1
志第十二歷六
Treatise Twelve: Calendrics, Part Six.
2
大統曆法三下推步
▲ Grand Concordance Calendar Method, Part Three (Lower): Stepwise Computation.
3
▲步交食
▲ Stepwise Computation: Nodes and Eclipses
4
交週日二十七日二十一刻二二二四。 半之爲交中日。
Nodal period in days: 27d 21 quarter-hours 2224 parts. Halve this for the half-nodal interval in days.
5
交終度三百六十三度七九三四一九六。 半之爲交中日度。
Nodal arc for a complete cycle: 363.7934196°. Halve this for the half-nodal arc in degrees.
6
正交度三百五十七度六四。
Major-node longitude: 357.64°.
7
中交度一百八十八度零五。
Middle-node longitude: 188.05°.
8
前準一百六十六度三九六八。
Forward criterion: 166.3968°.
9
後準一十五度五。
Backward criterion: 15.05°.
10
交差二日三一八三六九。
Nodal offset: 2d 318369 parts.
11
交望一十四日七六五二九六五。
Nodal half-cycle interval: 14d 7652965 parts.
12
日食陽曆限六度。 定法六十。
Solar eclipse, yang-calendar limit: 6°. Standardizing divisor: 60.
13
日食陰曆限八度。 定法八十。
Solar eclipse, yin-calendar limit: 8°. Standardizing divisor: 80.
14
月食十三度五分。 定法八十七。
Lunar eclipse limit: 13°5. Standardizing divisor: 87.
15
陽食限視定朔入交。
Solar-eclipse limit: inspect the true new moon's nodal entry.
16
零日六零已下一十三日一零已上在一十四日，不問小余，皆入食限。
From 0d60 parts downward, or from 13d10 upward, through day 14 inclusive: regardless of fractional remainder, all fall within the solar-eclipse limit.
17
一十五日二零已下二十五日六零已上在二十六日、二十七日，不問小余，皆入食限。
From 15d20 downward, or from 25d60 upward, on days 26–27 regardless of fractional remainder: all within the limit.
18
▲陰食限視定望入交。
▲ Lunar-eclipse limit: inspect the true full moon's nodal entry.
19
一日二零已下一十二日四零已上在零日一十三日，不問小余，皆入食限。 又視定朔小余在日出前、日入後二十分已上者，日食在夜。 定望小余在日入前、日出後八刻二十分已上者，月食在晝。 皆不必布算。
From 1d20 downward, or from 12d40 upward, on days 0 and 13 regardless of fractional remainder: all within the limit. Also: if the true new moon's fractional day is 20 parts or more before sunrise, or after sunset, the solar eclipse occurs at night. If the true full moon's fractional day is 8 quarter-hours 20 parts or more before sunset, or after sunrise, the lunar eclipse occurs in daytime. None of these require full tabular computation.
20
推日食用數
Procedure: Quantities for Solar-Eclipse Computation
21
推交常度置有食之朔入交凡分，以月平行度乘之，即得。
Derive the mean nodal longitude: set the general nodal-entry parts for the eclipse new moon, multiply by mean lunar motion, and the product is obtained.
22
推交定度置交常度，以朔下盈縮差盈加縮減之，即得。
Derive the true nodal longitude: set the mean value; add or subtract the sub-conjunction expansion–contraction difference (surplus add, deficit subtract).
23
推日食正交限度視交定度在七度已下，三百四十一度已上者，食在正交。 在一百七十五度已上，二百零二度已下者，食在中交。 不在限內不食。
Derive the major-node eclipse limit: if true nodal longitude is ≤7° or ≥341°, the eclipse occurs at the major node. If between 175° and 202° inclusive, the eclipse is at the middle node. Outside these bounds there is no eclipse.
24
推中前中後分視定朔小余，在半日周已下，用減半日周，餘爲中前分。 在半日周已上，減去半日周，餘爲中後分。
Derive pre-noon, noon, and post-noon fractions: inspect the true new moon's fractional day; if below half the diurnal circuit, subtract from half the diurnal circuit—the remainder is the pre-noon fraction. If above half the diurnal circuit, subtract half the diurnal circuit—the remainder is the post-noon fraction.
25
推時差置半日擊，以中前、中後分減之，餘以中後分乘之，所得以九千六百而一爲時差。 在中前爲減，中後爲加。
Derive the hour equation: set the half-day divisor, subtract the pre- or post-noon fraction, multiply the remainder by the post-noon fraction, and divide by 9600 for the hour equation. Apply as subtraction before noon, addition after noon.
26
推食甚定分置定朔小余，以時差加減之，即得。
Derive the greatest-eclipse fractional day: set the true new moon's fractional day and apply the hour equation.
27
推距午定分置中前、中後分，加時差即得。 但加不減。
Derive the meridian-distance fraction: set the pre- or post-noon fraction and add the hour equation. Add only—never subtract.
28
推食甚入盈縮歷置原得盈縮歷，加入定朔大餘及食甚定分，即得。
Derive greatest-eclipse entry into the expansion–contraction sequence: set the prior sequence entry, add the true new moon's day count and greatest-eclipse fraction.
29
推食甚盈縮差依步氣朔求之。
Derive the greatest-eclipse expansion–contraction difference by the solar-term and lunation procedure.
30
推食甚入盈縮歷行定度置食甚入盈縮歷，盈縮差，盈加縮減之，即得。
Derive the greatest-eclipse fixed solar longitude on the expansion–contraction track: set the sequence entry and apply the difference (surplus add, deficit subtract).
31
推南北凡差視食甚人盈縮歷行定度，在周天象限已下爲初限，已上與半歲周相減爲末限。 以初末限自之，如一千八百七十度而一，得數，置四度四十六分減之，餘爲南北凡差。
Derive the general north–south parallax: inspect the fixed solar longitude at greatest eclipse; below the circumpolar image limit is the initial limit, above it subtract from half the tropical year for the terminal limit. Square the active limit, divide by 1870; subtract the result from 4°46′—the remainder is the general north–south parallax.
32
推南北定差置南北凡差，以距午定分乘之，如半晝分而一，以減凡差，餘爲南北定差。 若凡差數少，即反減之。 盈初縮末食在正交爲減，中交爲加。 縮初盈末，食在正交爲加，中交爲減。 如系凡差反減而得者，則其加減反是。
Derive the true north–south parallax: multiply the general value by the meridian-distance fraction, halve by the half-daylight fraction, subtract from the general value—the remainder is the true parallax. When the general value is small, reverse the subtraction. At initial surplus and terminal deficit: subtract at the major node, add at the middle node. At initial deficit and terminal surplus: add at the major node, subtract at the middle node. When the general value required reverse subtraction, reverse the addition and subtraction rules accordingly.
33
推東西凡差置半歲周，減去食甚入盈縮歷行定度，餘食甚入盈縮歷行定度乘之，以一千八百七十除之爲度，即東西凡差。
Derive the general east–west parallax: set half the tropical year, subtract the fixed solar longitude at greatest eclipse, multiply the remainder by that longitude, and divide by 1870 for degrees.
34
推東西定差置東西凡差，以距午定分乘之，如二千五百度而一，視得數在東西凡差以下，即爲東西定差。 若在凡差已上，倍凡差減之，餘爲定差。 盈歷中前，縮歷後者，正交減，中交加。 盈歷中後，縮中前者，正交加，中交減。
Derive the true east–west parallax: multiply the general value by the meridian-distance fraction and divide by 2500; if the quotient does not exceed the general value, it is the true parallax. If above the general value, subtract the quotient from twice the general value—the remainder is the true parallax. Expansion sequence before noon, contraction after: subtract at the major node, add at the middle node. Expansion after noon, contraction before: add at the major node, subtract at the middle node.
35
推正交中定限度視日食在正交者置正交度，在中交者置中交度，以南北東西二定差加減之，即得。
Derive the true major- or middle-node limit: for major-node eclipses set the major-node degree, for middle-node set the middle-node degree, then apply the true north–south and east–west parallaxes.
36
推日食入陰陽曆去閃前後度視交定在正交定限度已下，減去交定度，餘爲陰曆交前度。 已上，減去正交定限度，餘爲陽曆交後度。 在中交定限度已下，減去交定度，餘爲陽曆閃前度。 已上，減去中交定限度，餘爲陰曆後度。 若交定在七度已下者加交終度，減去正交定限度，餘爲陽曆交後度。
Derive the solar eclipse's yang/yin-calendar offset before and after the penumbral gleam: if true nodal longitude is below the major-node true limit, subtract it—the remainder is the yin-calendar pre-node arc. If above, subtract the major-node true limit—the remainder is the yang-calendar post-node arc. If below the middle-node true limit, subtract the true nodal longitude—the remainder is the yang-calendar pre-gleam arc. If above, subtract the middle-node true limit—the remainder is the yin-calendar post-node arc. If true nodal longitude is ≤7°, add the nodal-cycle arc and subtract the major-node true limit—the remainder is the yang-calendar post-node arc.
37
推日食分秒在陽曆者，置陽食限六度，減去陽曆交前、交後度，不及減者，不食。 陰曆同。 餘以定法六十而一。 在陰曆者，置陰食限八度，減去陰曆交前、交後度，餘以定法八十而一，即得。
Derive the solar-eclipse magnitude: in the yang calendar, set the 6° yang limit and subtract the pre- and post-node yang arcs; if subtraction fails, there is no eclipse. The yin-calendar case is the same. Divide the remainder by the standardizing divisor 60. In the yin calendar, set the 8° yin limit, subtract the pre- and post-node yin arcs, and divide the remainder by 80 to obtain the magnitude.
38
推定用分置日食分秒與二十分相減相乘，爲開方積。 以平方法開之，爲開方數。 用五千七百四十分七因八百二十分也。 乘之，如定限行度而一，即得。
Derive the fixed duration fraction: set the eclipse magnitude and 20 parts, subtract and multiply to form the square-root product. Take the square root by the mean-square method to obtain the root value. Multiply by 5740 parts (7 × 820). Multiply, then divide by the fixed travel-limit degree to obtain the result.
39
推初虧復圓時刻置食甚定分，以定用分減爲初虧，加爲復圓。 各依發斂加時，即時刻。
Derive the first-contact and last-contact fractional hours: set the greatest-eclipse fraction; subtract the fixed duration for first contact, add for last contact. Apply emission-and-gathering to each to obtain the clock time.
40
推日食起復方位陽曆初虧西南，甚於正南，復於東南。 陰曆初虧西北，甚於正北，復於東北。 若在八分以上，不分陰陽曆皆虧正西，復東位。 據午地而論
Derive solar-eclipse contact azimuths: yang calendar—first contact southwest, greatest due south, last contact southeast. Yin calendar—first contact northwest, greatest due north, last contact northeast. If the magnitude reaches 0.8° or more, regardless of calendar branch, first contact is due west and last contact due east. Reckoned from the local meridian at noon.
41
推食甚日躔黃道宿次置食甚入盈縮歷行定度，在盈就爲定積度，在縮加半歲周爲定積度。 置定積度，以歲前冬至加時黃道日度加之，滿黃道積度鈐去之，至不滿宿次即食甚日躔。
Derive the Sun's ecliptic lodge at greatest eclipse: set the fixed solar longitude at greatest eclipse; in surplus it is the fixed accumulation, in deficit add half the tropical year. Set the fixed accumulation, add the ecliptic solar longitude at the pre-year winter solstice hour, discard the ecliptic cycle when full, and the lodge not yet filled is the Sun's position at greatest eclipse.
42
推日帶食視初虧食甚分，有在日出分已下，爲晨刻帶食。 食甚復圓分，有在日入分已上，爲昏刻帶食。 在晨置日出分，在昏昏置日入分，皆以食甚分與之相減，餘爲帶食差。 置帶帶差，以日食分秒乘之，以定用分而一，所得減日食分秒，餘爲所見帶食分秒。
Derive a solar eclipse at sunrise or sunset: if first contact or greatest eclipse falls below the sunrise fraction, it is a dawn eclipse at sunrise. If greatest eclipse or last contact falls above the sunset fraction, it is a dusk eclipse at sunset. At dawn set the sunrise fraction; at dusk set the sunset fraction; subtract the greatest-eclipse fraction in each case—the remainder is the horizon-eclipse offset. Set the horizon-eclipse offset, multiply by the eclipse magnitude, divide by the fixed duration fraction, and subtract from the magnitude—the remainder is the visible horizon-eclipse magnitude.
43
▲推月食用數
▲ Procedure: Quantities for Lunar-Eclipse Computation
44
盈縮歷
Expansion–contraction sequence
45
盈縮差
Expansion–contraction difference
46
遲疾歷
Slow–rapid sequence
47
遲疾差
Slow–rapid difference
48
加減差
Additive–subtractive difference
49
入交凡分
General nodal-entry parts
50
定入遲疾歷
Fixed entry into the slow–rapid sequence
51
定限行度
Fixed travel-limit degree
52
日出分
Sunrise fraction
53
日入分
Sunset fraction
54
▲歲前冬至加時黃道宿次
▲ Ecliptic lodge at the pre-year winter solstice hour of addition
55
推交常度置望下入交凡分，乘月平行，如日食法。
Derive the mean nodal longitude: set the general nodal-entry parts below the full moon, multiply by mean lunar motion, as for solar eclipses.
56
推交定度置交常度，以望下盈縮差盈加縮減之即得。 不及減者，加交終度減之。
Derive the true nodal longitude: set the mean value; add or subtract the sub-full-moon expansion–contraction difference (surplus add, deficit subtract). If subtraction fails, add the nodal-cycle arc and subtract again.
57
推食甚定分不用時差，即以定望分爲食甚分。
Derive the greatest-eclipse fraction: omit the hour equation; take the true full moon's fractional day directly as the greatest-eclipse fraction.
58
推食甚入盈縮歷行定度法同推日食。
Derive the greatest-eclipse fixed solar longitude on the expansion–contraction track: same procedure as for solar eclipses.
59
推月食入陰陽曆視交定度在交中度已下爲陽曆，已上減去交中度，餘爲隊歷。
Derive the lunar eclipse's yang/yin calendar branch: if true nodal longitude is below the middle-node degree it is the yang calendar; if above, subtract the middle-node degree—the remainder is the yin calendar.
60
推交前交後度視所得入陰陽曆，在後準已下爲交後，在前準已上置交中度減之，餘爲交前。
Derive the pre- and post-node arcs: inspect the yang/yin branch obtained; if below the backward criterion it is post-node; if above the forward criterion, set the middle-node degree and subtract—the remainder is pre-node.
61
推月食分秒置月食限一十三度零五，減去前交後度，不及減者不食。 餘以定法八十七分而一，即得。
To find the lunar eclipse magnitude: set the lunar eclipse limit at 13°0.5 and subtract the pre- or post-node degree; if subtraction fails, there is no eclipse. Divide the remainder by the fixed divisor of 87 parts to obtain the result.
62
推月食用分置三十分，與月食分秒相減相乘，爲開方積。 依平方法開之，爲開方數。 又以四千九百二十乃六因八百二十分數。 分乘之，如定限行度而一，即得。
To find the lunar eclipse usable parts: set 30 parts, then subtract and cross-multiply with the eclipse magnitude to form the square-root product. Extract the square root by the standard method to obtain the root number. Multiply again by 4920, which is six times the 820-part factor. Multiply by that value and scale by the fixed-limit travel degrees to obtain the result.
63
推月食三限初虧、食甚、復圓。 時刻置食甚分定分，以用分減爲初虧，加爲復圓。 依發斂得時刻如日食。
To find the three lunar eclipse limits: first contact, greatest eclipse, and last contact. For the times: set the greatest-eclipse fixed parts; subtract the usable parts for first contact and add them for last contact. Derive the clock time by emission and collection, as for a solar eclipse.
64
推月食五限時刻月食十分已上者，用五限推之，初虧、食既、食甚、生光、復圓也。 置月食分秒，減去十分，餘與十分相減相乘，爲開方積。 平方開之，爲開方數。 又以四千九百二十分乘之，如定限行度而一爲既內分。 與定用分相減，餘爲既外分。 置食甚定分，減既內分爲既分，又減既外分爲初虧分。 再置食甚定分，加既內分爲生光分，又加既外分爲復圓分。 各依以斂得時刻。
To find the five-limit times for a lunar eclipse: when the eclipse magnitude is 10 tenths or greater, apply the five limits—first contact, totality, greatest eclipse, emergence of light, and last contact. Set the lunar eclipse magnitude, subtract 10 tenths, and with the remainder cross-multiply by 10 tenths to form the square-root product. Take the square root to obtain the root number. Multiply again by 4920 parts and scale by the fixed-limit travel degrees to obtain the inner-totality parts. Subtract from the fixed usable parts; the remainder gives the outer-totality parts. Set the greatest-eclipse fixed parts; subtract the inner-totality parts for totality, then subtract the outer-totality parts for first contact. Set the greatest-eclipse fixed parts again; add the inner-totality parts for emergence of light, then add the outer-totality parts for last contact. Convert each limit to clock time by emission and collection.
65
推更點置晨分們之，五分之爲更法，又五分之爲點法。
To find the night watches and marks: set the dawn parts and halve them for the watch divisor, then halve again for the mark divisor.
66
推月食入更點各置三限或五限，在昏分已上減去昏分，在晨分已下加入晨分，不滿更法爲初更，不滿點法爲一點，以次求之，各得更點之數。
To find the eclipse's watch and mark: for each three- or five-limit time, subtract dusk parts if above dusk, add dawn parts if below dawn; the remainder modulo the watch divisor gives the watch, modulo the mark divisor gives the mark—continue in sequence for each limit.
67
推月食起復方位陽曆初虧東北，甚於正北，復於西北。 陰曆初虧東南，甚於正南，復於西南。 若食在八分已上者，皆初虧正東，復於正西。
To find the eclipse's beginning and ending azimuth: in the yang sequence, first contact lies northeast, greatest at due north, and last contact northwest. In the yin sequence, first contact lies southeast, greatest at due south, and last contact southwest. When the eclipse reaches 8 tenths or more, first contact is due east and last contact due west in every case.
68
推食甚月離黃道宿次置食甚入盈縮歷定度，在盈加半周天，在縮減去七十五秒爲定積度。 置定積度，加歲前冬至加時黃道日度，以黃道積度鈐去之，即得。
To find the Moon's ecliptic lodge at greatest eclipse: set the greatest-eclipse entry into the expansion–contraction ephemeris fixed degree; in expansion add half the celestial circuit, in contraction subtract 75 seconds to obtain the fixed accumulated degree. Set the fixed accumulated degree, add the ecliptic solar degree at the prior year's winter solstice moment, and subtract successive entries from the ecliptic accumulation tally to obtain the result.
69
推月帶食視初虧、食甚、復圓等分，在日入分以下，爲昏刻帶食。 在日出分已上，爲晨刻帶食。 推法同日食。
To determine whether the Moon rises or sets during the eclipse: if first contact, greatest eclipse, or last contact falls below the sunset parts, the eclipse accompanies dusk. If it falls above the sunrise parts, the eclipse accompanies dawn. The procedure matches that for solar eclipses.
70
▲步五星
▲ Stepping the Five Planets
71
歷度三百六十五度二五七五，半之爲歷中，又半之爲歷策。
Ephemeris arc: 365.2575°; halve it for the ephemeris midpoint, and halve again for the ephemeris seed.
72
△木星
△ Jupiter
73
合應二百四十三萬二三零一。 置中積三億七千六百一十九萬七七五，加辛巳合應一百一十九七二六，得三億七行七百三十七萬九五零一，滿木星周率去之，餘爲《大統》合應。
Conjunction correspondence: 2,430,2301. Set the central accumulation at 376,199,7775, add the xinsi conjunction correspondence of 119,726 to obtain 377,737,9501, and discard the Jupiter cycle rate; the remainder is the Grand Concordance conjunction correspondence.
74
歷應五百三十八萬二五七七二二一五。 置中積，加辛巳歷應一千八百九十九萬九四八一，得三億九千五百一十九萬娥二五六，滿木星曆率去之，餘爲《大統》歷應。
Ephemeris correspondence: 5,382,5772215. Set the central accumulation, add the xinsi ephemeris correspondence of 18,999,9481 to obtain 395,199,6256, and discard the Jupiter ephemeris rate; the remainder is the Grand Concordance ephemeris correspondence.
75
周率三百九十八萬八八。
Cycle rate: 3,988,8800.
76
歷率四千三百三十一萬二九六四八六五。
Ephemeris rate: 43,312,964865.
77
度率一十一萬八五八二。
Degree rate: 118,582.
78
伏見一十三度。
Arc of concealment and appearance: 13°.
79
初行率
Initial motion rate
80
一十六日八六三度八六
16 days 86 parts; 3°86 parts
81
二度九三
2°93 parts
82
二十三分
23 parts
83
晨疾初二十八日
Morning fast, initial: 28 days
84
六度二一
6°21 parts
85
四度六四
4°64 parts
86
二十二分
22 parts
87
晨疾末二十八日
Morning fast, final: 28 days
88
五度五一
5°51 parts
89
四度六四
4°64 parts
90
二十二分
22 parts
91
晨遲初二十八日
Morning slow, initial: 28 days
92
四度三一
4°31 parts
93
三度二八
3°28 parts
94
一十八分
18 parts
95
晨遲末二十八日
Morning slow, final: 28 days
96
一度九一
1°91 parts
97
一度四五
1°45 parts
98
一十二分
12 parts
99
二十四日
24 days
100
四十六日五八四度八八一二五零度三二八七五
46 days 58 parts; 4°88125 parts; 0°32875 parts
101
四十六日五八四度八八一二五零度三二八七五一十六分
46 days 58 parts; 4°88125 parts; 0°32875 parts; 16 parts
102
二十四日
24 days
103
夕遲初二十八日
Evening slow, initial: 28 days
104
一度九一
1°91 parts
105
一度四五
1°45 parts
106
夕遲末二十八日
Evening slow, final: 28 days
107
四度三一
4°31 parts
108
三度二八
3°28 parts
109
一十二分
12 parts
110
夕疾初二十八日
Evening fast, initial: 28 days
111
五度五一
5°51 parts
112
四度一九
4°19 parts
113
一十八分
18 parts
114
夕疾末二十八日
Evening fast, final: 28 days
115
六度一一
6°11 parts
116
四度六四
4°64 parts
117
二十一分
21 parts
118
一十六日八六三度八六
16 days 86 parts; 3°86 parts
119
二度九三
2°93 parts
120
二十二分
22 parts
121
△火星
△ Mars
122
合應二百四十零萬一四。 置中積，加辛巳合應五十六萬七五四五，得三億七千六百七十六萬七三二，滿火星周率去之，爲《大統》合應。 中積見木星，五星並同。
Conjunction correspondence: 2,400,1400. Set the central accumulation, add the xinsi conjunction correspondence of 567,545 to obtain 376,767,732, and discard the Mars cycle rate to obtain the Grand Concordance conjunction correspondence. The central accumulation follows the Jupiter procedure; the same method applies to all five planets.
123
歷應三百八十四萬五七八九三五。 置中積，加辛巳歷應五百四十七萬二九三八，得三億八千一百六十七萬二七一三，滿火星曆率去之。
Ephemeris correspondence: 3,845,78935. Set the central accumulation, add the xinsi ephemeris correspondence of 5,472,938 to obtain 381,672,713, and discard the Mars ephemeris rate.
124
周率七百七十九萬九二九。
Cycle rate: 7,799,9200.
125
歷率六百八十六萬九五八零四三。
Ephemeris rate: 6,869,58043.
126
度率一萬八八零七五。
Degree rate: 18,88075.
127
伏見一十九度。
Arc of concealment and appearance: 19°.
128
初行率
Initial motion rate
129
六十九日
69 days
130
五十度
50°
131
四十六度五零七十三分
46°50 parts; 73 parts
132
晨疾初五十九日
Morning fast, initial: 59 days
133
四十一度八零三十八度八七七十二分
41°80 parts; 38°87 parts; 72 parts
134
晨疾末五十七日
Morning fast, final: 57 days
135
三十九度零八三十六度三四七十分
39°08 parts; 36°34 parts; 70 parts
136
晨次疾初五十三日
Morning second-fast, initial: 53 days
137
三十四度一六三十一度七七六十七分
34°16 parts; 31°77 parts; 67 parts
138
晨次疾末四十七日
Morning second-fast, final: 47 days
139
二十七度零四二十五度一五六十二分
27°04 parts; 25°15 parts; 62 parts
140
晨遲初三十九日
Morning slow, initial: 39 days
141
一十七度七二一十六度四八五十三分
17°72 parts; 16°48 parts; 53 parts
142
晨初末二十九日
Morning beginning, final: 29 days
143
六度二零
6°20 parts
144
五度七七
5°77 parts
145
三十八分
38 parts
146
二十八日六九四五八度六五六七五六度四六三二五
28 days 6945 parts; 8°65675 parts; 6°46325 parts
147
二十八日九六四五八度六五六七五六度四六三二五四十四分
28 days 9645 parts; 8°65675 parts; 6°46325 parts; 44 parts
148
夕遲初二十九日
Evening slow, initial: 29 days
149
六度二零
6°20 parts
150
五度七七
5°77 parts
151
夕遲末三十九日
Evening slow, final: 39 days
152
一十七度七二一十六度四八三十八分
17°72 parts; 16°48 parts; 38 parts
153
夕次疾初四十七日
Evening second-fast, initial: 47 days
154
二十七度零四二十五度一五五十三分
27°04 parts; 25°15 parts; 53 parts
155
夕遲疾末五十三日
Evening slow-fast, final: 53 days
156
三十四度一六三十一度七七六十二分
34°16 parts; 31°77 parts; 62 parts
157
夕疾初五十七日
Evening fast, initial: 57 days
158
三十九度零八三十六度三四六十七分
39°08 parts; 36°34 parts; 67 parts
159
夕疾末五十九日
Evening fast, final: 59 days
160
四十一度八零三十八度八七七十分
41°80 parts; 38°87 parts; 70 parts
161
六十九日
69 days
162
五十度
50°
163
四十六度五零七十二分
46°50 parts; 72 parts
164
△土星
△ Saturn
165
合應二百零六萬四七三四。 置中積，加辛巳合應一十七萬五六四三，得三億七千六百三十七萬五四一八，滿土星周率去之。
Conjunction correspondence: 2,064,734. Set the central accumulation, add the xinsi conjunction correspondence of 175,643 to obtain 376,375,418, and discard the Saturn cycle rate.
166
歷應一億零六百零零萬三七九九零二。 置中積，加辛巳歷應五千二百二十四萬零五六一，得四億二千八百四十四萬零三三六，滿土星曆率去之。
Ephemeris correspondence: 106,003,79902. Set the central accumulation, add the xinsi ephemeris correspondence of 52,240,0561 to obtain 428,440,0336, and discard the Saturn ephemeris rate.
167
周率三百七十八萬零九一六。
Cycle rate: 3,780,916.
168
歷率一億零七百四十七萬八八四五六六。
Ephemeris rate: 107,478,84566.
169
度率二十九萬四二五五。
Degree rate: 294,255.
170
伏見一十八度。
Arc of concealment and appearance: 18°.
171
初行率
Initial motion rate
172
二十日四零
20 days 40 parts
173
二度四零
2°40 parts
174
一度四九
1°49 parts
175
一十二分
12 parts
176
三十一日
31 days
177
三度四零
3°40 parts
178
二度一一
2°11 parts
179
一十一分
11 parts
180
晨次疾二十九日
Morning second-fast: 29 days
181
二度七五
2°75 parts
182
一度七一
1°71 parts
183
一十分
10 parts
184
二十六日
26 days
185
一度五零
1°50 parts
186
零度八三
0°83 parts
187
三十日
30 days
188
晨退五十二日六四五八三度六二五四五零度二八四五五
Morning retrograde: 52 days 6458 parts; 3°62545 parts; 0°28455 parts
189
夕退五十二日六四五八三度六二五四五零度二八四五五一十分
Evening retrograde: 52 days 6458 parts; 3°62545 parts; 0°28455 parts; 10 parts
190
三十日
30 days
191
二十六日
26 days
192
一度五零
1°50 parts
193
零度八三
0°83 parts
194
夕次疾二十九日
Evening second-fast: 29 days
195
二度七五
2°75 parts
196
一度七一
1°71 parts
197
三十一日
31 days
198
三度四零
3°40 parts
199
二度一一
2°11 parts
200
一十分
10 parts
201
二十日四零
20 days 40 parts
202
二度四零
2°40 parts
203
一度四九
1°49 parts
204
一十一分
11 parts
205
△金星
△ Venus
206
合應二百三十七萬九四一五。 置中積，加辛巳合應五百七十一萬六三三零，得三億八千一百九十一萬六一零五，滿金星周率去之。
Conjunction correspondence: 2,379,415. Set the central accumulation, add the xinsi conjunction correspondence of 5,716,330 to obtain 381,916,105, and discard the Venus cycle rate.
207
歷應一十零萬四一八九。 置中積，加辛巳歷應一十一萬九六三九，得三億七千六百三十一萬九四一四，滿金星曆率去之。
Ephemeris correspondence: 1,004,189. Set the central accumulation, add the xinsi ephemeris correspondence of 119,639 to obtain 376,319,414, and discard the Venus ephemeris rate.
208
周率五百八十三萬九零二六。
Cycle rate: 5,839,026.
209
歷率三百六十五萬二五七五。
Ephemeris rate: 3,652,575.
210
度率一萬。
Degree rate: 10,000.
211
伏見一十度半
Arc of concealment and appearance: 10½°.
212
初行率
Initial motion rate
213
三十九日
39 days
214
四十九度五零四十七度六四一度二七五
49°50 parts; 47°64 parts; 1°275 parts
215
夕疾初五十二日
Evening fast, initial: 52 days
216
六十五度五零六十三度零四一度二七五
65°50 parts; 63°04 parts; 1°275 parts
217
夕疾末四十九日
Evening fast, final: 49 days
218
六十一度
61°
219
五十八度七一一度二五五
58°71 parts; 1°255 parts
220
夕次疾初四十二日
Evening second-fast, initial: 42 days
221
五十度二五
50°25 parts
222
四十八度三六一度二三五
48°36 parts; 1°235 parts
223
夕次疾末三十九日
Evening second-fast, final: 39 days
224
四十二度五零四十度九零
42°50 parts; 40°90 parts
225
一度一六
1°16 parts
226
夕遲初三十三日
Evening slow, initial: 33 days
227
二十七度
27°
228
二十五度九九一度零二
25°99 parts; 1°02 parts
229
夕初末一十六日
Evening beginning, final: 16 days
230
四度二五
4°25 parts
231
四度零九
4°09 parts
232
六十二分
62 parts
233
一十日九五三一三度六九八七一度五九一三
10 days 9531 parts; 3°6987 parts; 1°5913 parts
234
夕退伏六日
Evening retrograde concealment: 6 days
235
四度三五
4°35 parts
236
一度六三
1°63 parts
237
六十一分
61 parts
238
合退伏六日
Conjunction retrograde concealment: 6 days
239
四度三五
4°35 parts
240
一度六三
1°63 parts
241
八十二分
82 parts
242
一十日九五三一三度六九八七一度五九一三六十一分
10 days 9531 parts; 3°6987 parts; 1°5913 parts; 61 parts
243
晨遲初一十六日
Morning slow, initial: 16 days
244
四度二五
4°25 parts
245
四度零九
4°09 parts
246
晨遲末三十三日
Morning slow, final: 33 days
247
二十七度
27°
248
二十五度九九六十二分
25°99 parts; 62 parts
249
晨次疾初三十九日
Morning second-fast, initial: 39 days
250
四十二度五零四十度九零
42°50 parts; 40°90 parts
251
一度零二
1°02 parts
252
晨次疾末四十二日
Morning second-fast, final: 42 days
253
五十度二五
50°25 parts
254
四十八度三六一度一六
48°36 parts; 1°16 parts
255
晨疾初四十九日
Morning fast, initial: 49 days
256
六十一度
61°
257
五十八度七一一度二三五
58°71 parts; 1°235 parts
258
晨疾末五十二日
Morning fast, final: 52 days
259
六十五度五零六十三度零四一度二五五
65°50 parts; 63°04 parts; 1°255 parts
260
三十九日
39 days
261
四十九度五零四十七度六四一度二六五
49°50 parts; 47°64 parts; 1°265 parts
262
△水星
△ Mercury
263
合應三十零萬三二一二。 置中積，加辛巳合應七十零萬零四三七，得三億七千六百九十零萬零二一二，滿水星周率去之。
Conjunction correspondence: 3,003,212. Set the central accumulation, add the xinsi conjunction correspondence of 700,0437 to obtain 376,900,0212, and discard the Mercury cycle rate.
264
歷應二百零三萬九七一一。 置中積，加辛巳歷應二百零五萬五一六一，得三億七千八百二十五萬四九三六，滿水星曆率去之。
Ephemeris correspondence: 2,039,711. Set the central accumulation, add the xinsi ephemeris correspondence of 2,055,161 to obtain 378,254,936, and discard the Mercury ephemeris rate.
265
周率一百一十五萬八七六。
Cycle rate: 1,158,876.
266
歷率三百六十五萬二五七五。
Ephemeris rate: 3,652,575.
267
度率一萬。
Degree rate: 10,000.
268
晨伏夕見一十六度半。
Arc of morning concealment and evening appearance: 16½°.
269
夕伏晨見一十九度。
Arc of evening concealment and morning appearance: 19°.
270
段目段日
Segment titles; segment days
271
初行率
Initial motion rate
272
合伏一十七日七五三十四度二五二十九度零八二度一五五八
Conjunction concealment: 17 days 75 parts; 34°25 parts; 29°08 parts; 2°1558 parts
273
夕疾一十五日
Evening fast: 15 days
274
二十一度三八一十八度一六一度七零三四
21°38 parts; 18°16 parts; 1°7034 parts
275
夕遲一十二日
Evening slow: 12 days
276
一十度一二
10°12 parts
277
八度五九
8°59 parts
278
一度一四七二
1°1472 parts
279
夕留二日
Evening station: 2 days
280
夕退伏一十一日一八八七度八一二
Evening retrograde concealment: 11 days 188 parts; 7°812 parts
281
二度一零八
2°108 parts
282
合退伏一十一日一八八七度八一二
Conjunction retrograde concealment: 11 days 188 parts; 7°812 parts
283
二度一零八
2°108 parts
284
一度零三四六
1°0346 parts
285
晨留二日
Morning station: 2 days
286
晨遲一十二日
Morning slow: 12 days
287
一十度一二
10°12 parts
288
八度五九
8°59 parts
289
晨疾一十五日
Morning fast: 15 days
290
二十一度三八一十八度一六一度一四七二
21°38 parts; 18°16 parts; 1°1472 parts
291
晨伏一十七日七五三十四度二五二十九度零八一度七零三四
Morning concealment: 17 days 75 parts; 34°25 parts; 29°08 parts; 1°7034 parts
292
推五星前後合置中積，加合應，滿周率去之，餘爲前合。 再置周率，以前合減之，於爲後合。 如滿歲周去之，即其年無後合分。
Procedure to derive the five planets' prior and subsequent conjunctions: set the central accumulation, add the conjunction correspondence, and when full discard the cycle rate; the remainder is the prior conjunction. Set the cycle rate again and subtract the prior conjunction; the remainder is the subsequent conjunction. If the result fills the year circuit, discard it—then that year has no subsequent-conjunction remainder.
293
推五星中積日中星度置各星後合，既爲合伏下中積中星。 命爲日，曰中積。 命爲度，曰中星。 累加段日，爲各段中積。 皆滿歲周去之。 以各段下平度，累加各段下平度，滿歲周去。 退則減之，不及減，加歲周減之。 次復累加之，爲各段中星。
Procedure to derive the five planets' central-accumulation days and mean star positions: set each planet's subsequent conjunction, which already serves as the mean accumulation and mean star at conjunction concealment. Count it as days—this is the central accumulation. Count it as degrees—this is the mean star. Accumulate segment days to obtain each segment's central accumulation. Whenever full, discard the year circuit. Accumulate each segment's lower mean degree; when full, discard the year circuit. In retrograde motion subtract; if subtraction falls short, add the year circuit and subtract. Accumulate again in succession to obtain each segment's mean star.
294
推五星盈縮歷置中積，加歷應及生合，滿歷率去之，餘以度率而一爲度。 在歷中已下爲盈，已上減去歷中爲縮。 置各星合伏下盈縮歷，以段下限度累加之之滿歷中去之，盈交縮，縮交盈，即各段盈縮歷。
Procedure to derive the five planets' expansion–contraction sequence: set the central accumulation, add the ephemeris correspondence and natal conjunction, discard the ephemeris rate when full, and divide the remainder by the degree rate to obtain degrees. Below the ephemeris midpoint counts as expansion; above it, subtract the ephemeris midpoint to obtain contraction. Set each planet's expansion–contraction sequence at conjunction concealment, accumulate the limit beneath each segment, and discard the ephemeris rate when full; expansion crossing contraction becomes contraction and contraction crossing expansion becomes expansion—yielding each segment's expansion–contraction sequence.
295
推五星盈縮差置各段盈縮歷，以歷策除之爲策數，不盡，爲策餘。 以其下損益分見立成。 乘之，以歷策而一，所得益加損減其盈縮積分，即盈縮差。 金星倍之，水星三之。
Procedure to derive the five planets' expansion–contraction difference: set each segment's expansion–contraction sequence, divide by the ephemeris seed for the seed count, and treat the remainder as the seed remainder. Take the decrease-and-increase parts shown beneath it in the ready reckoner. Multiply, divide by the ephemeris seed, and add the product in expansion or subtract it in contraction from the expansion–contraction accumulated parts to obtain the expansion–contraction difference. Double the result for Venus; triple it for Mercury.
296
推定積日置各段中積，以其段盈縮差盈加縮減之，即得。 滿歲周去之，如中積不及減者，加歲周減之。 本段原無差者，借前段差加之，則金水二星，亦只用所得盈縮差，不用三之倍之。
Procedure to derive the fixed accumulated day: set each segment's central accumulation and, with that segment's expansion–contraction difference, add in expansion and subtract in contraction. Discard the year circuit when full; if the central accumulation cannot be reduced, add the year circuit and subtract. If a segment originally has no difference, borrow the prior segment's difference and add it; for Venus and Mercury, use only the expansion–contraction difference obtained, without doubling or tripling.
297
推加時定日置定積日，以歲前天正冬至分加之，滿紀法去之，餘命甲子算外，即爲定日。 視定積日會滿歲周去者，用本年冬至，會加歲周減者，用歲前冬至。
Procedure to derive the time-of-addition fixed day: set the fixed accumulated day, add the parts of the prior year's heavenly-standard winter solstice, discard the era rule when full, and count the remainder from jiazi outside the count to obtain the fixed day. If the fixed accumulated day fills the year circuit and is discarded, use this year's winter solstice; if it requires adding the year circuit and subtracting, use the prior year's winter solstice.
298
推所入月日置合伏下定積，以加天正閏餘滿朔策除之，爲月數。 起歲前十一月，其不滿朔策者，即入月已來日分也。 視其月定朔甲子，與加時定日甲子相去即合伏日，累加相距日，滿各月大小去之，即各段所入月日。
Procedure to derive the month and day entered: set the fixed accumulation at conjunction concealment, add the intercalary remainder of the heavenly standard, and when full divide by the new-moon stride to obtain the month count. Count from the eleventh month before the year; the remainder short of the new-moon stride is the days elapsed within the month entered. Compare that month's fixed-new-moon jiazi with the time-of-addition fixed day's jiazi to obtain the conjunction-concealment day; accumulate intervening days and discard each month's long-or-short month allowance when full to obtain the month and day entered for each segment.
299
推定星置各段中星，依推定積日法，以盈縮差加減之。
Procedure to derive the fixed star: set each segment's mean star and, by the fixed-accumulated-day procedure, add or subtract the expansion–contraction difference.
300
推加時定星置定星，以歲前冬至加時黃道日度加之，滿週歲天去之。 若定積日會加歲周者，用歲前黃道日度。 遇減歲周者，用本年黃道目度，如原無中星度，段下亦無定星星及加時定星度分。
Procedure to derive the time-of-addition fixed star: set the fixed star, add the ecliptic solar degree at the prior year's winter-solstice moment, and discard the circuit-of-year heaven when full. If the fixed accumulated day requires adding the year circuit, use the prior year's ecliptic solar degree. When the year circuit must be subtracted, use this year's ecliptic solar degree; if there was originally no mean-star degree, the segment lists neither fixed-star degrees nor time-of-addition fixed-star degrees and parts.
301
推加減定分置定日小余，以其段初行率乘之，滿萬爲分，所得諸段爲減分，退段爲加分。
Procedure to derive the additive-subtractive fixed parts: set the fixed day's small remainder, multiply by that segment's initial motion rate, scale by ten thousand to obtain parts; prograde segments yield subtractive parts, retrograde segments additive parts.
302
推夜半定星及宿次置加時定星，以加減定分加減之，爲夜半定星。 以黃道積度鈐減之，爲夜半宿次。 其留段即用時定星，爲夜半一星。
Procedure to derive the midnight fixed star and lodge ordinal: set the time-of-addition fixed star and add or subtract the additive-subtractive fixed parts to obtain the midnight fixed star. Subtract successive entries from the ecliptic accumulated-degree dial to obtain the midnight lodge ordinal. For station segments, take the time-of-addition fixed star directly as the midnight fixed star.
303
推日度率置各段定日，與次段定日相減爲日率。 次段不及減，加紀法減之。 置各段夜半－定星，與次段夜半定星相減爲度漲。 次段不及減，加周天減之。 凡近留之段，皆用留段加時定星，與本段夜半定星相減。 如星度逆者，以後段減前段，即各得度率。
Procedure to derive the daily degree rate: subtract each segment's fixed day from the next segment's fixed day to obtain the day rate. If the next segment cannot be subtracted, add the era rule and subtract. Set each segment's midnight fixed star and subtract the next segment's midnight fixed star to obtain the degree increment. If the next segment cannot be subtracted, add the celestial circuit and subtract. For segments near a station, subtract this segment's midnight fixed star from the station segment's time-of-addition fixed star. If the star's motion is retrograde, subtract the earlier segment from the later one to obtain each degree rate.
304
推平行分置度率，以日率除之，即得。
Procedure to derive parallel motion: set the degree rate and divide by the day rate.
305
推凡差及增減總差日差以本段前後之平行分相減，爲本段凡差。 凡五星之伏段及近留之遲段及退段，皆無凡差。 倍凡差，退一位爲增減差。 倍增減差爲總差。 置總差，以日率減一日除之爲日差。 初日行分多，爲減差。 末日行分多，爲加差。
Procedure to derive the general difference, increase–decrease difference, total difference, and daily difference: subtract this segment's prior and subsequent parallel motion; the remainder is this segment's general difference. For all five planets, concealment segments, slow segments near station, and retrograde segments have no general difference. Double the general difference and shift back one decimal place to obtain the increase–decrease difference. Double the increase–decrease difference to obtain the total difference. Set the total difference and divide by the day rate minus one day to obtain the daily difference. If the initial day's motion parts are greater, the correction is subtractive. If the final day's motion parts are greater, the correction is additive.
306
推初日行分末日行分以增減差加減其段平行分，爲初末日行分。 視本段平行分與次段平行分相較，前多後少者，加爲初，減爲末。 前少後多者，減爲初，加爲末。
Procedure to derive initial-day and final-day motion parts: add or subtract the increase–decrease difference from this segment's parallel motion. Compare this segment's parallel motion with the next segment's: if the prior exceeds the subsequent, add for the initial day and subtract for the final day. If the prior is less and the subsequent more, subtract for the initial day and add for the final day.
307
推撫心差諸段爲增減差總差日差合伏者，置次段初日行分，加其日差之半，亦次段日差。 爲末日行分。 晨伏、夕伏者，置前段本段之前。 末日行分，加其日差之半，亦前段日差。 爲二伏初日行分。 置伏段呼得初末日行分，皆與本段平行分相減，餘爲增減差。 又以增差加減平行分，爲初末日行分。 視合伏末日行全較平行分，少則加，多則減，爲初日行分。 晨伏、夕伏初日行分較平行分，亦少加多減，爲末日行分。 木、火之晨遲末，土之晨遲，金之夕遲末，水之夕遲，皆置其前末日行分，銳其日差減之，即前段日差。 餘爲初日行分。 木、火之夕遲初，土之夕遲，金之晨初，水之晨遲，皆置其後段初日行分，倍其日差減之，後段日差。 餘爲末日行分。 木、火、土之夕伏，金、水之晨伏，皆置其前段末日行分，內加其前段日差之半，爲鈦段初日行分，皆與平行分相減，餘爲增減差。 木、火之晨退、夕退，置其平行分，退一位、六因之，爲增減差。 晨退減爲初，加爲末。 夕退加爲初，減爲末。 晨加夕減，二段相比較。 金之夕退伏合伏，置其平行分，退一位，三因之折半。 水之夕退伏合退伏，以平行分折半，各爲增減差。 金之夕退，置其平分，退一位，三在之折半。 水之夕退伏合退伏，以平行分折半，各爲增減差。 金之夕退，置其後段祿日行分，減日差，後段日差。 爲末日行分。 金之晨退，置其前段末日行分，減日差，前段日差。 爲初日行分。 皆與平行分相減，餘爲增減差。 凡增減差，倍之爲總差，以相距日率減一除之，爲日差。 其初末日行分有其一者，以增減差加減，更求其一，如伏段法，餘依前後平行分相較增減之。 金、火之夕遲末，晨遲初，置其段平行分，以相距日率下不倫分乘之，不倫分之秒，與平行之分對。 即爲增減差。 置平行分，夕者以增減差，加爲初日行分，減爲末日行分。 晨者反是。
Procedure to derive heart-smoothing difference for segments as increase–decrease, total, and daily differences—at conjunction concealment: set the next segment's initial-day motion parts and add half its daily difference, which is also the next segment's daily difference. The result is the final-day motion parts. For morning concealment and evening concealment, set the segment before this one. Take the final-day motion parts and add half the daily difference, which is also the prior segment's daily difference. The result is the initial-day motion parts for the two concealments. Set the concealment segment's initial and final day motion parts obtained by calculation; subtract each from this segment's parallel motion; the remainder is the increase–decrease difference. Again add or subtract the increase–decrease difference from parallel motion to obtain initial and final day motion parts. Compare conjunction-concealment final-day motion parts with parallel motion: if less, add; if more, subtract—to obtain initial-day motion parts. For morning and evening concealment, compare initial-day motion parts with parallel motion and likewise add if less or subtract if more to obtain final-day motion parts. For Jupiter and Mars at morning slow final, Saturn at morning slow, Venus at evening slow final, and Mercury at evening slow: set the prior segment's final-day motion parts and subtract the refined daily difference to obtain the prior segment's daily difference. The remainder is the initial-day motion parts. For Jupiter and Mars at evening slow initial, Saturn at evening slow, Venus at morning initial, and Mercury at morning slow: set the subsequent segment's initial-day motion parts and subtract double the daily difference to obtain the subsequent segment's daily difference. The remainder is the final-day motion parts. For Jupiter, Mars, and Saturn at evening concealment and Venus and Mercury at morning concealment: set the prior segment's final-day motion parts, inwardly add half the prior segment's daily difference for the next segment's initial-day motion parts; subtract each from parallel motion; the remainder is the increase–decrease difference. For Jupiter and Mars at morning retrograde and evening retrograde: set parallel motion, shift back one place, multiply by six, and obtain the increase–decrease difference. Morning retrograde: subtract for the initial day and add for the final day. Evening retrograde: add for the initial day and subtract for the final day. Morning adds and evening subtracts—compare the two segments accordingly. For Venus at evening retrograde concealment and conjunction concealment: set parallel motion, shift back one place, triple, and halve. For Mercury at evening retrograde concealment and conjunction retrograde concealment: halve parallel motion; each yields the increase–decrease difference. For Venus at evening retrograde: set parallel motion, shift back one place, triple, and halve. For Mercury at evening retrograde concealment and conjunction retrograde concealment: halve parallel motion; each yields the increase–decrease difference. For Venus at evening retrograde: set the subsequent segment's initial-day motion parts and subtract the daily difference to obtain the subsequent segment's daily difference. The result is the final-day motion parts. For Venus at morning retrograde: set the prior segment's final-day motion parts and subtract the daily difference to obtain the prior segment's daily difference. The result is the initial-day motion parts. In each case subtract from parallel motion; the remainder is the increase–decrease difference. For any increase–decrease difference, double it for the total difference and divide by the separating day rate minus one to obtain the daily difference. When only one of the initial or final day motion parts is known, add or subtract the increase–decrease difference to find the other, as in the concealment-segment method; otherwise follow comparing prior and subsequent parallel motion for increase or decrease. For Venus and Mars at evening slow final and morning slow initial: set this segment's parallel motion and multiply by the irregular parts under the separating day rate; the irregular parts' arc-seconds pair with the parallel motion's parts. The result is the increase–decrease difference. Set parallel motion: for evening segments add the increase–decrease difference for initial-day motion parts and subtract for final-day motion parts. For morning segments, the reverse applies.
308
不倫分金、火星之夕遲末，與晨遲初，其增減差，多於平行分者，爲不倫分也。
Irregular parts: for Venus and Mars at evening slow final and morning slow initial, when the increase–decrease difference exceeds parallel motion, these are irregular parts.
309
十七日
17 days
310
八十八秒八八五
88″885
311
十六日
16 days
312
八十八秒二三一
88″231
313
十五日
15 days
314
八十七秒四九六
87″496
315
十四日
14 days
316
八十六秒七六一
86″761
317
推五星每日細行，置各段夜半宿次，以初日行分順加退減之，爲次日宿次。 又以日差加減其初日行分，爲每日行分，亦順加退減於次日宿次，滿黃道宿次去之，至次段宿次而止，爲每日夜半宿次。
Procedure to derive the five planets' daily fine motion: set each segment's midnight lodge ordinal and, with the initial day's motion parts, add in prograde motion or subtract in retrograde to obtain the next day's lodge ordinal. Again add or subtract the daily difference to the initial day's motion parts for the daily motion parts; likewise add or subtract prograde or retrograde motion to the next day's lodge ordinal; discard the ecliptic lodge ordinal when full; continue until the next segment's lodge ordinal to obtain each day's midnight lodge ordinal.
318
推五星順逆交宮時刻視逐日五星細行，與黃道十二宮界宿次同名，其度分又相近者以相減。 視其餘分，在本日行分以下者，爲交宮在本日也。 順行者，以本日夜半星行宿次度分減宮界度分。 退行者，以宮界度分減本日夜半星行宿次度分。 扣以日周乘之爲實，以本日行分爲法，法除實，得數，依發斂加時法，得交宮時刻。
Procedure to derive the five planets' prograde and retrograde palace-crossing times: inspect each day's fine planetary motion; when it shares a lodge name with an ecliptic twelve-palace boundary and the degrees and parts are near, subtract them. Inspect the remainder: if it falls below this day's motion parts, palace crossing occurs on this day. For prograde motion, subtract palace-boundary degrees and parts from this day's midnight star-motion lodge degrees and parts. For retrograde motion, subtract this day's midnight star-motion lodge degrees and parts from palace-boundary degrees and parts. Multiply by the day circuit for the dividend and use this day's motion parts as divisor; divide to obtain the number, then apply the emission-and-collection time-addition method to obtain the palace-crossing time.
319
推五星伏見凡取伏見，伏者要在已下，見者要在已上。 晨見晨伏者，置其日太陽行度，內減各星行度。 夕見夕伏者，置其日各星行度，內減太陽行度。 即爲其日晨昏伏見度。 置本日伏見度，與次日伏見度相減，餘四而一，即得晨昏伏見分。 視本日伏見度較次日伏見度爲多者減，少者加。 晨者，置本日伏見度，以伏見分加減之，爲晨伏見度。 夕者，三因伏見分，置伏見度加減之，爲夕伏見度。 視在各星伏見度上下取之。
Procedure to derive the five planets' concealment and appearance: in seeking concealment and appearance, concealment must fall at si or below and appearance at si or above. For morning appearance and morning concealment, set that day's solar motion in degrees and inwardly subtract each star's motion in degrees. For evening appearance and evening concealment, set that day's stellar motion in degrees and inwardly subtract solar motion in degrees. The result is that day's morning-and-evening concealment-and-appearance degree. Set this day's concealment-and-appearance degree and subtract the next day's; divide the remainder by four to obtain morning-and-evening concealment-and-appearance parts. Compare this day's concealment-and-appearance degree with the next day's: subtract if greater, add if less. For morning, set this day's concealment-and-appearance degree and add or subtract the concealment-and-appearance parts to obtain the morning concealment-and-appearance degree. For evening, triple the concealment-and-appearance parts and add or subtract them to the concealment-and-appearance degree to obtain the evening concealment-and-appearance degree. Inspect each star's concealment-and-appearance degree and take the value above or below accordingly.
320
△步四餘
△ Stepwise Computation: The Four Residuals
321
紫氣週日一萬零二百二十七日一七九二。
Purple Qi cycle days: 10,227d 1792 parts.
322
紫氣度率二十八日，日行三分五七一四二九。
Purple Qi degree rate: 28 days; daily motion 3 parts 571429.
323
紫氣至後策八千一百九十四萬九六二三。
Purple Qi after-solstice tally: 81,949,623.
324
月孛週日三千二百三十一日九六八四。
Moon Comet cycle days: 3,231d 9684 parts.
325
月孛度率八日八四八四九二，日行十一分三零一三六一。
Moon Comet degree rate: 8d 848492 parts; daily motion 11 parts 301361.
326
月孛至後策一千二百二十萬四六五九。
Moon Comet after-solstice tally: 12,204,659.
327
羅計週日六千七百九十三日四四三二。
Rahu–Ketu cycle days: 6,793d 4432 parts.
328
羅計度率一十八日五九九一零七七六，日行五分三七六六零二。
Rahu–Ketu degree rate: 18d 59910776 parts; daily motion 5 parts 376602.
329
羅至後策五千三百三十三萬六二一七。
Rahu after-solstice tally: 53,336,217.
330
計都至後策一千九百三十六萬九零零一。
Ketu after-solstice tally: 19,369,001.
331
推四餘至後策置中積，加各餘至後策，滿週日去之，即得。
Procedure to derive the four residuals' after-solstice tallies: set the central accumulation, add each residual's after-solstice tally, and discard the cycle days when full.
332
推四餘周后策以至後策，減立成內各宿初末度積日，即得。
Procedure to derive the four residuals' cycle tallies: subtract each lodge's initial- and final-degree accumulated days in the ready reckoner from the after-solstice tally.
333
推四餘入各宿次初末度積日置各餘周后策，加入其年冬至分，滿紀法去之，即各餘末度積日。 紫氣、月孛爲各宿初，羅喉、計都爲各宿末。 氣孛順行，羅計逆行。
Procedure to derive the four residuals' accumulated days at each lodge's initial and final degrees: set each residual's cycle tally, add this year's winter solstice parts, and discard the era rule when full to obtain each residual's final-degree accumulated days. Purple Qi and Moon Comet mark each lodge's initial degree; Rahu and Ketu mark each lodge's final degree. Purple Qi and Moon Comet move prograde; Rahu and Ketu move retrograde.
334
推四餘初末度積日所入月日置各餘周后策，加入天正閏餘滿期策減之，起十一月至不滿朔策，即所入月也。 其初末度積日即滿紀法去者。 命甲子算外，爲日辰小餘，以發斂求之爲時刻。 視定朔某甲女，即知入月已來日也。
Procedure to derive the month and day entered from initial- and final-degree accumulated days: set each residual's cycle tally, add the heavenly-standard intercalary remainder, subtract the period stride when full, and count from the eleventh month to what falls short of the new-moon stride for the month entered. For its initial- and final-degree accumulated days, discard the era rule when full. Count from jiazi outside the count for the day stem-branch and small remainder, and obtain the time by emission-and-collection. Inspect the fixed new moon's jiazi day to know the days elapsed since entering the month.
335
推四餘每日行度置各餘初末度積日，氣孛以度率日累加之，至末度加其宿零日及分，即次宿之初度。 羅計先加其宿零日及分，後以度率日累加之，即次宿之末度。 徊以其大餘，命甲子算外爲日辰。 其交次宿，以小餘以斂爲時刻。
Procedure to derive the four residuals' daily motion in degrees: set each residual's initial- and final-degree accumulated days; for Purple Qi and Moon Comet accumulate day by day at the degree rate, and at the final degree add that lodge's zero days and parts to reach the next lodge's initial degree. For Rahu and Ketu, first add that lodge's zero days and parts, then accumulate day by day at the degree rate to reach the next lodge's final degree. In each case, with its large remainder count from jiazi outside the count for the day stem-branch. When crossing to the next lodge, apply collection to the small remainder to obtain the time.
336
推四餘交宮以至後策減各宿交宮積日，餘爲入某宮積中天正閏餘，滿朔策去之，起十一月至不滿朔策，即所入月。 又置入宮積日，加冬至分，滿紀法去之，爲日辰，小餘以斂爲時刻。 視定朔甲子，即知交宮及時刻。
Procedure to derive the four residuals' palace crossing: subtract each lodge's palace-crossing accumulated days from the after-solstice tally; the remainder is the accumulated days entering a given palace; add the heavenly-standard intercalary remainder, discard the new-moon stride when full, and count from the eleventh month to what falls short of the new-moon stride for the month entered. Again set the palace-entry accumulated days, add winter solstice parts, discard the era rule when full for the day stem-branch, and obtain the time from the small remainder by collection. Inspect the fixed new moon's jiazi day to know the palace crossing and its time.
337
▲紫氣宿次日分立成入箕初度。
▲ Purple Qi: lodge-by-lodge daily minute-division ready reckoner, entering the initial degree of Ji.
338
以下表格略
Tables below omitted.
339
至後策少者用前氐下積日，多者用後氐下積日。
If the after-solstice tally is less, use the prior Di lower accumulated days; if more, use the posterior Di lower accumulated days.