1
高宗時，戊寅曆益疎，淳風作甲子元曆以獻。 詔太史起麟德二年頒用，謂之麟德曆。 古曆有章、蔀，有元、紀，有日分、度分，參差不齊，淳風為總法千三百四十以一之。 損益中晷術以考日至，為木渾圖以測黃道，餘因劉焯皇極曆法，增損所宜。 當時以為密，與太史令瞿曇羅所上經緯曆參行。
Under Emperor Gaozong the Wuyin calendar grew increasingly inaccurate; Li Chunfeng composed the Jiazi Origin calendar and presented it. An edict ordered the Grand Astrologer to implement it from Lindé year 2; it was called the Lindé Calendar. Ancient calendars used rules, obscurations, origins, eras, day parts, and degree parts in inconsistent units; Chunfeng unified them with a single general factor of 1,340. He revised the gnomon midday method to determine solstice arrival, built a wooden armillary to measure the ecliptic, and otherwise adapted Liu Chuo's Huangji calendar with suitable adjustments. It was then regarded as precise and was used concurrently with the Jingwei calendar that Grand Astrologer Qu Tanluo had submitted.
2
弘道元年十二月甲寅朔，壬午晦。 八月，詔二年元日用甲申，故進以癸未晦焉。
In Hongdao year 1 the twelfth month began on jiayin and ended on renchen. In the eighth month an edict fixed New Year's Day of year 2 on jiashen, so the prior month-end was advanced to guwei.
3
永昌元年十一月，改元載初，用周正，以十二月為臘月，建寅月為一月。 神功二年，司曆以臘為閏，而前歲之晦，月見東方，太后詔以正月為閏十月。 是歲，甲子南至，改元聖曆。 命瞿曇羅作光宅曆，將用之。 三年，罷作光宅曆，復行夏時，終開元十六年。
In Yongchang year 1, eleventh month, the era became Zaichu; the court adopted the Zhou calendar, making the twelfth month La month and the yin month the first month. In Shenggong year 2 the calendar office intercalated La month, but the moon appeared in the east on the prior year's last day; the Empress Dowager decreed that the first month become intercalary tenth month. That year, at jiazi winter solstice, the era name was changed to Shenglü. Qu Tanluo was ordered to compose the Guangzhai calendar for imminent use. In year 3 work on the Guangzhai calendar was halted and the Xia calendar restored, continuing through Kaiyuan year 16.
4
麟德曆麟德元年甲子，距上元積二十六萬九千八百八十算。
For the Lindé Calendar, Lindé year 1 jiazi stood 269,880 counts from the high origin.
5
總法千三百四十。
General factor: 1,340.
6
朞實四十八萬九千四百二十八。
Sequence constant: 489,428.
7
常朔實三萬九千五百七十一。 加三百六十二曰盈朔實，減三百五十一曰朒朔實。
Mean new-moon constant: 39,571. Add 362 for the waxing new-moon constant; subtract 351 for the waning new-moon constant.
8
辰率三百三十五。
Double-hour rate: 335.
9
以朞實乘積算，為朞總。 如總法得一，為日。 六十去之，命甲子算外，得冬至。 累加日十五、小餘二百九十二、小分六之五，得次氣。 六乘小餘，辰率而一，命子半算外，各其加時。
Multiply the accumulated count by the sequence constant to obtain the sequence total. Divide by the general factor; each quotient is one day. Cast out multiples of 60, count from beyond jiazi, and obtain winter solstice. Add cumulatively 15 days, 292 minor remainder, and 5/6 of a minor part to obtain the next qi. Multiply the minor remainder by 6, divide by the double-hour rate, and count from beyond mid-zi for each add-time.
10
以常朔實去朞總，不滿為閏餘。 以閏餘減朞總，為總實，如總法得一，為日。 以減冬至，得天正常朔。 又以常朔小餘并閏餘，以減朞總，為總實。 因常朔加日二十九、小餘七百一十一，得次朔。 因朔加日七、小餘五百一十二太，得上弦。 又加，得望及下弦。
Cast the mean new-moon constant out of the sequence total; what remains is the intercalation remainder. Subtract the intercalation remainder from the sequence total for total proper; divide by the general factor for days. Subtract from winter solstice to obtain the mean astronomical new moon. Also add the mean new-moon minor remainder to the intercalation remainder and subtract from the sequence total for total proper. From the mean new moon add 29 days and 711 minor remainder to obtain the next new moon. From new moon add 7 days and 512 3/4 minor remainder for first quarter. Add again for full moon and last quarter.
11
進綱十六。 秋分後。
Advance cord: 16. After autumn equinox.
12
退紀十七。 春分後。
Retreat era: 17. After spring equinox.
13
中節
Solar terms
14
躔差率
Lodging difference rate
15
消息總
Discharge-recovery total
16
先後率
Advance-retard rate
17
盈朒積
Wax-wane accumulation
18
冬至
Winter Solstice
19
益七百二十二
Increase 722
20
息初
Discharge initial
21
先五十四
Advance 54
22
盈初
Wax initial
23
小寒
Lesser Cold
24
益六百一十八
Increase 618
25
息七百二十二
Discharge 722
26
先四十六
Advance 46
27
盈五十四
Wax 54
28
大寒
Greater Cold
29
益五百一十四
Increase 514
30
息千三百四十
Discharge 1,340
31
先三十八
Advance 38
32
盈百
Wax 100
33
立春
Start of Spring
34
益五百一十四
Increase 514
35
息千八百五十四
Discharge 1,854
36
先三十八
Advance 38
37
盈百三十八
Wax 138
38
啟蟄
Awakening of Insects
39
益六百一十八
Increase 618
40
息二千三百六十八
Discharge 2,368
41
先四十六
Advance 46
42
盈百七十六
Wax 176
43
雨水
Rain Water
44
益七百二十二
Increase 722
45
息二千九百八十六
Discharge 2,986
46
先五十四
Advance 54
47
盈二百二十二
Wax 222
48
春分
Spring Equinox
49
益七百二十二
Increase 722
50
息三千七百八
Discharge 3,708
51
後五十四
Retard 54
52
盈二百七十六
Wax 276
53
清明
Clear and Bright
54
損六百一十八
Decrease 618
55
息二千九百八十六
Discharge 2,986
56
後四十六
Retard 46
57
盈二百二十二
Wax 222
58
穀雨
Grain Rain
59
損五百一十四
Decrease 514
60
息二千三百六十八
Discharge 2,368
61
後三十八
Retard 38
62
盈百七十六
Wax 176
63
立夏
Start of Summer
64
損五百一十四
Decrease 514
65
息千八百五十四
Discharge 1,854
66
後三十八
Retard 38
67
盈百三十八
Wax 138
68
小滿
Lesser Full Grain
69
損六百一十八
Decrease 618
70
息千三百四十
Discharge 1,340
71
後四十六
Retard 46
72
盈百
Wax 100
73
芒種
Grain in Ear
74
損七百二十二
Decrease 722
75
息七百二十二
Discharge 722
76
後五十四
Retard 54
77
盈五十四
Wax 54
78
夏至
Summer Solstice
79
益七百二十二
Increase 722
80
消初
Reduction initial
81
先五十四
Advance 54
82
朒初
Wane initial
83
小暑
Lesser Heat
84
益六百一十八
Increase 618
85
消七百二十二
Reduction 722
86
先四十六
Advance 46
87
朒五十四
Wane 54
88
大暑
Greater Heat
89
益五百一十四
Increase 514
90
消千三百四十
Reduction 1,340
91
先三十八
Advance 38
92
朒百
Wane 100
93
立秋
Start of Autumn
94
益五百一十四
Increase 514
95
消千八百五十四
Reduction 1,854
96
先三十八
Advance 38
97
朒百三十八
Wane 138
98
處暑
End of Heat
99
益六百一十八
Increase 618
100
消二千三百六十八
Reduction 2,368
101
先四十六
Advance 46
102
朒百七十六
Wane 176
103
白露
White Dew
104
益七百二十二
Increase 722
105
消二千九百八十六
Reduction 2,986
106
先五十四
Advance 54
107
朒二百二十二
Wane 222
108
秋分
Autumn Equinox
109
益七百二十二
Increase 722
110
消三千七百八
Reduction 3,708
111
後五十四
Retard 54
112
朒二百七十六
Wane 276
113
寒露
Cold Dew
114
損六百一十八
Decrease 618
115
消二千九百八十六
Reduction 2,986
116
後四十六
Retard 46
117
朒二百二十二
Wane 222
118
霜降
Frost Descent
119
損五百一十四
Decrease 514
120
消二千三百六十八
Reduction 2,368
121
後三十八
Retard 38
122
朒百七十六
Wane 176
123
立冬
Start of Winter
124
損五百一十四
Decrease 514
125
消千八百五十四
Reduction 1,854
126
後三十八
Retard 38
127
朒百三十八
Wane 138
128
小雪
Lesser Snow
129
損六百一十八
Decrease 618
130
消千三百四十
Reduction 1,340
131
後四十六
Retard 46
132
朒百
Wane 100
133
大雪
Greater Snow
134
損七百二十二
Decrease 722
135
消七百二十二
Reduction 722
136
後五十四
Retard 54
137
朒五十四
Wane 54
138
各以其氣率并後氣率而半之，十二乘之，綱紀除之，為末率。 二率相減，餘以十二乘之，綱紀除，為總差。 又以十二乘總差，綱紀除之，為別差。 以總差前少以減末率，前多以加末率，為初率。 累以別差，前少以加初率，前多以減初率，為每日躔差及先後率。 乃循積而損益之，各得其日定氣消息與盈朒積。 其後無同率，因前末為初率； 前少者加總差，前多者以總差減之，為末率。 餘依術入之。
For each solar term, add its rate and the next term's rate, halve the sum, multiply by 12, and divide by cord and era to obtain the final rate. Subtract the two rates, multiply the remainder by 12, and divide by cord and era for the total difference. Multiply the total difference by 12 again and divide by cord and era for the separate difference. If the prior term is deficient, subtract the total difference from the final rate; if excessive, add it—to obtain the initial rate. Accumulate the separate difference: add to the initial rate when the prior term is deficient, subtract when excessive—to obtain the daily lodging difference and advance-retard rate. Then adjust cumulatively to obtain each day's fixed qi discharge-recovery and wax-wane accumulation. When later terms have no matching rate, take the prior final rate as the initial rate; if the prior term is deficient add the total difference; if excessive subtract the total difference—for the final rate. Apply the remaining steps according to the method.
139
各以氣下消息積，息減、消加常氣，為定氣。 各以定氣大小餘減所近朔望大小餘，十二通其日，以辰率約其餘，相從為辰總。 其氣前多以乘末率，前少以乘初率，十二而一，為總率。 前多者，以辰總減綱紀，以乘十二，綱紀而一，以加總率，辰總乘之，二十四除之； 前少者，辰總再乘別差，二百八十八除之：皆加總率。 乃以先加、後減其氣盈朒積為定。 以定積盈加、朒減常朔弦望，得盈朒大小餘。
For each solar term, apply the discharge-recovery accumulation below it: subtract discharge and add reduction from the mean qi to obtain fixed qi. Subtract the nearest new-moon or full-moon large and minor remainders from each fixed qi's remainders; multiply days by 12, reduce the remainder by the double-hour rate, and combine for the double-hour total. If the qi is prior-excessive multiply by the final rate; if prior-deficient multiply by the initial rate; divide by 12 for the total rate. If prior-excessive: subtract the double-hour total from cord and era, multiply by 12, divide by cord and era, and add to the total rate; multiply by the double-hour total and divide by 24; if prior-deficient: multiply the double-hour total twice by the separate difference and divide by 288—in both cases add to the total rate. Then advance-add and retard-subtract the qi wax-wane accumulation to obtain the fixed value. Apply the fixed accumulation to wax-add and wane-subtract the mean new, first-quarter, full, and last-quarter moons, obtaining wax-wane large and minor remainders.
140
變周四十四萬三千七十七。
Change cycle: 443,077.
141
變日二十七，餘七百四十三，變奇一。
Change days: 27, remainder 743, change odd: 1.
142
變奇法十二。
Change-odd factor: 12.
143
月程法六十七。
Lunar circuit factor: 67.
144
以奇法乘總實，滿變周，去之； 不滿者，奇法而一，為變分。 盈總法從日，得天正常朔夜半入變。 加常朔小餘，為經辰所入。 因朔加七日、餘五百一十二、奇九，得上弦。 轉加，得望、下弦及次朔。 加之滿變日及餘，去之。 又以所入盈朒定積，盈加、朒減之，得朔、弦、望盈朒經辰所入。
Multiply total proper by the odd factor; cast out full change cycles; what remains, divide by the odd factor, for change parts. Carry excess general factor into days to obtain mean astronomical new moon at midnight entry into change. Add the mean new-moon minor remainder for the circuit double-hour entry. From new moon add 7 days, remainder 512, and odd 9 for first quarter. Continue adding for full moon, last quarter, and the next new moon. When the sum fills change days and remainder, cast it out. Again apply the wax-wane fixed accumulation at entry—wax-add and wane-subtract—to obtain wax-wane circuit double-hour entry for new moon and quarters.
145
變日
Change days
146
離程
Departure circuit
147
增減率
Increase-decrease rate
148
遲速積
Slow-fast accumulation
149
一日
Day 1
150
九百八十五
985
151
增百三十四
Increase 134
152
速初
Speed initial
153
二日
Day 2
154
九百七十四
974
155
增百一十七
Increase 117
156
速百三十四
Speed 134
157
三日
Day 3
158
九百六十二
962
159
增九十九
Increase 99
160
速二百五十一
Speed 251
161
四日
Day 4
162
九百四十八
948
163
增七十八
Increase 78
164
速三百五十
Speed 350
165
五日
Day 5
166
九百三十三
933
167
增五十六
Increase 56
168
速四百二十八
Speed 428
169
六日
Day 6
170
九百一十八
918
171
增三十三
Increase 33
172
速四百八十四
Speed 484
173
七日
Day 7
174
九百二
902
175
增九初增九末減隱
Increase of 9 at start and end, with hidden decrease
176
速五百一十七
Speed 517
177
八日
Day 8
178
八百八十六
886
179
減十四
Decrease 14
180
速五百二十六
Speed 526
181
九日
Day 9
182
八百七十
870
183
減三十八
Decrease 38
184
速五百一十二
Speed 512
185
十日
Day 10
186
八百五十四
854
187
減六十四
Decrease 64
188
速四百七十四
fast (moon ahead of mean motion) accumulation 474
189
十一日
variable-calendar day 11
190
八百三十九
separation progression 839
191
減八十五
decrease rate 85
192
速四百一十二
fast accumulation 412
193
十二日
variable-calendar day 12
194
八百二十六
separation progression 826
195
減百四
decrease rate 104
196
速三百二十七
fast accumulation 327
197
十三日
variable-calendar day 13
198
八百一十五
separation progression 815
199
減百二十一
decrease rate 121
200
速二百二十三
fast accumulation 223
201
十四日
variable-calendar day 14
202
八百八
separation progression 808
203
初減百二末增二十九
decrease rate 120 at day-start, increase 29 at day-end
204
速百二
fast accumulation 102
205
十五日
variable-calendar day 15
206
八百十
separation progression 810
207
增百二十八
increase rate 128
208
遲二十九
slow accumulation 29
209
十六日
variable-calendar day 16
210
八百一十九
separation progression 819
211
增百一十五
increase rate 115
212
遲百五十七
slow accumulation 157
213
十七日
variable-calendar day 17
214
八百三十二
separation progression 832
215
增九十五
increase rate 95
216
遲二百七十二
slow accumulation 272
217
十八日
variable-calendar day 18
218
八百四十六
separation progression 846
219
增七十四
increase rate 74
220
遲三百六十七
slow accumulation 367
221
十九日
variable-calendar day 19
222
八百六十一
separation progression 861
223
增五十二
increase rate 52
224
遲四百四十一
slow accumulation 441
225
二十日
variable-calendar day 20
226
八百七十七
separation progression 877
227
增二十八
increase rate 28
228
遲四百九十三
slow accumulation 493
229
二十一日
variable-calendar day 21
230
八百九十三
separation progression 893
231
增四初增四末減隱
increase rate 4; +4 at day-start, negligible decrease at day-end
232
遲五百二十一
slow accumulation 521
233
二十二日
variable-calendar day 22
234
九百九
separation progression 909
235
減二十
decrease rate 20
236
遲五百二十五
slow accumulation 525
237
二十三日
variable-calendar day 23
238
九百二十五
separation progression 925
239
減四十四
decrease rate 44
240
遲五百五
slow accumulation 505
241
二十四日
variable-calendar day 24
242
九百四十一
separation progression 941
243
減六十八
decrease rate 68
244
遲四百六十一
slow accumulation 461
245
二十五日
variable-calendar day 25
246
九百五十五
separation progression 955
247
減八十九
decrease rate 89
248
遲三百九十三
slow accumulation 393
249
二十六日
variable-calendar day 26
250
九百六十八
separation progression 968
251
減百八
decrease rate 108
252
遲三百四
slow accumulation 304
253
二十七日
variable-calendar day 27
254
九百七十九
separation progression 979
255
減百二十五
decrease rate 125
256
遲百九十六
slow accumulation 196
257
二十八日
variable-calendar day 28
258
九百八十五
separation progression 985
259
減百四十四初減七十一末增入後
decrease rate 144; −71 at day-start, carry increase into next day
260
遲七十一
slow accumulation 71
261
以離程與次相減，得進退差； 後多為進，後少為退，等為平。 各列朔、弦、望盈朒經辰所入日增減率，并後率而半之，為通率。 又二率相減，為率差。 增者以入變曆日餘減總法，餘乘率差，總法而一，并率差而半之； 減者半入餘乘率差，亦總法而一：皆加通率。 以乘入餘，總法除，為經辰變率。 半之，以速減、遲加入餘，為轉餘。 增者以減總法，減者因餘：皆乘率差，總法而一； 以加通率，變率乘之，總法除之，以速減、遲加變率，為定率。 乃以定率增減遲速積為定。 其後無同率，亦因前率。 應增者，以通率為初數，半率差而減之； 應損者，即為通率。 其曆率損益入餘進退日者，分為二日，隨餘初末，如法求之，所得并以加減變率為定。
Subtract the next day's separation progression from the current one to obtain the advance-retreat difference. If the next value is larger, the motion is advancing; if smaller, retreating; if equal, level. For each syzygy and quarter-moon excess-deficit canonical-time entry, list the day's increase-decrease rate, average it with the next day's rate, and take the result as the common rate. Subtract the two rates to obtain the rate difference. For increasing rates: subtract the variable-calendar day remainder from the total divisor, multiply the remainder by the rate difference and divide by the total divisor, then add half the rate difference; For decreasing rates: halve the entry remainder, multiply by the rate difference and divide by the total divisor; in both cases add the common rate. Multiply by the entry remainder and divide by the total divisor to obtain the canonical-time variable rate. Halve it, subtract the fast component and add the slow component to the entry remainder, yielding the transfer remainder. For increasing cases subtract from the total divisor; for decreasing cases use the remainder directly; multiply by the rate difference and divide by the total divisor; Add to the common rate, multiply the variable rate thereby and divide by the total divisor; subtract fast and add slow to the variable rate to fix the rate. Then apply the fixed rate to increase or decrease the slow-fast accumulation and obtain the fixed value. Where no matching rate follows, derive from the preceding rate likewise. When increase applies, take the common rate as the initial value and subtract half the rate difference; When decrease applies, the common rate itself is used. When a calendar rate's increase-decrease pushes the entry remainder across a day boundary, split the day in two; compute separately for the beginning and end portions, then combine and adjust the variable rate accordingly.
262
七日：初，千一百九十一； 末，百四十九。 十四日：初，千四十二； 末，二百九十八。 二十一日：初，八百九十二； 末，四百四十八。 二十八日：初，七百四十三； 末，五百九十七。 各視入餘初數，已下為初，已上以初數減之，餘為末。
Day 7: initial value 1,191; final value 149. Day 14: initial value 1,042; final value 298. Day 21: initial value 892; final value 448. Day 28: initial value 743; final value 597. For each case, compare the entry remainder with the initial threshold: at or below is the initial portion; above, subtract the threshold and treat the remainder as the final portion.
263
各以入變遲速定數，速減、遲加朔弦望盈朒小餘； 滿若不足，進退其日。 加其常日者為盈，減其常日者為朒。 各為定大小餘，命日如前。 乃前朔、後朔迭相推校，盈朒之課，據實為準； 損不侵朒，益不過盈。
Apply the fixed slow-fast value for variable-calendar entry: subtract fast, add slow to the syzygy/quarter-moon excess-deficit minor remainder; If the result overflows or falls short, advance or retreat the day accordingly. Adding to the mean day yields excess (yíng); subtracting yields deficit (nò). Fix the major and minor remainders for each and name the day as before. Then cross-check successive new moons iteratively, taking actual excess-deficit results as the standard; Decreases must not violate deficit limits; increases must not exceed excess limits.
264
定朔日名與次朔同者大，不同者小，無中氣者為閏月。 其元日有交、加時應見者，消息前後一兩月，以定大小，令虧在晦、二，弦、望亦隨消息。 月朔盈朒之極，不過頻三。 其或過者，觀定小餘近夜半者量之。
For the fixed new moon: same day-name as the next new moon means a major (30-day) month; different means a minor (29-day) month; no mid-term qi marks an intercalary month. When a crossing on the origin day should be visible at the computed hour, shift one or two months per the emission-absorption cycle to fix month length, placing deficiency on the last or second day; quarter and full moons follow the same cycle. Consecutive new-moon excess or deficit occurs at most three times in a row. If it exceeds this, adjust according to whether the fixed minor remainder falls near midnight.
265
黃道：南斗，二十四度三百二十八分。 牛，七度。 婺女，十一度。 虛，十度。 危，十六度。 營室，十八度。 東壁，十度。 奎，十七度。 婁，十三度。 胃，十五度。 昴，十一度。 畢，十六度。 觜觿，二度。 參，九度。 東井，三十度。 輿鬼，四度。 柳，十四度。 七星，七度。 張，十七度。 翼，十九度。 軫，十八度。 角，十三度。 亢，十度。 氐，十六度。 房，五度。 心，五度。 尾，十八度。 箕，十度。
Ecliptic: Southern Dipper (Nán Dǒu), width 24°328′. Ox (Niú), 7°. Maiden (Wǔ Nǚ), 11°. Emptiness (Xū), 10°. Rooftop (Wēi), 16°. Encampment (Yíng Shì), 18°. Eastern Wall (Dōng Bì), 10°. Legs (Kuí), 17°. Bond (Lóu), 13°. Stomach (Wèi), 15°. Hairy Head (Mǎo), 11°. Net (Bì), 16°. Turtle Beak (Zī Jī), 2°. Three Stars (Shēn), 9°. Eastern Well (Dōng Jǐng), 30°. Chariot Ghost (Yú Guǐ), 4°. Willow (Liǔ), 14°. Seven Stars (Qī Xīng), 7°. Extended Net (Zhāng), 17°. Wings (Yì), 19°. Chariot Crossboard (Zhěn), 18°. Horn (Jiǎo), 13°. Gullet (Kàng), 10°. Root (Dī), 16°. Chamber (Fáng), 5°. Heart (Xīn), 5°. Tail (Wěi), 18°. Winnowing Basket (Jī), 10°.
266
冬至之初日躔定在南斗十二度。 每加十五度二百九十二分、小分五，依宿度去之，各得定氣加時日度。
At Winter Solstice, the daily solar longitude is fixed at 12° within the Dipper lodge. Add 15°292′5″ each time and reduce by lodge widths to obtain each fixed qi's hour-of-occurrence solar longitude.
267
各以初日躔差乘定氣小餘，總法而一，進加、退減小餘，為分； 以減加時度，為氣初夜半度。 乃日加一度，以躔差進加、退減之，得次日。 以定朔弦望小餘副之； 以乘躔差，總法而一，進加、退減其副，各加夜半日躔，為加時宿度。
Multiply the first-day station difference by the fixed-qi minor remainder, divide by the total divisor, and advance-add or retreat-subtract to obtain parts; Subtract from the hour-of-occurrence degree to obtain the qi-initial midnight solar longitude. Then add 1° per day, applying the station difference additively or subtractively, to obtain the next day. Set the fixed syzygy/quarter-moon minor remainder alongside; Multiply by the station difference, divide by the total divisor, advance-add or retreat-subtract the auxiliary value, and add to midnight solar longitude to obtain the hour-of-occurrence lodge position.
268
合朔度，即月離也。 上弦，加度九十一度、分四百一十七。 望，加度百八十二度、分八百三十四。 下弦，加度二百七十三度、分千二百五十一。 訖，半其分，降一等，以同程法，得加時月離。 因天正常朔夜半所入變日及餘，定朔有進退日者，亦進退一日，為定朔夜半所入。 累加一日，得次日。
The conjunction degree is the Moon's ecliptic longitude. First quarter: add 91°417′. Full moon: add 182°834′. Last quarter: add 273°1251′. Halve the parts, reduce one rank, and normalize by the progression divisor to obtain the hour-of-occurrence lunar longitude. From the canonical new-moon midnight variable-calendar day and remainder: if the fixed new moon advances or retreats a day, shift the midnight entry likewise. Accumulate by adding one day to obtain the next day.
269
各以夜半入變餘乘進退差，總法而一，進加、退減離程，為定程。 以定朔弦望小餘乘之，總法而一，以減加時月離，為夜半月離。 求次日，程法約定程，累加之。 若以定程乘夜刻，二百除，為晨分。 以減定程，為昏分。 其夜半月離，朔後加昏為昏度，望後加晨為晨度。 其注曆，五乘弦望小餘，程法而一，為刻。 不滿晨前刻者，退命算上。
Multiply the midnight variable-entry remainder by the advance-retreat difference, divide by the total divisor, and advance-add or retreat-subtract the separation progression to fix the progression. Multiply by the fixed syzygy/quarter-moon minor remainder, divide by the total divisor, and subtract from the hour-of-occurrence lunar longitude to obtain midnight lunar longitude. For the next day, divide the fixed progression by the progression divisor and accumulate. Multiply the fixed progression by night quarters and divide by 200 to obtain the dawn portion. Subtract from the fixed progression to obtain the dusk portion. From midnight lunar longitude: after new moon add the dusk portion for dusk longitude; after full moon add the dawn portion for dawn longitude. For calendar annotation, multiply the quarter-moon minor remainder by 5 and divide by the progression divisor to obtain day-quarters. If less than the pre-dawn quarters, retreat one unit on the counting board.
270
辰刻八，分二十四。
Canonical time unit: 8 quarters, 24 parts.
271
刻分法七十二。
Quarter-part divisor: 72.
272
定氣
Fixed solar term (dìng qì)
273
晨前刻
Day-quarters before dawn
274
黃道去極度
Ecliptic departure from the celestial pole (declination proxy)
275
屈伸率
Daily day-length stretch/compression rate
276
發斂差
Emission-absorption correction difference
277
冬至
Winter Solstice
278
三十刻
30 day-quarters before dawn
279
百一十五度三分
115°3′ ecliptic departure from pole
280
伸一三分
daily stretch rate 1.3
281
益十六
emission correction +16
282
小寒
Lesser Cold
283
二十九刻五十四分
29 day-quarters 54/72 part before dawn
284
百一十三度一分
113°1′ ecliptic departure from pole
285
伸三七分
daily stretch rate 3.7
286
益十六
emission correction +16
287
大寒
Major Cold
288
二十九刻十八分
29 day-quarters 18/72 part before dawn
289
百一十度七分
110°7′ ecliptic departure from pole
290
伸六一分
daily stretch rate 6.1
291
益二十二
emission correction +22
292
立春
Start of Spring
293
二十八刻三十三分
28 day-quarters 33/72 part before dawn
294
百七度九分
107°9′ ecliptic departure from pole
295
伸九四分
daily stretch rate 9.4
296
益九
emission correction +9
297
啟蟄
Awakening of Insects
298
二十七刻三十分
27 day-quarters 30/72 part before dawn
299
百二度九分
102°9′ ecliptic departure from pole
300
伸十七分半
daily stretch rate 17.5
301
益七
emission correction +7
302
雨水
Rain Water
303
二十六刻十八分
26 day-quarters 18/72 part before dawn
304
九十七度三分
97°3′ ecliptic departure from pole
305
伸十一八分
daily stretch rate 11.8
306
益三
emission correction +3
307
春分
Spring Equinox
308
二十五刻
25 day-quarters before dawn
309
九十一度三分
91°3′ ecliptic departure from pole
310
伸十二二分半
daily stretch rate 12.25
311
損三
absorption correction −3
312
清明
Clear and Bright
313
二十三刻五十四分
23 day-quarters 54/72 part before dawn
314
八十五度三分
85°3′ ecliptic departure from pole
315
伸十一八分
daily stretch rate 11.8
316
損七
absorption correction −7
317
穀雨
Grain Rain
318
二十二刻四十二分
22 day-quarters 42/72 part before dawn
319
七十九度七分
79°7′ ecliptic departure from pole
320
伸十七分半
daily stretch rate 17.5
321
損九
absorption correction −9
322
立夏
Start of Summer
323
二十一刻三十九分
21 day-quarters 39/72 part before dawn
324
七十四度七分
74°7′ ecliptic departure from pole
325
伸九四分
daily stretch rate 9.4
326
損二十二
absorption correction −22
327
小滿
Lesser Fullness
328
二十刻五十四分
20 day-quarters 54/72 part before dawn
329
七十度九分
70°9′ ecliptic departure from pole
330
伸六一分
daily stretch rate 6.1
331
損十六
absorption correction −16
332
芒種
Grain in Ear
333
二十刻十八分
20 day-quarters 18/72 part before dawn
334
六十八度五分
68°5′ ecliptic departure from pole
335
伸三七分
daily stretch rate 3.7
336
損十六
absorption correction −16
337
夏至
Summer Solstice
338
二十刻
20 day-quarters before dawn
339
六十七度三分
67°3′ ecliptic departure from pole
340
屈一三分
daily compression rate 1.3
341
益十六
emission correction +16
342
小暑
Lesser Heat
343
二十刻十八分
20 ke 18 fen.
344
六十八度五分
68 degrees 5 fen.
345
屈三七分
Contraction: 3.7 fen.
346
益十六
Increase: 16.
347
大暑
Major Heat.
348
二十刻五十四分
20 ke 54 fen.
349
七十度九分
70 degrees 9 fen.
350
屈六一分
Contraction: 6.1 fen.
351
益二十二
Increase: 22.
352
立秋
Start of Autumn.
353
二十一刻三十九分
21 ke 39 fen.
354
七十四度七分
74 degrees 7 fen.
355
屈九四分
Contraction: 9.4 fen.
356
益九
Increase: 9.
357
處暑
End Heat.
358
二十二刻四十二分
22 ke 42 fen.
359
七十九度七分
79 degrees 7 fen.
360
屈十七分半
Contraction: 17.5 fen.
361
益七
Increase: 7.
362
白露
White Dew.
363
二十三刻五十四分
23 ke 54 fen.
364
八十五度三分
85 degrees 3 fen.
365
屈十一八分
Contraction: 11.8 fen.
366
益三
Increase: 3.
367
秋分
Autumn Equinox.
368
二十五刻
25 ke.
369
九十一度三分
91 degrees 3 fen.
370
屈十二二分半
Contraction: 12.25 fen.
371
損三
Decrease: 3.
372
寒露
Cold Dew.
373
二十六刻十八分
26 ke 18 fen.
374
九十七度三分
97 degrees 3 fen.
375
屈十一八分
Contraction: 11.8 fen.
376
損七
Decrease: 7.
377
霜降
Frost's Descent.
378
二十七刻三十分
27 ke 30 fen.
379
百二度九分
102 degrees 9 fen.
380
屈十七分半
Contraction: 17.5 fen.
381
損九
Decrease: 9.
382
立冬
Start of Winter.
383
二十八刻三十三分
28 ke 33 fen.
384
百七度九分
107 degrees 9 fen.
385
屈九四分
Contraction: 9.4 fen.
386
損二十二
Decrease: 22.
387
小雪
Minor Snow.
388
二十九刻十八分
29 ke 18 fen.
389
百一十度七分
110 degrees 7 fen.
390
屈六一分
Contraction: 6.1 fen.
391
損十六
Decrease: 16.
392
大雪
Major Snow.
393
二十九刻五十四分
29 ke 54 fen.
394
百一十三度一分
113 degrees 1 fen.
395
屈三七分
Contraction: 3.7 fen.
396
損十六
Decrease: 16.
397
置其氣屈伸率，各以發斂差損益之，為每日屈伸率。 差滿十，從分； 分滿十，為率。 各累計其率為刻分。 百八十乘之，十一乘綱紀除之，為刻差。 各半之，以伸減、屈加晨前刻分，為每日晨前定刻。 倍之，為夜刻。 以減一百，為晝刻。 以三十四約刻差，為分； 分滿十，為度。 以伸減、屈加氣初黃道去極，得每日。 以晝刻乘朞實，二百乘，總法除，為昏中度。 以減三百六十五度三百二十八分，餘為旦中度。 各以加日躔，得昏旦中星，赤道計之。 其赤道同太初星距。
Set each solar term's flexion and extension rate, then apply the opening-closing difference to obtain the daily flexion and extension rate. When the difference reaches 10, carry to the fen; when fen reach 10, that becomes the rate. Accumulate each day's rate to obtain ke-fen. Multiply by 180 and divide by 11 times the cycle constant to obtain the ke-difference. Halve each; where extended subtract and where contracted add to the morning-before ke-fen to obtain the daily fixed morning-before ke. Double it to obtain night ke. Subtract from 100 to obtain day ke. Divide the ke-difference by 34 to obtain fen; when fen reach 10, that becomes degrees. Where extended subtract and where contracted add to the ecliptic departure from the pole at the start of the qi, to obtain each day's value. Multiply day ke by the period constant, multiply by 200, and divide by the general divisor to obtain dusk mid-degrees. Subtract from 365 degrees 328 fen; the remainder is dawn mid-degrees. Add each to the daily solar position to obtain dusk and dawn culminating stars, reckoned on the equator. Its equatorial coordinates match the Taichu star distances.
398
遊交終率千九十三萬九千三百一十三。
Nodal-travel completion rate: 10,939,313.
399
奇率三百。
Odd rate: 300.
400
約終三萬六千四百六十四，奇百一十三。
Reduced completion: 36,464; odd 113.
401
交中萬八千二百三十二，奇五十六半。
Node center: 18,232; odd 56.5.
402
交終日二十七，餘二百八十四，奇百一十三。
Nodal period: 27 days; remainder 284; odd 113.
403
交中日十三，餘八百一十二，奇五十六半。
Mid-node day: 13 days; remainder 812; odd 56.5.
404
虧朔三千一百六，奇百八十七。
New-moon eclipse: 3,106; odd 187.
405
實望萬九千七百八十五，奇百五十。
True full moon: 19,785; odd 150.
406
後準千五百五十三，奇九十三半。
Posterior limit: 1,553; odd 93.5.
407
前準萬六千六百七十八，奇二百六十三。
Prior limit: 16,678; odd 263.
408
置總實，以奇率乘之，滿終率去之； 不滿，以奇率約，為入交分。 加天正常朔小餘，得朔汎交分。 求次朔，以虧朔加之。 因朔求望，以實望加之。 各以朔望入氣盈朒定積，盈加、朒減之； 又六十乘遲速定數，七百七十七除，為限數； 以速減、遲加，為定交分。 其朔，月在日道裏者，以所入限數減遲速定數，餘以速減、遲加其定交分。 而出日道表者，為變交分。 不出表者，依定交分。 其變交分三時半內者，依術消息，以定蝕不。 交中已下者，為月在外道； 已上者，去之，餘為月在內道。 其分如後準已下，為交後分； 前準已上者，反減交中，餘為交前分。 望則月蝕，朔在內道則日蝕。 百一十二約前後分，為去交時。 置定朔小餘，副之。 辰率約之，以艮、巽、坤、乾為次，命算外。 其餘，半法已下為初； 已上者，去之，為末。 初則因餘，末則減法，各為差率。 月在內道者，益去交時十而三除之。 以乘差率，十四而一，為差。 其朔，在二分前後一氣內，即以差為定； 近冬至以去寒露、雨水，近夏至以去清明、白露氣數倍之，又三除去交時增之； 近冬至艮巽以加、坤乾以減，近夏至艮巽以減、坤乾以加其差，為定差。 艮、巽加副，坤、乾減副。 月在外道者，三除去交時數，以乘差率，十四而一，為差。 艮、坤以減副，巽、乾以加副，為食定小餘。 望即因定望小餘，即所在辰； 近朝夕者，以日出沒刻校前後十二刻半內候之。
Set the total constant, multiply by the odd rate, and discard full multiples of the completion rate; the remainder, reduced by the odd rate, gives the entry-into-node parts. Add the heaven-normal mean new-moon minor remainder to obtain the new-moon general node parts. To find the next new moon, add the new-moon eclipse parts. From new moon to find full moon, add the true full moon parts. For each, take the new-moon or full-moon entry-qi elongation fixed accumulation; add if elongating, subtract if contracting; also multiply the slow-fast fixed number by 60 and divide by 777 to obtain the limit number; subtract if slow and add if fast to obtain the fixed node parts. At new moon, if the moon is inside the solar path, subtract the limit number entered from the slow-fast fixed number; with the remainder, subtract if slow and add if fast to the fixed node parts. If it exits outside the solar path, that is the variable node parts. If it does not exit outside, use the fixed node parts. If the variable node parts fall within 3.5 double-hours, apply the waxing-waning procedure to determine whether an eclipse occurs. Below the mid-node, the moon is on the outer path; above it, subtract; the remainder is the moon on the inner path. If the parts are at or below the posterior limit, that is post-node parts; if at or above the prior limit, subtract inversely from the node center; the remainder is pre-node parts. At full moon there is a lunar eclipse; at new moon, if on the inner path, there is a solar eclipse. Divide pre- and post-node parts by 112 to obtain the distance-from-node in double-hours. Set the fixed new-moon minor remainder aside. Divide by the double-hour rate; in sequence gen, xun, kun, qian, count beyond the tally. Of the remainder, at or below half the divisor is initial; above that, subtract to obtain final. If initial, use the remainder; if final, subtract the divisor—each gives the difference-rate. If the moon is on the inner path, increase the distance-from-node in double-hours by three-tenths and divide. Multiply by the difference-rate and divide by 14 to obtain the difference. At new moon, if within one qi before or after either equinox, use the difference as fixed; near winter solstice use the distance from Cold Dew and Rain Water; near summer solstice use the distance from Clear and Bright and White Dew; multiply the qi count, and again increase by three-tenths of the distance-from-node; near winter solstice gen and xun add and kun and qian subtract; near summer solstice gen and xun subtract and kun and qian add to the difference, as the fixed difference. Gen and xun add to the aside; kun and qian subtract from the aside. If the moon is on the outer path, reduce the distance-from-node count by three-tenths, multiply by the difference-rate, and divide by 14 to obtain the difference. Gen and kun subtract from the aside; xun and qian add to the aside, as the eclipse fixed minor remainder. At full moon, follow the fixed full-moon minor remainder—that is the double-hour of occurrence; if near dawn or dusk, check within 12.5 ke before and after using sunrise and sunset ke.
409
月在外道，朔不應蝕。 夏至初日，以二百四十八為初準。 去交前後分如初準已下、加時在午正前後七刻內者，蝕。 朔去夏至前後，每一日損初準二分，皆畢於九十四日，為每日變準。 交分如變準已下、加時如前者，亦蝕。 又以末準六十減初準及變準，餘以十八約之，為刻準。 以幷午正前後七刻內數，為時準。 加時準內交分，如末準已下，亦蝕。 又置末準，每一刻加十八，為差準。 加時刻去午前後如刻準已上、交分如差準已下者，亦蝕。 自秋分至春分，去交如末準已下、加時巳、午、未者，亦蝕。
If the moon is on the outer path, at new moon there should be no eclipse. On the first day of summer solstice, use 248 as the initial limit. If pre- and post-node parts are at or below the initial limit and the add-time is within 7 ke before or after noon, there is an eclipse. Before and after new moon from summer solstice, diminish the initial limit by 2 fen each day, completing at 94 days, as the daily variable limit. If node parts are at or below the variable limit and the add-time is as above, there is also an eclipse. Also subtract the final limit of 60 from the initial and variable limits; divide the remainder by 18 to obtain the ke-limit. Combine with the count within 7 ke before and after noon, as the time-limit. If node parts are within the add-time limit and at or below the final limit, there is also an eclipse. Also set the final limit; add 18 for each ke, as the difference-limit. If the add-time ke from before or after noon is at or above the ke-limit and node parts are at or below the difference-limit, there is also an eclipse. From autumn equinox through spring equinox, if the distance from node is at or below the final limit and the add-time is si, wu, or wei, there is also an eclipse.
410
月在內道，朔應蝕。 若在夏至初日，以千三百七十三為初準。 去交如初準已上、加時在午正前後十八刻內者，或不蝕。 夏至前後每日益初準一分半，皆畢於九十四日，為每日變準。 以初準減變準，餘十而一，為刻準。 以減午正前後十八刻，餘為時準。 其去交在變準已上、加時在準內，或不蝕。
If the moon is on the inner path, at new moon there should be an eclipse. If at the first day of summer solstice, use 1,373 as the initial limit. If the distance from node is at or above the initial limit and the add-time is within 18 ke before or after noon, there may be no eclipse. Before and after summer solstice, increase the initial limit by 1.5 fen each day, completing at 94 days, as the daily variable limit. Subtract the variable limit from the initial limit; one-tenth of the remainder is the ke-limit. Subtract from 18 ke before and after noon; the remainder is the time-limit. If the distance from node is above the variable limit and the add-time is within the limit, there may be no eclipse.
411
望去交前後定分，冬，減二百二十四； 夏，減五十四； 春，交後減百，交前減二百； 秋，交後減二百，交前減百。 不足減者，蝕既。 有餘者，以減後準，百四而一，得月蝕分。
Remove the fixed pre- and post-node parts: in winter subtract 224; in summer subtract 54; in spring, post-node subtract 100 and pre-node subtract 200; in autumn, post-node subtract 200 and pre-node subtract 100. If subtraction would go negative, the eclipse is total. If there is a remainder, subtract from the posterior limit; divide by 104 to obtain lunar eclipse parts.
412
朔交，月在內道，入冬至畢定雨水，及秋分畢大雪，皆以五百五十八為蝕差。 入春分，日損六分，畢芒種。 以蝕差減去交分； 不足減者，反減蝕差，為不蝕分。 其不蝕分，自小滿畢小暑，加時在午正前後七刻外者，皆減一時； 三刻內者，加一時。 大寒畢立春交前五時外、大暑畢立冬交後五時外者，皆減一時； 五時內者，加一時。 諸加時蝕差應減者，交後減之，交前加之； 應加者，交後加之，交前減之。 不足減者，皆既； 加減入不蝕限者，或不蝕。 月在外道，冬至初日，無蝕差。 自後日益六分，畢於雨水。 入春分，畢白露，皆以五百二十二為差。 入秋分，日損六分，畢大雪。 以差加去交分，為蝕分。 以減後準，餘為不蝕分。 十五約蝕差，以百四，為定法。 其不蝕分，如定法得一，以減十五，餘得日蝕分。
At new-moon node with the moon on the inner path, entering winter solstice through fixed Rain Water, and autumn equinox through Major Snow, all use 558 as the eclipse-difference. Entering spring equinox, diminish by 6 fen each day, completing at Grain in Ear. Subtract the eclipse-difference from the distance-from-node parts; if subtraction would go negative, reverse-subtract the eclipse-difference to obtain the non-eclipse threshold. For the non-eclipse threshold, from Lesser Full Grain through Minor Heat, if the add-time is outside 7 ke before or after noon, subtract 1 double-hour in all cases; if within 3 ke, add 1 double-hour. From Major Cold through Start of Spring, pre-node beyond 5 double-hours, and from Major Heat through Start of Winter, post-node beyond 5 double-hours, subtract 1 double-hour in all cases; if within 5 double-hours, add 1 double-hour. For all add-time eclipse-differences that call for subtraction, subtract at post-node and add at pre-node; those that call for addition, add at post-node and subtract at pre-node. If subtraction would go negative, the eclipse is total; if the adjustment enters the non-eclipse threshold, there may be no eclipse. If the moon is on the outer path, on the first day of winter solstice, there is no eclipse-difference. Thereafter increase by 6 fen each day, completing at Rain Water. Entering spring equinox through White Dew, all use 522 as the difference. After autumn equinox, diminish by 6 fen each day, completing at Major Snow. Add the difference to the distance-from-node parts to obtain eclipse parts. Subtract from the posterior limit; the remainder is the non-eclipse threshold. Divide the eclipse-difference by 15 and multiply by 104 to obtain the fixed divisor. For the non-eclipse threshold, if the fixed divisor yields 1, subtract from 15; the remainder is solar eclipse magnitude.
413
歲星總率五十三萬四千四百八十三，奇四十五。 伏分二萬四千三十一，奇七十二半。
Jupiter total rate: 534,483; odd 45. Invisibility parts: 24,031; odd 72.5.
414
終日三百九十八，餘千一百六十三，奇四十五。
Total days: 398; remainder 1,163; odd 45.
415
平見，入冬至，畢小寒，均減六日。 入大寒，日損六十七分。 入春分，依平。 乃日加八十九分。 入立夏，畢小滿，均加六日。 入芒種，日損八十九分。 入夏至，畢立秋，均加四日。 入處暑，日損百七十八分。 入白露，依平。 自後日減五十二分。 入小雪，畢大雪，均減六日。
At mean appearance, entering winter solstice through Minor Cold, uniformly subtract 6 days. Entering Major Cold, diminish by 67 fen each day. Entering spring equinox, use the mean rate. Then add 89 fen each day. Entering Start of Summer through Lesser Full Grain, uniformly add 6 days. Entering Grain in Ear, diminish by 89 fen each day. Entering summer solstice through Start of Autumn, uniformly add 4 days. Entering End Heat, diminish by 178 fen each day. Entering White Dew, use the mean rate. Thereafter diminish by 52 fen each day. Entering Minor Snow through Major Snow, uniformly subtract 6 days.
416
初順，百一十四日行十八度五百九分，日益遲一分。 前留，二十六日。 旋退，四十二日，退六度十二分，日益疾二分。 又退，四十二日，退六度十二分，日益遲二分。 後留，二十五日。 後順，百一十四日行十八度五百九分，日益疾一分。 日盡而夕伏。
Initial direct motion: 114 days covering 18 degrees 509 parts, slowing by 1 part each day. First station: 26 days. Then retrograde: 42 days, retreating 6 degrees 12 parts, speeding by 2 parts each day. Retrograde again: 42 days, retreating 6 degrees 12 parts, slowing by 2 parts each day. Second station: 25 days. Later direct motion: 114 days covering 18 degrees 509 parts, speeding by 1 part each day. When its days are exhausted it disappears at evening.
417
熒惑總率百四萬五千八十，奇六十。 伏分九萬七千九十，奇三十。
Mars total rate: 1,045,080; odd 60. Invisibility parts: 97,090; odd 30.
418
終日七百七十九，餘千二百二十，奇六十。
Total days: 779; remainder 1,220; odd 60.
419
平見，入冬至，減二十七日。 自後日損六百三分。 入大寒，日加四百二分。 入雨水，畢穀雨，均加二十七日。 入立夏，日損百九十八分。 入立秋，依平。 入處暑，日減百九十八分。 入小雪，畢大雪，均減二十七日。
At mean appearance, entering winter solstice, subtract 27 days. Thereafter diminish by 603 fen each day. Entering Major Cold, add 402 fen each day. Entering Rain Water through Grain Rain, uniformly add 27 days. Entering Start of Summer, diminish by 198 fen each day. Entering Start of Autumn, use the mean rate. Entering End Heat, diminish by 198 fen each day. Entering Minor Snow through Major Snow, uniformly subtract 27 days.
420
初順，入冬至，率二百四十三日行百六十五度。 乃三日損日度各二。 小寒初日，率二日三十三日行百五十五度。 乃二日損一。 入穀雨四日，平，畢小滿九日，率百七十八日行百度。 乃三日損一。 夏至初日，平，畢六日，率百七十一日行九十三度。 乃三日益一。 入立秋初日，百八十四日行百六度。 乃每日益一。 入白露初日，率二百一十四日行百三十六度。 乃五日益六。 入秋分初日，率二百三十二日行百五十四度。 又每日益一。 入寒露初日，率二百四十七日行百六十九度。 乃五日益三。 入霜降五日，平，畢立冬十三日，率二百五十九日行百八十一度。 乃二日損日一。 入冬至，復初。
Initial direct motion, entering winter solstice: rate 243 days for 165 degrees of travel. Then every 3 days diminish the day-rate and degree-rate by 2 each. On the first day of Minor Cold, rate 233 days for 155 degrees of travel. Then diminish by 1 every 2 days. Entering Grain Rain for 4 days, use the mean rate; through Lesser Full Grain for 9 days, rate 178 days for 100 degrees of travel. Then diminish by 1 every 3 days. On the first day of summer solstice, use the mean rate; through 6 days, rate 171 days for 93 degrees of travel. Then increase by 1 every 3 days. On the first day of Start of Autumn, 184 days for 106 degrees of travel. Then increase by 1 each day. On the first day of White Dew, rate 214 days for 136 degrees of travel. Then increase by 6 every 5 days. On the first day of autumn equinox, rate 232 days for 154 degrees of travel. Again increase by 1 each day. On the first day of Cold Dew, rate 247 days for 169 degrees of travel. Then increase by 3 every 5 days. Entering Frost's Descent for 5 days, use the mean rate; through Start of Winter for 13 days, rate 259 days for 181 degrees of travel. Then diminish the day-rate by 1 every 2 days. Entering winter solstice, return to the initial rate.
421
各依所入常氣，平者依率，餘皆計日損益，為前疾日度定率。 其前遲及留退，入氣有損益日、度者，計日損益，皆準此法。 疾行日率，入大寒，六日損一； 入春分，畢立夏，均減十日； 入小滿，三日損所減一； 畢芒種，依平； 入立秋，三日益一； 入白露，畢秋分，均加十日； 入寒露，一日半損所加一； 畢氣盡，依平：為變日率。 疾行度率，入大寒畢啟蟄，立夏畢夏至，大暑畢氣盡，霜降畢小雪，皆加四度； 清明畢穀雨，加二度：為變度率。
For each regular qi entered: if mean, follow the rate; otherwise calculate the daily increase or decrease to obtain the initial-rapid day-degree fixed rate. For initial slow motion and station-retreat, if the qi entered calls for day or degree adjustments, calculate daily increase and decrease—all follow this method. Rapid-travel day-rate: entering Major Cold, diminish by 1 every 6 days; entering spring equinox through Start of Summer, uniformly subtract 10 days; entering Lesser Full Grain, diminish the subtracted amount by 1 every 3 days; through Grain in Ear, use the mean rate; entering Start of Autumn, increase by 1 every 3 days; entering White Dew through autumn equinox, uniformly add 10 days; entering Cold Dew, diminish the added amount by 1 every 1.5 days; when the qi is exhausted, use the mean rate—this is the variable day-rate. Rapid-travel degree-rate: entering Major Cold through Awakening of Insects, Start of Summer through summer solstice, Major Heat through qi exhaustion, and Frost's Descent through Minor Snow, all add 4 degrees; Clear and Bright through Grain Rain, add 2 degrees—this is the variable degree-rate.
422
初行入處暑，減日率六十，度率三十； 入白露，畢秋分，減日率四十四，度率二十二：皆為初遲半度之行。 盡此日、度，乃求所減之餘日、度率，續之，為疾。 初行入大寒畢大暑，差行，日益遲一分。 其前遲、後遲，日率既有增損，而益遲、益疾，差分皆檢括前疾末日行分，為前遲初日行分。 以前遲平行分減之，餘為前遲總差。 後疾初日行分，為後遲末日行分，以後遲初日行分減之，餘為後遲總差。 相減，為前後別日差分。 其不滿者皆調為小分。 遲疾之際，行分衰殺不倫者，依此。
At initial travel entering End Heat, subtract 60 from the day-rate and 30 from the degree-rate; entering White Dew through autumn equinox, subtract 44 from the day-rate and 22 from the degree-rate—all as initial slow half-degree travel. When these days and degrees are exhausted, calculate the remaining day- and degree-rates subtracted and continue as rapid motion. Initial travel entering Major Cold through Major Heat uses differential motion, slowing by 1 part each day. For initial slow and later slow phases, since the day-rate already has increase and decrease, the increasingly slow and increasingly fast difference-parts are all checked against the initial-rapid last-day motion-parts to obtain the initial-slow first-day motion-parts. Subtract the initial-slow uniform-motion parts; the remainder is the initial-slow total difference. The later-rapid first-day motion-parts are the later-slow last-day motion-parts; subtract the later-slow first-day motion-parts; the remainder is the later-slow total difference. Subtract the two to obtain the before-and-after separate day-difference parts. Any remainder is adjusted as minor parts. At the slow-rapid transition, where motion-parts decay irregularly, follow this method.
423
前遲，入冬至，率六十日行二十五度； 先疾，日益遲二分。 入小寒，三日損一。 大寒初日，率五十五日行二十度。 乃三日益一。 立春初日，平，畢清明，率六十日行二十五度。 入穀雨，每氣別減一度。 立夏初日，平，畢小滿，率六十日行二十二度。 入芒種，每氣別益一度。 夏至初日，平，畢處暑，率六十日行二十五度。 入白露，三日損一。 秋分初日，率六十日行二十五度。 乃每日益日一，三日益度二。 寒露初日，率七十五日行三十度。 乃每日損日一，三日損度一。 霜降初日，率六十日行二十五度。 乃二日損一度。 入立冬一日，平，畢氣盡，率六十日行十七度。 入小雪，五日益一度。 大雪初日，率六十日行二十度。 乃三日益一度。 入冬至，復初。
Initial slow motion, entering winter solstice: rate 60 days for 25 degrees of travel; Initially fast, slowing by 2 parts each day. Entering Lesser Cold, subtract 1 every 3 days. On the first day of Great Cold, the rate is 55 days for 20 degrees. Then increase by 1 every 3 days. From the first day of Start of Spring through Clear and Bright, use uniform motion at 25 degrees in 60 days. Entering Grain Rain, subtract 1 degree per solar term. From the first day of Start of Summer through Lesser Full Grain, use uniform motion at 22 degrees in 60 days. Entering Grain in Ear, add 1 degree per solar term. From the first day of summer solstice through End Heat, use uniform motion at 25 degrees in 60 days. Entering White Dew, subtract 1 every 3 days. On the first day of Autumn Equinox, the rate is 60 days for 25 degrees. Then add 1 to the day-count each day and 2 to the degree-count every 3 days. On the first day of Cold Dew, the rate is 75 days for 30 degrees. Then subtract 1 from the day-count each day and 1 from the degree-count every 3 days. On the first day of Frost's Descent, the rate is 60 days for 25 degrees. Then subtract 1 degree every 2 days. One day after entering Start of Winter through the end of the qi, use uniform motion at 17 degrees in 60 days. Entering Lesser Snow, add 1 degree every 5 days. On the first day of Great Snow, the rate is 60 days for 20 degrees. Then add 1 degree every 3 days. Entering winter solstice, return to the initial values.
424
前留，十三日。 前疾減日率一者，以其數分益此留及後遲日率。 前疾加日率者，以其數分減此留及後遲日率。 旋退，西行。 入冬至初日，率六十三日退二十一度。 乃四日益度一。 小寒一日，率六十三日退二十六度。 乃三日半損度一。 立春三日，平，畢啟蟄，率六十三日退十七度。 乃二日益日、度各一。 雨水八日，平，畢氣盡，率六十七日退二十一度。 入春分，每氣損日、度各一。 大暑初日，平，畢氣盡，率五十八日退十二度。 立秋初日，平，畢氣盡，率五十七日退十一度。 乃二日益日一。 寒露九日，平，畢氣盡，率六十六日退二十度。 乃二日損一。 霜降六日，平，畢氣盡，率六十三日退十七度。 乃三日益一。 立冬十一日，平，畢氣盡，率六十七日退二十一度。 乃二日損一。 入冬至，復初。
First stationary period: 13 days. If the initial rapid phase subtracted 1 from the day-rate, distribute that amount to increase this stationary and later slow day-rates. If the initial rapid phase added to the day-rate, distribute that amount to decrease this stationary and later slow day-rates. Then retrograde, moving westward. On the first day of winter solstice, the rate is 63 days retreating 21 degrees. Then add 1 degree every 4 days. One day into Lesser Cold, the rate is 63 days retreating 26 degrees. Then subtract 1 degree every 3.5 days. Three days into Start of Spring through Awakening of Insects, use uniform motion at 17 degrees retreated in 63 days. Then add 1 each to the day-count and degree-count every 2 days. Eight days into Rain Water through the end of the qi, use uniform motion at 21 degrees retreated in 67 days. Entering Spring Equinox, subtract 1 each from the day-count and degree-count per solar term. On the first day of Great Heat through the end of the qi, use uniform motion at 12 degrees retreated in 58 days. On the first day of Start of Autumn through the end of the qi, use uniform motion at 11 degrees retreated in 57 days. Then add 1 to the day-count every 2 days. Nine days into Cold Dew through the end of the qi, use uniform motion at 20 degrees retreated in 66 days. Then subtract 1 every 2 days. Six days into Frost's Descent through the end of the qi, use uniform motion at 17 degrees retreated in 63 days. Then add 1 every 3 days. Eleven days into Start of Winter through the end of the qi, use uniform motion at 21 degrees retreated in 67 days. Then subtract 1 every 2 days. Entering winter solstice, return to the initial values.
425
後留，冬至初，留十三日。 乃二日半益一。 大寒初日，平，畢氣盡，留二十五日。 乃二日半損一。 雨水初日，留十三日。 乃三日益一。 清明初日，留二十三日。 乃日損一。 清明十日，平，畢處暑，留十三日。 乃二日損一。 秋分十一日，無留。 乃每日益一。 霜降初日，留十九日。 乃三日損一。 立冬畢大雪，留十三日。
Later stationary period: at winter solstice, stationary for 13 days. Then add 1 every 2.5 days. On the first day of Great Cold through the end of the qi, stationary for 25 days. Then subtract 1 every 2.5 days. On the first day of Rain Water, stationary for 13 days. Then add 1 every 3 days. On the first day of Clear and Bright, stationary for 23 days. Then subtract 1 each day. Ten days into Clear and Bright through End Heat, use the mean rate; stationary for 13 days. Then subtract 1 every 2 days. Eleven days into Autumn Equinox, no stationary period. Then add 1 each day. On the first day of Frost's Descent, stationary for 19 days. Then subtract 1 every 3 days. From Start of Winter through Great Snow, stationary for 13 days.
426
後遲，順，六十日行二十五度，日益疾二分。 前疾加度者，此遲依數減之，為定度。 前疾無加度者，此遲入秋分至立冬減三度，入冬至減五度。 後留定日朒十三日者，以所朒日數加此遲日率。
Later slow phase: direct motion, 25 degrees in 60 days, accelerating by 2 parts each day. If the initial rapid phase added degrees, subtract that amount in this slow phase to get the fixed total. If the initial rapid phase added no degrees, subtract 3 degrees from this slow phase between Autumn Equinox and Start of Winter, and 5 degrees entering winter solstice. If the later stationary fixed days fall short by 13, add the shortfall in days to this slow day-rate.
427
後疾，冬至初日，率二百一十日行百三十二度。 乃每日損一。 大寒八日，率百七十二日行九十四度。 乃二日損一。 啟蟄，平，畢氣盡，率百六十一日行八十三度。 乃二日益一。 芒種十四日，平，畢夏至，率二百三十三日行百五十五度。 乃每日益一。 大暑初日，平，畢處暑，率二百六十三日行百八十五度。 乃二日損一。 秋分一日，率二百五十五日行百七十七度。 乃一日半損一。 大雪初日，率二百五日行百二十七度。 乃三日益一。 入冬至，復初。
Later rapid phase: on the first day of winter solstice, the rate is 210 days for 132 degrees. Then subtract 1 each day. Eight days into Great Cold, the rate is 172 days for 94 degrees. Then subtract 1 every 2 days. At Awakening of Insects through the end of the qi, use uniform motion at 83 degrees in 161 days. Then add 1 every 2 days. Fourteen days into Grain in Ear through summer solstice, use uniform motion at 155 degrees in 233 days. Then add 1 each day. On the first day of Great Heat through End Heat, use uniform motion at 185 degrees in 263 days. Then subtract 1 every 2 days. One day into Autumn Equinox, the rate is 255 days for 177 degrees. Then subtract 1 every 1.5 days. On the first day of Great Snow, the rate is 205 days for 127 degrees. Then add 1 every 3 days. Entering winter solstice, return to the initial values.
428
其入常氣日度之率有損益者，計日損益，為後疾定日度率。 疾行日率，其前遲定日朒六十及退行定日朒六十三者，皆以所朒日數加疾行定日率； 前遲定日盈六十、退行定日盈六十三、後留定日盈十三者，皆以所盈日數減此疾定日率：各為變日率。 疾行度率，其前遲定度朒二十五、退行定度盈十七、後遲入秋分到冬至減度者，皆以所盈朒度數加此疾定率； 前遲定度盈二十五及退行定度朒十七者，皆以所盈朒度數減此疾定度率：各為變度率。
Where the day- and degree-rates for entering regular qi have adjustments, calculate the daily increase or decrease to obtain the later rapid fixed day- and degree-rates. For the rapid-motion day-rate: if the earlier slow fixed days fall short by 60 or the retrograde fixed days fall short by 63, add the shortfall in days to the rapid-motion fixed day-rate; if the earlier slow fixed days exceed by 60, retrograde fixed days exceed by 63, or later stationary fixed days exceed by 13, subtract the surplus in days from this rapid fixed day-rate—each yields a variable day-rate. For the rapid-motion degree-rate: if the earlier slow fixed degrees fall short by 25, retrograde fixed degrees exceed by 17, or the later slow phase subtracts degrees between Autumn Equinox and winter solstice, add the surplus or shortfall in degrees to this rapid fixed rate; if the earlier slow fixed degrees exceed by 25 or retrograde fixed degrees fall short by 17, subtract the surplus or shortfall in degrees from this rapid fixed degree-rate—each yields a variable degree-rate.
429
初行入春分畢穀雨，差行，日益疾一分。 初行入立夏畢夏至，日行半度，六十六日行三十三度。 小暑畢大暑，五十日行二十五度。 立秋畢氣盡，二十日行十度。 減率續行，並同前。 盡日度而夕伏。
If the initial motion falls between Spring Equinox and Grain Rain, use variable motion, accelerating by 1 part each day. If the initial motion falls between Start of Summer and summer solstice, travel half a degree per day, covering 33 degrees in 66 days. From Lesser Heat through Great Heat, 25 degrees in 50 days. From Start of Autumn through the end of the qi, 10 degrees in 20 days. Continue with the diminished rates as before. When the allotted days and degrees are used up, enter evening invisibility.
430
鎮星總率五十萬六千六百二十三，奇二十九。
Saturn total cycle rate: 506,623; remainder 29.
431
伏分二萬二千八百三十一，奇六十四半。
Invisibility fraction: 22,831; remainder 64½.
432
終日三百七十八，餘一百三，奇二十九。
Total days: 378; remainder 103; odd part 29.
433
平見，入冬至，初減四日。 乃日益八十九分。 入大寒，畢春分，均減八日。 入清明，日損五十九分。 入小暑初，依平。 自後日加八十九分。 入白露初，加八日。 自後日損百七十八分。 入秋分，均加四日。 入寒露，日損五十九分。 入小雪初日，依平。 乃日減八十九分。
For mean appearance at winter solstice, initially subtract 4 days. Then each day add 89 parts. Entering Great Cold through Spring Equinox, subtract an even 8 days. Entering Clear and Bright, reduce by 59 parts each day. On entering Lesser Heat, use the mean rate. Thereafter add 89 parts each day. On entering White Dew, add 8 days. Thereafter reduce by 178 parts each day. Entering Autumn Equinox, add an even 4 days. Entering Cold Dew, reduce by 59 parts each day. On the first day of Lesser Snow, use the mean rate. Then reduce by 89 parts each day.
434
初順，八十三日行七度二百九十分，日益遲半分。 前留，三十七日。 旋退，五十一日退二度四百九十一分，日益疾少半。 又退，五十一日退二度四百九十一分，日益遲少半。 後留，三十七日。 後順，八十三日，行七度二百九十分，日益疾半分。 日盡而夕伏。
At first direct motion, 7 degrees 290 parts in 83 days, slowing by half a part each day. First stationary period: 37 days. Then retrograde, retreating 2 degrees 491 parts in 51 days, accelerating by half a part less each day. Again retrograde, retreating 2 degrees 491 parts in 51 days, slowing by half a part less each day. Later stationary period: 37 days. Later direct motion, 7 degrees 290 parts in 83 days, accelerating by half a part each day. When the days are used up, enter evening invisibility.
435
太白總率七十八萬四千四百四十九，奇九。 伏分五萬六千二百二十四，奇五十四半。
Venus total cycle rate: 784,449; remainder 9. Invisibility fraction: 56,224; remainder 54½.
436
終日五百八十三，餘千二百二十九，奇九。
Total days: 583; remainder 1,229; odd part 9.
437
夕見伏日二百五十六。
Evening visibility and invisibility: 256 days.
438
晨見伏日三百二十七，餘千二百二十九，奇九。
Morning visibility and invisibility: 327 days; remainder 1,229; odd part 9.
439
夕平見，入冬至，初依平，乃日減百分。 入啟蟄，畢春分，均減九日。 入清明，日損百分。 入芒種，依平。 入夏至，日加百分。 入處暑，畢秋分，均加九日。 入寒露，日損百分。 入大雪，依平。
For evening mean appearance at winter solstice, initially use the mean rate, then reduce by 100 parts each day. Entering Awakening of Insects through Spring Equinox, subtract an even 9 days. Entering Clear and Bright, reduce by 100 parts each day. Entering Grain in Ear, use the mean rate. Entering summer solstice, add 100 parts each day. Entering End Heat through Autumn Equinox, add an even 9 days. Entering Cold Dew, reduce by 100 parts each day. Entering Great Snow, use the mean rate.
440
夕順，入冬至畢立夏，入立秋畢大雪，率百七十二日行二百六度。 入小滿後，十日益一度，為定度。 入白露，畢春分，差行，益遲二分。 自餘平行。 夏至畢小暑，率百七十二日行二百九度。 入大暑，五日損一度，畢氣盡。 平行，入冬至，大暑畢氣盡，率十三日行十三度。 入冬至，十日損一，畢立春。 入立秋，十日益一，畢秋分。 啟蟄畢芒種，七日行七度。 入夏至後，五日益一，畢於小暑。 寒露初日，率二十三日行二十二度，乃六日損一，畢小雪。 順遲，四十二日，行三十度，日益遲八分。 前疾加過二百六度者，準數損此度。 夕留，七日。 夕退，十日退五度。 日盡而夕伏。
Evening direct motion from winter solstice through Start of Summer and from Start of Autumn through Great Snow: 206 degrees in 172 days. After Lesser Full Grain, add 1 degree every 10 days as the fixed total. Entering White Dew through Spring Equinox, use variable motion, slowing by an additional 2 parts. In all other seasons, use uniform motion. From summer solstice through Lesser Heat, 209 degrees in 172 days. Entering Great Heat, subtract 1 degree every 5 days through the end of the qi. Uniform motion from winter solstice and from Great Heat through the end of the qi: 13 degrees in 13 days. After winter solstice, subtract 1 every 10 days through Start of Spring. After Start of Autumn, add 1 every 10 days through Autumn Equinox. From Awakening of Insects through Grain in Ear, 7 degrees in 7 days. After summer solstice, add 1 every 5 days through Lesser Heat. On the first day of Cold Dew, 22 degrees in 23 days; then subtract 1 every 6 days through Lesser Snow. Direct slow motion, 30 degrees in 42 days, slowing by 8 parts each day. If the initial rapid phase added more than 206 degrees, subtract that amount from this phase. Evening stationary period: 7 days. Evening retrograde: 5 degrees in 10 days. When the days are used up, enter evening invisibility.
441
晨平見，入冬至，依平。 入小寒，日加六十七分。 入立春，畢立夏，均加三日。 入小滿，日損六十七分。 入夏至，依平。 入小暑，日減六十七分。 入立秋，畢立冬，均減三日。 入小雪，日損六十七分。
For morning mean appearance at winter solstice, use the mean rate. Entering Lesser Cold, add 67 parts each day. Entering Start of Spring through Start of Summer, add an even 3 days. Entering Lesser Full Grain, reduce by 67 parts each day. Entering summer solstice, use the mean rate. Entering Lesser Heat, reduce by 67 parts each day. Entering Start of Autumn through Start of Winter, subtract an even 3 days. Entering Lesser Snow, reduce by 67 parts each day.
442
晨退，十日退五度。 晨留，七日。 順遲，冬至畢立夏，大雪畢氣盡，率四十二日行三十度，日益疾八分。 入小滿，率十日損一度，畢芒種。 夏至畢寒露，率四十二日行二十七度。 入霜降，每氣益一度，畢小雪。 平行，冬至畢氣盡，立夏畢氣盡，十三日行十三度。 入小寒後，六日益日、度各一，畢啟蟄。 小滿後，七日損日、度各一，畢立秋。 雨水初日，率二十三日行二十三度。 自後六日損日、度各一，畢穀雨。 處暑畢寒露，無平行。 入霜降後，五日益日、度各一，畢大雪。 疾行，百七十二日，行二百六度。 前遲行損度不滿三十度者，此疾依數益之。 處暑畢寒露，差行，日益疾一分。 自餘平行。 日盡而晨伏。
Morning retrograde: 5 degrees in 10 days. Morning stationary period: 7 days. Direct slow motion from winter solstice through Start of Summer and from Great Snow through the end of the qi: 30 degrees in 42 days, accelerating by 8 parts each day. After Lesser Full Grain, subtract 1 degree every 10 days through Grain in Ear. From summer solstice through Cold Dew, 27 degrees in 42 days. Entering Frost's Descent, add 1 degree per solar term through Lesser Snow. Uniform motion from winter solstice through the end of the qi and from Start of Summer through the end of the qi: 13 degrees in 13 days. After Lesser Cold, add 1 each to the day-count and degree-count every 6 days through Awakening of Insects. After Lesser Full Grain, subtract 1 each from the day-count and degree-count every 7 days through Start of Autumn. On the first day of Rain Water, 23 degrees in 23 days. Thereafter subtract 1 each from the day-count and degree-count every 6 days through Grain Rain. From End Heat through Cold Dew, there is no uniform-motion phase. After Frost's Descent, add 1 each to the day-count and degree-count every 5 days through Great Snow. Rapid motion: 206 degrees in 172 days. If the earlier slow phase subtracted fewer than 30 degrees, add that amount in this rapid phase. From End Heat through Cold Dew, use variable motion, accelerating by 1 part each day. In all other seasons, use uniform motion. When the days are used up, enter morning invisibility.
443
辰星總率十五萬五千二百七十八，奇六十六。
Mercury total cycle rate: 155,278; remainder 66.
444
伏分二萬二千六百九十九，奇三十三。
Invisibility fraction: 22,699; remainder 33.
445
終日百一十五，餘千一百七十八，奇六十六。
Total days: 115; remainder 1,178; odd part 66.
446
夕見伏日五十二。
Evening visibility and invisibility: 52 days.
447
晨見伏日六十三，餘千一百七十八，奇六十六。
Morning visibility and invisibility: 63 days; remainder 1,178; odd part 66.
448
夕平見，入冬至，畢清明，依平。 入穀雨，畢芒種，均減二日。 入夏至，畢大暑，依平。 入立秋，畢霜降，應見不見。 其在立秋、霜降氣內，夕去日十八度外、三十六度內有木、火、土、金星者，亦見。
For evening mean appearance from winter solstice through Clear and Bright, use the mean rate. Entering Grain Rain through Grain in Ear, subtract an even 2 days. Entering summer solstice through Great Heat, use the mean rate. Entering Start of Autumn through Frost's Descent, it should appear but does not. If within Start of Autumn or Frost's Descent, the evening distance from the sun is between 18 and 36 degrees and Jupiter, Mars, Saturn, or Venus is present, Mercury also appears.
449
入立冬，畢大雪，依平。
Entering Start of Winter through Great Snow, use the mean rate.
450
順疾，十二日行二十一度六分，日行一度五百三分。 大暑畢處暑，十二日行十七度二分，日行一度二百八十分。 平行，七日行七度。 入大暑後，二日損日、度各一。 入立秋，無此平行。 順遲，六日行二度四分，日行二百二十四分。 前疾行十七度者，無此遲行。 夕留，五日。 日盡而夕伏。
Direct rapid motion: 21 degrees 6 parts in 12 days, at 1 degree 503 parts per day. From Great Heat through End Heat, 17 degrees 2 parts in 12 days, at 1 degree 280 parts per day. Uniform motion: 7 degrees in 7 days. After Great Heat, subtract 1 each from the day-count and degree-count every 2 days. Entering Start of Autumn, omit this uniform-motion phase. Direct slow motion: 2 degrees 4 parts in 6 days, at 224 parts per day. If the earlier rapid phase covered 17 degrees, omit this slow-motion phase. Evening stationary period: 5 days. When the days are used up, enter evening invisibility.
451
晨平見，入冬至，均減四日。 入小寒，畢大寒，依平。 入立春，畢啟蟄，均減三日。 其在啟蟄氣內，去日度如前，晨無木、火、土、金星者，不見。 入雨水，畢立夏，應見不見。 其在立夏氣內，去日度如前，晨有木、火、土、金星者，亦見。 入小滿，畢寒露，依平。 入霜降，畢立冬，均加一日。 入小雪，畢大雪，依平。
For morning mean appearance at winter solstice, subtract an even 4 days. Entering Lesser Cold through Great Cold, use the mean rate. Entering Start of Spring through Awakening of Insects, subtract an even 3 days. If within Awakening of Insects, at the same solar distance as described above, and no Jupiter, Mars, Saturn, or Venus is visible in the morning, it does not appear. Entering Rain Water through Start of Summer, it should appear but does not. If within Start of Summer, at the same solar distance as described above, and Jupiter, Mars, Saturn, or Venus is visible in the morning, it also appears. Entering Lesser Full Grain through Cold Dew, use the mean rate. Entering Frost's Descent through Start of Winter, add an even 1 day. Entering Lesser Snow through Great Snow, use the mean rate.
452
晨見，留，五日。 順遲，六日行二度四分，日行二百二十四分。 入大寒，畢啟蟄，無此遲行。 平行，七日行七度。 入大寒後，二日損日、度各一。 入立春，無此平行。 順疾，行十二日行二十一度六分，日行一度五百三分。 前無遲行者，十二日行十七度一十分，日行一度二百八十分。 日盡而晨伏。
Morning appearance: stationary for 5 days. Direct slow motion: 2 degrees 4 parts in 6 days, at 224 parts per day. Entering Great Cold through Awakening of Insects, omit this slow-motion phase. Uniform motion: 7 degrees in 7 days. After Great Cold, subtract 1 each from the day-count and degree-count every 2 days. Entering Start of Spring, omit this uniform-motion phase. Direct rapid motion: 21 degrees 6 parts in 12 days, at 1 degree 503 parts per day. If there was no slow phase earlier, 17 degrees 10 parts in 12 days, at 1 degree 280 parts per day. When the days are used up, enter morning invisibility.
453
各以伏分減總實，以總率去之； 不盡，反以減總率，如總法，為日。 天正定朔與常朔有進退者，亦進減、退加一日。 乃隨次月大小去之，命日算外，得平見所在。 各半見餘以同半總。 太白、辰星以夕見伏日加之，得晨平見。 各依所入常氣加減日及應計日損益者，以損益所加減； 訖，餘以加減平見，為常見。 又以常見日消息定數之半，息減、消加常見，為定見日及分。
For each planet, subtract the invisibility fraction from the accumulated remainder and divide by the total cycle rate; take the remainder, subtract it from the rate, divide by the degree-factor to get days. If the celestial corrected new moon and mean new moon differ in advance or retreat, also subtract a day for advance and add a day for retreat. Then cast out full months according to their lengths, count the day beyond the starting count, and obtain the mean appearance date. Halve each planet's appearance remainder to match the half-total. For Venus and Mercury, add the evening visibility-and-invisibility days to get the morning mean appearance. For each planet, apply the daily increments or decrements according to the solar term entered and the scheduled daily adjustments; when complete, apply the remainder to adjust the mean appearance to get the regular appearance. Then take half the waxing-waning fixed number for the regular appearance day, subtract at waning and add at waxing, to get the fixed appearance day and parts.
454
置定見夜半日躔，半其分，以其日躔差乘定見餘，總法而一，進加、退減之，乃以其星初見去日度，歲星十四，太白十一，熒惑、鎮星、辰星十七，晨減、夕加，得初見定辰所在宿度。 其初見消息定數，亦半之，以息加、消減其星初見行留日率。 其歲星、鎮星不須加減。 其加減不滿日者，與見通之，過半從日，乃依行星日度率，求初日行分。
Set the fixed appearance midnight solar longitude, halve the parts, multiply the fixed appearance remainder by the solar-longitude difference and divide by the total factor, advance to add or retreat to subtract; then subtract (morning) or add (evening) the planet's solar distance—Jupiter 14°, Venus 11°, Mars, Saturn, and Mercury 17° each—to get the first-appearance fixed lodge and degree. Halve the first-appearance waxing-waning fixed number and use waning to add and waxing to diminish the planet's first-appearance motion and stationary day-rates. Jupiter and Saturn require no such adjustment. When the adjustment does not fill a day, combine it with the appearance date; if over half, count as a full day; then use the planet's day- and degree-rates to find the initial day's motion-parts.
455
置定見餘，以減半總，各以初日行分乘之，半總而一，順加、逆減星初見定辰所在度分，得星見後夜半宿度。 以所行度分，順加、逆減之。 其差行益疾益遲者，副置初日行分，各以其差遲損、疾加之，留者因前，逆則依減，以程法約行分為度分，得每日所至。
Set the fixed appearance remainder, subtract it from the half-total, multiply each by the initial day's motion-parts and divide by the half-total; add in direct motion or subtract in retrograde from the first-appearance fixed lodge and degree to get the midnight lodge and degree after appearance. Add or subtract the degrees and parts traveled, forward in direct motion and backward in retrograde. When motion varies between faster and slower phases, set aside the initial day's motion-parts and apply the daily differential—subtracting for deceleration, adding for acceleration; during stationary, continue from the prior value; during retrograde, subtract accordingly; reduce motion-parts to degrees and parts by the circuit method to find each day's position.
456
求行分者，皆以半總乘定度率，有分者從之。 日率除，為平行度分。 置定日率，減一，以所差分乘之，二而一，為差率。 以疾減、遲加平行，為初日所行度及分。
To find motion-parts, multiply the fixed degree-rate by the half-total, carrying any fractional parts. Divide by the day-rate to get the uniform-motion degrees and parts. Set the fixed day-rate, subtract 1, multiply by the differential parts, and halve the result to get the differential rate. Subtract for acceleration and add for deceleration to the uniform motion to get the initial day's degrees and parts.
457
中宗反正，太史丞南宮說以麟德曆上元，五星有入氣加減，非合璧連珠之正，以神龍元年歲次乙巳，故治乙巳元曆。 推而上之，積四十一萬四千三百六十算，得十一月甲子朔夜半冬至，七曜起牽牛之初。 其術有黃道而無赤道，推五星先步定合，加伏日以求定見。 他與淳風術同。 所異者，惟平合加減差。 既成，而睿宗即位，罷之。
When Zhongzong restored the throne, Chief Clerk of the Astrological Office Nangong Shuo held that the Lindé calendar's origin epoch, with its five-planet qi-entry adjustments, did not match the true alignment of combined disks and strung pearls; using Shenlong 1, year yi-si, he therefore compiled the Yisi Origin Calendar. Extrapolating upward, with an accumulated count of 414,360, he obtained eleventh-month jiazi new moon at midnight on winter solstice, with the seven luminaries beginning at the start of Ox. Its method used the ecliptic but not the equator; computing the five planets began with stepping out the fixed conjunction and adding invisibility days to obtain fixed appearance. The rest was the same as Li Chunfeng's method. What differed was only the add-subtract difference at mean conjunction. Once it was completed, Ruizong took the throne and it was abolished.
458
校勘記0.85em|columns=2
Collation notes for this chapter.