1
開元大衍曆演紀上元閼逢困敦之歲，距開元十二年甲子，積九千六百九十六萬一千七百四十算。
In computing the evolving era of the Kaiyuan Dayan calendar, the high origin falls in the year Yān-féng Kùn-dūn; reckoned back from Kaiyuan 12 (jiazi), the accumulated count is 96,961,740.
2
一曰步中朔術
I. Method for Determining Central Qi and New Moons
3
通法三千四十。
Universal divisor: 3,040.
4
策實百一十一萬三百四十三。
Year-fraction constant: 1,110,343.
5
揲法八萬九千七百七十三。
New-moon constant: 89,773.
6
滅法九萬一千二百。
Reduction constant: 91,200.
7
策餘萬五千九百四十三。
Stalk remainder: 15,943.
8
用差萬七千一百二十四。
Applied difference: 17,124.
9
掛限八萬七千一十八。
Intercalation threshold: 87,018.
10
三元之策十五，餘六百六十四，秒七。
Three-origin interval: 15 stalks, remainder 664, 7 parts.
11
四象之策二十九，餘千六百一十三。
Four-image interval: 29 stalks, remainder 1,613.
12
中盈分千三百二十八，秒十四。
Central-qi surplus fraction: 1,328, 14 parts.
13
朔虛分千四百二十七。
New-moon void fraction: 1,427.
14
爻數六十。
Hexagram line count: 60.
15
象統二十四。
Image cycle: 24.
16
以策實乘積算，曰中積分。 盈通法得一，為積日。 爻數去之，餘起甲子算外，得天正中氣。 凡分為小餘，日為大餘。 加三元之策，得次氣。 凡率相因加者，下有餘秒，皆以類相從。 而滿法迭進，用加上位。 日盈爻數去之。
Multiply the era count by the year-fraction constant to obtain the central accumulated parts. Divide by the universal divisor; the quotient is accumulated days. Remove full cycles of sixty lines; from the remainder, counting outward from jiazi, obtain the celestial central qi. The fractional parts become the minor remainder; the days become the major remainder. Add the three-origin interval to obtain the next qi. Whenever linked rates are added and fractional remainders remain below, combine them by kind. When a divisor is filled, carry upward step by step and add to the higher place. Cast out full cycles of sixty when days overflow.
17
以揲法去中積分，不盡曰歸餘之掛。 以減中積分，為朔積分。 如通法為日，去命如前，得天正經朔。 加一象之日七、餘千一百六十三少，得上弦。 倍之，得望。 參之，得下弦。 四之，是謂一揲，得後月朔。 凡四分，一為少，三為太。 綜中盈、朔虛分，累益歸餘之掛，每其月閏衰。 凡歸餘之掛五萬六千七百六十以上，其歲有閏。 因考其閏衰，滿掛限以上，其月合置閏。 或以進退，皆以定朔無中氣裁焉。
Divide the central accumulated parts by the new-moon constant; the remainder is the intercalation pendulum. Subtract that amount from the central accumulated parts to obtain the new-moon accumulated parts. Convert by the universal divisor into days; fix the era as before to obtain the canonical winter new moon. Add one four-image interval—seven days, remainder 1,163 lesser—to obtain the first quarter. Double it to obtain full moon. Triple it to obtain last quarter. Quadruple it—one full sorting—to obtain the next month’s new moon. In quartering remainders, one part is “lesser,” three are “greater.” Combine the central surplus and new-moon void fractions, steadily increasing the intercalation pendulum; each month the intercalation fraction diminishes. Whenever the intercalation pendulum reaches 56,760 or more, the year receives an intercalary month. Track the intercalation wane; when it exceeds the intercalation threshold, assign an intercalary month. Whether advancing or retreating, the rule is the fixed new moon in a month without central qi.
18
凡常氣小餘不滿通法、如中盈分之半已下者，以象統乘之，內秒分，參而伍之，以減策實； 不盡，如策餘為日。 命常氣初日算外，得沒日。 凡經朔小餘不滿朔虛分者，以小餘減通法，餘倍參伍乘之，用減滅法； 不盡，如朔虛分為日。 命經朔初日算外，得滅日。
Whenever a regular qi’s minor remainder is less than the universal divisor and no more than half the central surplus fraction, multiply by the image cycle, include seconds, combine and quintuple, and subtract from the stalk substance; The remainder, divided by the stalk remainder, gives the days. Count outward from the first day of the regular qi; the result is the extinction day. Whenever a canonical new moon’s minor remainder is less than the new-moon void fraction, subtract the minor remainder from the universal divisor; double and quintuple the remainder; The remainder, divided by the new-moon void fraction, gives the days. Count outward from the first day of the canonical new moon; the result is the lunar extinguishing day.
19
二曰發斂術
II. Method for Issuing and Gathering In
20
天中之策五，餘二百二十一，秒三十一； 秒法七十二。
Heaven-central interval: five stalks, remainder 221, 31 parts; Second divisor: 72.
21
地中之策六，餘二百六十五，秒八十六； 秒法百二十。
Earth-central interval: six stalks, remainder 265, 86 parts; Second divisor: 120.
22
貞悔之策三，餘百三十二，秒百三。
Zhen-hui interval: three stalks, remainder 132, 103 parts.
23
辰法七百六十。
Chronogram divisor: 760.
24
刻法三百四。
Clepsydra divisor: 304.
25
各因中節命之，得初候。 加天中之策，得次候。 又加，得末候。 因中氣命之，得公卦用事。 以地中之策累加之，得次卦，若以貞悔之策加侯卦，得十有二節之初外卦用事。 因四立命之，得春木、夏火、秋金、冬水用事。 以貞悔之策減季月中氣，得土王用事。 凡相加減而秒母不齊，當令母互乘子，乃加減之； 母相乘為法。
For each, start from the mid-node of the central qi; obtain the first phenological hou. Add the heaven-central interval to obtain the next hou. Add again to obtain the last hou. Fix from the central qi; obtain when the duke hexagram governs. Keep adding the earth-central interval for successive hexagrams; or add the zhen-hui interval to the marquis hexagram to obtain when the outer hexagram of the twelve nodes governs. Fix from the four establishment days; obtain when spring Wood, summer Fire, autumn Metal, and winter Water govern. Subtract the zhen-hui interval from the last seasonal month’s central qi; obtain when Earth the King governs. Whenever addition or subtraction leaves unequal fractional denominators, cross-multiply numerators by the opposite denominators, then add or subtract; The product of denominators is the common divisor.
26
常氣月中節四正卦
Regular qi · Month · Mid-node · Four cardinal hexagrams
27
初候
First hou
28
次候
Second hou
29
末候
Last hou
30
始卦
Opening hexagram
31
中卦
Middle hexagram
32
終卦
Closing hexagram
33
冬至十一月中坎初六
Winter Solstice · eleventh month, mid-month · Kan ☵, first line
34
丘蚓結
Earthworms coil up
35
麋角解
Elk shed their antlers
36
水泉動
Springs begin to stir
37
公中孚
Duke · Zhōng Fú
38
辟復
Sovereign · Fù
39
侯屯內
Marquis · Zhūn (inner)
40
小寒十二月節坎九二
Lesser Cold · twelfth month, node · Kan ☵, second line
41
鴈北鄉
Wild geese fly north
42
鵲始巢
Magpies begin to nest
43
野雞始雊
Pheasants begin to call
44
侯屯外
Marquis · Zhūn (outer)
45
大夫謙
Great officer · Qiān
46
卿睽
Minister · Kuí
47
大寒十二月中坎六三
Greater Cold · twelfth month, mid-month · Kan ☵, third line
48
雞始乳
Hens begin to brood
49
鷙鳥厲疾
Birds of prey grow fierce
50
水澤腹堅
Ice hardens at the depths of the waters
51
公升
Duke · Shēng
52
辟臨
Sovereign · Lín
53
侯小過內
Marquis · Xiǎo Guò (inner)
54
立春正月節坎六四
Start of Spring · first month, node · Kan ☵, fourth line
55
東風解凍
The east wind melts the ice
56
蟄蟲始振
Creatures in hibernation begin to stir
57
魚上冰
Fish appear beneath the ice
58
侯小過外
Marquis · Xiǎo Guò (outer)
59
大夫蒙
Great officer · Méng
60
卿益
Minister · Yì
61
雨水正月中坎九五
Rain Water · first month, mid-month · Kan ☵, fifth line
62
獺祭魚
Otters present fish
63
鴻鴈來
Wild geese return
64
草木萌動
Plants put forth buds
65
公漸
Duke · Jiàn
66
辟泰
Sovereign · Tài
67
侯需內
Marquis · Xū (inner)
68
驚蟄二月節坎上六
Awakening of Insects · second month, node · Kan ☵, sixth line
69
桃始華
Peaches begin to blossom
70
倉庚鳴
Orioles begin to sing
71
鷹化為鳩
Hawks transform into turtledoves
72
侯需外
Marquis · Xū, outer
73
大夫隨
Great officer · Suí
74
卿晉
Minister · Jìn
75
春分二月中震初九
Spring Equinox · second month, mid-month · Zhèn ☳, first line
76
玄鳥至
Swallows return
77
雷乃發聲
Thunder begins to sound
78
始電
Lightning first appears
79
公解
Duke · Xiè
80
辟大壯
Sovereign · Dà Zhuàng
81
侯豫內
Marquis · Yù, inner
82
清明三月節震六二
Pure Brightness · third month, node · Zhèn ☳, second line
83
桐始華
Paulownia begins to flower
84
田鼠化為鴽
Field mice transform into quails
85
虹始見
Rainbows first appear
86
侯豫外
Marquis · Yù, outer
87
大夫訟
Great officer · Sòng
88
卿蠱
Minister · Gǔ
89
穀雨三月中震六三
Grain Rain · third month, mid-month · Zhèn ☳, third line
90
萍始生
Duckweed begins to grow
91
鳴鳩拂其羽
Cuckoos smooth their feathers
92
戴勝降于桑
Hoopoes descend upon the mulberries
93
公革
Duke · Gé
94
辟史
Sovereign · Shǐ
95
侯旅內
Marquis · Lǚ, inner
96
立夏四月節震九四
Start of Summer · fourth month, node · Zhèn ☳, fourth line
97
螻蟈鳴
Mole-crickets begin to chirp
98
丘蚓出
Earthworms come forth
99
王瓜生
Snake-gourds begin to grow
100
侯旅外
Marquis · Lǚ, outer
101
大夫師
Great officer · Shī
102
卿比
Minister · Bǐ
103
小滿四月中震六五
Lesser Fullness · fourth month, mid-month · Zhèn ☳, fifth line
104
苦菜秀
Bitter greens come into flower
105
靡草死
Tender grasses wither
106
小暑至
Lesser heat arrives
107
公小畜
Duke · Xiǎo Xù
108
辟乾
Sovereign · Qián
109
侯大有內
Marquis · Dà Yǒu, inner
110
芒種五月節震上六
Grain in Ear · fifth month, node · Zhèn ☳, sixth line
111
螳蜋生
Praying mantises hatch
112
鵙始鳴
Shrikes begin to call
113
反舌無聲
The mockingbirds fall silent
114
侯大有外
Marquis · Dà Yǒu, outer
115
大夫家人
Great officer · Jiā Rén
116
卿井
Minister · Jǐng
117
夏至五月中離初九
Summer Solstice · fifth month, mid-month · Lí ☲, first line
118
鹿角解
Deer shed their antlers
119
蜩始鳴
Cicadas begin to chirp
120
半夏生
Pinellia comes into growth
121
公咸
Duke · Xián
122
辟姤
Sovereign · Gòu
123
侯鼎內
Marquis · Dǐng, inner
124
小暑六月節離六二
Lesser Heat · sixth month, node · Lí ☲, second line
125
溫風至
Warm winds arrive
126
蟋蟀居壁
Crickets take to the walls
127
鷹乃學習
Hawks begin to learn flight
128
侯鼎外
Marquis · Dǐng, outer
129
大夫豐
Great officer · Fēng
130
卿渙
Minister · Huàn
131
大暑六月中離九三
Greater Heat · sixth month, mid-month · Lí ☲, third line
132
腐草為螢
Rotting grass turns into fireflies
133
土潤溽暑
The earth is damp with sultry heat
134
大雨時行
Heavy rains fall at intervals
135
公履
Duke · Lǚ
136
辟遯
Sovereign · Dùn
137
侯恆內
Marquis · Héng, inner
138
立秋七月節離九四
Start of Autumn · seventh month, node · Lí ☲, fourth line
139
涼風至
Cool winds arrive
140
白露降
White dew falls
141
寒蟬鳴
Cold cicadas begin to chirp
142
侯恆外
Marquis · Héng, outer
143
大夫節
Great officer · Jié
144
卿同人
Minister · Tóng Rén
145
處暑七月中離六五
End of Heat · seventh month, mid-month · Lí ☲, fifth line
146
鷹祭鳥
Hawks present their catch
147
天地始肅
Heaven and earth begin to turn austere
148
禾乃登
The grain ripens
149
公損
Duke · Sǔn
150
辟否
Sovereign · Pǐ
151
侯巽內
Marquis · Xùn, inner
152
白露八月節離上九
White Dew · eighth month, node · Lí ☲, sixth line
153
鴻鴈來
Wild geese return
154
玄鳥歸
Swallows depart
155
羣鳥養羞
Birds gather stores of food
156
候巽外
Marquis · Xùn, outer
157
大夫萃
Great officer · Cuì
158
卿大畜
Minister · Dà Chù
159
秋分八月中兌初九
Autumn Equinox · eighth month, mid-month · Duì ☱, first line
160
雷乃收聲
Thunder falls silent
161
蟄蟲培戶
Creatures in hibernation close their burrows
162
水始涸
The waters begin to recede
163
公賁
Duke · Bì
164
辟觀
Sovereign · Guān
165
侯歸妹內
Marquis · Guī Mèi, inner
166
寒露九月節兌九二
Cold Dew · ninth month, node · Duì ☱, second line
167
鴻鴈來賓
The guest geese arrive
168
雀入大水為蛤
Sparrows dive into the waters and become clams
169
菊有黃華
Chrysanthemums show yellow blossoms
170
侯歸妹外
Marquis · Guī Mèi, outer
171
大夫無妄
Great officer · Wú Wàng
172
卿明夷
Minister · Míng Yí
173
霜降九月中兌六三
Frost Descent · ninth month, mid-month · Duì ☱, third line
174
豺乃祭獸
Wolves present their kill
175
草木黃落
Grasses and trees turn yellow and shed their leaves
176
蟄蟲咸俯
Creatures in hibernation all lie low
177
公困
Duke · Kùn
178
辟剝
Sovereign · Bō
179
侯艮內
Marquis · Gèn, inner
180
立冬十月節兌九四
Start of Winter · tenth month, node · Duì ☱, fourth line
181
水始冰
The waters begin to freeze
182
地始凍
The ground begins to harden with frost
183
野雞入水為蜃
Pheasants enter the waters and become great clams
184
侯艮外
Marquis · Gèn, outer
185
大夫既濟
Great officer · Jì Jì
186
卿噬嗑
Minister · Shì Kè
187
小雪十月中兌九五
Lesser Snow · tenth month, mid-month · Duì ☱, fifth line
188
虹藏不見
Rainbows are no longer seen
189
天氣上騰地氣下降
The qi of heaven ascends and the qi of earth descends
190
閉塞而成冬
All is shut in, and winter takes hold
191
公大過
Duke · Dà Guò
192
辟坤
Sovereign · Kūn
193
侯未濟內
Marquis · Wèi Jì, inner
194
大雪十一月節兌上六
Greater Snow · eleventh month, node · Duì ☱, sixth line
195
鶡鳥不鳴
Otters fall silent
196
虎始交
Tigers begin to mate
197
荔挺生
Litchi sprouts break forth
198
侯未濟外
Marquis · Wèi Jì, outer
199
大夫蹇
Great officer · Jiǎn
200
卿頤
Minister · Yí
201
各以通法約其月閏衰，為日，得中氣去經朔日算。 求卦、候者，各以天、地之策，累加減之。 凡發斂加時，各置其小餘，以六爻乘之，如辰法而一，為半辰之數。 不盡者，三約為分。 分滿刻法為刻。 若令滿象積為刻者，即置不盡之數，十之，十九而一，為分。 命辰起子半算外。
For each month, divide the intercalation wane by the universal divisor to obtain days, yielding the interval from central qi to canonical new moon. To determine hexagrams and phenological hou, cumulatively add or subtract the heaven- and earth-central intervals. When fixing the time of issuing and gathering in, set each minor remainder, multiply by six lines, and divide by the chronogram divisor to obtain the half-chronogram count. The remainder, divided by three, gives fractional parts. When the parts fill the clepsydra divisor, they become clepsydra notches. If the image accumulation is to fill clepsydra marks directly, take the remainder, multiply by ten, and divide by nineteen for the parts. Fix the chronogram from the half-period of zi, counting outward from the tally.
202
三曰步日躔術
III. Method for Determining Solar Motion
203
乾實百一十一萬三百七十九太。
Circuit-of-heaven constant: 1,110,379 and greater.
204
周天度三百六十五，虛分七百七十九太。
Degrees in the circuit of heaven: 365, void fraction 779 and greater.
205
歲差三十六太。
Precession per year: 36 and greater.
206
定氣
Fixed qi
207
盈縮分
Expansion and contraction fractions
208
先後數
Lead and lag counts
209
損益率
Rates of increase and decrease
210
朓朒積
Lunar elongation accumulation
211
冬至
Winter Solstice
212
盈二千三百五十三
Expansion: 2,353
213
先端
Leading edge
214
益百七十六
Increase rate: 176
215
朒初
Waning phase, initial
216
小寒
Lesser Cold
217
盈千八百四十五
Expansion: 1,845
218
先二千三百五十三
Lead count: 2,353
219
益百三十八
Increase rate: 138
220
朒百七十六
Waning accumulation: 176
221
大寒
Greater Cold
222
盈千三百九十
Expansion: 1,390
223
先四千一百九十八
Lead count: 4,198
224
益百四
Increase rate: 104
225
朒三百一十四
Waning accumulation: 314
226
立春
Start of Spring
227
盈九百七十六
Expansion: 976
228
先五千五百八十八
Lead count: 5,588
229
益七十三
Increase rate: 73
230
朒四百一十八
Waning accumulation: 418
231
雨水
Rain Water
232
盈五百八十八
Expansion: 588
233
先六千五百六十四
Lead count: 6,564
234
益四十四
Increase rate: 44
235
朒四百九十一
Waning accumulation: 491
236
驚蟄
Awakening of Insects
237
盈二百一十四
Expansion: 214
238
先七千一百五十二
Lead count: 7,152
239
益十六
Increase rate: 16
240
朒五百三十五
Waning accumulation: 535
241
春分
Spring Equinox
242
縮二百一十四
Contraction: 214
243
先七千三百六十六
Lead count: 7,366
244
損十六
Decrease rate: 16
245
朒五百五十一
Waning accumulation: 551
246
清明
Pure Brightness
247
縮五百八十八
Contraction: 588
248
先七千一百五十二
Lead count: 7,152
249
損四十四
Decrease rate: 44
250
朒五百三十五
Waning accumulation: 535
251
穀雨
Grain Rain
252
縮九百七十六
Contraction: 976
253
先六千五百六十四
Lead count: 6,564
254
損七十三
Decrease rate: 73
255
朒四百九十一
Waning accumulation: 491
256
立夏
Start of Summer
257
縮千三百九十
Contraction: 1,390
258
先五千五百八十八
Lead count: 5,588
259
損百四
Decrease rate: 104
260
朒四百一十八
Waning accumulation: 418
261
小滿
Lesser Fullness
262
縮千八百四十五
Contraction: 1,845
263
先四千一百九十八
Lead count: 4,198
264
損百三十八
Decrease rate: 138
265
朒三百一十四
Waning accumulation: 314
266
芒種
Grain in Ear
267
縮二千三百五十三
Contraction: 2,353
268
先二千三百五十三
Lead count: 2,353
269
損百七十六
Decrease rate: 176
270
朒百七十六
Waning accumulation: 176
271
夏至
Summer Solstice
272
縮二千三百五十三
Contraction: 2,353
273
後端
Trailing edge
274
益百七十六
Increase rate: 176
275
朓初
Waxing phase, initial
276
小暑
Lesser Heat
277
縮千八百四十五
Contraction: 1,845
278
後二千三百五十三
Lag count: 2,353
279
益百三十八
Increase rate: 138
280
朓百七十六
Waxing accumulation: 176
281
大暑
Greater Heat
282
縮千三百九十
Contraction: 1,390
283
後四千一百九十八
Lag count: 4,198
284
益百四
Increase rate: 104
285
朓三百一十四
Waxing accumulation: 314
286
立秋
Start of Autumn
287
縮九百七十六
Contraction: 976
288
後五千五百八十八
Lag count: 5,588
289
益七十三
Increase rate: 73
290
朓四百一十八
Waxing accumulation: 418
291
處暑
End of Heat
292
縮五百八十八
Contraction: 588
293
後六千五百六十四
Lag count: 6,564
294
益四十四
Increase rate: 44
295
朓四百九十一
Waxing accumulation: 491
296
白露
White Dew
297
縮二百一十四
Contraction: 214
298
後七千一百五十二
Lag count: 7,152
299
益十六
Increase rate: 16
300
朓五百三十五
Waxing accumulation: 535
301
秋分
Autumn Equinox
302
盈二百一十四
Expansion: 214
303
後七千三百六十六
Lag count: 7,366
304
損十六
Decrease rate: 16
305
朓五百五十一
Waxing accumulation: 551
306
寒露
Cold Dew
307
盈五百八十八
Expansion: 588
308
後七千一百五十二
Lag count: 7,152
309
損四十四
Decrease rate: 44
310
朓五百三十五
Waxing accumulation: 535
311
霜降
Frost Descent
312
盈九百七十六
Expansion: 976
313
後六千五百六十四
Lag count: 6,564
314
損七十三
Decrease rate: 73
315
朓四百九十一
Waxing accumulation: 491
316
立冬
Start of Winter
317
盈千三百九十
Expansion: 1,390
318
後五千五百八十八
Lag count: 5,588
319
損百四
Decrease rate: 104
320
朓四百一十八
Waxing accumulation: 418
321
小雪
Lesser Snow
322
盈千八百四十五
Expansion: 1,845
323
後四千一百九十八
Lag count: 4,198
324
損百三十八
Decrease rate: 138
325
朓三百一十四
Waxing accumulation: 314
326
大雪
Greater Snow
327
盈二千三百五十三
Expansion: 2,353
328
後二千三百五十三
Lag count: 2,353
329
損百七十六
Decrease rate: 176
330
朓百七十六
Waxing accumulation: 176
331
以盈縮分盈減、縮加三元之策，為定氣所有日及餘。 乃十二乘日，又三其小餘，辰法約而一，從之，為定氣辰數。 不盡，十之，又約為分。 以所入氣幷後氣盈縮分，倍六爻乘之，綜兩氣辰數除之，為末率。 又列二氣盈縮分，皆倍六爻乘之，各如辰數而一； 以少減多，餘為氣差。 至後以差加末率，分後以差減末率，為初率。 倍氣差，亦倍六爻乘之，復綜兩氣辰數除，為日差。 半之，以加減初末，各為定率。 以日差至後以減、分後以加氣初定率，為每日盈縮分。 乃馴積之，隨所入氣日加減氣下先後數，各其日定數。 其求朓朒倣此。 冬至後為陽復，在盈加之，在縮減之； 夏至後為陰復，在縮加之，在盈減之。 距四正前一氣，在陰陽變革之際，不可相幷，皆因前末為初率。 以氣差至前加之，分前減之，為末率。 餘依前術，各得所求。 其分不滿全數，母又每氣不同，當退法除之。 以百為母，半已上，收成一。 冬至、夏至偕得天地之中，無有盈縮。 餘各以氣下先後數先減、後加常氣小餘，滿若不足，進退其日，得定大小餘。 凡推日月度及軌漏、交蝕，依定氣； 注曆，依常氣； 以減經朔、弦、望，各其所入日算。 若大餘不足減，加爻數，乃減之。 減所入定氣日算一，各以日差乘而半之； 前少以加、前多以減氣初定率，以乘其所入定氣日算及餘秒。 凡除者，先以母通全，內子，乃相乘； 母相乘除之。 所得以損益朓朒積，各其入朓朒定數。 若非朔望有交者，以十二乘所入日算； 三其小餘，辰法除而從之； 以乘損益率，如定氣辰數而一。 所得以損益朓朒積，各為定數。
Apply the expansion–contraction fractions: subtract where expansion and add where contraction from the three-origin tally, yielding the days and remainder for fixed qi. Multiply the days by twelve, triple the minor remainder, divide by the chronogram divisor and add the quotient, to obtain the fixed-qi chronogram count. The remainder, multiplied by ten and reduced again, gives fractional parts. Combine the expansion–contraction fractions of the current and next qi, double and multiply by six lines, divide by the sum of their chronogram counts, and obtain the terminal rate. Set out both qi’s expansion–contraction fractions, each doubled and multiplied by six lines, and divide each by its chronogram count. Subtract the smaller from the larger; the remainder is the qi difference. After a solstice, add the difference to the terminal rate; after an equinox, subtract it—this yields the initial rate. Double the qi difference, double and multiply by six lines, divide again by the combined chronogram counts of both qi, and obtain the day difference. Halve the day difference and add or subtract from the initial and terminal rates to obtain the fixed rates. Using the day difference, subtract from the initial fixed rate after a solstice and add after an equinox, yielding the daily expansion–contraction fraction. Accumulate stepwise: for each day within the entered qi, add or subtract the lead and lag counts listed under that qi to obtain each day’s fixed tally. To determine waxing and waning accumulations, follow the same procedure. After the Winter Solstice is yang recovery: add in expansion and subtract in contraction. After the Summer Solstice is yin recovery: add in contraction and subtract in expansion. For the qi immediately before each cardinal solstice or equinox, at the yin–yang transition the rates cannot be merged; use the prior qi’s terminal rate as the initial rate. Before a solstice add the difference to obtain the terminal rate; before an equinox subtract it. For the rest, follow the method above; each quantity sought is thereby obtained. When fractional parts do not make a full unit and each qi has a different denominator, reduce by retreating the divisor. Use one hundred as the denominator; at one-half or above, round up to one. At the Winter and Summer Solstices alike the sun reaches the cosmological mean; there is neither expansion nor contraction. For the remaining qi, first subtract then add the lead and lag counts under each qi to the regular qi’s minor remainder; carry or borrow days as needed to obtain fixed major and minor remainders. All calculations of solar and lunar longitude, orbital motion, clepsydra marks, and eclipses use fixed qi. Published calendars follow regular qi. Reduce the canonical new, quarter, and full moons by each entered day count. If the major remainder is too small to subtract, add sixty lines, then subtract. Subtract one from the entered fixed-qi day count, multiply by the day difference, and halve. If the prior fraction is smaller, add; if larger, subtract from the qi’s initial fixed rate; then multiply by the entered fixed-qi day count, remainder, and seconds. In every division, first unify whole units through the denominator, include the numerator, then multiply. Multiply and divide by the denominators. Use the result to adjust the waxing–waning accumulation, obtaining each entered waxing–waning fixed tally. When new or full moon is not an eclipse syzygy, multiply the entered day count by twelve. Triple the minor remainder, divide by the chronogram divisor, and add the quotient; Multiply by the rate of increase or decrease, and divide by the fixed-qi chronogram count. Use the result to adjust the anomalistic accumulation, yielding a fixed value for each entry.
332
南斗二十六，牛八，婺女十二，虛十，虛分七百七十九太。 危十七，營室十六，東壁九，奎十六，婁十二，胃十四，昴十一，畢十七，觜觿一，參十，東井三十三，輿鬼三，柳十五，七星七，張十八，翼十八，軫十七，角十二，亢九，氐十五，房五，心五，尾十八，箕十一，為赤道度。 其畢、觜觿、參、輿鬼四宿度數，與古不同。 依天以儀測定，用為常數。 紘帶天中，儀極攸憑，以格黃道。
Southern Dipper: 26°; Ox: 8°; Maid: 12°; Emptiness: 10°, void fraction 779 and greater. Rooftop 17°, Encampment 16°, Eastern Wall 9°, Stride 16°, Bond 12°, Stomach 14°, Hairy Head 11°, Net 17°, Turtle Beak 1°, Three Stars 10°, Eastern Well 33°, Ghost Cart 3°, Willow 15°, Seven Stars 7°, Extended Net 18°, Wings 18°, Chariot Shaft 17°, Horn 12°, Neck 9°, Root 15°, Room 5°, Heart 5°, Tail 18°, Winnowing Basket 11° — the equatorial lodge degrees. The arc-degrees assigned to Net, Turtle Beak, Three Stars, and Ghost Cart differ from the ancient reckoning. They were fixed by armillary measurement against the sky and adopted as standard constants. The celestial girdle runs through heaven’s center; the polar axis of the instrument is the reference for laying out the ecliptic.
333
推冬至歲差所在，每距冬至前後各五度為限，初數十二，每限減一。 盡九限，數終於四。 當二立之際，一度少強，依平。 乃距春分前、秋分後，初限起四，每限增一，盡九限，終於十二，而黃道交復。 計春分後、秋分前，亦五度為限。 初數十二，盡九限，數終於四。 當二立之際，一度少強，依平。 乃距夏至前後，初限起四，盡九限，終於十二。 皆累裁之，以數乘限度，百二十而一，得度。 不滿者，十二除，為分。 若以十除，則大分，十二為母，命太、半、少及強、弱。 命曰黃赤道差數。 二至前後各九限，以差減赤道度，二分前後各九限，以差加赤道度，各為黃道度。
To find where the winter-solstice precession applies: take five degrees on either side of the solstice as one band; begin at twelve and subtract one for each successive band. After nine bands the tally reaches four. At the two Establishment qi, treat one degree as slightly strong and use the mean value. From before the spring equinox and after the autumn equinox, begin the first band at four and add one per band; after nine bands the tally is twelve, and the ecliptic obliquity cycle completes. For the interval after the spring equinox and before the autumn equinox, use the same five-degree bands. Begin at twelve; after nine bands the tally reaches four. At the two Establishment qi, treat one degree as slightly strong and use the mean value. Before and after the summer solstice, begin the first band at four; after nine bands the tally is twelve. Accumulate the trims in order, multiply the band index by the limit value, and divide by 120 to obtain degrees. The remainder, divided by twelve, gives fractional parts. If divided by ten instead, use twelve as the denominator for major parts, naming greater, half, lesser, strong, and weak fractions. This is called the ecliptic–equator difference. Within nine bands on either side of each solstice, subtract the difference from the equatorial longitude; within nine bands on either side of each equinox, add it — yielding ecliptic longitude in each case.
334
開元十二年，南斗二十三半，牛七半，婺女十一少，虛十，六虛之差十九太。 危十七太，營室十七少，東壁九太，奎十七半，婁十二太，胃十四太，昴十一，畢十六少，觜觿一，參九少，東井三十，輿鬼二太，柳十四少，七星六太，張十八太，翼十九少，軫十八太，角十三，亢九半，氐十五太，房五，心四太，尾十七，箕十少，為黃道度，以步日行。 月與五星出入，循此。 求此宿度，皆有餘分，前後輩之成少、半、太，準為全度。 若上考往古，下驗將來，當據歲差，每移一度，各依術算，使得當時度分，然後可以步三辰矣。
Kaiyuan year 12: Southern Dipper 23½°, Ox 7½°, Maid 11° lesser, Emptiness 10°, six Emptiness-difference parts 19 greater. Rooftop 17° greater, Encampment 17° lesser, Eastern Wall 9° greater, Stride 17½°, Bond 12° greater, Stomach 14° greater, Hairy Head 11°, Net 16° lesser, Turtle Beak 1°, Three Stars 9° lesser, Eastern Well 30°, Ghost Cart 2° greater, Willow 14° lesser, Seven Stars 6° greater, Extended Net 18° greater, Wings 19° lesser, Chariot Shaft 18° greater, Horn 13°, Neck 9½°, Root 15° greater, Room 5°, Heart 4° greater, Tail 17°, Winnowing Basket 10° lesser — the ecliptic lodge degrees used to pace the sun’s daily course. The moon and the five planets are reckoned by the same ecliptic framework. These lodge longitudes all carry fractional remainders; successive reckonings round them to lesser, half, or greater parts to align with whole degrees. To verify against past ages and test future ones, apply the precession: for each degree of shift, recalculate by the method to obtain contemporary longitudes — only then may one pace the sun, moon, and planets.
335
以乾實去中積分； 不盡者，盈通法為度。 命起赤道虛九，宿次去之，經虛去分，至不滿宿算外，得冬至加時日度。 以三元之策累加之，得次氣加時日度。
Subtract the circuit-of-heaven constant from the central accumulation; The remainder, divided by the universal divisor, gives degrees. Count from equatorial Emptiness 9°, subtracting whole lodges and the Emptiness fraction in turn, until less than one lodge remains beyond the tally — this yields the solar longitude at winter-solstice hour-addition. Add the three-origin interval repeatedly to obtain the hour-added solar degree for each successive qi.
336
以度餘減通法； 餘以冬至日躔距度所入限數乘之，為距前分。 置距度下黃赤道差，以通法乘之，減去距前分； 餘滿百二十除，為定差。 不滿者，以象統乘之，復除，為秒分。 乃以定差減赤道宿度，得冬至加時黃道日度。
Subtract the degree remainder from the universal divisor; Multiply the remainder by the band index for the winter-solstice solar station’s distance entry to obtain the pre-distance fraction. Take the ecliptic–equator difference for the distance band, multiply by the universal divisor, and subtract the pre-distance fraction; When the remainder fills 120, divide to obtain the fixed difference. If it does not fill, multiply by the image cycle and divide again to obtain seconds and parts. Subtract the fixed difference from the equatorial lodge longitude to obtain the ecliptic solar degree at winter-solstice hour-addition.
337
又置歲差，以限數乘之，滿百二十除，為秒分。 不盡為小分。 以加三元之策，因累裁之。 命以黃道宿次，各得定氣加時日度。
Set the annual precession, multiply by the band index, and divide by 120 for seconds and parts. The remainder becomes minor parts. Add this to the three-origin interval and accumulate the trims in order. Count through the ecliptic lodges in sequence to obtain the hour-added solar degree for each fixed qi.
338
置其氣定小餘，副之。 以乘其日盈縮分，滿通法而一，盈加、縮減其副。 用減其日加時度餘，得其夜半日度。 因累加一策，以其日盈縮分盈加、縮減度餘，得每日夜半日度。
Set the fixed minor remainder for that qi and keep a duplicate tally. Multiply by its daily equation of time, divide by the universal divisor, and add or subtract from the duplicate according to surplus or deficit. Subtract this from the day’s hour-added degree remainder to obtain the solar longitude at midnight. Add one interval at each step; apply each day’s equation to the degree remainder by addition or subtraction — yielding the midnight solar degree day by day.
339
四曰步月離術
IV. Method for Determining Lunar Motion
340
轉終六百七十萬一千二百七十九。
Rotation cycle constant: 670,1279.
341
轉終日二十七，餘千六百八十五，秒七十九。
Rotation cycle: 27 days, remainder 1,685, 79 seconds.
342
轉法七十六。
Rotation divisor: 76.
343
轉秒法八十。
Rotation second divisor: 80.
344
以秒法乘朔積分，盈轉終去之； 餘復以秒法約，為入轉分； 滿通法，為日。 命日算外，得天正經朔加時所入。 因加轉差日一、餘二千九百六十七、秒一，得次朔。 以一象之策，循變相加，得弦、望。 盈轉終日及餘秒者，去之。 各以經朔、弦、望小餘減之，得其日夜半所入。
Multiply the new-moon accumulation by the second divisor and cast out full rotation cycles; Reduce the remainder again by the second divisor to obtain rotation entry parts; When the parts fill the universal divisor, they become days. Count outward from the day tally to obtain the rotation entry at the hour-added canonical new moon of the celestial first month. Add the rotation increment of one day, remainder 2,967, and one second to obtain the next new moon. Add one four-image interval in turn to obtain the first and last quarters. When days and remainder-seconds fill the rotation cycle, cast them out. Subtract each new-moon, quarter, or full-moon minor remainder to obtain the rotation entry at that night’s midnight.
345
轉日
Rotation day
346
轉分
Rotation parts
347
列衰
Column decrement
348
轉積度
Accumulated rotation degrees
349
損益率
Rates of increase and decrease
350
朓朒積
Anomalistic accumulation
351
一日
Day 1
352
九百一十七
917
353
進十三
Tabular advance: 13
354
度初
Rotation degrees, initial
355
益二百九十七
Increase rate: 297
356
朒初
Waning accumulation, initial
357
二日
Day 2
358
九百三十
930
359
進十三
Tabular advance: 13
360
十二度五分
12° 5 parts
361
益二百五十九
Increase rate: 259
362
朒二百九十七
Waning accumulation: 297
363
三日
Day 3
364
九百四十三
943
365
進十三
Tabular advance: 13
366
二十四度二十三分
24° 23 parts
367
益二百二十
Increase rate: 220
368
朒五百五十六
Waning accumulation: 556
369
四日
Day 4
370
九百五十六
956
371
進十四
Tabular advance: 14
372
三十六度五十四分
36° 54 parts
373
益百八十
Increase rate: 180
374
朒七百七十六
Waning accumulation: 776
375
五日
Day 5
376
九百七十
970
377
進十四
Tabular advance: 14
378
四十九度二十二分
49° 22 parts
379
益百三十九
Increase rate: 139
380
朒九百五十六
Waning accumulation: 956
381
六日
Day 6
382
九百八十四
984
383
進十六
Tabular advance: 16
384
六十二度四分
62° 4 parts
385
益九十七
Increase rate: 97
386
朒千九十五
Waning accumulation: 1,095
387
七日
Day 7
388
千
1,000
389
進十八
Tabular advance: 18
390
七十五度空
75°, void
391
初益四十八末損六
Initial increase rate: 48, terminal decrease rate: 6
392
朒千一百九十二
Slow accumulation: 1,192
393
八日
Day 8
394
千一十八
1,018
395
進十九
Tabular advance: 19
396
八十八度十二分
88° 12 parts
397
損六十四
Decrease rate: 64
398
朒千二百三十四
Slow accumulation: 1,234
399
九日
Day 9
400
千三十七
1,037
401
進十四
Tabular advance: 14
402
百一度四十二分
101° 42 parts
403
損百六
Decrease rate: 106
404
朒千一百七十
Slow accumulation: 1,170
405
十日
Day 10
406
千五十一
1,051
407
進十四
Tabular advance: 14
408
百一十五度十五分
115° 15 parts
409
損百四十八
Decrease rate: 148
410
朒千六十四
Slow accumulation: 1,064
411
十一日
Day 11
412
千六十五
1,065
413
進十四
Tabular advance: 14
414
百二十九度二分
129° 2 parts
415
損百八十九
Decrease rate: 189
416
朒九百一十六
Slow accumulation: 916
417
十二日
Day 12
418
千七十九
1,079
419
進十三
Tabular advance: 13
420
百四十三度三分
143° 3 parts
421
損二百二十九
Decrease rate: 229
422
朒七百二十七
Slow accumulation: 727
423
十三日
Day 13
424
千九十二
1,092
425
進十三
Tabular advance: 13
426
百五十七度十八分
157° 18 parts
427
損二百六十七
Decrease rate: 267
428
朒四百九十八
Slow accumulation: 498
429
十四日
Day 14
430
千一百五
1,105
431
進十退三
Tabular advance: 10, retreat: 3
432
百七十一度四十六分
171° 46 parts
433
初損二百三十一末益六十六
Initial decrease rate: 231, terminal increase rate: 66
434
朒二百三十一
Slow accumulation: 231
435
十五日
Day 15
436
千一百一十二
1,112
437
退十三
Tabular retreat: 13
438
百八十六度十一分
186° 11 parts
439
益二百八十九
Increase rate: 289
440
朓六十六
Fast accumulation: 66
441
十六日
Day 16
442
千九十九
1,099
443
退十三
Tabular retreat: 13
444
二百度五十九分
200° 59 parts
445
益二百五十
Increase rate: 250
446
朓三百五十五
Fast accumulation: 355
447
十七日
Day 17
448
千八十六
1,086
449
退十三
Tabular retreat: 13
450
二百一十五度十八分
215° 18 parts
451
益二百一十一
Increase rate: 211
452
朓六百五
Fast accumulation: 605
453
十八日
Day 18
454
千七十三
1,073
455
退十四
Tabular retreat: 14
456
二百二十九度四十分
229° 40 parts
457
益百七十一
Increase rate: 171
458
朓八百一十六
Fast accumulation: 816
459
十九日
Day 19
460
千五十九
1,059
461
退十四
Tabular retreat: 14
462
二百四十三度四十九分
243° 49 parts
463
益百三十
Increase rate: 130
464
朓九百八十七
Fast accumulation: 987
465
二十日
Day 20
466
千四十五
1,045
467
退十七
Tabular retreat: 17
468
二百五十七度四十四分
257° 44 parts
469
益八十七
Increase rate: 87
470
朓千一百一十七
Fast accumulation: 1,117
471
二十一日
Day 21
472
千二十八
1,028
473
退十八
Tabular retreat: 18
474
二百七十一度二十五分
271° 25 parts
475
初益三十六末損十八
Initial increase rate: 36, terminal decrease rate: 18
476
朓千二百四
Fast accumulation: 1,204
477
二十二日
Day 22
478
千一十
1,010
479
退十八
Tabular retreat: 18
480
二百八十四度六十五分
284° 65 parts
481
損七十三
Decrease rate: 73
482
朓千二百二十二
Fast accumulation: 1,222
483
二十三日
Day 23
484
九百九十二
992
485
退十四
Tabular retreat: 14
486
二百九十八度十一分
298° 11 parts
487
損百一十六
Decrease rate: 116
488
朓千一百四十九
Fast accumulation: 1,149
489
二十四日
Day 24
490
九百七十八
978
491
退十四
Tabular retreat: 14
492
三百一十一度十五分
311° 15 parts
493
損百五十七
Decrease rate: 157
494
朓千三十三
Fast accumulation: 1,033
495
二十五日
Day 25
496
九百六十四
964
497
退十四
Tabular retreat: 14
498
三百二十四度五分
324° 5 parts
499
損百九十八
Decrease rate: 198
500
朓八百七十六
Fast accumulation: 876
501
二十六日
Day 26
502
九百五十
950
503
退十三
Tabular retreat: 13
504
三百三十六度五十七分
336° 57 parts
505
損二百三十七
Decrease rate: 237
506
朓六百七十八
Fast accumulation: 678
507
二十七日
Day 27
508
九百三十七
937
509
退十三
Tabular retreat: 13
510
三百四十九度十九分
349° 19 parts
511
損二百七十六
Decrease rate: 276
512
朓四百四十一
Fast accumulation: 441
513
二十八日
Day 28
514
九百二十四
924
515
退七進六
Tabular retreat: 7, advance: 6
516
三百六十一度四十四分
361° 44 parts
517
初損百六十五末益入後
Initial decrease rate: 165, terminal increase rate enters thereafter
518
朓百六十五
Fast accumulation: 165
519
各置朔、弦、望所入轉日損益率，幷後率而半之，為通率。 又二率相減，為率差。 前多者，以入餘減通法，餘乘率差，盈通法得一，幷率差而半之； 前少者，半入餘，乘率差，亦以通法除之：為加時轉率。 乃半之，以損益加時所入，餘為轉餘。 其轉餘，應益者，減法； 應損者，因餘：皆以乘率差，盈通法得一，加於通率，轉率乘之，通法約之，以朓減、朒加轉率，為定率。 乃以定率損益朓朒積，為定數。 其後無同率者，亦因前率。 應益者，以通率為初數，半率差而減之； 應損者，即為通率。 其損益入餘進退日，分為二日，隨餘初末，如法求之。 所得並以損益轉率。 此術本出皇極曆，以究算術之微變。 若非朔望有交者，直以入餘乘損益率，如通法而一，以損益朓朒，為定數。
For each syzygy, set the increase-decrease rates for the entered rotation days at new, quarter, and full moon; average them with the following rate to obtain the universal rate. Subtract the two rates; the remainder is the rate difference. If the prior rate is larger, subtract the entry remainder from the universal divisor, multiply the remainder by the rate difference, divide by the universal divisor and round up, then halve together with the rate difference; If the prior rate is smaller, halve the entry remainder, multiply by the rate difference, and divide by the universal divisor likewise: this yields the hour-added rotation rate. Halve this and apply increase or decrease to the hour entry; the remainder is the rotation remainder. For the rotation remainder, when increase applies, subtract from the divisor; when decrease applies, use the remainder as basis: multiply by the rate difference, divide by the universal divisor and add to the universal rate; multiply by the rotation rate and reduce by the universal divisor; subtract for fast and add for slow to the rotation rate, obtaining the fixed rate. Apply the fixed rate to adjust the anomalistic accumulation, yielding the fixed number. When no matching rate follows, proceed from the prior rate in the same way. When increase applies, take the universal rate as the initial value and subtract half the rate difference; when decrease applies, the universal rate itself serves. When adjustment of the entry remainder carries or borrows a day, apportion the parts over two days and compute by initial or terminal remainder as the method requires. Use the result together to adjust the rotation rate. This procedure derives from the Huangji calendar, refining the subtle variations of computational astronomy. When new or full moon has no eclipse, multiply the entry remainder directly by the increase-decrease rate, divide by the universal divisor, and adjust fast-slow accordingly to obtain the fixed number.
520
七日：初數二千七百一，末數三百三十九。 十四日：初數二千三百六十三，末數六百七十七。 二十一日：初數二千二十四，末數千一十六。 二十八日：初數千六百八十六，末數千三百五十四。 以四象約轉終，均得六日二千七百一分。 就全數約為九分日之八。 各以減法，餘為末數。 乃四象馴變相加，各其所當之日初末數也。 視入轉餘，如初數已下者，加減損益，因循前率。 如初數以上，則反其衰，歸于後率云。
Day 7: initial value 2,701, terminal value 339. Day 14: initial value 2,363, terminal value 677. Day 21: initial value 2,024, terminal value 1,016. Day 28: initial value 1,686, terminal value 1,354. Divide the rotation cycle by the four-image divisor; each segment is six days, 2,701 parts. Reduce the full count to approximate eight parts in nine of a day. For each segment, subtract from the divisor; the remainder is the terminal value. Accumulate the four-image transitions in order, yielding each corresponding day’s initial and terminal values. If the entered rotation remainder falls below the initial value, apply increase or decrease following the prior rate. If above the initial value, reverse the decline and revert to the posterior rate.
521
各置朔、弦、望大小餘，以入氣、入轉朓朒定數，朓減、朒加之，為定朔、弦、望大小餘。 定朔日名與後朔同者，月大； 不同者，小； 無中氣者，為閏月。 凡言夜半，皆起晨前子正之中。 若注曆，觀弦、望定小餘，不盈晨初餘數者，退一日。 其望有交、起虧在晨初已前者，亦如之。 又月行九道遲疾，則有三大二小。 以日行盈縮累增損之，則容有四大三小，理數然也。 若俯循常儀，當察加時早晚，隨其所近而進退之，使不過三大三小。 其正月朔有交、加時正見者，消息前後一兩月，以定大小，令虧在晦、二。 定朔、弦、望夜半日度各隨所直日度及餘分命之。 乃列定朔、望小餘，副之。 以乘其日盈縮分，如通法而一，盈加、縮減其副。 以加夜半日度，各得加時日度。
Set the major and minor remainders for each new, quarter, and full moon; apply the fixed fast-slow numbers for entered qi and rotation—subtract for fast, add for slow—to obtain the fixed syzygy remainders. When the fixed new-moon day name matches the next new moon, the month is long; when they differ, it is short; when no central qi falls in the month, it is intercalary. All references to midnight begin from true midnight at the start of zi before dawn. In calendar annotation, if the fixed minor remainder at quarter or full moon does not reach the early-morning initial remainder, set the date back one day. The same applies when a full-moon eclipse begins before early morning. Because the moon’s nine-path motion varies in speed, months naturally run three long and two short. Cumulative adjustment by the sun’s daily equation of time can occasionally yield four long and three short months; the arithmetic permits it. In practice, inspect whether the hour-added time is early or late and adjust accordingly, keeping within three long and three short months. When the first-month new moon has an eclipse at exact visibility, adjust long-short assignment a month or two on either side so that waning falls on the last or second day. For each fixed syzygy at midnight, name the solar longitude from that day’s degree and remainder. Array the fixed new- and full-moon minor remainders and keep duplicate tallies. Multiply by that day’s equation of time, divide by the universal divisor, and add or subtract from the duplicate according to surplus or deficit. Add to the midnight solar longitude to obtain the hour-added solar degree for each.
522
凡合朔所交，冬在陰曆、夏在陽曆，月行青道； 冬至、夏至後，青道半交在春分之宿，當黃道東。 立冬、立夏後，青道半交在立春之宿，當黃道東南。 至所衝之宿，亦如之。 冬在陽曆，夏在陰曆，月行白道； 冬至、夏至後，白道半交在秋分之宿，當黃道西。 立冬、立夏後，白道半交在立秋之宿，當黃道西北。 至所衝之宿，亦如之。 春在陽曆，秋在陰曆，月行朱道； 春分、秋分後，朱道半交在夏至之宿，當黃道南。 立春、立秋後，朱道半交在立夏之宿，當黃道西南。 至所衝之宿，亦如之。 春在陰曆，秋在陽曆，月行黑道。 春分、秋分後，黑道半交在冬至之宿，當黃道北。 立春、立秋後，黑道半交在立冬之宿，當黃道東北。 至所衝之宿，亦如之。 四序離為八節，至陰陽之所交，皆與黃道相會，故月有九行。 各視月交所入七十二候距交初中黃道日度，每五度為限，亦初數十二，每限減一，數終於四，乃一度強，依平。 更從四起，每限增一，終於十二，而至半交，其去黃道六度。 又自十二，每限減一，數終於四，亦一度強，依平。 更從四起，每限增一，終於十二，復與日軌相會。 各累計其數，以乘限度，二百四十而一，得度。 不滿者，二十四除，為分，若以二十除之，則大分，以十二為母。 為月行與黃道差數。 距半交前後各九限，以差數為減； 距正交前後各九限，以差數為加。 此加減出入六度，單與黃道相較之數。 若較之赤道，則隨氣遷變不常。 計去冬至、夏至以來候數，乘黃道所差，十八而一，為月行與赤道差數。 凡日以赤道內為陰，外為陽； 月以黃道內為陰，外為陽。 故月行宿度，入春分交後行陰曆、秋分交後行陽曆，皆為同名。 若入春分交後行陽曆、秋分交後行陰曆，皆為異名。 其在同名，以差數為加者，加之； 減者，減之。 若在異名，以差數為加者，減之； 減者，加之。 皆以增損黃道度，為九道定度。
At syzygy, when winter falls in yin months and summer in yang months, the moon follows the green path; After the Winter and Summer Solstices, the green path’s half-intersection lies at the spring-equinox lodge, east of the ecliptic. After Start of Winter and Start of Summer, the green path’s half-intersection lies at the Start-of-Spring lodge, southeast of the ecliptic. At the opposing lodge, the same applies. When winter is in yang months and summer in yin months, the moon follows the white path; After the Winter and Summer Solstices, the white path’s half-intersection lies at the autumn-equinox lodge, west of the ecliptic. After Start of Winter and Start of Summer, the white path’s half-intersection lies at the Start-of-Autumn lodge, northwest of the ecliptic. At the opposing lodge, the same applies. When spring is in yang months and autumn in yin months, the moon follows the vermilion path; After the Spring and Autumn Equinoxes, the vermilion path’s half-intersection lies at the summer-solstice lodge, south of the ecliptic. After Start of Spring and Start of Autumn, the vermilion path’s half-intersection lies at the Start-of-Summer lodge, southwest of the ecliptic. At the opposing lodge, the same applies. When spring is in yin months and autumn in yang months, the moon follows the black path. After the Spring and Autumn Equinoxes, the black path’s half-intersection lies at the winter-solstice lodge, north of the ecliptic. After Start of Spring and Start of Autumn, the black path’s half-intersection lies at the Start-of-Winter lodge, northeast of the ecliptic. At the opposing lodge, the same applies. The four seasons yield eight nodes; at each yin–yang crossing the moon meets the ecliptic — hence nine paths of lunar motion. For each conjunction entry, take the ecliptic solar degree at the initial and middle of the seventy-two hou; every five degrees is one band, beginning at twelve and subtracting one per band until four, then one degree slightly strong—use the mean. Begin again at four; add one per band until twelve at half-intersection, six degrees from the ecliptic. From twelve again, subtract one per band until four, likewise one degree slightly strong—use the mean. Begin at four again; add one per band until twelve, reuniting with the solar track. Accumulate the counts in order, multiply by the band index, and divide by 240 to obtain degrees. The remainder, divided by twenty-four, gives parts; if divided by twenty instead, major parts use twelve as denominator. This is the lunar ecliptic latitude difference. Within nine bands on either side of half-intersection, subtract the difference; within nine bands on either side of true conjunction, add the difference. This adjustment shifts latitude by six degrees—the value compared directly with the ecliptic alone. Compared with the equator, it varies with the seasons and is not constant. Count hou elapsed since the Winter or Summer Solstice, multiply the ecliptic difference, and divide by eighteen to obtain the lunar equatorial latitude difference. For the sun, inside the equator is yin and outside is yang; for the moon, inside the ecliptic is yin and outside is yang. Hence after the spring-equinox crossing the moon follows yin months, and after the autumn-equinox crossing yang months — both are same-named. If after the spring-equinox crossing it follows yang months, or after the autumn-equinox crossing yin months, both are differently named. Under same name, where the difference is for increase, add it; where for decrease, subtract it. Under different name, where the difference is for increase, subtract it; where for decrease, add it. Apply these rules to adjust ecliptic longitude and obtain the fixed nine-path degree.
523
各以中氣去經朔日算，加其入交汎，乃以減交終，得平交入中氣日算。 滿三元之策去之，餘得入後節日算。 因求次交者，以交終加之，滿三元之策去之，得後平交入氣日算。
For each central qi, subtract days from mean new moon, add the entered conjunction general term, and subtract from the conjunction cycle to obtain the mean conjunction’s day-count within that qi. Cast out full three-origin intervals; the remainder is the day-count entering the next node. To find the next conjunction, add the conjunction cycle and cast out full three-origin intervals, yielding the next mean conjunction’s day-count within qi.
524
各以氣初先後數先加、後減之，得平交入定氣日算。 倍六爻乘之，三其小餘，辰法除而從之； 以乘其氣損益率，如定氣辰數而一； 所得以損益其氣朓朒積，為定數。
For each qi, apply the initial prior-posterior numbers by addition then subtraction to obtain the mean conjunction’s day-count within fixed qi. Double and multiply by six lines; triple the minor remainder and divide by the chronogram divisor; multiply by that qi’s increase-decrease rate and divide by the fixed-qi chronogram count; use the result to adjust that qi’s anomalistic accumulation as the fixed number.
525
又置平交所入定氣餘，加其日夜半入轉餘，以乘其日損益率，滿通法而一，以損益其日朓朒積，交率乘之，交數而一，為定數。 乃以入氣入轉朓朒定數，朓減、朒加平交入氣餘，滿若不足，進退日算，為正交入定氣日算。 其入定氣餘，副之，乘其日盈縮分，滿通法而一，以盈加、縮減其副，以加其日夜半日度，得正交加時黃道日度。 以正交加時度餘減通法，餘以正交之宿距度所入限數乘之，為距前分。 置距度下月道與黃道差，以通法乘之，減去距前分，餘滿二百四十除，為定差。 不滿者一退為秒。 以定差及秒加黃道度、餘，仍計去冬至、夏至已來候數乘定差，十八而一； 所得依名同異而加減之，滿若不足，進退其度，得正交加時月離九道宿度。
Set the mean conjunction’s remainder within fixed qi; add the day’s midnight rotation entry remainder; multiply by that day’s increase-decrease rate, divide by the universal divisor, and adjust that day’s anomalistic accumulation; multiply by the conjunction rate and divide by the conjunction number for the fixed value. Apply the fixed fast-slow numbers for entered qi and rotation—subtract for fast, add for slow—to the mean conjunction’s qi remainder; carry or borrow days to obtain the true conjunction’s day-count within fixed qi. Keep a duplicate of the remainder within fixed qi; multiply by that day’s equation of time, divide by the universal divisor, and add or subtract from the duplicate; add to that night’s midnight solar longitude to obtain the true-conjunction hour-added ecliptic degree. Subtract the true-conjunction hour-added degree remainder from the universal divisor; multiply the remainder by the band index for the true-conjunction lodge’s distance entry to obtain the pre-distance fraction. Take the lunar ecliptic latitude difference for the distance band, multiply by the universal divisor, subtract the pre-distance fraction, and when the remainder fills 240 divide to obtain the fixed difference. If it does not fill, retreat one place for seconds. Add the fixed difference and seconds to the ecliptic degree and remainder; still count hou from the Winter or Summer Solstice, multiply the fixed difference, and divide by eighteen; apply the result according to same or different name; carry or borrow degrees to obtain the true-conjunction hour-added lunar nine-path lodge longitude.
526
各置定朔、弦、望加時日度，從九道循次相加。 凡合朔加時，月行潛在日下，與太陽同度，是謂離象。 先置朔、弦、望加時黃道日度，以正交加時所在黃道宿度減之； 餘以加其正交九道宿度，命起正交宿度算外，即朔、弦、望加時所當九道宿度也。 其合朔加時，若非正交，則日在黃道，月在九道，各入宿度雖多少不同，考其去極，若應繩準。 故云：月行潛在日下，與太陽同度。 以一象之度九十一、餘九百五十四、秒二十二半為上弦，兌象。 倍之，而與日衝，得望，坎象。 參之，得下弦，震象。 各以加其所當九道宿度，秒盈象統從餘，餘滿通法從度，得其日加時月度。 綜五位成數四十，以約度餘，為分。 不盡者，因為小分。
Set the hour-added solar degree for each fixed syzygy and accumulate along the nine paths in sequence. At conjunction hour-addition the moon lies hidden beneath the sun at the same longitude—the departure image. Set the ecliptic solar degree at hour-added new, quarter, or full moon; subtract the ecliptic lodge degree at true-conjunction hour-addition; add the remainder to the nine-path lodge at true conjunction; count outward from that lodge to obtain the nine-path longitude at hour-added syzygy. At conjunction hour-addition, if not true conjunction, the sun remains on the ecliptic and the moon on the nine paths; though lodge longitudes differ, their polar distances align to the plumb line. Hence: the moon moves hidden beneath the sun at the same longitude. One image interval: 91°, remainder 954, 22½ seconds—first quarter, Duì ☱. Double it and oppose the sun to obtain full moon, Kǎn ☵. Triple it to obtain last quarter, Zhèn ☳. Add each to the corresponding nine-path lodge; carry seconds into the image cycle and remainder into degrees via the universal divisor to obtain that day’s hour-added lunar longitude. The five positions sum to forty; reduce the degree remainder thereby for parts. The remainder becomes minor parts.
527
視經朔夜半入轉，若定朔大餘有進退者，亦加減轉日。 否則因經朔為定。 累加一日，得次日。 各以夜半入轉餘乘列衰，如通法而一； 所得以進加、退減其日轉分，為月轉定分。 滿轉法，為度。
Inspect midnight rotation entry at mean new moon; if the fixed new-moon major remainder advances or retreats, adjust the rotation day likewise. Otherwise take mean new moon as the fixed value. Add one day at each step to obtain the next day. Multiply each midnight rotation entry remainder by the column decline and divide by the universal divisor; use the result to advance or retreat that day’s rotation parts, yielding the lunar rotation fixed parts. When the parts fill the rotation divisor, they become degrees.
528
視定朔、弦、望夜半入轉，各半列衰以減轉分。 退者，定餘乘衰，以通法除，幷衰而半之； 進者，半餘乘衰，亦以通法除：皆加所減。 乃以定餘乘之，盈通法得一，以減加時月度，為夜半月度。 各以每日轉定分累加之，得次日。 若以入轉定分，乘其日夜漏，倍百刻除，為晨分。 以減轉定分，餘為昏分。 望前以昏、望後以晨加夜半度，各得晨昏月。
At fixed syzygy midnight rotation entry, halve the column decline and subtract from rotation parts. If retreating, multiply the fixed remainder by the decline, divide by the universal divisor, and halve together with the decline; if advancing, halve the remainder, multiply by the decline, divide by the universal divisor likewise, and add to the amount subtracted. Multiply by the fixed remainder; divide by the universal divisor and subtract from the hour-added lunar degree to obtain midnight lunar longitude. Add the daily rotation fixed parts at each step to obtain the next day. Multiply the entered rotation fixed parts by the day-night clepsydra and divide by double the hundred marks for dawn parts. Subtract from the rotation fixed parts; the remainder is dusk parts. Before full moon add dusk parts, after full moon add dawn parts to midnight longitude to obtain dawn and dusk lunar positions.
529
交日
Intersection day
530
屈伸率
Latitude bend-stretch rates
531
屈伸積
Latitude bend-stretch accumulation
532
一日
Day 1
533
屈二十七
Bend rate: 27
534
積初
Accumulation, initial
535
二日
Day 2
536
屈十九
Bend rate: 19
537
積二十七
Accumulation: 27
538
三日
Day 3
539
屈十三
Bend rate: 13
540
積四十六
Accumulation: 46
541
四日
Day 4
542
屈八
Bend rate: 8
543
積五十九
Accumulation: 59
544
五日
Day 5
545
屈十三
Bend rate: 13
546
積六十七
Accumulation: 67
547
六日
Day 6
548
屈十九
Bend rate: 19
549
積一度四
Accumulation: 1° 4
550
七日
Day 7
551
初屈二十末伸七
Initial bend rate: 20, terminal stretch rate: 7
552
積一度二十三
Accumulation: 1° 23
553
八日
Day 8
554
伸十九
Stretch rate: 19
555
積一度三十六
Accumulation: 1° 36
556
九日
Day 9
557
伸十三
Stretch rate: 13
558
積一度十七
Accumulation: 1° 17
559
十日
Day 10
560
伸八
Stretch rate: 8
561
積一度四
Accumulation: 1° 4
562
十一日
Day 11
563
伸十三
Stretch rate: 13
564
積七十二
Accumulation: 72
565
十二日
Day 12
566
伸十九
Stretch rate: 19
567
積五十九
Accumulation: 59
568
十三日
Day 13
569
伸二十七
Stretch rate: 27
570
積四十
Accumulation: 40
571
十四日
Day 14
572
初伸十三末屈入後
Initial stretch rate: 13, terminal bend rate enters thereafter
573
積十三
Accumulation: 13
574
各視每日夜半入陰陽曆交日數，以其下屈伸積，月道與黃道同名者，加之； 異名者，減之。 各以加減每日晨昏黃道月度，為入宿定度及分。
For each day, take the yin-yang month intersection-day count at midnight; where the moon-path and ecliptic are same-named, add the bend-stretch accumulation below; where differently named, subtract it. Apply these adjustments to each day’s dawn and dusk ecliptic lunar longitudes to obtain fixed lodge degrees and parts.
575
五曰步軌漏術
V. Method for Determining Orbital Clepsydra
576
爻統千五百二十。
Line cycle: 1,520
577
象積四百八十。
Image accumulation: 480.
578
辰八刻百六十分。
Chronogram interval: 8 ke, 160 fen.
579
昏、明二刻二百四十分。
Twilight interval: 2 ke, 240 fen.
580
定氣
Fixed qi
581
陟降率
Rates of ascension and descent
582
消息衰
Message wane
583
陽城日晷
Yangcheng gnomon table
584
漏刻
Clepsydra marks
585
黃道去極度
Ecliptic polar distance
586
距中星度
Distance to culminating star
587
冬至
Winter Solstice
588
降七十八
Descent rate: 78
589
息空六十四
Message wane: void 64
590
丈二尺七寸一分五十
Gnomon shadow: 1 zhang 2 chi 7 cun 1 fen 50
591
二十七刻二百三十分
27 ke 230 fen
592
百一十七度二十分
117° 20 parts
593
八十二度二十六分
82° 26 parts
594
小寒
Lesser Cold
595
降七十二
Descent rate: 72
596
息十一九十一
Message wane: 11, 91
597
丈二尺二寸二分七十七
Gnomon shadow: 1 zhang 2 chi 2 cun 2 fen 77
598
二十七刻百三十五分
27 ke 135 fen
599
百一十四度三十五分
114° 35 parts
600
八十二度九十一分
82° 91 parts
601
大寒
Greater Cold
602
降五十三
Descent rate: 53
603
息二十二四十二
Message wane: 22, 42
604
丈一尺二寸一分八十二
Gnomon shadow: 1 zhang 1 chi 2 cun 1 fen 82
605
二十六刻三百八十分
26 ke 380 fen
606
百一十一度九十分
111° 90 parts
607
八十四度七十七分
84° 77 parts
608
立春
Start of Spring
609
降三十四
Descent rate: 34
610
息三十二十五
Message wane: 30, 25
611
九尺七寸三分五十一
Gnomon shadow: 9 chi 7 cun 3 fen 51
612
二十五刻四百七十五分
25 ke 475 fen
613
百八度五分
108° 5 parts
614
八十七度七十分
87° 70 parts
615
雨水
Rain Water
616
降初限七十八
Descent, initial limit: 78
617
息三十五七十八
Message wane: 35, 78
618
八尺二寸一分六
Gnomon shadow: 8 chi 2 cun 1 fen 6
619
二十四刻四百七十分
24 ke 470 fen
620
百三度二十分
103° 20 parts
621
九十一度三十九分
91° 39 parts
622
驚蟄
Awakening of Insects
623
降一
Descent rate: 1
624
息三十九五十
Message wane: 39, 50
625
六尺七寸三分八十四
Gnomon shadow: 6 chi 7 cun 3 fen 84
626
二十三刻三百六十分
23 ke 360 fen
627
九十七度三十分
97° 30 parts
628
九十五度八十八分
95° 88 parts
629
春分
Spring Equinox
630
陟五
Ascension rate: 5
631
息三十九六十五
Message wane: 39, 65
632
五尺四寸三分十九
Gnomon shadow: 5 chi 4 cun 3 fen 19
633
二十二刻二百三十分
22 ke 230 fen
634
九十一度三十分
91° 30 parts
635
百度四十四分五十
100° 44 parts 50
636
清明
Pure Brightness
637
陟初限一
Ascension, initial limit: 1
638
息三十八八十九
Message wane: 38, 89
639
四尺三寸二分十一
Gnomon shadow: 4 chi 3 cun 2 fen 11
640
二十一刻百二十分
21 ke 120 fen
641
八十五度三十分
85° 30 parts
642
百五度一分
105° 1 parts
643
穀雨
Grain Rain
644
陟三十二
Ascension rate: 32
645
息三十三五十六
Message wane: 33, 56
646
三尺三寸四十七
Gnomon shadow: 3 chi 3 cun 47
647
二十刻十分
20 ke 10 fen
648
七十九度三十分
79° 30 parts
649
百九度五十分
109° 50 parts
650
立夏
Start of Summer
651
陟五十二
Ascension rate: 52
652
息二十八三十八
Message wane: 28, 38
653
二尺五寸三分三十一
Gnomon shadow: 2 chi 5 cun 3 fen 31
654
十九刻五分
19 ke 5 fen
655
七十四度五十五分
74° 55 parts
656
百十三度十九分
113° 19 parts
657
小滿
Lesser Fullness
658
陟六十三
Ascension rate: 63
659
息二十十二
Message wane: 20, 12
660
尺九寸五分七十六分
Gnomon shadow: 1 chi 9 cun 5 fen 76
661
十八刻百分
18 ke 100 fen
662
七十度七十分
70° 70 parts
663
百一十六度十二分
116° 12 parts
664
芒種
Grain in Ear
665
陟六十四
Ascension rate: 64
666
息十十二
Message wane: 10, 12
667
尺六寸三
Gnomon shadow: 1 chi 6 cun 3
668
十七刻三百三十五分
17 ke 335 fen
669
六十八度二十五分
68° 25 parts
670
百一十七度九十八分
117° 98 parts
671
夏至
Summer Solstice
672
降六十四
Descent rate: 64
673
消空五十二
Wane: void 52
674
尺四寸七分七十九
Gnomon shadow: 1 chi 4 cun 7 fen 79
675
十七刻二百五十分
17 ke 250 fen
676
六十七度四十分
67° 40 parts
677
百一十八度六十三分
118° 63 parts
678
小暑
Lesser Heat
679
降六十三
Descent rate: 63
680
消十七十六
Wane: 17, 16
681
尺六寸三
Gnomon shadow: 1 chi 6 cun 3
682
十七刻三百三十五分
17 ke 335 fen
683
六十八度二十五分
68° 25 parts
684
百一十七度九十八分
117° 98 parts
685
大暑
Greater Heat
686
降五十二
Descent rate: 52
687
消二十七十五
Wane: 27, 15
688
尺九寸五分七十六
Gnomon shadow: 1 chi 9 cun 5 fen 76
689
十八刻百分
18 ke 100 fen
690
七十度七十分
70° 70 parts
691
百一十六度十二分
116° 12 parts
692
立秋
Start of Autumn
693
降三十二
Descent rate: 32
694
消二十八九十
Wane: 28, 90
695
二尺五寸三分三十一
Gnomon shadow: 2 chi 5 cun 3 fen 31
696
十九刻五分
19 ke 5 fen
697
七十四度五十五分
74° 55 parts
698
百十一三度十九分
113° 19 parts
699
處暑
End of Heat
700
降初限九十九
Descent, initial limit: 99
701
消三十四五十五分
Wane: 34, 55
702
三尺三寸四十七
Gnomon shadow: 3 chi 3 cun 47
703
二十刻十分
20 ke 10 fen
704
七十九度三十分
79° 30 parts
705
百九度五十分
109° 50 parts
706
白露
White Dew
707
降五
Descent rate: 5
708
消三十八九十
Wane: 38, 90
709
四尺三寸二分十一
Gnomon shadow: 4 chi 3 cun 2 fen 11
710
二十一刻百二十分
21 ke 120 fen
711
八十五度三十分
85° 30 parts
712
百五度一分
105° 1 parts
713
秋分
Autumn Equinox
714
陟一
Ascension rate: 1
715
消三十九六十六
Wane: 39, 66
716
五尺四寸三分十九
Gnomon shadow: 5 chi 4 cun 3 fen 19
717
二十二刻二百三十分
22 ke 230 fen
718
九十一度三十分
91° 30 parts
719
百度四十四分五十
100° 44 parts 50
720
寒露
Cold Dew
721
陟初限一
Ascension, initial limit: 1
722
消三十九五十
Wane: 39, 50
723
六尺七寸三分八十四
Gnomon shadow: 6 chi 7 cun 3 fen 84
724
二十三刻三百六十分
23 ke 360 fen
725
九十七度三十分
97° 30 parts
726
九十五度八十八分
95° 88 parts
727
霜降
Frost Descent
728
陟三十四
Ascension rate: 34
729
消廿四九十八
Wane: 24, 98
730
八尺二寸一分六
Gnomon shadow: 8 chi 2 cun 1 fen 6
731
二十四刻四百七十分
24 ke 470 fen
732
百三度二十分
103° 20 parts
733
九十一度三十九分
91° 39 parts
734
立冬
Start of Winter
735
陟五十三
Ascension rate: 53
736
消二十九七十二
Wane: 29, 72
737
九尺七寸三分五十一
Gnomon shadow: 9 chi 7 cun 3 fen 51
738
二十五刻四百七十五分
25 ke 475 fen
739
百八度五分
108° 5 parts
740
八十七度七十分
87° 70 parts
741
小雪
Lesser Snow
742
陟七十二
Ascension rate: 72
743
消二十一七十
Wane: 21, 70
744
丈一尺二寸一分八十二
Gnomon shadow: 1 zhang 1 chi 2 cun 1 fen 82
745
二十六刻三百八十分
26 ke 380 fen
746
百一十一度九十分
111° 90 parts
747
八十四度七十七分
84° 77 parts
748
大雪
Greater Snow
749
陟七十八
Ascension rate: 78
750
消十一十三
Wane: 11, 13
751
丈二尺二寸二分七十七
Gnomon shadow: 1 zhang 2 chi 2 cun 2 fen 77
752
二十七刻百三十五分
27 ke 135 fen
753
百一十四度三十五分
114° 35 parts
754
八十二度九十一分
82° 91 parts
755
各置其氣消息衰，依定氣所有日，每以陟降率陟減、降加其分，滿百從衰，各得每日消息定衰。 其距二分前後各一氣之外，陟降不等，皆以三日為限。 雨水初日，降七十八。 初限，日損十二。 次限，日損八。 次限，日損三。 次限，日損二。 次限，日損一。 清明初日，陟一。 初限，日益一。 次限，日益二。 次限，日益三。 次限，日益八。 末限，日益十九。 處暑初日，降九十九。 初限，日損十九。 次限，日損八。 次限，日損三。 次限，日損二。 末限，日損一。 寒露初日，陟一。 初限，日益一。 次限，日益二。 次限，日益三。 次限，日益八。 末限，日益十二。 各置初日陟降率，依限次損益之，為每日率。 乃遞以陟減、降加氣初消息衰，各得每日定衰。
For each seasonal node, set its message-wane constant; for each day in that fixed qi, apply the ascension-descent rate—subtract on ascension, add on descent—to the parts; carry hundreds into the wane tally to obtain the daily fixed message wane. Outside the single qi bracketing each equinox, where ascension and descent are unequal, a three-day band limit applies throughout. Rain Water, day 1: descent rate 78. First band: decrease 12 per day. Second band: decrease 8 per day. Third band: decrease 3 per day. Fourth band: decrease 2 per day. Fifth band: decrease 1 per day. Pure Brightness, day 1: ascension rate 1. First band: increase 1 per day. Second band: increase 2 per day. Third band: increase 3 per day. Fourth band: increase 8 per day. Final band: increase 19 per day. End of Heat, day 1: descent rate 99. First band: decrease 19 per day. Second band: decrease 8 per day. Third band: decrease 3 per day. Fourth band: decrease 2 per day. Final band: decrease 1 per day. Cold Dew, day 1: ascension rate 1. First band: increase 1 per day. Second band: increase 2 per day. Third band: increase 3 per day. Fourth band: increase 8 per day. Final band: increase 12 per day. Set each first-day ascension-descent rate; step through the limit bands by increase or decrease to obtain the daily rate. Then successively subtract ascension and add descent from the qi’s opening message wane to obtain the daily fixed wane.
756
南方戴日之下，正中無晷。 自戴日之北一度，乃初數千三百七十九。 自此起差，每度增一，終於二十五度，計增二十六分。 又每度增二，終於四十度。 又每度增六，終於四十四度，增六十八。 又每度增二，終於五十度。 又每度增七，終於五十五度。 又每度增十九，終於六十度，增百六十。 又每度增三十三，終於六十五度。 又每度增三十六，終於七十度。 又每度增三十九，終於七十二度，增二百六十。 又度增四百四十。 又度增千六十。 又度增千八百六十。 又度增二千八百四十。 又度增四千。 又度增五千三百四十。 各為每度差。 因累其差，以遞加初數，滿百為分，分十為寸，各為每度晷差。 又累其晷差，得戴日之北每度晷數。
Directly beneath the sub-solar point in the south, at true noon there is no shadow. One degree north of the sub-solar point, the initial count is 1,379. From here the increment begins: add 1 per degree up to 25°, for a total increase of 26 parts. Then add 2 per degree up to 40°. Then add 6 per degree up to 44°, increase 68. Then add 2 per degree up to 50°. Then add 7 per degree up to 55°. Then add 19 per degree up to 60°, increase 160. Then add 33 per degree up to 65°. Then add 36 per degree up to 70°. Then add 39 per degree up to 72°, increase 260. Then add 440 per degree. Then add 1,060 per degree. Then add 1,860 per degree. Then add 2,840 per degree. Then add 4,000 per degree. Then add 5,340 per degree. Each value is the per-degree difference. Sum the differences cumulatively onto the initial count; carry hundreds into parts and tens of parts into cun to obtain the per-degree gnomon difference. Sum those differences again to obtain the gnomon reading for each degree north of the sub-solar point.
757
各置其氣去極度，以極去戴日度五十六及分八十二半減之，得戴日之北度數。 各以其消息定衰所直度之晷差，滿百為分，分十為寸，得每日晷差。 乃遞以息減、消加其氣初晷數，得每日中晷常數。
Set the qi’s polar distance and subtract 56° 82½ parts (the pole’s offset from the sub-solar point) to obtain degrees north of the sub-solar point. Take the per-degree gnomon difference at the degree of the fixed message wane; carry hundreds into parts and tens into cun for the daily gnomon difference. Then successively subtract on waning breath and add on growing breath from the qi’s opening gnomon count to obtain the daily mean noon gnomon constant.
758
以其日所在氣定小餘，爻統減之，餘為中後分。 不足減，反相減，為中前分。 以其晷差乘之，如通法而一，為變差。 以加減中晷常數，冬至後，中前以差減，中後以差加。 夏至後，中前以差加，中後以差減。 冬至一日，有減無加。 夏至一日，有加無減。 得每日中晷定數。
Take the day’s fixed minor remainder within its qi, subtract the line divisor, and the remainder is the after-noon fraction. If it will not subtract, reverse the subtraction to obtain the before-noon fraction. Multiply by the gnomon difference, divide by the universal divisor, and obtain the variation difference. Apply it to the mean noon gnomon constant: after Winter Solstice, subtract before noon and add after noon. After Summer Solstice, add before noon and subtract after noon. On Winter Solstice day, only decrease applies—no increase. On Summer Solstice day, only increase applies—no decrease. This yields the daily fixed noon gnomon count.
759
又置消息定衰，滿象積為刻，不滿為分。 各遞以息減、消加其氣初夜半漏，得每日夜半漏定數。 其全刻，以九千一百二十乘之，十九乘刻分從之，如三百而一，為晨初餘數。
Set the fixed message wane again; carry image accumulation into clepsydra marks, leaving the remainder as parts. Successively subtract on waning breath and add on growing breath from the qi’s opening midnight leak to obtain the daily fixed midnight leak. For whole marks, multiply by 9,120, add 19 times the fractional marks, divide by 300, and obtain the dawn-opening remainder.
760
各倍夜半漏，為夜刻。 以減百刻，餘為晝刻。 減晝五刻以加夜，即晝為見刻，夜為沒刻。 半沒刻加半辰，起子初算外，得日出辰刻。 以見刻加而命之，得日入。 置夜刻，五而一，得每更差刻。 又五除之，得每籌差刻。 以昏刻加日入辰刻，得甲夜初刻。 又以更籌差加之，得五夜更籌所當辰。 其夜半定漏，亦名晨初夜刻。
Double the midnight leak to obtain night clepsydra marks. Subtract from 100 marks; the remainder is day marks. Move five marks from day to night: day marks become appearance marks, night marks disappearance marks. Add half disappearance marks to half a chronogram; count outward from the zi-opening tally to obtain sunrise chronogram marks. Add appearance marks and assign the count to obtain sunset. Set night marks, divide by five, and obtain the per-watch mark difference. Divide by five again for the per-stave mark difference. Add dusk marks to the sunset chronogram mark to obtain the first watch of night A. Add the watch-stave difference repeatedly to obtain the chronogram for each watch and stave of the five night watches. The fixed midnight leak is also called dawn-opening night marks.
761
又置消息定衰，滿百為度，不滿為分。 各遞以息減、消加氣初去極度，各得每日去極定數。
Set the fixed message wane again; carry hundreds into degrees, leaving the remainder as parts. Successively subtract on waning breath and add on growing breath from the qi’s opening polar distance to obtain the daily fixed polar distance.
762
又置消息定衰，以萬二千三百八十六乘之，如萬六千二百七十七而一，為度差。 差滿百為度。 各遞以息加、消減其氣初距中度，得每日距中度定數。 倍之，以減周天，為距子度。
Set the fixed message wane, multiply by 12,386, divide by 16,277, and obtain the degree difference. When the difference fills 100, carry into degrees. Successively add on waning breath and subtract on growing breath from the qi’s opening distance to culmination to obtain the daily fixed culminating distance. Double it, subtract from the circuit of heaven, and obtain the distance from zi.
763
置其日赤道日度，加距中度，得昏中星。 倍距子度，以加昏中星，得曉中星。 命昏中星為甲夜中星，加每更差度，得五夜中星。
Take the day’s equatorial solar longitude, add culminating distance, and obtain the dusk culminating star. Double the distance from zi, add to the dusk star, and obtain the dawn culminating star. Take the dusk culminating star as night A’s culminating star; add the per-watch degree difference for each of the five night watches.
764
凡九服所在，每氣初日中晷常數不齊。 使每氣去極度數相減，各為其氣消息定數。 因測其地二至日晷，測一至可矣，不必兼要冬夏。 於其戴日之北每度晷數中，較取長短同者，以為其地戴日北度數及分。 每氣各以消息定數加減之，因冬至後者，每氣以減。 因夏至後者，每氣以加。 得每氣戴日北度數。 各因所直度分之晷數，為其地每定氣初日中晷常數。 其測晷有在表南者，亦據其晷尺寸長短與戴日北每度晷數同者，因取其所直之度，去戴日北度數。 反之，為去戴日南度。 然後以消息定數加減之。
Across the nine domains, the mean noon gnomon constants at each qi’s opening are not uniform. Subtract each qi’s polar distances pairwise; each difference is that qi’s fixed message count. Measure the locality’s solstitial gnomon; one solstice suffices—both winter and summer need not be taken. Among per-degree gnomon counts north of the sub-solar point, match equal shadow lengths to fix that place’s degrees and parts north of the sub-solar point. For each qi, apply its fixed message count; after Winter Solstice, subtract for each qi. After Summer Solstice, add for each qi. This yields each qi’s degrees north of the sub-solar point. Take the gnomon count at each degree and part as that locality’s fixed mean noon gnomon constant at each qi’s opening. If the measured shadow falls south of the table, match its length to the north-of-sub-solar per-degree table, take the corresponding degree, and subtract from degrees north of the sub-solar point. Reverse the sign to obtain degrees south of the sub-solar point. Then apply the fixed message count by addition or subtraction.
765
二至各於其地下水漏以定當處晝夜刻數。 乃相減，為冬夏至差刻。 半之，以加減二至晝夜刻數，為定春秋分初日晝夜刻數。 乃置每氣消息定數。 以當處差刻數乘之，如二至去極差度四十七分，八十而一，所得依分前後加減初日晝夜漏刻，各得餘定氣初日晝夜漏刻。
At each solstice, calibrate the local water clock to fix the day-and-night clepsydra marks for that place. Subtract the two to obtain the solstitial mark difference. Halve the difference and apply it to the solstitial day-night marks to fix the spring and autumn equinox opening day-night marks. Then set each qi’s fixed message count. Multiply by the local mark difference, divide by the solstitial polar-distance span (47 parts 80), and add or subtract the opening day-night leak marks by fractional rank to obtain each remaining fixed qi’s opening day-night leaks.
766
置每日消息定衰，亦以差刻乘之，差度而一，所得以息減、消加其氣初漏刻，得次日。 其求距中度及昏明中星日出入，皆依陽城法求之。 仍以差刻乘之，差度而一，為今有之數。 若置其地春秋定日中晷常數與陽城每日晷數，較其同者，因其日夜半漏亦為其地定春秋分初日夜半漏。 求餘定氣初日，亦以消息定數依分前後加減刻分，春分後以減，秋分後以加。 滿象積為刻。 求次日，亦以消息定衰，依陽城術求之。 此術究理，大體合通。 然高山平川，視日不等。 較其日晷，長短乃同。 考其水漏，多少殊別。 以茲參課，前術為審。
Take the daily fixed message wane, multiply by the mark difference, divide by the degree difference, and step the qi’s opening leaks by waning breath and growing breath to obtain the next day. Distance to culmination, dusk and dawn culminating stars, and sunrise and sunset are all found by the Yangcheng method. Multiply by the mark difference and divide by the degree difference for the present-part correction. Match the locality’s fixed equinox mean noon gnomon to Yangcheng’s daily table; on the matching day, its midnight leak becomes the locality’s fixed equinox opening midnight leak. For each remaining fixed qi’s opening day, apply the fixed message count to marks and parts by fractional rank—subtract after spring equinox, add after autumn equinox. Carry image accumulation into clepsydra marks. For the next day, step the fixed message wane by the Yangcheng procedure. This method plumbs the principle and is broadly sound. Yet mountain heights and level plains do not present the sun equally. Only when their noon shadows are compared do lengths agree. Water-clock flows differ greatly in rate. On this comparison, the earlier method is the more reliable.