1
志第十四曆三
Treatise 14: Calendar 3
2
開元《大衍歷經》
The Kaiyuan Canon of the Dayan Calendar
3
演紀上元閼逢困敦之歲，距今開元十二年甲子歲，歲積九千六百六十六萬一千七百四十算。
In computing the evolving era, the high origin falls in the year Yān-féng Kùn-dūn; reckoned back from Kaiyuan 12 (jiazi), the accumulated count is 96,661,740.
4
大衍步中朔第一
Dayan Calendar: Method for Central Qi and New Moons, I
5
大衍通法：三千四十。
Dayan universal divisor: 3,040.
6
策實：一百一十一萬三百四十三。
Year-fraction constant: 1,110,343.
7
揲法：八萬九千七百七十三。
New-moon constant: 89,773.
8
滅法：九萬一千三百。
Reduction constant: 91,300.
9
策餘：一萬五千九百四十三。
Stalk remainder: 15,943.
10
用差：一萬七千一百二十四。
Applied difference: 17,124.
11
掛限：八萬七千一十八。
Intercalation threshold: 87,018.
12
三元之策：一十五； 餘，六百六十四； 秒，七。
Three-origin interval: 15 stalks; remainder 664; 7 parts.
13
四象之策：二十九； 餘，一千六百一十三。
Four-image interval: 29 stalks; remainder 1,613.
14
中盈分：一千三百二十八； 秒，十四。
Central-qi surplus fraction: 1,328; 14 parts.
15
爻數：六十。
Hexagram line count: 60.
16
象統：二十四。
Image cycle: 24.
17
推天正中氣以策實乘入元距所求積算，命曰中積分。 盈大衍通法得一，為積日。 不盈者，為小餘。 爻數去積日，不盡日為大餘。 數從甲子起算外，即所求年天正中氣冬至日及小餘也。
To determine the celestial central qi, multiply the era count from the high origin to the year sought by the year-fraction constant; this yields the central accumulated parts. Divide by the Dayan universal divisor; the quotient is accumulated days. The remainder is the minor remainder. Cast out full cycles of sixty from the accumulated days; the days left are the major remainder. Count outward from jiazi; the result is the winter solstice day of the celestial central qi for the year sought, with its minor remainder.
18
求次氣因天正中氣大小餘，以三元之策及餘秒加之。 其秒盈象統，從小餘。 小餘滿大衍通法，從大餘。 大餘滿爻數，去之。 命如前，即次氣恆日及餘秒。 凡率相因加者，下有餘秒，皆以類相從。 而滿其法，則迭進之，用加上位。 日盈爻數，去之也。
To find the next qi, add the three-origin interval and its remainder and parts to the major and minor remainders of the celestial central qi. When the fractional parts fill the image cycle, carry into the minor remainder. When the minor remainder fills the Dayan universal divisor, carry into the major remainder. When the major remainder fills sixty, cast it out. Fix the era as before; the result is the constant day of the next qi with remainder and parts. Whenever linked rates are added and fractional remainders remain below, combine them by kind. When a divisor is filled, carry upward step by step and add to the higher place. Cast out full cycles of sixty when days overflow.
19
推天正合朔以揲法去中積分。 其所不盡，曰歸餘之卦。 以減積積分，餘為朔積分。 乃如大衍通法而一，為日。 不盡，為小餘。 日盈爻數，去之。 不盈者，為大餘。 命以甲子算外，即所求年天正合朔經日及小餘也。
To determine the winter new moon, divide the central accumulated parts by the new-moon constant. The remainder is the intercalation pendulum. Subtract that amount from the accumulated parts; the remainder is the new-moon accumulated parts. Divide by the Dayan universal divisor; the quotient is days. The remainder is the minor remainder. Cast out full cycles of sixty when days overflow. What remains is the major remainder. Count outward from jiazi; the result is the canonical winter new-moon day for the year sought, with its minor remainder.
20
求次朔及弦望因天正經朔大小餘，以四象之策及餘加之。 數除如法，即次朔經日及餘也。 又自經朔加一象之日七及餘一千一百六十三少，得上弦。 倍之，得望。 參之，得下弦。 四之，是謂一揲，復得後月之朔。 凡四分一為少，二為半，三為太，四為全。 加滿其前數，去之，從上位。 綜中朔盈虛分，累益歸餘之卦，每其月閏衰。 凡歸餘之卦五萬六千七百六十以上，其歲有閏。 因考其閏衰，滿卦限以上，其月及合置閏。 或有進退，皆以定朔無中氣裁焉。
To find successive new moons and the quarter and full moons, add the four-image interval and remainder to the major and minor remainders of the winter canonical new moon. Reduce the numbers by the usual rules; the result is the canonical day of the next new moon with remainder. From the canonical new moon, add one four-image interval—seven days and remainder 1,163 lesser—to obtain the first quarter. Double it to obtain full moon. Triple it to obtain last quarter. Quadruple it—one full sorting—to obtain the next month’s new moon. In quartering remainders, one part is lesser, two are half, three are greater, and four are full. When addition fills the prior place, cast it out and carry upward. Combine the central surplus and new-moon void fractions, steadily increasing the intercalation pendulum; each month the intercalation fraction diminishes. Whenever the intercalation pendulum reaches 56,760 or more, the year receives an intercalary month. Track the intercalation wane; when it exceeds the intercalation threshold, assign an intercalary month. Whether advancing or retreating, the rule is the fixed new moon in a month without central qi.
21
推沒日置有沒之氣恆小餘，以象統乘之，內秒分，參而伍之，以減策實。 餘滿策餘，為日。 不滿，為沒餘。 命起也。 凡恆氣小餘，不滿大衍通法，如中盈分半法已下，為有沒之氣。
To determine the extinction day, take the constant minor remainder of the qi subject to extinction, multiply by the image cycle, include seconds, combine and quintuple, and subtract from the year-fraction constant. When the remainder fills the stalk remainder, the quotient is days. What does not fill is the extinction remainder. Fix the count from the starting day. Whenever a constant qi’s minor remainder is less than the Dayan universal divisor and no more than half the central surplus fraction, that qi is subject to extinction.
22
推滅日以有滅之朔經小餘，減大衍通法。 餘，倍參伍乘之，用減滅法。 餘，滿朔虛分，為日。 不滿，為滅餘。 命起經朔初日算外，即合朔後滅日也。 凡經朔小餘不滿朔虛分者，為有滅之朔。
To determine the lunar extinguishing day, take the canonical minor remainder of the new moon subject to extinguishing and subtract the Dayan universal divisor. Double and quintuple the remainder, then subtract from the reduction constant. When the remainder fills the new-moon void fraction, the quotient is days. What does not fill is the extinguishing remainder. Count outward from the first day of the canonical new moon; the result is the lunar extinguishing day after conjunction. Whenever a canonical new moon’s minor remainder is less than the new-moon void fraction, that new moon is subject to extinguishing.
23
大衍步發斂術第二
Dayan Calendar: Method for Issuing and Gathering In, II
24
天中之策：五； 餘，二百二十二； 秒，三十一。 秒法：七十二。
Heaven-central interval: five stalks; remainder 222; 31 parts. Second divisor: 72.
25
地中之策：十八； 餘，一百六十五； 秒，八十六。 秒法：一百二十。
Earth-central interval: eighteen stalks; remainder 165; 86 parts. Second divisor: 120.
26
貞晦之策：三； 餘，一百三十二； 秒，一百三。 秒法：如前。
Zhen-hui interval: three stalks; remainder 132; 103 parts. Second divisor: as before.
27
辰法：七百六十。
Chronogram divisor: 760.
28
刻法：三百四。
Clepsydra divisor: 304.
29
推七十二候各因中節大小餘命之，即初候日也。 以天中之策及餘秒加之，數除如法，即次候日。 又加，得末候日。 凡發斂，皆以恆氣。
To determine the seventy-two phenological hou, for each start from the major and minor remainders of the central qi’s mid-node; the result is the first hou day. Add the heaven-central interval with remainder and parts, reduce by the usual rules, and obtain the next hou day. Add again to obtain the last hou day. All issuing-and-gathering-in calculations use constant qi.
30
推六十卦各因中氣大小餘命之，公卦用事日也。 以地之策及餘秒累加之，數除如法，各次卦用事日。 若以貞晦之策加諸候卦，得十二節之初外卦用事日。
To determine the sixty hexagrams, for each start from the major and minor remainders of the central qi; the result is the day when the duke hexagram governs. Keep adding the earth-central interval with remainder and parts, reduce by the usual rules, and obtain each successive hexagram’s governing day. Alternatively, add the zhen-hui interval to the marquis hexagrams to obtain when the outer hexagram of the twelve nodes governs at their opening.
31
推五行用事各因四立大小餘命之，即春木、夏火、秋金、冬水首用事日也。 以貞晦之策及餘秒，減四季中氣大小餘，即其月土始用事日。 凡抽加減而有秒者，母若不齊，當令母互乘子。 乃加減之。 母相乘為法。
To determine when the Five Phases govern, for each start from the major and minor remainders of the four establishment days; the result is the first governing day of spring Wood, summer Fire, autumn Metal, and winter Water. Subtract the zhen-hui interval and its remainder and parts from the last seasonal month’s central qi; the result is the day when Earth the King first governs. Whenever addition or subtraction involves seconds and the denominators differ, cross-multiply numerators by the opposite denominators. Then add or subtract. The product of denominators is the common divisor.
32
推發斂去朔各置其月閏衰，以大衍通法約之，為日。 不盡為餘，即其月中氣去經朔日算及餘秒也。 求卦候者，各以天地之策及餘秒累加減之，中氣之前以減，中氣之後以加。 得去經朔日算及餘秒。
To determine issuing and gathering in from the new moon, for each month divide the intercalation wane by the Dayan universal divisor to obtain days. The remainder gives the interval from central qi to canonical new moon in that month, with remainder and parts. To determine hexagrams and phenological hou, cumulatively add or subtract the heaven- and earth-central intervals with remainder and parts—subtract before central qi, add after. The result is the interval from canonical new moon with remainder and parts.
33
推發斂加時各置其小餘，以六爻乘之，如辰法而一，為半辰之數。 不盡者，五之，三刻法除之，為刻。 又不盡者，三約為分。 此分滿刻法為刻，若令滿象積為刻者，即置不盡之數，十之，十九而一，為分。 命起子半算外，各其加時所在辰刻及分也。
To fix the time of issuing and gathering in, set each minor remainder, multiply by six lines, and divide by the chronogram divisor to obtain the half-chronogram count. The remainder, multiplied by five and divided by three times the clepsydra divisor, gives clepsydra notches. Any remainder still left, divided by three, gives fractional parts. When the parts fill the clepsydra divisor they become notches; if the image accumulation is to fill clepsydra marks directly, take the remainder, multiply by ten, and divide by nineteen for the parts. Count outward from the midpoint of zi; the result is the chronogram, clepsydra notch, and fractional parts for each added time.
34
大衍步日躔術第三
Dayan Calendar: Method for Solar Motion, III
35
乾實：一百一十一萬三百七十九太。 周天度：三百六十五。 虛分七百七十九太。
Circuit-of-heaven constant: 1,110,379 and greater. Degrees in the circuit of heaven: 365. Void fraction: 779 and greater.
36
歲差：三十六太。
Precession per year: 36 and greater.
37
求每日先後定數以所入氣並後氣盈縮分，倍六爻乘之，綜兩氣辰數除，入之，為末率。 又列二氣盈縮分，皆倍六爻乘之，各如辰數而一，以少減多，餘為氣差。 加減末率，至後以差加，分後以差減。 為初率。 倍氣差，亦六爻乘之，復綜兩氣辰數以除之，為日差。 半之，以加減初末，各為定率。 以日差累加減氣初定率，至後以差減，分後以差加。 為每日盈縮分。 乃馴積之，隨所入氣日加減氣下先後數，各其日定。 冬至後為陽復，在盈加之，在縮減之。 夏至後為陰復，在縮加之，在盈減之。 距四正前一氣，在陰陽變革之際，不可相並，皆因前末為初率。 以氣差至前加之，分前減之，為末率。 餘依前率，各得所求。 其朓朒亦放此求之，各得每日定數。 其分不滿全數，母又每氣不同，當退法除之，用百為母，半已上從一，已下棄之。 下求軌漏，餘分不滿准此。
To determine each day’s lead-and-lag fixed tally, combine the expansion–contraction fractions of the current and next qi, double and multiply by six lines, divide by the sum of their chronogram counts, and obtain the terminal rate. Set out both qi’s expansion–contraction fractions, each doubled and multiplied by six lines and divided by its chronogram count; subtract the smaller from the larger; the remainder is the qi difference. Adjust the terminal rate: after a solstice add the difference; after an equinox subtract it. This yields the initial rate. Double the qi difference, multiply by six lines, divide again by the combined chronogram counts of both qi, and obtain the day difference. Halve the day difference and add or subtract from the initial and terminal rates to obtain the fixed rates. Using the day difference, adjust the qi’s initial fixed rate step by step: subtract after a solstice and add after an equinox. This yields the daily expansion–contraction fraction. Accumulate stepwise: for each day within the entered qi, add or subtract the lead and lag counts listed under that qi to fix each day’s tally. After the Winter Solstice is yang recovery: add in expansion and subtract in contraction. After the Summer Solstice is yin recovery: add in contraction and subtract in expansion. For the qi immediately before each cardinal solstice or equinox, at the yin–yang transition the rates cannot be merged; use the prior qi’s terminal rate as the initial rate. Before a solstice add the qi difference to obtain the terminal rate; before an equinox subtract it. For the rest, follow the rates above; each quantity sought is thereby obtained. Waxing and waning accumulations are determined by the same procedure, yielding each day’s fixed tally. When fractional parts do not make a full unit and each qi has a different denominator, reduce by retreating the divisor; use one hundred as the denominator; at one-half or above, round up to one; below that, discard. In lower steps for orbital motion and clepsydra marks, apply the same rule to remainders that do not fill a unit.
38
推二十四氣定日冬夏至皆在天地之中，無有盈縮。 餘各以氣下先後數，先減後加恆氣小餘。 滿若不足，進退其日。 命從甲子算外，各其定日及餘秒也。 凡推日月行度及軌漏交蝕，並依定氣。 若注曆即依恆氣也。
To fix the days of the twenty-four qi: at the Winter and Summer Solstices the sun reaches the cosmological mean, with neither expansion nor contraction. For the remaining qi, first subtract then add the lead and lag counts under each qi to the constant qi’s minor remainder. Carry or borrow days as needed. Count outward from jiazi; the result is each qi’s fixed day with remainder and parts. All calculations of solar and lunar longitude, orbital motion, clepsydra marks, and eclipses use fixed qi. Published calendars, however, follow constant qi.
39
推平朔四象以定氣相距置朔弦望經日大小餘，以所入定氣大小餘及秒分減之，各其所入定氣日算及餘秒也。 若大餘少不足減者，加爻數，然後減之。 其弦望小餘有少半太，當以爻乘之，乃以氣秒分減，退一加象統。 小餘不足減，退日算一，加大衍通法也。
To find mean new moons and the four phases: set the canonical new-, quarter-, and full-moon major and minor remainders at the fixed-qi interval, subtract the entered fixed qi’s major and minor remainders with seconds and parts, and obtain each entered fixed qi’s day count with remainder and parts. If the major remainder is too small to subtract, add sixty lines, then subtract. When a quarter or full moon’s minor remainder includes lesser, half, or greater fractions, multiply by the line count, subtract the qi’s seconds and parts, retreat one place, and add the image cycle. If the minor remainder is too small to subtract, borrow one day and add the Dayan universal divisor.
40
求朔弦望經日入朓朒各置其所入定氣日算及餘秒。 減日算一，各以日差乘而半之，以加減其氣初定率，前少，加之； 前多，減之。 以乘其所入定氣日算及餘秒。 凡除者，先以母通全，內子，乃相乘，母相乘除之也。 若忽微之數煩多而不甚相校者，過半收為全，不盈半法，棄之。 所得以損益朓朒積，各為其日所入朓朒定數。 若非朔望有交者，以十二乘所入日算。 三其小餘，辰法除而從之。 以乘損益率，如定氣辰數而一。 所得以損益朓朒積，各為定數也。
To find each canonical syzygy’s waxing–waning entry, set the entered fixed qi’s day count with remainder and parts. Subtract one from the day count, multiply by the day difference and halve, then adjust the qi’s initial fixed rate: if the prior fraction is smaller, add; if larger, subtract. Multiply by the entered fixed qi’s day count, remainder, and parts. In every division, first unify whole units through the denominator, include the numerator, then multiply and divide by the denominators. When minute quantities are too numerous to compare closely, round up at more than half a unit and discard what falls below half the divisor. Use the result to adjust the waxing–waning accumulation, obtaining each day’s entered waxing–waning fixed tally. When new or full moon is not an eclipse syzygy, multiply the entered day count by twelve. Triple the minor remainder, divide by the chronogram divisor, and add the quotient. Multiply by the rate of increase or decrease, and divide by the fixed-qi chronogram count. Use the result to adjust the waxing–waning accumulation, yielding a fixed value for each entry.
41
赤道宿度
Equatorial Lodge Degrees
42
右北方七宿九十八度虛分七百七十九太
Northern quarter (seven lodges): 98°, void fraction 779 and greater.
43
右西方七宿八十一度
Western quarter (seven lodges): 81°.
44
右東方七宿七十五度
Eastern quarter (seven lodges): 75°.
45
前皆赤道度。 其畢、觜、參及輿鬼四宿度數，與古不同，今並依天以儀測定，用為常數。 紘帶天中，儀極攸憑，以格黃道也。 推黃道，准冬至歲差所在，每距冬至前後各五度為限。 初數十二，每限減一，盡九限，數終於四。 殷二立之際，一度少強，依平。 乃距春分前、秋分後，初限起四，每限增一，盡九限，終於十二，而黃道交復。 計春分後、秋分前，亦五度為限，初數十二，盡九限，數終於四。 殷二立之際，一度少強，依平。 乃距夏至前後，初限起四，盡九限，終於十二。 皆累裁之，以數乘限度，百二十而一，得度。 不滿者，十二除為分。 若以十除，則大分。 十二為母，命乙太半少及強弱。 命曰黃赤道差數。 二至前後，各九限，以差減赤道度，為黃道度。 二分前後，各九限，以差加赤道度，為黃道度。 若從黃道度反推赤道，二至前後各加之，二分前後須減之。
These are all equatorial longitudes. The arc-degrees assigned to Net, Turtle Beak, Three Stars, and Ghost Cart differ from the ancient reckoning; they are now fixed by armillary measurement against the sky and adopted as standard constants. The celestial girdle runs through heaven’s center; the polar axis of the instrument is the reference for laying out the ecliptic. To determine the ecliptic: find where the winter-solstice precession applies, taking five degrees on either side of the solstice as one band. Begin at twelve and subtract one for each successive band; after nine bands the tally reaches four. At the two Establishment qi, treat one degree as slightly strong and use the mean value. From before the spring equinox and after the autumn equinox, begin the first band at four and add one per band; after nine bands the tally is twelve, and the ecliptic obliquity cycle completes. For the interval after the spring equinox and before the autumn equinox, use the same five-degree bands; begin at twelve; after nine bands the tally reaches four. At the two Establishment qi, treat one degree as slightly strong and use the mean value. Before and after the summer solstice, begin the first band at four; after nine bands the tally is twelve. Accumulate the trims in order, multiply the band index by the limit value, and divide by 120 to obtain degrees. The remainder, divided by twelve, gives fractional parts. If divided by ten instead, the major parts become too large. Use twelve as the denominator, naming greater, half, lesser, strong, and weak fractions. This is called the ecliptic–equator difference. Within nine bands on either side of each solstice, subtract the difference from the equatorial longitude to obtain ecliptic longitude. Within nine bands on either side of each equinox, add the difference to the equatorial longitude to obtain ecliptic longitude. To convert ecliptic longitude back to equatorial, add the difference near solstices and subtract it near equinoxes.
46
黃道宿度
Ecliptic Lodge Degrees
47
右北方九十七度六虛之差十九太
Northern quarter: 97°, six void-difference parts 19 and greater.
48
右西方八十二度半
Western quarter: 82½°.
49
右南方一百一十度半
Southern quarter: 110½°.
50
右東方七十五度少
Eastern quarter: 75° lesser.
51
前皆黃道度。 其步日行月與五星出入，循此。 求此宿度，皆有餘分。 前後輩之成少、半、太，准為全度。 若上考古下驗將來，當據歲差。 每移一度，各依術算，使得當時宿度及分，然可步日月五星，知其犯守也。
These are all ecliptic longitudes. The sun’s daily motion, the moon, and the five planets are all reckoned by this ecliptic framework. These lodge longitudes all carry fractional remainders. Successive reckonings round them to lesser, half, or greater parts to align with whole degrees. To verify against past ages and test future ones, apply the precession. For each degree of shift, recalculate by the method to obtain contemporary longitudes — only then may one pace the sun, moon, and planets and know their stations and transgressions.
52
推日度以乾實去中積分。 不盡者，盈大衍通法為度。 不滿，為度餘。 命起赤道虛九，去分。 不滿宿算外，即所求年天正冬至加時日所在度及餘也。 以三元之策累加之，命宿次如前，各得氣初日加時赤道宿度。
To determine solar longitude, subtract the circuit-of-heaven constant from the central accumulation. The remainder, divided by the Dayan universal divisor, gives degrees. What remains is the degree remainder. Count from equatorial Emptiness 9°, subtracting the fraction. When less than one lodge remains beyond the tally, the result is the solar longitude at winter-solstice hour-addition for the year sought, with its remainder. Add the three-origin interval repeatedly and count through the lodges as before to obtain the equatorial longitude at hour-addition on the first day of each qi.
53
求黃道日度以度餘減大衍通法。 餘以冬至日躔之宿距度所入限乘之，為距前分。 置距度下黃赤道差，以大衍通法乘之，減去距前分。 餘，滿百二十除，為定差。 不滿者，以象統乘之。 復除，為秒分。 乃以定差及秒減赤道宿度。 餘，依前命之，即天正冬至加時所在黃道宿度及餘也。
To determine ecliptic solar longitude, subtract the degree remainder from the Dayan universal divisor. Multiply the remainder by the band index for the winter-solstice solar station’s distance entry to obtain the pre-distance fraction. Take the ecliptic–equator difference for the distance band, multiply by the Dayan universal divisor, and subtract the pre-distance fraction. When the remainder fills 120, divide to obtain the fixed difference. If it does not fill, multiply by the image cycle. Divide again to obtain seconds and parts. Subtract the fixed difference and seconds from the equatorial lodge longitude. Fix the remainder as before; the result is the ecliptic longitude at winter-solstice hour-addition, with its remainder.
54
求次定氣置歲差，以限數乘之，滿百二十除，為秒分。 不盡為小分。 以加於三元之策秒分，因累而裁之，命以黃道宿次去之，各得定氣加時日躔所在宿及餘也。
To find successive fixed qi, set the annual precession, multiply by the band index, and divide by 120 for seconds and parts. The remainder becomes minor parts. Add this to the three-origin interval’s seconds and parts, accumulate the trims in order, and count through the ecliptic lodges to obtain the solar station at hour-addition for each fixed qi, with remainder.
55
求定氣初日夜半日所在度各置其氣定小餘，副之，以乘其日盈縮分，滿大衍通法而一，盈加縮減其副，用減其日時度餘，命如前，各其日夜半日躔行在。 求次日，各因定氣初日夜半度，累加一策，乃以其日盈縮分，盈加縮減度餘，命以宿次，即半日所在度及餘也。
To find midnight solar longitude on the first day of each fixed qi, set the fixed minor remainder and keep a duplicate; multiply by the daily equation of time, divide by the Dayan universal divisor, add or subtract from the duplicate according to surplus or deficit, subtract from the hour-added degree remainder, and fix as before to obtain midnight solar longitude. For each following day, start from the first-day midnight longitude, add one interval at each step, apply that day’s equation to the degree remainder by addition or subtraction, and count through the lodges to obtain midnight solar longitude with remainder.
56
大衍步月離術第四
Dayan Calendar: Method for Lunar Motion, IV
57
轉終分：六百七十萬一千二百七十九。
Rotation cycle constant: 670,1279.
58
轉終日：二十七； 餘，一千六百八十五； 秒，七十九。
Rotation cycle: 27 days; remainder 1,685; 79 parts.
59
轉法：七十六。
Rotation divisor: 76.
60
轉秒法：八十。
Rotation second-fraction divisor: 80.
61
推天正經朔入轉以轉終分去朔積分，不盡，以秒法乘，盈轉終分又去之，餘如秒法一而入轉分。 不盡為秒。 入轉分滿大衍通法，為日。 不滿為餘。 命日算外，即所求年天正經朔加時入轉日及餘秒。
To calculate rotation entry at the celestial first month's canonical new moon, divide the new-moon accumulated fraction by the rotation terminal fraction; multiply the remainder by the second-fraction divisor and cast out full rotation terminal fractions again; divide the remainder by the second-fraction divisor to obtain rotation entry parts. What is not exhausted are second-fractions. When rotation entry parts fill the Dayan universal divisor, each full unit counts as one day. What does not fill is the remainder. Count outward from the day tally to obtain the rotation entry day and second-fractions at the hour of occurrence for the celestial first month's canonical new moon of the year sought.
62
求次朔入轉因天正所入轉差日一、轉餘二千九百六十七、秒分一，盈轉終日餘秒者去之。 數除如前，即次日經朔加時所入。 考上下弦望，如求經朔四象術，循變相加，若以經朔望小餘減之，各其日夜半所入轉日及餘秒。
To find the next new moon's rotation entry, add to the celestial first month's entry a rotation increment of one day, remainder 2,967, and one second-fraction; cast out full rotation terminal days and remainder seconds. Divide and assign as before to obtain the entry at the next day's canonical new moon hour of occurrence. To verify the first and last quarters and full moon, apply the four-phase method for canonical new moons and add each variation in turn; subtracting each canonical new-moon or full-moon small remainder yields the rotation entry day and second-fractions at that night's midnight.
63
求朔弦望入朓朒定數各朔其所入日損益而半之，為通率。 又二率相減為率差。 前多者，以入餘減大衍通法，餘乘率差，盈大衍通法得一，並率差而半之。 前少者，半入餘，乘率差，亦以大衍通法除之，為加時轉率。 乃半之，以損益加時所入，餘為轉餘。 其轉餘，應益者，減法； 應損者，因餘。 皆以乘率差，盈大衍通法得一，加於通率。 轉率乘之，大衍通法約之，以朓減朒加轉率為定率。 乃以定率損益朓朒積為定數。 其後無同率者，亦因前率，益者以通率為初數，半率差而減之。 應通率，其損益入餘，進退日者，分為二日，隨餘初末如法求之，所得並以損益轉率。 此術本出《皇極曆》，以究算術之微變。 若非朔望有交者，直以入餘乘損益，如大衍通法而一，以損益朓朒為定數，各得所求。
To find the fixed waxing-waning numbers for each syzygy, halve the increase-decrease rate of the entered day at each new moon to obtain the universal rate. Subtract the two rates; the remainder is the rate difference. When the prior rate is larger, subtract the entry remainder from the Dayan universal divisor, multiply the remainder by the rate difference, divide by the Dayan universal divisor and round up, then halve together with the rate difference. When the prior rate is smaller, halve the entry remainder, multiply by the rate difference, and divide by the Dayan universal divisor likewise—this yields the hour-added rotation rate. Halve this and apply increase or decrease to the hour entry; the remainder is the rotation remainder. For the rotation remainder, when increase applies, subtract from the divisor; when decrease applies, use the remainder as basis. In each case multiply by the rate difference, divide by the Dayan universal divisor, and add the result to the universal rate. Multiply by the rotation rate and reduce by the Dayan universal divisor; subtract for waxing and add for waning to the rotation rate—this yields the fixed rate. Apply the fixed rate to adjust the waxing-waning accumulation, yielding the fixed number. When no matching rate follows, proceed from the prior rate in the same way; when increase applies, take the universal rate as the initial value and subtract half the rate difference. When the universal rate applies and adjustment of the entry remainder carries or borrows a day, apportion the parts over two days and compute by initial or terminal remainder as the method requires; use the result together to adjust the rotation rate. This procedure derives from the Huangji calendar, refining the subtle variations of computational astronomy. When new or full moon has no eclipse, multiply the entry remainder directly by the increase-decrease rate, divide by the Dayan universal divisor, and adjust waxing-waning accordingly to obtain the fixed number for each syzygy.
64
七日初：二千七百一，約為大分八。 末：三百三十九，約為大分一。
Day 7, initial value: 2,701, approximating to eight major parts. Terminal value: 339, approximating to one major part.
65
十四日初：二千三百六十三，約為大分七。 末：六百七十七，約為大分二。
Day 14, initial value: 2,363, approximating to seven major parts. Terminal value: 677, approximating to two major parts.
66
二十一日初：二千二十四，約為大分六。 末：一千一十六，約為大分三。
Day 21, initial value: 2,024, approximating to six major parts. Terminal value: 1,016, approximating to three major parts.
67
二十八日初：一千六百八十六，約為大分五。 末：一千三百五十四，約為大分四。
Day 28, initial value: 1,686, approximating to five major parts. Terminal value: 1,354, approximating to four major parts.
68
右以四象約轉終日及餘，均得六日二千七百一分。 就全數約為大分，是為之八分。 以減法，餘為末數。 乃四象馴變相加，各其所當之日初末數也。 視入轉餘，如初數以下者，加減損益，因循前率； 如初數以上，則反其衰，歸於後率云。
Above, divide the rotation terminal days and remainder by the four-phase divisor; each segment is six days, 2,701 parts. Reduce the full count to major parts—this is eight parts in nine of a day. For each segment, subtract from the divisor; the remainder is the terminal value. Accumulate the four-phase transitions in order, yielding each corresponding day's initial and terminal values. If the entered rotation remainder falls at or below the initial value, apply increase or decrease following the prior rate; if above the initial value, reverse the decline and revert to the posterior rate.
69
求朔弦望定日及餘以入氣、入轉朓朒定數，同名相從，異名相消。 乃以朓減朒加四象經小餘。 滿若不足，進大餘。 命以甲子算外，各其定日及小餘。 幹名與後朔葉同者，月大。 不同者，小； 無中氣者，為閏月。 凡言夜半者，皆起晨前子正之中。 若注曆觀弦望定小餘，不盈晨初餘數者，退一日。 其望，小餘雖滿此數，若有交蝕，虧初起在晨初已前者，亦如之。 又月行九道遲疾，則三大二小。 以日行盈縮，累增損之，則容有四大三小，理數然也。 若俯循常儀，當察加時早晚，隨其所近而進退之，使不過三小。 其正月朔，若有交加時正見者，消息前後一兩月，以定大小，令虧在晦二。
To find the fixed days and remainders for each syzygy, apply the fixed waxing-waning numbers for entered qi and rotation entry—same name combine, different name cancel. Then subtract for waxing and add for waning to the four-phase canonical small remainder. When full or insufficient, carry into the large remainder. Assign from jiazi outside the reckoning—each gives its fixed day and small remainder. When the stem name matches the next new moon's branch, the month is long. When they differ, the month is short; when no central qi falls in the month, it is intercalary. All references to midnight begin from true midnight at the start of zi before dawn. In calendar annotation, if the fixed minor remainder at quarter or full moon does not reach the early-morning initial remainder, set the date back one day. The same applies at full moon when the small remainder fills this number but an eclipse begins before early morning. Because the moon's nine-path motion varies in speed, months naturally run three long and two short. Cumulative adjustment by the sun's daily equation of time can occasionally yield four long and three short months; the arithmetic permits it. In practice, inspect whether the hour of occurrence is early or late and adjust accordingly, keeping within three short months. When the first-month new moon has an eclipse at exact visibility, adjust long-short assignment a month or two on either side so that waning falls on the last or second day.
70
推定朔弦望夜半日所在度各隨定氣次日以所直日度及餘分命焉。 若以五星相加減者，以四約度餘。 乃列朔弦望小餘，副之，以乘其日盈縮分，如大衍通法而一，盈加縮減其副，以加其日夜半度餘，命如前，各其日加時日躔所次。
To find the solar longitude at midnight for each fixed syzygy, follow the day after the fixed solar term and assign from that day's degree and remainder parts. If adding or subtracting the five planets, divide the degree remainder by four. Array the small remainders for each syzygy and keep duplicate tallies; multiply by that day's equation of time, divide by the Dayan universal divisor, and add or subtract from the duplicate according to surplus or deficit; add to that night's midnight degree remainder and assign as before—each gives where the sun's progression stands at the hour of occurrence.
71
推月九道度凡合朔所交，冬在陰曆，夏在陽曆，月行青道。 冬、夏至後，青道半交在春分之宿，殷黃道東。 立冬、夏後，青道半交在立春之宿，殷黃道東南。 至所沖之宿亦如之也。 冬在陽曆，夏在陰曆，月行白道。 冬至夏至後，白道半交在秋分之宿，殷黃道西。 立北。 至所沖之宿亦如之也。 春在陽曆，秋在陰曆，月行硃道。 春、秋分後，硃道半交在夏至之宿，殷黃道南。 立春立秋後，硃道半交在立夏之宿，殷黃道西南。 至所沖之宿亦如之也。 春在陰曆，秋在陽曆，月行黑道。 春、秋分後，黑道半交在冬至之宿，殷黃道北。 立春立秋後，黑道半交在立冬之宿，殷黃道東北。 至所沖之宿亦如之也。 四序離為八節，至陰陽之始交，皆以黃道相會，故月有九行。 各視月交所入七十二候，距交初黃道日每五度為限。 交初交中同。 亦初數十二，每限減一，數終於四，乃一度強，依平。 更從四起，每限增一，終於十二，而至半交，其去黃道六度。 又自十二，每限減一，數終於四，亦一度強，依平。 更從四起，每限增一，終於十二，復與日軌相會。 各累計其數，以乘限度，二百四十而一，得度。 不滿者，二十四除，為分。 若以二十除之，則大分。 十二為母，命以半太及強弱也。 為月行與黃道差數。 距半交前後各九限，以差數為減； 距正交前後各九限，以差數為加。 此加減是出入六度，單與黃道相交之數也。 若交赤道，則隨氣遷變不恆。 計去冬至夏至以來候數，乘黃道所差，十八而一，為月行與赤道差數。 凡日以赤道內為陰，赤道外為陽； 月以黃道內為陰，黃道外為陽。 故月行宿度入春分交後行陰曆，秋分交後行陽曆，皆為同名； 若入春分交後行陽曆，秋分交後行陰曆，皆為異名。 其在同名，以差數為加者加之，減者減之； 若在異名，以差數為加者減之，減者加之。 皆以增損黃道度為九道定數。
To derive the moon's nine-path degrees: at syzygy, when winter falls in yin months and summer in yang months, the moon follows the green path. After the Winter and Summer Solstices, the green path's half-intersection lies at the spring-equinox lodge, near the ecliptic to the east. After Start of Winter and Start of Summer, the green path's half-intersection lies at the Start-of-Spring lodge, near the ecliptic to the southeast. At the opposing lodge, the same applies. When winter is in yang months and summer in yin months, the moon follows the white path. After the Winter and Summer Solstices, the white path's half-intersection lies at the autumn-equinox lodge, near the ecliptic to the west. After Start-of-Winter and Start-of-Summer, the white path's half-crossing is at the Start-of-Autumn lodge, near the ecliptic northwest. At the opposing lodge, the same applies. When spring is in yang months and autumn in yin months, the moon follows the vermilion path. After the Spring and Autumn Equinoxes, the vermilion path's half-intersection lies at the summer-solstice lodge, near the ecliptic to the south. After Start of Spring and Start of Autumn, the vermilion path's half-intersection lies at the Start-of-Summer lodge, near the ecliptic to the southwest. At the opposing lodge, the same applies. When spring is in yin months and autumn in yang months, the moon follows the black path. After the Spring and Autumn Equinoxes, the black path's half-intersection lies at the winter-solstice lodge, near the ecliptic to the north. After Start of Spring and Start of Autumn, the black path's half-intersection lies at the Start-of-Winter lodge, near the ecliptic to the northeast. At the opposing lodge, the same applies. The four seasons yield eight nodes; at each yin-yang crossing the moon meets the ecliptic—hence nine paths of lunar motion. For each conjunction entry, take the seventy-two hou from the initial crossing; every five degrees of ecliptic solar longitude is one band. Initial and middle conjunction follow the same rule. Begin at twelve and subtract one per band until four, then one degree slightly strong—use the mean. Begin again at four; add one per band until twelve at half-intersection, six degrees from the ecliptic. From twelve again, subtract one per band until four, likewise one degree slightly strong—use the mean. Begin at four again; add one per band until twelve, reuniting with the solar track. Accumulate the counts in order, multiply by the band index, and divide by 240 to obtain degrees. The remainder, divided by twenty-four, gives parts. If divided by twenty instead, the result is major parts. With twelve as denominator, assign using half, major, and strong/weak fractions. This is the lunar ecliptic latitude difference. Within nine bands on either side of half-intersection, subtract the difference; within nine bands on either side of true conjunction, add the difference. This adjustment shifts latitude by six degrees—the value compared directly with the ecliptic alone. If measured against the equator, it varies with the seasons and is not constant. Count hou elapsed since the Winter or Summer Solstice, multiply the ecliptic difference, and divide by eighteen to obtain the lunar equatorial latitude difference. For the sun, inside the equator is yin and outside is yang; for the moon, inside the ecliptic is yin and outside is yang. Hence after the spring-equinox crossing the moon follows yin months, and after the autumn-equinox crossing yang months—both are same-named; if after the spring-equinox crossing it follows yang months, or after the autumn-equinox crossing yin months, both are differently named. Under same name, where the difference is for increase, add it; where for decrease, subtract it; under different name, where the difference is for increase, subtract it; where for decrease, add it. Apply these rules to adjust ecliptic longitude and obtain the fixed nine-path number.
72
推月九道平交入氣各以其月恆中氣，去經朔日算及餘秒，加其月經朔加時入交泛日及餘秒，乃以減交終日及餘秒，其餘即各平交入其月恆中氣日算及餘秒也。 滿三元之策及餘秒則去之，其餘即平交入後月恆節氣日算及餘秒。 因求次交者，以交終日及餘秒加之。 滿三元之策及餘秒，去之。 不滿者，為平交入其氣日算及餘秒。 各以其氣初先後數先加、後減其入餘。 滿若不足，進退日算，即平交入定氣日算及餘秒也。
To calculate mean conjunction entry into qi for the moon's nine paths, for each month's constant central solar term subtract the mean new moon day count and second-fractions, add that month's mean new moon hour-added entered-conjunction general day and second-fractions, then subtract from the conjunction terminal days and second-fractions—the remainder is the mean conjunction's entry day count and second-fractions within that month's constant central solar term. Cast out full three-origin rod counts and second-fractions; the remainder is the mean conjunction's entry day count and second-fractions into the next month's constant solar term. To find the next conjunction, add the conjunction terminal days and second-fractions. Cast out full three-origin rod counts and second-fractions. What does not fill is the mean conjunction's entry day count and second-fractions within that solar term. For each solar term, apply the lead and lag counts at its beginning—first add, then subtract—to the entry remainder. When the result fills or falls short, advance or retreat the day count—that yields the mean conjunction's entry day count and second-fractions within the fixed solar term.
73
求平交入氣朓朒定數置所入定氣日算，倍六爻乘之，三其小餘，辰法除而從之，以乘其氣損益率，如定氣辰數而一，所得以損益其氣朓朒積為定數也。
To find the fixed waxing-waning number for mean conjunction entry into a solar term, set the entered fixed solar term day count; double and multiply by six lines; triple the small remainder and divide by the chronogram divisor with carry; multiply by the solar term's increase-decrease rate and divide by the fixed solar term's chronogram count; use the result to adjust the solar term's waxing-waning accumulation—the fixed number.
74
求平交入轉朓朒定數置所入定氣餘，加其日夜半入轉餘，以乘其日損益率，滿大衍通法而一，所得以損益其日朓朒積，乃以交率乘之，交數而一，為定數。
To find the fixed waxing-waning number for mean conjunction rotation entry, set the entered fixed solar term remainder and add that night's midnight rotation entry remainder; multiply by that day's increase-decrease rate and divide by the Dayan universal divisor; adjust that day's waxing-waning accumulation accordingly; then multiply by the conjunction rate and divide by the conjunction number—the fixed number.
75
求正交入氣置平交入氣及入轉朓朒定數，同名相從，異名相消。 乃以朓減、朒加平交入氣餘，滿若不足，進退日算，即為正交入定氣日算及餘也。
To find true conjunction entry into a solar term, set the mean conjunction's qi entry and the fixed waxing-waning numbers for rotation entry—same-named quantities combine, differently named quantities cancel. Then subtract for waxing and add for waning to the mean conjunction's qi entry remainder; when the result fills or falls short, advance or retreat the day count—that gives the true conjunction's entry day count and remainder within the fixed solar term.
76
求正交加時黃道宿度置正交入定氣餘，副之，乘其日盈縮分，滿大衍通法而一，所得以盈加縮減其副，以加其日夜半日度，即正交加時所在黃度及餘也。
To find the ecliptic lodge degree at the true conjunction hour of occurrence, set the true conjunction's fixed solar term entry remainder as an auxiliary tally; multiply by that day's equation-of-time fraction and divide by the Dayan universal divisor; add for surplus and subtract for deficit from the auxiliary; add to that night's midnight solar degree—the ecliptic degree and remainder at the true conjunction hour.
77
求正交加時月離九道宿度以正交加時度餘，減大衍通法。 餘以正交之宿距度所入限數乘之，為距前分。 置距度下月道與黃道差，以大衍通法乘之，減去距前分，餘滿二百四十除，為定差。 不滿者，一退為秒。 以定差及秒加黃道度，餘，仍計去冬至夏至以來候數，乘定差，十八而一，所得依名同異而加減之，滿若不足，進退其度，命如前，即正交加時月離所在九道宿度及餘也。
To find the moon's nine-path lodge degree at the true conjunction hour of occurrence, subtract the true conjunction hour's degree remainder from the Dayan universal divisor. Multiply the remainder by the limit number entered in the true conjunction's lodge distance degree—the result is the pre-distance fraction. Set the moon-path and ecliptic difference below the distance degree; multiply by the Dayan universal divisor and subtract the pre-distance fraction; when the remainder fills 240, divide—the fixed difference. What does not fill: retreat one place to obtain second-fractions. Add the fixed difference and second-fractions to the ecliptic degree and remainder; still count the pentads elapsed since the Winter or Summer Solstice; multiply the fixed difference and divide by eighteen; add or subtract according to same or different name; when the result fills or falls short, advance or retreat the degree and assign as before—the nine-path lodge degree and remainder where the moon stands at the true conjunction hour.
78
推定朔弦望加時月所在度各置其日加時日躔所在，變從九道，循次相加。 凡合朔加時月行潛在日下，與太陽同度，是為離象。 凡置朔弦望加時黃道日度，以正交加時所在黃道宿度減之，餘以加其正交九道宿度，命起正交宿度算外，即朔弦望加時所當九道宿度也。 其合朔加時若非正交，則日在黃道，月在九道，各入宿度，雖多少不同，考其去極，若應準繩，故云月行潛在日下，與太陽同度。
To find the moon's lodge degree at the hour of occurrence for each fixed syzygy, set where the sun's progression stands at that day's hour of occurrence, convert to the nine paths, and add each increment in turn. At every new-moon hour of occurrence the moon moves hidden beneath the sun at the same degree—this is the Li trigram image. Set the ecliptic solar degree at each syzygy hour of occurrence; subtract the ecliptic lodge degree at the true conjunction hour; add the remainder to the true-conjunction nine-path lodge degree and assign from that lodge outside the reckoning—the nine-path lodge degree at each syzygy hour. When the new-moon hour of occurrence is not at true conjunction, the sun lies on the ecliptic and the moon on the nine paths—each at its own lodge degree. Though the longitudes differ, their polar distances match the plumb-line standard; hence the moon is said to move hidden beneath the sun at the same degree.
79
以一象之度九十一、餘九百五十四、秒二十二半為上弦，兌象。 倍之而與日沖，得望，坎象。 參之，得下弦，震象。 各以加其所當九道宿度，秒盈象統從餘，餘滿大衍通法從度。 命如前，各其日加時月所在度及餘秒也。 綜五位成數四十，以約度餘，為分。 不盡者，因為小分也。
Take one image's arc—91 degrees, remainder 954, and 22½ second-fractions—as the first quarter, the Dui trigram image. Double it to oppose the sun—obtaining full moon, the Kan trigram image. Take a third—obtaining last quarter, the Zhen trigram image. Add each to the corresponding nine-path lodge degree; when second-fractions fill the image cycle carry into the remainder; when the remainder fills the Dayan universal divisor carry into degrees. Assign as before—each gives the degree and second-fractions where the moon stands at that day's hour of occurrence. Combine the five positions to make 40 and divide the degree remainder—the result is minutes. What is not exhausted thus becomes minor fractions.
80
推定朔夜半入轉恆視經朔夜半所入，若定朔大餘有進退者，亦加減轉日，否則因經朔為定。 徑求次定朔夜半入轉，因前定朔夜半所入，大月加轉差日二，小月加日一，轉餘皆一千三百五十四秒分一。 數除如前，即次月定朔夜半所入。
To derive fixed-new-moon midnight rotation entry, always start from the mean new moon's midnight entry; if the fixed new moon's large remainder requires advance or retreat, likewise adjust the rotation days; otherwise take the mean new moon as fixed. To find the next fixed new moon's midnight rotation entry directly, from the prior fixed new moon's midnight entry: add two rotation-difference days in a long month and one in a short month; the rotation remainder is always 1,354 plus one second-fraction. Divide and assign as before—that gives the next month's fixed-new-moon midnight rotation entry.
81
求次日累加一日，去命如，各其夜半所入轉日及餘秒。
To find the next day, accumulate by adding one day, cast out and assign as before—each gives the rotation entry day and second-fractions at midnight.
82
求每日月轉定度各以夜半入轉餘，乘列衰，如大衍通法而一，所得以進加退減其日轉分，為月每所轉定分，滿轉法為度也。
To find the moon's daily fixed rotation degree, multiply each night's midnight rotation entry remainder by the column decline and divide by the Dayan universal divisor; add for advance and subtract for retreat from that day's rotation minutes—the moon's fixed rotation minutes; when they fill the rotation divisor, that makes degrees.
83
求朔弦望定日前夜半月所在度各半列衰，減轉分。 退者，定餘乘衰，以大衍通法除，並衰而半之； 進者，半定餘乘衰，定以大衍通法除，皆加所減。 乃以定餘乘之，盈大衍通法得一，以減加時月度及分。 因夜半准此求轉分以加之，亦得加時月度。 若非朔望有交，直以定小餘乘所入日轉交分，如大衍通法而一，以減其日時月度，亦得所求。
To find the moon's midnight position on the night before each fixed syzygy, halve the column decline and subtract from the rotation minutes. When retreating, multiply the fixed remainder by the decline, divide by the Dayan universal divisor, combine with the decline, and halve; When advancing, multiply half the fixed remainder by the decline and divide by the Dayan universal divisor—both add to what was subtracted. Then multiply by the fixed remainder; when it fills the Dayan universal divisor obtain one; subtract from the hour-of-occurrence lunar degree and minutes. From midnight, apply the same method to find rotation minutes and add—this also yields the hour-of-occurrence lunar degree. When new or full moon has no crossing, multiply the fixed small remainder directly by that day's rotation crossing minutes, divide by the Dayan universal divisor, and subtract from that day's hour lunar degree—this also yields the result sought.
84
求次日夜半月度各以其日轉定分加之，分滿轉法從度，命如前，即次日夜半月所在度及分。
To find the next night's midnight lunar degree, add that day's fixed rotation minutes; when minutes fill the rotation divisor carry into degrees and assign as before—the next night's midnight lunar degree and minutes.
85
推月晨昏度各以所入轉定分乘其日夜漏，倍百刻除，為晨分。 以減轉定分，餘為昏分。 分滿轉法，從度。 以加夜半度，望前以昏加，望後以晨加。 各得其日晨昏月所在度及分。
To derive the moon's dawn and dusk degrees, multiply the entered fixed rotation minutes by that night's clepsydra drain and divide by twice one hundred clepsydra marks—the dawn minutes. Subtract from the fixed rotation minutes—the remainder is the dusk minutes. When minutes fill the rotation divisor, carry into degrees. Add to the midnight degree; before full moon add the dusk minutes, after full moon add the dawn minutes. Each gives that day's dawn and dusk lunar degree and minutes.
86
大衍步軌漏第五
Dayan Calendar: Method for Orbital Motion and Clepsydra, V
87
爻統：一千五百二十。
Line unity: 1,520.
88
象積：四百八十。
Image accumulation: 480.
89
辰刻：八； 刻分，一百六十。
Chronogram clepsydra marks: 8; Clepsydra mark minutes: 160.
90
昏明刻：各二； 刻分，二百四十。
Dusk and dawn clepsydra marks: 2 each; Clepsydra mark minutes: 240.
91
求每日消息定衰各置其氣消息衰，依定氣日數，每日以陟降率陟減降加其分，滿百從衰，不滿為分。 各得每日消息定衰及分。 其距二分前後各一氣之外，陟降不等，各每以三日為一限，損益如後。
To find the daily message decay fixed decline, set each solar term's message decline; according to the fixed solar term's day count, each day apply the ascending-descending rate—subtract for ascending and add for descending to its minutes; when minutes fill 100 carry into the decline; what does not fill remains as minutes. Each day yields its message decay fixed decline and minutes. Outside the single solar term before and after each equinox, ascending and descending rates are unequal; each interval uses three days as one limit—increase and decrease as follows.
92
雨水初日：降七十八。 初限每日損十二，次限每日損八，次限每日損三，次限每日損二，末限每日損一。
Rain Water, first day: descending 78. First limit: decrease 12 daily; second limit: decrease 8 daily; third limit: decrease 3 daily; fourth limit: decrease 2 daily; final limit: decrease 1 daily.
93
清明初日：陟一。 初限每日益一，次限每日益二，次限每日益三，次限每日益八，末限每日益十九。
Clear and Bright, first day: ascending 1. First limit: increase 1 daily; second limit: increase 2 daily; third limit: increase 3 daily; fourth limit: increase 8 daily; final limit: increase 19 daily.
94
處暑初日：降九十九。 初限每日損十九，次限每日損八，次限每日損三，次限每日損二，末限每日損一。
End of Heat, first day: descending 99. First limit: decrease 19 daily; second limit: decrease 8 daily; third limit: decrease 3 daily; fourth limit: decrease 2 daily; final limit: decrease 1 daily.
95
寒露初日：陟一。 初限每日益一，次限每日益二，次限每日益三，次限每日益八，末限每日益十二。
Cold Dew, first day: ascending 1. First limit: increase 1 daily; second limit: increase 2 daily; third limit: increase 3 daily; fourth limit: increase 8 daily; final limit: increase 12 daily.
96
求前件四氣置初日陟降率，每日依限次損益之，各為每日率。 乃遞以陟減降加其氣初日消息衰分，亦得每日定衰及分也。
For the four solar terms above, set the first-day ascending-descending rate and increase or decrease daily according to each limit in turn—each yields a daily rate. Then in sequence subtract for ascending and add for descending to each solar term's first-day message decline minutes—this also yields the daily fixed decline and minutes.
97
推戴日之北每度晷數南方戴日之下，正中無晷。 自戴日之北一度，乃初數一千三百七十九。 從此起差，每度增一，終於二十五度。 又每度增二，終於四十度。 又每度增六，終於四十四度，增六十八。 每度增二，終於五十五度。 又每度增十九，終於六十度，度增一百六十。 又每度增三十三，終於六十五度。 又每度增三十六，終於七十度。 又每度增三十九，終於七十二度，增二百六十。 又度增四百四十，又度增一千六十，又度增一千八百六十，又度增二千八百四十，又度增四千，又度增五千三百四十，而各為每度差。 因累其差以遞加初數，滿百為分，分滿十為寸，各為每度晷差。 又每度晷差數。
To derive the gnomon count for each degree north of the subsolar point: directly beneath the subsolar point in the south, at the center there is no shadow. From one degree north of the subsolar point, the initial number is 1,379. From here take the difference; each degree increases by one, ending at twenty-five degrees. Again each degree increases by two, ending at forty degrees. Again each degree increases by six, ending at forty-four degrees; the increment becomes sixty-eight. Each degree increases by two, ending at fifty-five degrees. Again each degree increases by nineteen, ending at sixty degrees; the per-degree increment becomes one hundred sixty. Again each degree increases by thirty-three, ending at sixty-five degrees. Again each degree increases by thirty-six, ending at seventy degrees. Again each degree increases by thirty-nine, ending at seventy-two degrees; the increment becomes two hundred sixty. Again the per-degree increment is four hundred forty; then one thousand sixty; then one thousand eight hundred sixty; then two thousand eight hundred forty; then four thousand; then five thousand three hundred forty—each is the per-degree difference. Accumulate the differences and add them in sequence to the initial number; when the total fills 100 that makes minutes; when minutes fill 10 that makes inches—each yields the gnomon difference for each degree. Also the gnomon-difference number for each degree.
98
求陽城日晷每日中常數各置其氣去極度，以極去戴日下度五十六，盈分八十二減半之，各得戴日之北度數及分。 各以其消息定衰戴日北所直度分之晷差，滿百為分，分滿十為寸，各為每日晷差。 乃遞以息減消加其氣初晷數，得每日中晷常數也。
To find the Yangcheng gnomon daily midday constant, set each solar term's polar distance; subtract from the pole-to-subsolar-point value of 56 degrees and 82 surplus minutes and halve—each gives the degrees and minutes north of the subsolar point. For each day, take the message decay fixed decline at the gnomon difference for the degree and minutes due north of the subsolar point; when the total fills 100 that makes minutes; when minutes fill 10 that makes inches—each yields the daily gnomon difference. Then in sequence subtract for wax and add for wane to each solar term's initial gnomon count—obtaining the daily midday gnomon constant.
99
求每日中晷定數各置其日所在氣定小餘，以爻統減之，餘為中後分。 置前後分，以其日晷差乘之，如大衍通法而一，為變差。 乃以變差加減其日中晷常數，冬至後，中前以差減，中後以差加。 夏至後，中前以差加，中後以差減。 冬至一日有減無加，夏至一日有加無減。 各得每日中晷定數。
To find the daily fixed midday gnomon count, set that day's entered solar term fixed small remainder; subtract the line unity—the remainder is the after-midday fraction. Set the before-and-after fractions; multiply by that day's gnomon difference and divide by the Dayan universal divisor—the variation difference. Apply the variation difference to adjust the midday gnomon constant; after the Winter Solstice, subtract the difference before midday and add it after midday. After the Summer Solstice, add the difference before midday and subtract it after midday. On the Winter Solstice day there is subtraction but no addition; on the Summer Solstice day there is addition but no subtraction. Each day yields its fixed midday gnomon count.
100
求每日夜半漏定數置消息定衰，滿象積為刻，不滿為分。 各遞以息減消加其氣初夜半漏，各得每日夜半漏定數。
To find the daily fixed midnight clepsydra drain, set the message decay fixed decline; when it fills the image accumulation that makes clepsydra marks; what does not fill remains as minutes. For each day in sequence subtract for wax and add for wane to the solar term's initial midnight clepsydra drain—each yields the daily fixed midnight clepsydra drain.
101
求晨初餘數置夜半定漏全刻，以九千一百二十乘之，十九乘刻分從之，如三百而一，所得為晨初餘數，不盡為小分。
To find the early-morning initial remainder, set the full clepsydra marks of the fixed midnight drain; multiply by 9,120 with 19 times the clepsydra mark minutes carried; divide by 300—the result is the early-morning initial remainder; what is not exhausted is minor fractions.
102
求每日晝夜漏及日出入所在辰刻各倍夜半之漏，為夜刻。 以減百刻，餘為晝刻。 減晝五刻以加夜，即晝為見刻，夜為沒刻。 半沒刻以半辰刻加之，命起子初刻算外，即日出辰刻。 以見刻加之，命如前，即日入辰刻。 置夜刻以五除之，得每更差刻，又五除之，得每籌差刻。 以昏刻加日入辰刻，得甲夜初刻。 又以更籌差加之，得次更一籌之數。 以次累加，滿辰刻去之，命如前，即得五夜更籌所當辰及分也。 其夜半定漏，亦名晨初夜刻。
To find the daily day-and-night clepsydra drain and the chronogram clepsydra marks of sunrise and sunset, double each night's midnight drain—the night clepsydra marks. Subtract from one hundred clepsydra marks—the remainder is the day clepsydra marks. Subtract five day clepsydra marks and add them to the night—day clepsydra marks become visibility clepsydra marks and night clepsydra marks become disappearance clepsydra marks. Add half the disappearance clepsydra marks and half the chronogram clepsydra marks; assign from the zi initial clepsydra mark outside the reckoning—the sunrise chronogram clepsydra marks. Add the visibility clepsydra marks and assign as before—the sunset chronogram clepsydra marks. Set the night clepsydra marks and divide by five to obtain the difference clepsydra marks per watch; divide again by five to obtain the difference clepsydra marks per tally. Add the dusk clepsydra marks to the sunset chronogram clepsydra marks—obtaining the first watch initial clepsydra mark of night A. Add the watch-and-tally difference again—obtaining the count for the next watch and first tally. Accumulate in sequence; when the total fills chronogram clepsydra marks cast out and assign as before—obtaining the double-hour and minutes for each watch and tally through five nights. The fixed midnight clepsydra drain is also called the early-morning initial night clepsydra marks.
103
求每日黃道去極定數置消息定衰，滿百為度，不滿為分，各遞以息減消加其氣初去極度，各得每日去極定數。
To find the daily fixed ecliptic polar distance, set the message decay fixed decline; when it fills 100 that makes degrees and what does not fill remains as minutes; for each day in sequence subtract for wax and add for wane to the solar term's initial polar distance—each yields the daily fixed polar distance.
104
求每日距中度定數置消息定衰，以一萬二千三百八十六乘之，如一萬六千二百七十七而一，為每日度差。 差滿百為度，不滿為分。 各遞以息加消減其氣初距中度，各得每日距中度定數。 倍距中度以減周天度，五而一，所得為每更度差。
To find the daily fixed center-distance, set the message decay fixed decline; multiply by 12,386 and divide by 16,277—the daily degree difference. When the difference fills 100 that makes degrees; what does not fill remains as minutes. For each day in sequence add for wax and subtract for wane to the solar term's initial center-distance—each yields the daily fixed center-distance. Double the center-distance and subtract from the circuit-of-heaven degrees; divide by five—the degree difference per watch.
105
求每日昏明及每更中宿度所臨置其日所在赤道宿度，以距中度加之，命宿次如前，即得其日昏中所臨宿度。 以每更差度加之，命如前，即乙夜初中所臨宿度及分也。
To find which lodge degrees stand at dusk and dawn each day and at the midpoint of each watch, set that day's equatorial lodge degree, add the center-distance, and assign the lodge sequence as before—the lodge degree at dusk that day. Add the per-watch degree difference and assign as before—that yields the lodge degree and minutes occupied at the opening of the second watch of night B.
106
求九服所在每氣初日中晷常數置氣去極度數相減，各為生氣消息定數，因測所在冬夏至日晷長短，但測至即得，不必要須冬至。 于其戴日之北度及分晷數中，校取長短，同者便為所在戴日北度數及分。 氣各以消定數加減之，因冬至後者每氣以減，因夏至後者每氣以加。 各得每氣戴日北度數及分。 各因其氣所直度分之晷數長短，即各為所在每定氣初日中晷常數。 其測晷有在表南者，亦據其晷尺寸長短，與戴日北每度晷數同者，因取其所直之度，去戴日北度數，反之，為去戴日南度，然後以消息定數加減。
To find the midday gnomon constant on the first day of each solar term at each nine-domain locale, set each solar term and subtract the polar-distance values—each difference is a generating-qi message decay fixed number. Then measure the solstice gnomon lengths at the locale; either solstice suffices—the Winter Solstice is not strictly required. Among the gnomon counts for each degree and minute north of the subsolar point, compare shadow lengths; whichever matches gives the local degrees and minutes north of the subsolar point. For each solar term adjust by the waning fixed number: after the Winter Solstice subtract for each term; after the Summer Solstice add for each term. Each solar term yields the degrees and minutes north of the subsolar point. From the gnomon length at the degrees and minutes directly under each solar term, each becomes the local midday gnomon constant on the first day of each fixed solar term. When measured shadows fall south of the gnomon table, match shadow length against the per-degree gnomon count north of the subsolar point; take the indicated degree, reverse from the north-of-subsolar-point value to obtain south-of-subsolar-point degrees, then adjust by the message decay fixed number.
107
求九服所在晝夜漏刻冬夏至各於所在下水漏，以定當處晝夜刻數。 乃相減，為冬夏至差刻。 半之，以加減二至晝夜刻數，加夏至、減冬至。 為春秋分定日晝夜刻數。 乃置每氣消息定數，以當處二至差刻數乘之，如二至去極差度四十七分，八十而一，所得依分前後加減二分初日晝夜漏刻，春分前秋分後，加夜減晝； 春分後秋分前，加晝減夜。 各得所在定氣初日晝夜漏刻數。 求次日者，置每日消息定衰，亦以差刻乘之，差度而一，所得以息減消加其氣初漏刻，各得所求。 其求距中度及昏明中宿日出入所在，皆依陽城法求，仍以差度而今有之，即得也。
To find day-and-night clepsydra quarters for each nine-domain locale, at the Winter and Summer Solstices measure the water clepsydra at the locale to fix the local day and night quarter counts. Subtract the two—the difference is the solstice quarter-difference. Halve it and use the result to adjust the solstice day-and-night quarter counts—add for the Summer Solstice and subtract for the Winter Solstice. This yields the fixed day-and-night quarter counts for the equinoxes. Then set each solar term's message decay fixed number, multiply by the local solstice quarter-difference, and divide by the solstice polar-distance difference of 47 minutes at one per 80; use the result to adjust day-and-night clepsydra quarters on each equinox first day—before the Spring Equinox and after the Autumn Equinox, add night and subtract day; After the Spring Equinox and before the Autumn Equinox, add day and subtract night. Each locale yields the day-and-night clepsydra quarter count for the first day of each fixed solar term. To find the next day, set the daily message decay fixed decline, multiply by the quarter-difference, divide by the degree-difference, and subtract for wax and add for wane from the solar term's initial clepsydra quarters—each day yields the value sought. Center-distance and the lodges occupied at dusk, dawn, and sunrise and sunset all derive by the Yangcheng method, with the present degree difference applied—thus obtained.
108
又術置所在春秋分定日中晷常數，與陽城每日晷數校取同者，因其日夜半漏，即為所在定春秋分初日夜半漏。 求餘氣定日，每以消息定數，依分前後加減刻分。 春分前以加，分後以減； 秋分前以減，分後以加。 滿象積為刻，不滿為分，各為所在定氣初日夜半定漏。
Another procedure: set the local fixed equinox midday gnomon constant, compare it with the Yangcheng daily gnomon count and take the match; that day's midnight drain becomes the locale's fixed midnight drain on the equinox first day. For the remaining fixed solar-term days, apply the message decay fixed number each time, adding and subtracting quarter-minutes before and after the equinox. Before the Spring Equinox add; after the equinox subtract; Before the Autumn Equinox subtract; after the equinox add. When the total fills the image accumulation that makes quarters and what does not fill remains as minutes—each is the locale's fixed midnight drain on the first day of each fixed solar term.
109
求次日以消息定衰依陽城法求之，即得。 此術究理，大體合通。 但高山平川，視日不等。 校其日晷，長短乃同。 考其日漏，多少懸別。 以茲參課，前術為審也。
To find the next day, derive the message decay fixed decline by the Yangcheng method—that suffices. This procedure probes the underlying principle and is broadly sound. Yet on high mountains and level plains, the apparent sun differs. Adjust the daily gnomon and the shadow lengths align. Examine the daily clepsydra drain and the amounts diverge sharply. On this comparison, the prior procedure proves the sounder one.
110
大衍步交會術第六
Dayan Procedure for Conjunction and Eclipse, Part Six
111
交終：八億二千七百二十五萬一千三百二十二。
Conjunction terminal: 827,251,322.
112
交中：四萬一千三百六十二； 秒，五千六百六十一。
Conjunction middle: 41,362; second-fractions, 5,661.
113
終日：二十七； 餘，六百四十五； 秒，一千三百二十二。
Terminal days: 27; remainder, 645; second-fractions, 1,322.
114
中日：十三； 餘，一千八百四十二； 秒，五千六百六十一。
Mid-days: 13; remainder, 1,842; second-fractions, 5,661.
115
朔差日：二； 餘，九百六十七； 秒，八千六百七十八。
New-moon difference days: 2; remainder, 967; second-fractions, 8,678.
116
望差日：一； 餘，四百八十三； 秒，九千三百三十九。
Full-moon difference days: 1; remainder, 483; second-fractions, 9,339.
117
望數日：十四； 餘，二千三百二十六； 秒，五十。
Full-moon count days: 14; remainder, 2,326; second-fractions, 50.
118
交限日：十二； 餘，一千三百五十八； 秒，六千三百二十二。
Conjunction limit days: 12; remainder, 1,358; second-fractions, 6,322.
119
交率：三百四十三。
Conjunction rate: 343.
120
交數：四千三百六十九。
Conjunction number: 4,369.
121
辰法：七百六十。
Double-hour divisor: 760.
122
秒分法：一萬。
Second-fraction divisor: 10,000.
123
推天正經朔入交以交終去朔積分，不盡，以秒分法乘。 盈交終，又去之。 餘如秒法而一，為入交分。 不盡，為秒。 入交分滿大衍通法，為日； 不滿，為餘。 命日算外，即所求年天正經朔加時入交泛日及餘秒。
To derive the celestial first month's mean new moon entered conjunction, subtract the accumulated new-moon fraction from the conjunction terminal; multiply what remains by the second-fraction divisor. When it fills the conjunction terminal, cast out again. Divide the remainder by the second-fraction divisor—the quotient is the entered-conjunction parts. What is not exhausted is second-fractions. When entered-conjunction parts fill the Dayan universal divisor, each full unit counts as one day; What does not fill is remainder. Assign from the day count outside the reckoning—that gives the sought year's celestial first month mean new moon hour-of-occurrence entered-conjunction general days and remainder second-fractions.
124
求次朔入交因天正所入，加朔差日及餘秒，盈終日及餘秒者，去之。 數除如前，即次月經朔加時所入。
To find the next month's entered conjunction, from the celestial first month's entry add the new-moon difference days and remainder second-fractions; when full of terminal days and remainder second-fractions, cast out. Count and divide as before—that is the next month's mean new moon hour-of-occurrence entry.
125
求望以望數日及餘秒加之，去命如前，即得所求。 若以經朔望小餘減之，各其日夜半所入交泛日及餘秒。
To find full moon, add the full-moon count days and remainder second-fractions, cast out, and assign as before—thus obtained. If the mean new- or full-moon small remainder is subtracted, each night yields midnight entered-conjunction general days and remainder second-fractions.
126
求定朔夜半入交恆視經朔望夜半所入，定朔望大餘。 有進退者，亦加減交日。 否則，因經為定，各得所求。 求次定朔夜半入交：因前定朔夜半所入，大月加交差日二，月小加日一，餘皆二千三百九十四、秒八千六百七十八。 求次日：累加一百，數除如前，各其夜半所入交泛日及餘秒。
To find fixed new moon midnight entered conjunction, always observe the mean new or full moon midnight entry together with the fixed new- or full-moon large remainder. When there is advance or retreat, likewise add or subtract conjunction days. Otherwise take the mean as fixed—each yields what is sought. To find the next fixed new moon midnight entry, from the prior fixed new moon midnight entry add 2 conjunction-difference days in a large month and 1 day in a small month; remainders are all 2,394 and second-fractions 8,678. To find the next day, accumulate by adding 100, then count and divide as before—each night yields midnight entered-conjunction general days and remainder second-fractions.
127
求朔望入交常日各以其日入氣朓朒定數，朓減朒加其入交泛，餘滿大衍通法從日，即為入交常及餘秒。
To find new and full moon entered-conjunction mean days, take each day's entered-qi waxing-waning fixed number; subtract for waxing and add for waning to the entered-conjunction general; when the remainder fills the Dayan universal divisor carry into days—the entered-conjunction mean and remainder second-fractions.
128
求朔望入交定日各置其日入轉朓朒定數，以交率乘之，如交數而一。 所得以朓減朒加入交常，餘數如前，即為入交定日及餘秒。
To find new and full moon entered-conjunction fixed days, set each day's entered-rotation waxing-waning fixed number, multiply by the conjunction rate, and divide by the conjunction number. Subtract for waxing and add for waning to the entered-conjunction mean; treat the remainder as before—the entered-conjunction fixed days and remainder second-fractions.
129
求月交入陰陽曆恆視其朔望入交定日及餘秒，如中日及餘秒已下者，為月入陽曆，已上者，以中日及餘秒去之，餘為月入陰曆。
To find the moon's conjunction entry into the yin-yang calendar, observe the new- or full-moon entered-conjunction fixed days and remainder second-fractions; if at or below the mid-days and remainder second-fractions, the month enters the yang calendar; if above, subtract the mid-days and remainder second-fractions—the remainder is entry into the yin calendar.
130
求四象六爻每度加減分及月去黃道定數以其爻加減率與後爻加減率相減，為前差。 又以後爻率與次後爻率相減，為後差。 二差相減，為中差。 置所在爻並後爻加減率，半中差以加而半之，十五而一，為爻末率，國為後爻初率。 每以本爻初末率相減，為爻差。 十五而一，為度差。 半之，以加減初率，少象減之，老象加之。 為定初率。 每次度差累加減之，少象以差減，老象以差加。 各得每度加減定分。 乃修積其分，滿百二十為度，各為每度月去黃道度數及分。 其四象，初爻無初率，上爻無末率，皆倍本爻加減率，十五而一。 所得各以初末率減之，皆互得其率。 餘依術算，各得所求。
To find the four images' six lines' per-degree increase-decrease parts and the moon's fixed distance from the ecliptic, subtract this line's increase-decrease rate from the next line's rate—the forward difference. Again subtract the next line's rate from the one after—the backward difference. Subtract the two differences—the middle difference. Set the occupied line and the next line's increase-decrease rates, add half the middle difference and halve, divide by 15—the line terminal rate, which becomes the next line's initial rate. Subtract each line's initial and terminal rates—the line difference. Divide by 15—the degree difference. Halve it and use the result to adjust the initial rate—the young image subtracts and the old image adds. This is the fixed initial rate. Each time accumulate the degree difference in subtraction and addition—the young image subtracts the difference and the old image adds it. Each yields the fixed increase-decrease parts per degree. Then accumulate the parts; when they fill 120 that makes degrees—each is the moon's degrees and minutes from the ecliptic per degree. For the four images, the first line has no initial rate and the top line no terminal rate; in each case double that line's increase-decrease rate and divide by 15. Subtract the result from each initial and terminal rate—in each case obtaining the other's rate. The remainder follows the procedure—each yields what is sought.
131
求朔望夜半月行入陰陽度數各置其日夜半入轉日及餘秒，餘以其日夜半入交定日及餘秒減之也，其秒母不等，當循率相通，然後減之，如不足減，即轉終日及一餘秒，然後減之。 餘為定交初日夜半入轉日及餘秒。 乃以定交初日夜半入餘與其日夜半入餘，各乘其日轉定分，如大衍通法而一。 所得滿轉法為度，不滿為分。 各以加其日轉積度及分，乃相減，其餘即為其夜半月行入陰陽度數及分也。 轉求次日，但以其日轉定分加之，滿轉法為度，即得。
To find the moon's motion entered into yin-yang degrees at new- and full-moon midnight, set each night's midnight entered-rotation days and remainder second-fractions and subtract that night's midnight entered-conjunction fixed days and remainder second-fractions; when second denominators differ, convert by rate correspondence before subtracting; if insufficient to subtract, add rotation terminal days and one remainder second-fraction, then subtract. The remainder is the fixed-conjunction first night midnight entered-rotation days and remainder second-fractions. Then take the fixed-conjunction first night midnight entry remainder and that night's midnight entry remainder; multiply each by that day's rotation fixed parts and divide by the Dayan universal divisor. When the result fills the rotation divisor that makes degrees; what does not fill is minutes. Add each to that day's rotation accumulated degrees and minutes, then subtract—the remainder is the moon's motion entered into yin-yang degrees and minutes at that midnight. To find the next day in turn, simply add that day's rotation fixed parts; when they fill the rotation divisor that makes degrees—thus obtained.
132
求朔望夜半月行入四象度數置其日夜半入陰陽度數及分，以一象之度九十除之。 若以小象除之，則兼除差度一、度分一百六、大分十三、小分十四，訖，然以次象除之。 所得以少陽、老陽、少陰、老陰為次，命起少陽算外，即其日夜半所入象度數及分也。 先以三十乘陰陽度分，十九而一，為度分。 乘又除，為小分。 然以象度及分除之。
To find the moon's motion entered into four-image degrees at new- and full-moon midnight, set that night's midnight entered yin-yang degrees and minutes and divide by one image's 90 degrees. If dividing by the young image, also divide out difference degree 1, degree-minutes 106, major-fractions 13, and minor-fractions 14; when finished, divide by the next image in sequence. Take the result in order as lesser yang, greater yang, lesser yin, and greater yin; assign from lesser yang outside the reckoning—that night's midnight entered image degrees and minutes. First multiply yin-yang degree-minutes by 30 and divide by 19—the degree-minutes. Multiply and divide again—the minor fractions. Then divide by image degrees and minutes.
133
求朔望夜半月行入六爻度數置其日夜半所入象度數及分，以一爻之度一十五除之。 所得命起其象初爻算外，即以其日夜半所入爻度數及分也。 其月行入少象初爻之內，皆為沾近黃道度。 當朔望則有虧蝕。 求入蝕限：其入交定日及餘秒，如望差已下交限已上者，為入蝕限。 望入蝕限，則月蝕； 朔入蝕限，月在陰曆則日蝕。 入限，如望差已下，為交後。 交限已上者，以減中日及餘，為交前。 置交前後定日及餘秒通之，為去交前後定分。 置去交定分，以十一乘之，如二千六百四十三除之，為去交度數。 不盡，以大衍通法乘之，復除為餘。 大抵去交十三度以上，雖入蝕限，為涉交數微，光影相接，或不見蝕。
To find the moon's motion entered into six-line degrees at new- and full-moon midnight, set that night's midnight entered image degrees and minutes and divide by one line's fifteen degrees. Take the result and assign from that image's first line outside the reckoning—that night's midnight entered line degrees and minutes. When the moon's motion enters within the young image's first line, it is always in degrees close to the ecliptic. At new or full moon there may be an eclipse. To find entered eclipse limit: when the entered-conjunction fixed days and remainder second-fractions are at or below the full-moon difference and at or above the conjunction limit, it has entered the eclipse limit. If full moon has entered the eclipse limit, there is a lunar eclipse; If new moon has entered the eclipse limit while the moon is in the yin calendar, there is a solar eclipse. Within the limit, if at or below the full-moon difference, it is after conjunction. If at or above the conjunction limit, subtract from the mid-days and remainder—it is before conjunction. Set the before- and after-conjunction fixed days and remainder second-fractions and reduce to a common denominator—the fixed parts from before or after conjunction. Set the fixed distance-from-conjunction parts, multiply by 11, and divide by 2,643—the distance-from-conjunction in degrees. What remains, multiply by the Dayan universal divisor and divide again for the remainder. Generally, when the distance from conjunction exceeds thirteen degrees, although the eclipse limit is entered, the transit is slight, light and shadow merely touch, and the eclipse may not be visible.
134
求月蝕分其去交定分七百七十九已下者，皆蝕既。 已上者，以交定分減望差，餘以一百八十三約之。 盡半已下，為半弱； 已上，為半強。 命以十五為限，得月蝕之大分。
To find the lunar eclipse magnitude: if the fixed distance-from-conjunction parts are 779 or below, the eclipse is total. If above, subtract the entered-conjunction fixed parts from the full-moon difference and divide the remainder by 183. If the quotient is half or below, it is a weak half; If above half, it is a strong half. Assign with 15 as the limit to obtain the lunar eclipse major fraction.
135
求月蝕所起月在陰曆，初起東南，甚于正南，復於西南。 月在陽曆，初起東北，甚於正北，復於西北。 其蝕十二分已上者，皆起於正東，復於正西。 此皆據南方正午而論之，若蝕于餘方者，各隨方面所在，准此取正，而定其蝕起復也。
To find where the lunar eclipse begins: when the moon is in the yin calendar, the shadow first appears in the southeast, reaches maximum at true south, and returns toward the southwest. When the moon is in the yang calendar, it first appears in the northeast, reaches maximum at true north, and returns toward the northwest. For eclipses of twelve parts or more, the shadow always begins at true east and returns at true west. All of this is reckoned from due south at noon; for eclipses visible in other regions, adjust according to the local direction to determine where the eclipse begins and ends.
136
求月蝕用刻置月蝕之大分。 五已下，因增三。 十已下，因增四。 十已上，因增五。 其去交定分五百二十已下，又增半。 二百六十已下，又增半。 各為泛用刻率。
To find the duration of a lunar eclipse in ke, set the lunar eclipse major fraction. If five or below, add three. If ten or below, add four. If ten or above, add five. If the fixed distance-from-conjunction parts are 520 or below, add another half. If 260 or below, add another half. Each result is the general-use ke rate.
137
求每日差積定數以所入氣並後氣增損差，倍六爻乘之，綜兩氣辰數除之，為氣末率。 又列二氣增損差，皆倍六爻乘之，各如辰數而一。 少減多，餘為氣差。 加減末率，冬至後以差減，夏至後以差加。 為初率。 倍氣差，亦倍六爻乘之，復綜兩氣辰數以除之，為日差。 半之，以加減初末，各為定率。 以日差累加減氣初定率，冬至後以差加，夏至後以差減。 為每日增損差。 乃循積之，隨所入氣日加減氣下差積，各其日定數。 其二至之前一氣，皆後無同差，不可相並，各因前末為初率。 以氣差冬至前減，夏至前加，為末率。 餘依算術，各得所求也。
To find the daily difference-accumulation fixed numbers, take the entered qi and following qi increase-decrease differences, double and multiply by six lines, sum the two qi double-hour numbers and divide—the qi terminal rate. Again set out the two qi increase-decrease differences, double and multiply each by six lines, and divide each by its double-hour number. Subtract the smaller from the larger—the remainder is the qi difference. Adjust the terminal rate: after the winter solstice subtract the difference; after the summer solstice add it. This is the initial rate. Double the qi difference, double and multiply by six lines, sum the two qi double-hour numbers again and divide—the day difference. Halve it and use the result to adjust the initial and terminal rates—each becomes a fixed rate. Accumulate the day difference to adjust the qi's initial fixed rate: after the winter solstice add the difference; after the summer solstice subtract it. This is the daily increase-decrease difference. Then accumulate in sequence; on each day within the qi, add and subtract the qi's difference accumulation—each day's fixed number. For the qi before each solstice, there is no matching difference afterward and they cannot be combined; each takes the prior terminal rate as its initial rate. Apply the qi difference—subtract before the winter solstice, add before the summer solstice—to obtain the terminal rate. The remainder follows the procedure—each yields what is sought.
138
陰曆：
Yin calendar:
139
蝕差：一千二百七十五。
Eclipse difference: 1,275.
140
蝕限：二千五百二十四。
Eclipse limit: 2,524.
141
或限：三千六百五十九。
Optional limit: 3,659.
142
陽曆：
Yang calendar:
143
蝕限：一百三十五。
Eclipse limit: 135.
144
或限：九百七十四。
Optional limit: 974.
145
求蝕差及諸限定數各置其差、限，以蝕朔所入氣日下差積，陰曆減之，陽曆加之，各為蝕定差及定限。
To find the fixed eclipse difference and fixed limits, set each difference and limit, subtract the eclipse new-moon entered-qi day's difference accumulation for the yin calendar and add it for the yang calendar—each yields the fixed eclipse difference and fixed limit.
146
求陰曆陽曆的蝕或蝕其陰曆去交定分滿蝕定差已上，為陰曆蝕。 不滿者，雖在陰曆，皆類同陽曆蝕也。 其去交定分滿蝕定限已下者，其蝕的見。 或限以下者，其蝕或見或不見。
To find yin-calendar and yang-calendar eclipses and or-limits: when the yin-calendar fixed distance-from-conjunction parts fill or exceed the fixed eclipse difference, it is a yin-calendar eclipse. If they do not fill that amount, although in the yin calendar, all are treated like yang-calendar eclipses. If the fixed distance-from-conjunction parts fill or fall below the fixed eclipse limit, the eclipse is certainly visible. If below the optional limit, the eclipse may or may not be visible.
147
求日蝕分陰曆蝕者，置去交定分，以蝕定差減之，餘一百四已下者，皆蝕既。 已上者，以一百四減之，其餘以一百四十三約之，其入或限者，以一百五十二約之。 半已下為半弱，半已上為半強，以減十五，餘為日蝕之大分。 其同陽曆蝕者，但去交定分，少於蝕定差六十已下者，皆蝕既。 六十已上者，置去交定分，以陽曆蝕定限加之，以九十約之。 其陽曆蝕者，直置去交定分，亦以九十約之。 其入或限者，以一百四十三約之。 半已下為半弱，半已上為半強，命以十五為限，亦得日蝕之大分。
To find the solar eclipse magnitude for a yin-calendar eclipse, set the fixed distance-from-conjunction parts and subtract the fixed eclipse difference; if the remainder is 104 or below, the eclipse is total. If above, subtract 104; divide the remainder by 143, or by 152 if within the optional limit. Half or below is a weak half, above half is a strong half; subtract from 15—the remainder is the solar eclipse major fraction. For those treated like yang-calendar eclipses, if the fixed distance-from-conjunction parts are less than the fixed eclipse difference by 60 or below, the eclipse is total. If above 60, set the fixed distance-from-conjunction parts, add the yang-calendar fixed eclipse limit, and divide by 90. For yang-calendar eclipses, set the fixed distance-from-conjunction parts directly and divide by 90. If within the optional limit, divide by 143. Half or below is a weak half, above half is a strong half; assign with 15 as the limit to obtain the solar eclipse major fraction.
148
求日蝕所起月在陰曆，初起西北，甚於正北，復于東北。 月在陽曆，初起西南，甚于正南，復于東南。 其蝕十二分已上，皆起正西，復於正東。 此亦據南方正午而論之。
To find where the solar eclipse begins: when the moon is in the yin calendar, the shadow first appears in the northwest, reaches maximum at true north, and returns toward the northeast. When the moon is in the yang calendar, it first appears in the southwest, reaches maximum at true south, and returns toward the southeast. For eclipses of twelve parts or more, the shadow always begins at true west and returns at true east. This is also reckoned from due south at noon.
149
求日蝕用刻置所蝕之大分，皆因增二。 其陰曆去交定分多於蝕定差七十已上者，又增三十五； 已下者，又增半。 其同陽曆去交定分少於蝕定差二十已下者，又增半； 四十已下者，又增半少。 各為泛月刻半率。
To find the duration of a solar eclipse in ke, set the eclipse major fraction and add two in each case. If the yin-calendar fixed distance-from-conjunction parts exceed the fixed eclipse difference by 70 or more, add another 35; If below that, add another half. For those treated like the yang calendar, if the fixed distance-from-conjunction parts are less than the fixed eclipse difference by 20 or below, add another half; If 40 or below, add another weak half. Each result is the general moon-ke half rate.
150
求日月蝕甚所在辰置去交定分，以交率乘之，二十乘交數除之，所得為差。 其月道與黃道同名者，以差加朔望定小餘； 異名，以差減朔望定小餘，置餘定餘。 如求發斂加時術入之，即蝕甚所在辰刻及分也。 其望甚辰月當沖蝕。
To find the double-hour of greatest eclipse, set the fixed distance-from-conjunction parts, multiply by the conjunction rate, and divide by twenty times the conjunction number—the result is the difference. If the moon's path and the ecliptic share the same name, add the difference to the new- or full-moon fixed small remainder; If they have different names, subtract the difference from the new- or full-moon fixed small remainder and set the result as the fixed remainder. Apply the procedure for finding activation-retraction hour-of-addition—the double-hour, ke, and minutes where greatest eclipse occurs. At full moon, when the greatest-eclipse double-hour arrives, the moon undergoes the opposing eclipse.
151
求虧初復末置日月蝕泛用刻率，副之，以乘其日入轉損益率，如大衍通法而一。 所得應朒者，依其損益； 應朓者，損加益減其副，為定用刻數。 半之，以減蝕甚辰刻，為虧初； 以加蝕甚辰刻，為復末。 其月蝕求入更籌者，置月蝕定用刻數，以其日每更差刻除，為更數； 不盡，以每籌差刻除，為籌數。 綜之為定用更籌。 乃累計日入至蝕甚辰刻置之，以昏刻加日入辰刻減之，餘以更籌差刻除之。 所得命以初更籌外，即蝕甚籌。 半定用更籌減之，為虧初； 以加之，為復末。 按天竺僧俱摩羅所傳斷日蝕法，其蝕朔日度躔于郁車宮者，的蝕。 諸斷不得其蝕，據日所在之宮，有火星在前三後一之宮並伏在日下，並不蝕。 若五星總出，並水見，又水在陰曆，及三星已上同聚一宿，亦不蝕。 凡星與日別宮或別宿則易斷，若同宿則難斷。 更有諸斷，理多煩碎，略陳梗概，不復具詳者。 其天竺所雲十二宮，則中國之十二次也。 曰郁車宮者，即中國降婁之次也。 十二次宿度，首尾具載「曆儀分野」卷中也。
To find first contact and last contact, set the general-use ke rate for the solar or lunar eclipse and duplicate it; multiply by that day's entered-rotation increase-decrease rate and divide by the Dayan universal divisor. If the result corresponds to waning, apply the increase-decrease accordingly; If it corresponds to waxing, reverse the increase-decrease on the duplicate—the fixed-use ke number. Halve it and subtract from the greatest-eclipse double-hour ke—first contact; Add to the greatest-eclipse double-hour ke—last contact. To find the entered watch and tally for a lunar eclipse, set the lunar eclipse fixed-use ke number and divide by that day's ke difference per watch for the watch number; What remains, divide by the ke difference per tally—the tally number. Sum them—the fixed-use watch-tally. Then accumulate from day entry to the greatest-eclipse double-hour ke and set it; add dusk ke to the day-entry double-hour ke and subtract; divide the remainder by the watch-tally ke difference. Take the result and assign from the first watch tally outside the reckoning—the greatest-eclipse tally. Halve the fixed-use watch-tally and subtract—first contact; Add—last contact. According to the solar-eclipse determination method transmitted by the Indian monk Kumara, if at the eclipse new moon the sun's position lodges in the Uttara-kuru palace, an eclipse is certain. When other determinations fail to yield an eclipse, if according to the palace where the sun lodges Mars occupies a palace three before or one after and is concealed below the sun, there is no eclipse. If all five planets are present, Mercury is visible, Mercury is in the yin calendar, and three or more planets gather in one lodge, likewise there is no eclipse. In general, when the planets and sun are in different palaces or lodges the determination is easy; when they share a lodge it is difficult. There are further determinations, mostly tedious in detail; only an outline is given here, not the full particulars. The twelve palaces spoken of in India correspond to China's twelve celestial stations. The palace called Uttara-kuru is China's Descending Harvest station. The lodge degrees of the twelve stations, from beginning to end, are fully recorded in the Calendar Instrument and Field Allocation fascicle.
152
求九服所在蝕差先測所在冬、夏至及春分定日中晷長短、陽城每日中晷常數，校取同者，各因其日蝕差，即為所在冬、夏至及春秋分定日蝕差。
To find the eclipse difference at each of the nine domains, first measure the local winter and summer solstice and spring equinox fixed-day noon gnomon lengths against Yangcheng's daily noon gnomon constants and select matching values; take the eclipse difference for each matching day as the local fixed-day eclipse difference for winter and summer solstice and spring and autumn equinox.
153
求九服所在每氣蝕差以夏至差減春分差，以春分差減冬至差，各為率。 並二率半之，六而一，為夏率。 二率相減，六一為差。 置總差，六而一，為氣。 半氣差，以加夏率，又以總差減之，為冬率。 冬率即是冬至之率也。 每以氣差加之各氣，為每氣定率。 乃循其率，以減冬至蝕差，各得每氣初日蝕差。 求每日，如陽城求之，若戴日之北，當計其所在，皆反之，即得。
To find the eclipse difference for each qi at the nine domains, subtract the spring equinox difference from the summer solstice difference and the winter solstice difference from the spring equinox difference—each yields a rate. Halve the sum of the two rates and divide by six—the summer rate. Subtract the two rates and divide by six—the difference. Set the total difference and divide by six—the qi interval. Halve the qi difference, add it to the summer rate, then subtract the total difference—the winter rate. The winter rate is the Winter Solstice rate. Add the qi difference to each solar term in turn—the fixed rate for each qi. Then apply these rates in sequence to subtract from the Winter Solstice eclipse difference—each yields the first-day eclipse difference for that qi. To find each day, derive by the Yangcheng method; if the locale lies north of the subsolar point, reckon its position and reverse all values—thus obtained.
154
大衍步五星術第七
Dayan Procedure for the Five Planets, Part Seven
155
終率：一百二十一萬二千三百七十九； 秒，十八。
Terminal rate: 1,212,379; second-fractions, 18.
156
終日：三百九十八； 餘，二千六百五十九； 秒，六。
Terminal days: 398; remainder, 2,659; second-fractions, 6.
157
變差算：空； 餘，三十四； 秒，十四。
Variation-difference rod count: nil; remainder, 34; second-fractions, 14.
158
象算：九十一； 餘，二百三十八； 秒，五十七十二。
Image rod count: 91; remainder, 238; second-fractions, 57; micro-fractions, 12.
159
爻算：十五； 餘，一百六十六； 秒，四十六十二。
Line rod count: 15; remainder, 166; second-fractions, 46; micro-fractions, 12.
160
終率：一百一十四萬九千三百九十九； 秒，九十八。
Terminal rate: 1,149,399; second-fractions, 98.
161
終日：三百七十八； 餘，二百七十九； 秒，九十八。
Terminal days: 378; remainder, 279; second-fractions, 98.
162
變差算：空； 餘，二十二； 秒，九十二。
Variation-difference rod count: nil; remainder, 22; second-fractions, 92.
163
象算：九十二； 餘，二百三十七； 秒，八十七。
Image rod count: 92; remainder, 237; second-fractions, 87.
164
爻算：十五； 餘，一百六十六； 秒，三十一。
Line rod count: 15; remainder, 166; second-fractions, 31.
165
終率：一百七十七萬五千三十； 秒，十二。
Terminal rate: 1,775,030; second-fractions, 12.
166
終日：五百八十三； 餘，二千七百一十一； 秒，十二。
Terminal days: 583; remainder, 2,711; second-fractions, 12.
167
中合日：二百九十一； 餘，二千八百七十五； 秒，六。
Mid-conjunction days: 291; remainder, 2,875; second-fractions, 6.
168
變差算：空； 餘，三十； 秒，五十三。
Variation-difference rod count: nil; remainder, 30; second-fractions, 53.
169
象算：九十二； 餘，二百三十八； 秒，三十四五十四。
Image rod count: 92; remainder, 238; second-fractions, 34; micro-fractions, 54.
170
爻算：十五； 餘，一百六十六； 秒，三十九九。
Line rod count: 15; remainder, 166; second-fractions, 39; micro-fractions, 9.
171
終率：三十五萬二千二百七十九； 秒，七十二。
Terminal rate: 352,279; second-fractions, 72.
172
終日：一百一十五； 餘，二千六百七十九； 秒，七十二。
Terminal days: 115; remainder, 2,679; second-fractions, 72.
173
中合日：五十七； 餘，二千八百五十九； 秒，八十六。
Mid-conjunction days: 57; remainder, 2,859; second-fractions, 86.
174
變差算：空； 餘，一百三十六； 秒，七十八六十。
Variation-difference rod count: nil; remainder, 136; second-fractions, 78; micro-fractions, 60.
175
象算：九十一； 餘，二百四十四； 秒，九十八六十。
Image rod count: 91; remainder, 244; second-fractions, 98; micro-fractions, 60.
176
爻算：十五； 餘，一百六十七； 秒，三十九七十四。
Line rod count: 15; remainder, 167; second-fractions, 39; micro-fractions, 74.
177
辰法：七百六十。
Double-hour divisor: 760.
178
秒法：一百。
Second-fraction divisor: 100.
179
微分法：九十六。
Micro-fraction divisor: 96.
180
推五星平合置中積分，以天正冬至小餘減之，各以其星終率去之，不盡者，返以減終率，滿大衍通法為日，不滿為餘，即所求年天正冬至夜半後星平合日算及餘秒也。
To calculate the five planets' mean conjunctions, set the central accumulated fraction and subtract the celestial first month's Winter Solstice small remainder; for each planet remove by its terminal rate, and what does not exhaust in turn subtract from the terminal rate; each full Dayan universal divisor counts as one day and the remainder is the residue—the mean conjunction day count and remainder second-fractions after midnight at the celestial first month's Winter Solstice for the year sought.
181
求平合入爻象曆置積年，各以其星變以差乘之，滿乾實去之，不滿者，以大衍通法約之，為日。 不盡為餘秒。 以減其星冬至夜半後平合日算及餘秒，即平合入曆算數及餘秒也。 各四約其餘，同其辰法也。
To find mean conjunction entry into the line-and-image calendar, set the accumulated years; for each planet multiply its variation-by-difference; cast out full Qian actuality cycles; what does not fill, reduce by the Dayan universal divisor—the days. What is not exhausted is the remainder second-fractions. Subtract from that planet's mean conjunction day count and remainder second-fractions after midnight at the Winter Solstice—the mean conjunction entry calendar rod count and remainder second-fractions. For each, quarter the remainder to align with the double-hour divisor.
182
求平合入四象置曆算數及秒，以一象之算及餘秒除之，所得，依入爻象次命起少陽算外，即平合所入象算數及餘秒也。
To find mean conjunction entry into the four images, set the calendar rod count and second-fractions and divide by one image's rod count and remainder second-fractions; assign from Lesser Yang outside the reckoning in line-and-image sequence—the mean conjunction's entered image rod count and remainder second-fractions.
183
求平合入六爻置所入象算數及餘秒，以一爻之算及餘秒除之，所得，命起其象初爻算外，即平合所入爻算數及餘秒也。
To find mean conjunction entry into the six lines, set the entered image rod count and remainder second-fractions and divide by one line's rod count and remainder second-fractions; assign from that image's first line outside the reckoning—the mean conjunction's entered line rod count and remainder second-fractions.
184
求四象六爻每算損益及進退定數以所入爻與後爻損益率相減為前差，又以後爻與次後爻損益率相減為後差，前後差相減為中差。 置所入爻並後爻損益率，半中差以加之，九之，二百七十四而一，為爻末率，因為後爻初率。 皆因前爻末率，以為後爻初率。 初末之率相減，為爻差。 倍爻差，九之，二百七十四而一為算差。 半之，加減初末，各為定率。 以算差累加減爻初定率，少象以差減，老象以差加。 為每損益率。 循累其率，隨所入爻，損益其下進退，即各得其算定。 其四象初爻無初率，上爻無末率，皆置本爻損益，四而九之，二百七十四而一，各以初末率減之，皆互得其率。 餘依術算，各得所求。
To find the four images' six lines' per-rod increase-decrease and advance-retreat fixed numbers, subtract the entered line's increase-decrease rate from the next line's rate—the forward difference; subtract the next line's rate from the one after—the backward difference; subtract the two differences—the middle difference. Set the entered line and the next line's increase-decrease rates, add half the middle difference, multiply by nine and divide by 274—the line terminal rate, which becomes the next line's initial rate. Each line takes the prior line's terminal rate as the next line's initial rate. Subtract the initial and terminal rates—the line difference. Double the line difference, multiply by nine and divide by 274—the rod difference. Halve it and use the result to adjust the initial and terminal rates—each is a fixed rate. Accumulate the rod difference to adjust each line's initial fixed rate—the young image subtracts the difference and the old image adds it. These are the per-step increase-decrease rates. Apply these rates in sequence; according to the entered line, adjust the advance and retreat below—each rod count is thereby fixed. For the four images, the first line has no initial rate and the top line no terminal rate; in each case set that line's increase-decrease rate, multiply by four and nine and divide by 274, and subtract the result from each initial and terminal rate—in each case obtaining the other's rate. The remainder follows the procedure—each yields what is sought.
185
求平合入進退定數各置其星平合所入爻之算差，半之，以減其所入算損益率。 損者，以所入餘乘限差，辰法除，並差而半之； 益者，半入餘乘差，亦辰法除。 加所減之率，乃以入餘乘之，辰法而一，所得以損益其算下進退，各為平合所入進退定數。 此法微密，用算稍繁。 若從省求之，亦可置其所入算餘，以乘其下損益率，如辰法而一，所得以損益其算下進退，各為定數。
To find the mean conjunction's entered advance-retreat fixed number, for each planet set its mean conjunction entered line rod difference, halve it, and subtract from that entered rod's increase-decrease rate. For a decrease rate, multiply the entered remainder by the limit difference, divide by the double-hour divisor, combine with the difference and halve; For an increase rate, multiply half the entered remainder by the difference and again divide by the double-hour divisor. Add to the reduced rate, multiply by the entered remainder and divide by the double-hour divisor, and use the result to adjust the advance and retreat below—each is the mean conjunction's entered advance-retreat fixed number. This method is subtle and precise, though the reckoning is somewhat elaborate. If one seeks it by the abbreviated method, one may instead set the entered rod remainder, multiply by the increase-decrease rate below it, and divide by the double-hour divisor; use the result to adjust the advance and retreat below the rod—each yields a fixed number.
186
求常合置平合所入進退定數，金星則倍置之。 各以合下乘數乘之，除數除之，所得滿辰法為日，不滿為餘，以進加退減平合日算及餘秒，先以四約平合餘，然以進加退減也。 即為冬至夜半後常合日算及餘也。
To find the regular conjunction, set the mean conjunction's entered advance-retreat fixed number; for Venus, double it. For each planet multiply by the conjunction-below multiplier and divide by the removal divisor; what fills the double-hour divisor counts as one day and the remainder is the residue; add for advance and subtract for retreat from the mean conjunction day count and second-fractions—first quarter the mean conjunction remainder, then apply advance and retreat. That gives the regular conjunction day count and remainder after midnight at the Winter Solstice.
187
求定合置常合日先後定數，四而一，所得滿辰法為日，不滿為餘。 乃以先減後加常合算及餘，即為冬至夜半後定合日算及餘也。
To find the fixed conjunction, set the regular conjunction day's fixed advance-and-retard number and divide by four; what fills the double-hour divisor counts as one day and what does not fill is the remainder. Then first subtract and then add to the regular conjunction count and remainder—that gives the fixed conjunction day count and remainder after midnight at the Winter Solstice.
188
求定合度置其日盈縮分，四而一以定合餘乘之，滿辰法而一，所得以盈加縮減其定餘，以加其日夜半日度餘，先四約夜半日度餘以加之。 滿辰法從度。 依前命之算外，即為定合加時度及餘也。
To find the fixed conjunction degree, set that day's equation-of-time fraction, divide by four and multiply by the fixed conjunction remainder, and divide by the double-hour divisor; add for surplus and subtract for deficit from the fixed remainder; add to that night's midnight solar degree remainder—first quarter the midnight solar degree remainder before adding. When it fills the double-hour divisor, carry into degrees. Assign outside the reckoning as before—that gives the fixed conjunction hour-of-occurrence degree and remainder.
189
求定合月日置冬至夜半後定合日算及餘秒，以天正冬至大小餘加之，天正經朔大小餘減之。 其至、朔小餘，皆以四約之，然用加減。 若至大餘少於經朔大餘者，又以爻數加之，然以經朔大小餘減之。 其餘滿四象之策及餘，除之，為月數，不盡者，為入朔日算及餘。 命月數起天正日算起經朔算外，即定所在日月也。 其定朔大餘有進退，進減退加一日，為在其日月定及餘也。
To find the fixed conjunction month and day, set the fixed conjunction day count and second-fractions after midnight at the Winter Solstice; add the celestial first month's Winter Solstice large and small remainders and subtract the celestial first month's canonical new moon large and small remainders. Quarter the solstice and new-moon small remainders, then use them to add and subtract. If the solstice large remainder is less than the canonical new moon large remainder, add the line count and then subtract the canonical new moon large and small remainders. Divide the remainder by the four-images rod count and remainder to obtain the month count; what is not exhausted is the new-moon entry day count and remainder. Count the months from the celestial first month day count outside the canonical new moon reckoning—that gives the fixed month and day. If the fixed new moon large remainder requires advance or retreat, subtract one day for advance and add one day for retreat—that gives the fixed month and day with its remainder.
190
求定合入爻置常合及定合應加減定數，同名相從，異名相消。 乃以加減其平合入爻算餘，滿若不足，進退其算，即為定合入爻算數及餘也。
To find fixed conjunction entry into the line, set the fixed numbers to be added and subtracted for the regular and fixed conjunctions—same-named quantities combine, differently named quantities cancel. Then add and subtract from the mean conjunction's entered line rod remainder; when it fills or falls short, advance or retreat the rod count—that gives the fixed conjunction entered line rod count and remainder.
191
求變行初日入爻置定合入爻算數及餘，以合後伏下變行度常率加之，滿爻率去之，命爻次如前，即次變初日入爻算數及餘也。 更求次變入爻變入，但以其下行度常加之，去命如上節。
To find the first day of a variation motion's line entry, set the fixed conjunction's entered line rod count and remainder and add the post-conjunction pre-visibility variation-motion degree mean rate; cast out full line rates and assign the line sequence as before—the next variation's first-day entered line rod count and remainder. To find each subsequent variation's line entry, simply add that station's degree mean rate; cast out and assign as in the prior section.
192
求變行初日入進退定數各置其變行初日入爻算數及餘，如平合求進退術入之，即得變行初日所入進退定數也。 置進退定數，各以其下乘數乘之，除數除之，所得各為進退變率。
To find the first-day advance-retreat fixed number for each variation motion, set that variation's first-day entered line rod count and remainder and apply the mean conjunction advance-retreat method—thus obtaining the first-day entered advance-retreat fixed number. Set the advance-retreat fixed numbers; for each multiply by its below-multiplier and divide by the removal divisor—each result is an advance-retreat variation rate.
193
求變行日度率置其本進退變率與後變率，同名者，相消為差。 在進前少，在退前多，各以差為加； 在進前多，在退前少，各以差為減。 異名者，相從謂並。 前退後進，各以並為加； 前進後退，各以並為減。 逆行度率則反之。 皆以差及並，加減日度中率，各為日度變率。 其水星疾行，直以差以並加減度之中率，為變率。 其日直因中率為變率，不煩加減也。
To find variation-motion day-and-degree rates, set the native advance-retreat variation rate and the next variation rate; when they share the same name, cancel them to obtain the difference. When in advance the prior rate is less, when in retreat the prior rate is more—in each case add the difference; When in advance the prior rate is more, when in retreat the prior rate is less—in each case subtract the difference. When differently named, combine them—this is called the sum. Prior retreat and subsequent advance—in each case add the sum; Prior advance and subsequent retreat—in each case subtract the sum. For retrograde degree rates, reverse the rule. In all cases add and subtract the difference and sum from the day-and-degree mean rate—each yields a day-and-degree variation rate. For Mercury's swift motion, directly add and subtract the difference and sum from the degree mean rate to obtain the variation rate. Its days directly take the mean rate as the variation rate, without further addition or subtraction.
194
求變行日度定率以定合日與後變初日先後定數，同名相消為差，異名者相從為並。 四而一，所得滿辰法為度。 乃以盈加縮減其合後伏度之變率及合前伏日之變率。 金水夕合日度，加減反之。 其二留日之變率，若差於中率者，即以所差之數為度，各加減本遲度之變率。 謂以多於中率之數加之，少於中率之數減之。 以下加減准此。 退行度變率，若差於中率者，即倍所差之數，各加減本疾度之變率。 其木土二星，既無遲疾，即加減前後順行度之變率。 其水星疾行度之變率，若差於中率者，即以所差之數為日，各加減留日變率。 其留日變率若少不足減者，即侵減遲日變率也。 各加減變率訖，皆為日度定率。 其日定率有分者，前後輩之。 輩，配也。 以少分配多分，滿全為日，有餘轉配。 其諸變率不加減者，皆依變率為定率。
To find variation-motion day-and-degree fixed rates, take the fixed conjunction day's and the next variation first day's fixed advance-and-retard numbers; same-named quantities cancel to form the difference, differently named quantities combine to form the sum. Divide by four; what fills the double-hour divisor counts as degrees. Then add for surplus and subtract for deficit from the post-conjunction pre-visibility degree variation rate and the pre-conjunction pre-visibility day variation rate. For Venus and Mercury at evening conjunction, reverse the additions and subtractions in day-degree terms. If the variation rate of its two stationary days differs from the mean rate, take the amount of difference as degrees and add or subtract it from the native slow-degree variation rate. That is, add when the amount exceeds the mean rate and subtract when it falls short. For the additions and subtractions below, follow this rule. If the retrograde-degree variation rate differs from the mean rate, double the amount of difference and add or subtract it from the native swift-degree variation rate. Jupiter and Saturn, having no slow and swift phases, simply adjust the before-and-after direct-motion degree variation rates. If Mercury's swift-motion degree variation rate differs from the mean rate, take the amount of difference as days and add or subtract it from the stationary-day variation rate. If the stationary-day variation rate is too small to subtract from, borrow from and subtract the slow-day variation rate. When all variation rates have been adjusted, each becomes a day-and-degree fixed rate. When day fixed rates have fractional parts, allocate them between the prior and subsequent rates. "Match" means allocate. Allocate the smaller fraction to the larger; when it fills a whole unit that makes whole days; if any remainder remains, carry it forward and allocate again. Variation rates that receive no adjustment all follow the variation rate as the fixed rate.
195
求定合後夜半星所在度置其星定合餘，以減辰法，餘以其星初日行分乘之，辰法而一，以加定合加時度餘，滿辰法為度。 依前命之算外，即定合後夜半星所在宿及餘。 自此以後，各依其星，計日行度所至，皆從夜半為始也。 轉求次日夜半星行至：各以其星一日所行度分，順加退減之。 其行有小分者，各滿其法從行分一。 行分滿辰法，從度一。 合之前後，伏不注度，留者因前，退則依減。 順行出虛，去六虛之差； 退行入虛，先加此差。 先置六虛之差，四而一，然用加減。 訖，皆以轉法約行分為度分，各得每日所至。 其三星之行日度定率，或加或減，益疾益遲，每日漸差，難為預定，今且略據日度中率商量置之。 其定率既有盈縮，即差數合隨而增損，當先檢括諸變定率與中率相近者，因用其差，求其初末之日行分為主。 自餘變因此消息，加減其差，各求初末行分。 迴圈比校，使際會參合，衰殺相循。 其金水皆以平行為主，前後諸變，亦准此求之。 其合前伏雖有日度定率，如至合而與後算計卻不葉者，皆從後算為定。 其五星初見伏之度，去日不等，各以日度與星度相校。 木去日十四度，金十一度，火土水各十七度，皆見； 各減一度皆伏。 其木火土三星前順之初，後順之末，又金水疾行、留、退初末，皆是見伏之初日，注曆消息定之。 其金水及日月等度，並棄其分也。
To find the planet's degree after midnight following fixed conjunction, set that planet's fixed conjunction remainder and subtract it from the double-hour divisor; multiply the remainder by that planet's first-day motion parts and divide by the double-hour divisor; add to the fixed conjunction hour-of-occurrence degree remainder; when the result fills the double-hour divisor, that makes degrees. Assign outside the reckoning as before—that gives the lodge and remainder where the planet stands after midnight following fixed conjunction. From here onward, for each planet reckon where the daily degree motion reaches—all reckoning begins from midnight. To find in turn the planet's position at the next night's midnight, for each planet add in direct motion and subtract in retrograde motion according to its one-day degree motion. When the motion has small parts, carry one into motion parts whenever they fill their divisor. When motion parts fill the double-hour divisor, carry one into degrees. Before and after conjunction, do not register degrees during invisibility; at station continue from the prior position; in retrograde subtract accordingly. In direct motion exiting the void, remove the six-void difference; In retrograde motion entering the void, first add this difference. First set the six-void difference and divide by four; then apply the addition and subtraction. When done, in all cases reduce motion parts to degree parts by the rotation divisor—each yields the daily position reached. The three planets' day-and-degree fixed rates may be increased or decreased, growing swifter or slower with daily gradual difference—hard to predetermine; for now we provisionally set them by consulting the day-and-degree mean rate. Since the fixed rates already carry surplus and deficit, the conjunction difference numbers follow with increase and decrease; one should first examine the variation fixed rates nearest the mean rate, use their difference accordingly, and take the first and last days' daily motion parts as the basis. From the remaining variations derive increase and decrease from this basis; add and subtract their difference to find the first and last motion parts for each. Compare in cycles round and round so that junctures meet and combine and waxing and waning follow in sequence. Venus and Mercury both take uniform motion as primary; the variations before and after likewise follow this method. Although the pre-conjunction invisibility period has day-and-degree fixed rates, if upon reaching conjunction the forward reckoning from the posterior count does not fit, all take the posterior reckoning as fixed. The five planets' first appearance and invisibility stand at unequal angular distance from the sun; in each case compare the sun's degree with the planet's degree. Jupiter fourteen degrees from the sun, Venus eleven degrees, Mars, Saturn, and Mercury each seventeen degrees—all are visible; Reduce each by one degree—all are hidden. For Jupiter, Mars, and Saturn, the start of the first direct segment and the end of the last direct segment; and for Venus and Mercury, the start and end of swift motion, station, and retrograde—all mark the first days of appearance and invisibility; register them fixed by the calendar's waxing-and-waning reckoning. For Venus, Mercury, and the sun and moon at equal degrees, discard the fractional parts altogether.
196
求每日差置所差分為實，以所差日為法。 實如法而一，所得為行分，不盡者為小分。 即是也每日差所行分及小分也。 其差若全，不用此術。
To find the daily difference, set the amount differed as dividend and the days differed as divisor. Divide the dividend by the divisor; the quotient is motion parts and the remainder is small parts. These are precisely the daily difference in motion parts and small parts. If the difference is a whole number, do not use this method.
197
求平行度及分置度定率，以辰法乘之，有分者從之，如日定率而一，為平行分。 不盡，為小分。 其行分滿辰法為度，即是一日所行度及分。
To find uniform-motion degrees and parts, set the degree fixed rate, multiply by the double-hour divisor with fractional parts carried along, and divide by the day fixed rate—the result is uniform-motion parts. What is not exhausted is small parts. When motion parts fill the double-hour divisor that makes degrees—this is the one-day motion in degrees and parts.
198
求差行初末日行度及分置日定率減一，以差分乘之。 二而一，為差率，以加減平行分。 益疾者，以差率減平為初日，加平為末日。 益遲者，以差率加平為初日，減平為末日也。 加減訖，即是初末日所行度及分。 其差不全而與日相合者，先置日定率減一，以所差分乘之，為實。 倍所差日為法。 實如法而一，為行分。 不盡者，因為小分，然為差率。
To find differential-motion first and last day motion degrees and parts, set the day fixed rate minus one and multiply by the difference parts. Halve it to obtain the difference rate, then add and subtract from uniform-motion parts. When increasing swiftness, subtract the difference rate from uniform motion for the first day and add uniform motion for the last day. When increasing slowness, add the difference rate to uniform motion for the first day and subtract uniform motion for the last day. When the additions and subtractions are done, that gives the first and last days' motion degrees and parts. When the difference is not whole but agrees with the day count, first set the day fixed rate minus one and multiply by the difference parts—the product is the dividend. Double the days differed—the result is the divisor. Divide the dividend by the divisor—the quotient is motion parts. What is not exhausted becomes small parts and then serves as the difference rate.
199
求差行次日行度及分置初日行分，益遲者，以每日差減之； 益疾者，以每日差加之，即為次日行度及分也。 其每日差、初日行皆有小分，母既不同，當令同之。 然用加減，轉求次日，准此各得所求也。
To find differential-motion next-day motion degrees and parts, set the first-day motion parts; when increasing slowness, subtract the daily difference; When increasing swiftness, add the daily difference—that gives the next day's motion degrees and parts. The daily difference and first-day motion each have small parts; since the denominators differ, make them the same. Then add and subtract and find each next day in turn; follow this rule to obtain each result sought.
200
徑求差行餘日行度及分置所求日減一，以每日差乘之，以加減初日行分，益遲減之，益疾加之。 滿辰法為度，不滿為行分，即是所求日行度及分也。
To find directly the motion degrees and parts for the remaining days of differential motion, set the day sought minus one, multiply by the daily difference, and add or subtract from the first-day motion parts—subtract for increasing slowness, add for increasing swiftness. When the result fills the double-hour divisor that makes degrees; when it does not fill, the remainder is motion parts—this is the daily motion degrees and parts sought.
201
求差行，先定日數，徑求積度及分置所求日減一，次每日差乘之，二而一，所得，以加減初日行分。 益遲減之，益疾加之。 以所求日乘之，如辰法而一，為積度。 不盡者，為行分。 即是從初日至所求日積度及分也。
To find differential motion when the day count is fixed first and to find directly the accumulated degrees and parts, set the day sought minus one, multiply by the daily difference and halve, and use the result to add or subtract from the first-day motion parts. Subtract for increasing slowness; add for increasing swiftness. Multiply by the day sought and divide by the double-hour divisor—that gives accumulated degrees. What is not exhausted is motion parts. That is the accumulated degrees and parts from the first day to the day sought.
202
求差行，先定度數，徑求日數置所求行度，以辰法乘之，有分者從之。 八之，如每日差而一，為積。 倍初日行分，以每日差加減之。 益遲者加之，益疾者減之。 如每日差而一，為率。 今自乘，以積加減之，益遲者以積減之，益疾者以積加之。 開方除之。 所得，以率加減之。 益遲者以率加之，益疾者以率減之。 乃半之，即所求日數也。 其開方除者，置所開之數為實，借一算於實之下，名曰下法。 步之，超一位，置商于上方，副商於下法之上，名曰方法。 命上商以除實，畢，倍方法一折，下法再折，乃置後商於下法之上，名曰隅法。 副隅並方，命後商以除實，畢，隅從方法折下就除，如前開之。 訖除，依上術求之即得也。
To find differential motion when the degree count is fixed first and to find directly the day count, set the degree motion sought and multiply by the double-hour divisor, with fractional parts carried along. Multiply by eight and divide by the daily difference—that gives the accumulation. Double the first-day motion parts and add or subtract the daily difference. Add for increasing slowness; subtract for increasing swiftness. Divide by the daily difference to obtain one—the result is the rate. Now square it; add or subtract the accumulation—when motion grows slower subtract the accumulation; when motion grows faster add the accumulation. Extract the square root and divide. Take the result and add or subtract the rate. When motion grows slower add the rate; when motion grows faster subtract the rate. Then halve it—that gives the number of days sought. For extracting the square root and dividing, set the number to be rooted as the dividend; place one count below the dividend—this is called the lower divisor. Step through it; skip one place; set the quotient above; place a secondary quotient on the lower divisor—this is called the square divisor. Use the upper quotient to divide the dividend; when finished, double the square divisor and fold once, fold the lower divisor twice; then set the next quotient on the lower divisor—this is called the corner divisor. Combine the secondary corner with the square divisor; use the next quotient to divide the dividend; when finished, the corner follows the square divisor, folds down, and divides as before—continue extracting the root in the same way. When division is finished, apply the procedure above and the result is obtained.
203
求星行黃道南北各視其星變行入陰陽爻而定之。 其前變入陽爻為黃道北，入陰爻為黃道南； 後變入陽爻為黃道南，入陰爻為黃道北。 其金水二星，以爻變為前變，各計其變行，起初日入爻之算，盡老象上爻末算之數，不滿變行度常率者，因置其數，以變行日定率乘之，如變行度常率而一，為日。 其入變日數，與此日數以下者，星在黃道南北，依本所入陰陽爻為定。 過此日數之外者，黃道南北則返之。
To find whether a star's motion lies north or south of the ecliptic, observe which yin or yang line its variation motion enters and determine it accordingly. In the preceding variation, entering a yang line means north of the ecliptic; entering a yin line means south of the ecliptic; in the following variation, entering a yang line means south of the ecliptic; entering a yin line means north of the ecliptic. For Venus and Mercury, take the line variation as the preceding variation; for each compute its variation motion from the day count entered at the first day of line entry through the last count of the old image's upper line; when it does not fill the variation motion's degree constant rate, set the number, multiply by the variation motion's daily fixed rate, and divide by the variation motion's degree constant rate to obtain days. When the days entered in variation fall at or below this day count, the star's position north or south of the ecliptic is fixed according to the yin or yang line originally entered. Beyond this day count, north and south of the ecliptic are reversed.