner Xu Baoyi and Nangong Jiyou, to further revise the Yisi Origin Calendar. By the Jinglong era the calendar was completed, and an edict ordered its adoption. Soon afterward Emperor Ruizong acceded to the throne, and the Jinglong Calendar was set aside and not used. The Linde Calendar Canon is here briefly recorded in the main outlines of its method.
189
母法一百。 兩大衍之數為母法。
Mother divisor: 100. Twice the Dayan number serves as the mother divisor.
190
旬週六十。 六甲之終數為旬周。
Ten-day cycle: 60. The terminal number of the six jia cycles serves as the ten-day cycle.
191
辰法八刻； 分，三十三少半。 以十二辰數除一百刻，得辰法。
Chronogram divisor: 8 double-hours; parts, 33 and a little less than half. Divide 100 marks by the twelve chronogram count to obtain the chronogram divisor.
192
期週三百六十五日； 餘，二十四； 奇，四十八。 一期之總日及餘奇數為期周。
Period cycle: 365 days; remainder, 24; odd, 48. The total days and remainder-and-odd numbers of one period serve as the period cycle.
193
氣法十五日； 餘，二十一； 奇，八十五少半。 以二十四氣分期周，得氣法。
Qi divisor: 15 days; remainder, 21; odd, 85 and a little less than half. Divide the period cycle by the twenty-four qi to obtain the qi divisor.
194
候法五日； 餘，七； 奇，二十八； 小分，四。 以七十二候分期周，得候法。
Pentad divisor: 5 days; remainder, 7; odd, 28; minor parts, 4. Divide the period cycle by the seventy-two pentads to obtain the pentad divisor.
195
月法二十九日； 餘，十三； 奇。 為月法。
Month divisor: 29 days; remainder, 13; Odd parts. This serves as the month divisor.
196
日法日舒月遠乃舒一合朔之及餘奇為日法。
The day divisor is the interval to one new-moon conjunction and its remainder and odd parts, obtained from the sun's elongation and the moon's recession.
197
望法十四日； 餘，七十六； 奇，五十三。 因為陰後限。 二分月法得望法。 亦是月行陰曆，後與朔望會交限。
Full-moon divisor: 14 days; remainder, 76; odd, 53. This also serves as the yin posterior limit. Halve the month divisor to obtain the full-moon divisor. It is also the limit after the moon, traveling the yin calendar, meets new and full moon crossings.
198
弦法七日； 餘，三十八； 奇，二十六半。 四分月法，得弦法。
Quarter-moon divisor: 7 days; remainder, 38; odd, 26 and a half. Take one quarter of the month divisor to obtain the quarter-moon divisor.
199
閏差十日； 餘，八十七； 奇，七十六。 月法去期周，餘得閏差。
Intercalation difference: 10 days; remainder, 87; odd, 76. Cast month divisors out of the period cycle; the remainder is the intercalation difference.
200
沒數九十一； 餘，三十一； 奇，十二。 四分期周，餘四分之得沒數。
Submergence number: 91; remainder, 31; odd, 12. Quarter the period cycle; the quartered remainder gives the submergence number.
201
沒法一； 餘，三十一； 奇，十二。 以旬周去期周，餘四分之，得沒法。
Submergence method: 1; remainder, 31; odd, 12. Subtract the ten-day cycle from the period cycle and quarter the remainder to obtain the submergence method.
202
月周法二十七日； 餘，五十五； 奇，四十五； 小分，五十九。 月行遲疾一周之數，為月周法。
Lunar circuit method: 27 days; remainder, 55; odd, 45; minor parts, 59. The count for the moon's fast-and-slow circuit of one cycle serves as the lunar circuit method.
203
月差法一日； 餘，九十七； 奇，六十； 小分，四十一。 以月周減月法，餘得月差。
Lunar difference method: 1 day; remainder, 97; odd, 60; minor parts, 41. Subtract the lunar circuit method from the month divisor; the remainder is the lunar difference.
204
周天法三百六十五度； 餘，二十五； 奇，七十一； 小分，十三。 二十八宿總度數、相距總數及餘奇，為周天法。
Circuit-of-heaven method: 365 degrees; remainder, 25; odd, 71; minor parts, 13. The total degrees of the twenty-eight lodges, their total interstitial distances, and the remainder and odd parts serve as the circuit-of-heaven method.
205
交周法二十七日； 餘，二十一； 奇，二十二； 小分，十六七分。 日行陰陽一周交於是日之數，為交周法。
Nodal-cycle method: 27 days; remainder, 21; odd, 22; minor parts, 16 and 7 parts. The day count for the sun's circuit through yin and yang to one nodal crossing serves as the nodal-cycle method.
206
交差法二日； 餘，三十一； 奇，八十三； 小分，八十三分。 以交周法減月法，得交差法。
Crossing-difference method: 2 days; remainder, 31; odd, 83; minor parts, 83 parts. Subtract the nodal-cycle method from the month divisor to obtain the crossing-difference method.
207
交中法十三日； 餘，六十； 奇，六十一； 小分，三分半。 二分交周，得交中法。
Mid-crossing method: 13 days; remainder, 60; odd, 61; minor parts, 3 and a half. Halve the nodal cycle to obtain the mid-crossing method.
208
陽前限十二日； 餘，四十四； 奇，六十九； 小分，十六七分。 月行陽曆，與朔望會之限。
Yang anterior limit: 12 days; remainder, 44; odd, 69; minor parts, 16 and 7 parts. The limit within which the moon, traveling the yang calendar, meets new and full moons.
209
陽後限一日； 餘，十五； 奇，九十一； 小分，九十一六分半。 月行陽曆，後與朔望會之限。
Yang posterior limit: 1 day; remainder, 15; odd, 91; minor parts, 91, 6 parts, and a half. The limit after the moon, traveling the yang calendar, meets new and full moons.
210
陰前限二十六日； 餘，五； 奇，三十； 小分，二十五半分。 月行陰曆，先與朔望會之限。
Yin anterior limit: 26 days; remainder, 5; odd, 30; minor parts, 25 and a half part. The limit before the moon, traveling the yin calendar, meets new and full moons.
211
木歲星合法三百九十八日； 餘，八十六； 奇，七十九； 小分，八十。
Jupiter accord method: 398 days; remainder, 86; odd, 79; minor parts, 80.
212
火熒惑合法七百七十九日； 餘，九十； 奇； 五十五； 小分，四十五。
Mars (Fire-Brilliance) synodic constant: 779 days; remainder, 90; odd parts; 55; minor parts, 45.
213
土鎮星合法三百七十八日； 餘，八； 奇，四； 小分，八十。
Saturn (Queller-Star) synodic constant: 378 days; remainder, 8; odd parts, 4; minor parts, 80.
214
金太白合法五百八十三日； 餘，九十一； 奇，七十七； 小分，七十。
Venus (Great White) synodic constant: 583 days; remainder, 91; odd parts, 77; minor parts, 70.
215
水辰星合法一百一十五日； 餘，八十七； 奇，九十五； 小分，七十。
Mercury (Chronogram-Star) synodic constant: 115 days; remainder, 87; odd parts, 95; minor parts, 70.
216
太極上元，歲次乙巳，十一月甲子朔旦冬至之日，黃鐘之始，夜半之時，鬥衡之末建於子中，日月如合璧，五星若連珠，俱起于星紀牽牛之初蹤。 今大唐，復歲次於乙巳，積四十一萬四千三百六十算外。 上驗往古，年減一算。 下求將來，年加一算。 《乙巳元曆》法積數，大約如此。 其算經不錄。
At the Supreme Ultimate Superior Origin, in the yisi year, on the eleventh month's jiazi day when new moon and winter solstice coincided—the start of the Yellow Bell, at midnight, with the Dipper's balance-beam tail set at the center of zi—sun and moon lay like a joined jade disk and the five planets like strung pearls, all rising from the first tracks of Xingji and Qianniu. Now Great Tang again places the year in the yisi cycle, with an accumulated count of 414,360 beyond the origin. To verify backward into antiquity, subtract one count per year. To project forward into the future, add one count per year. The accumulated numbers of the Yisi Origin Calendar method are broadly as described above. The computational classic does not record them.