1
紀元曆
Era Origin Calendar (Jiyuan li).
2
崇甯《紀元曆》
Chongning 《Era Origin Calendar》.
3
演紀上元上章執徐之歲，距元符三年庚辰，歲積二千八百六十一萬三千四百六十算； 至崇寧五年丙戌，歲積二千八百六十一萬三千四百六十六算。
Era computation (yan ji): from upper origin in the year shangzhang zhixu, the accumulated count to the third year of Yuanfu (gengchen) is 28,613,460 counts; to the fifth year of Chongning (bingxu), the accumulated count is 28,613,466 counts.
4
步氣朔第一
Procedures for qi and new moon (bu qi shuo), part one.
5
日法：七千二百九十。
Day divisor (ri fa): 7,290.
6
期實：二百六十六萬二千六百二十六。
Period dividend (qi shi): 2,662,626.
7
朔實：二十一萬五千二百七十八。
New-moon dividend (shuo shi): 215,278.
8
歲周：三百六十五日、餘一千七百七十六。
Year circuit (sui zhou): 365 days, remainder 1,776 parts.
9
氣策：一十五、餘一千五百九十二太。
Qi stride (qi ce): 15 days, remainder 1,592 parts major.
10
朔策：二十九、餘三千八百六十八。
New-moon stride (shuo ce): 29 days, remainder 3,868 parts.
11
望策：一十四、餘五千五百七十九。
Full-moon stride (wang ce): 14 days, remainder 5,579 parts.
12
弦策：七、餘二千七百八十九半。
Quarter-moon stride (xian ce): 7 days, remainder 2,789½ parts.
13
中盈分：三千一百八十五半。
Mid-qi surplus parts (zhong ying fen): 3,185½.
14
朔虛分：三千四百二十二。
New-moon void fraction (shuo xu fen): 3,422.
15
沒限：五千六百九十七少。
Submergence limit (mo xian): 5,697 parts minor.
16
旬周：四十三萬七千四百。
Ten-day circuit (xun zhou): 437,400.
17
紀法：六十。
Era divisor (ji fa): 60.
18
求天正冬至：置上元距所求積年，以期實乘之，為天正冬至氣積分； 滿旬周去之，不滿，如日法而一為大餘，不盡為小餘。 其大餘命己卯，算外，即所求年天正冬至日辰及餘。
To find the winter solstice of the celestial standard year (tian zheng dongzhi): set the accumulated years from upper origin to the target year, multiply by the period dividend (qi shi) to obtain the qi accumulated parts for the celestial-standard winter solstice; cast out full ten-day circuits (xun zhou); divide the remainder by the day divisor (ri fa) for the greater remainder (da yu), with the unfilled portion as the lesser remainder (xiao yu). count the greater remainder from jimao (outside the count) to obtain the day, double-hour, and remainder of the celestial-standard winter solstice for the target year.
19
求次氣：置天正冬至大、小餘，以氣策加之，四分之一為少，之二為半，之三為太。 如滿秒母，收從小余，小余滿日法從大余，大余盈紀法乃去之。
To find successive qi (ci qi): set the winter solstice greater and lesser remainders and add the qi stride (qi ce); one-fourth counts as minor (shao), two-fourths as half (ban), three-fourths as major (tai). when seconds fill the second parent (miao mu), carry into the lesser remainder; when the lesser remainder fills the day divisor, carry into the greater remainder; when the greater remainder exceeds the era divisor (ji fa), cast it out.
20
去命如前，即次氣日辰及餘。
name and cast out as before to obtain the day, double-hour, and remainder of the next qi.
21
求天正經朔：置天正冬至氣積分，以朔實去之，不盡，為天正閏餘； 用減氣積分，餘為天正十一月經朔加時積分。 滿旬周去之，不滿，如日法而一為大餘，不盡為小餘。 其大餘命己卯，算外，即所求年天正十一月經朔日辰及餘。
To find the canonical new moon of the celestial standard year (tian zheng jing shuo): set the winter solstice qi accumulated parts, divide by the new-moon dividend (shuo shi); the remainder is the celestial-standard intercalary remainder (run yu); subtract it from the qi accumulated parts; the remainder is the accumulated parts at the canonical new-moon moment of celestial-standard month 11. cast out full ten-day circuits; divide the remainder by the day divisor for the greater remainder, with the unfilled portion as the lesser remainder. count the greater remainder from jimao (outside the count) to obtain the day, double-hour, and remainder of the celestial-standard canonical new moon of month 11 for the target year.
22
求弦望及次朔經日：置天正經朔大、小餘，以弦策累加之，去命如前，即各得弦、望及次朔經日辰及餘。
To find quarter-moon, full-moon, and successive canonical days: set the celestial-standard canonical new moon greater and lesser remainders and add the quarter-moon stride (xian ce) cumulatively; name and cast out as before to obtain the day, double-hour, and remainder of each quarter-moon, full moon, and successive canonical new moon.
23
求沒日：置有沒常氣小余，凡常氣小餘在沒限已上者，為有沒之氣。
To find submergence days (mo ri): set the lesser remainder of the constant qi that bears submergence; whenever a constant qi's lesser remainder reaches the submergence limit (mo xian) or above, that qi has a submergence day.
24
六十乘之，用減四十四萬三千七百七十一，余滿六千三百七十一而一為日，不滿為餘。 命日起其氣初日辰，算外，即為氣內沒日辰。
multiply by 60, subtract from 443,771; divide the remainder by 6,371 for days, with the unfilled portion as the remainder. count the days outward from the opening day and double-hour of that qi (outside the count) to obtain the submergence day and hour within the qi.
25
求滅日：置有滅經朔小餘，凡經朔小餘不滿朔虛分者，為有滅之朔。
To find extinction days (mie ri): set the lesser remainder of the canonical new moon that bears extinction; whenever a canonical new moon's lesser remainder falls short of the new-moon void fraction (shuo xu fen), that month has an extinction day.
26
三十乘之，滿朔虛分而一為日，不滿為餘。 命日起其月經朔日辰，算外，即為月內滅日辰。
multiply by 30; divide by the new-moon void fraction for days, with the unfilled portion as the remainder. count the days outward from the canonical new moon day and double-hour of that month (outside the count) to obtain the extinction day and hour within the month.
27
步發斂
Procedures for expansion and contraction (bu fa lian).
28
候策：五、餘五百三十、秒五十五。
Hou stride (hou ce): 5 days, remainder 530 parts, 55 seconds.
29
卦策：六、餘六百三十七、秒六。
Hexagram stride (gua ce): 6 days, remainder 637 parts, 6 seconds.
30
土王策：三、餘三百一十八、秒三十三。
Earth ruler stride (tu wang ce): 3 days, remainder 318 parts, 33 seconds.
31
歲閏：七萬九千二百九十。
Year intercalation (sui run): 79,290.
32
月閏：六千六百七半。
Month intercalation (yue run): 6,607½.
33
閏限：二十萬八千六百七十半。
Intercalation limit (run xian): 208,670½.
34
辰法：一千二百一十五。
Chen divisor (chen fa): 1,215.
35
半辰法：六百七半。
Half-chen divisor (ban chen fa): 607½.
36
刻法：七百二十九。
Ke divisor (ke fa): 729.
37
秒法：六十。
Second divisor (miao fa): 60.
38
求七十二候：各置中節大、小餘命之，為初候； 以候策加之為次候； 又加之為末候。 各命己卯，算外，即得所求日辰。
To find the seventy-two hou: for each mid-node, set its major and minor remainders and designate them as the first hou; add the hou stride (hou ce) to obtain the next hou; add again to obtain the final hou. for each, count from jimao (outside the count) to obtain the requested day and double-hour.
39
求六十四卦：各置中氣大、小餘命之，為公卦用事日； 以卦策加之，得辟卦用事日； 又加之，得諸侯內卦用事日； 以土王策加之，得十有二節之初諸侯外卦用事日； 又加之，得大夫卦用事日； 復以卦策加之，得卿卦用事日。 各命己卯，算外，即得所求日辰。
To find the sixty-four hexagrams: for each mid-qi, set its major and minor remainders and designate them as the day when the duke hexagram (gong gua) takes effect; add the hexagram stride (gua ce) to obtain the day when the lord hexagram (bi gua) takes effect; add again to obtain the day when the marquis inner hexagram (zhuhou nei gua) takes effect; add the earth ruler stride (tu wang ce) to obtain the day when the marquis outer hexagram (zhuhou wai gua) takes effect at the opening of the twelve qi-periods; add again to obtain the day when the grandee hexagram (dafu gua) takes effect; add the hexagram stride again to obtain the day when the minister hexagram (qing gua) takes effect. for each, count from jimao (outside the count) to obtain the requested day and double-hour.
40
求五行用事：各因四立之節大、小餘命之，即春木、夏火、秋金、冬水首用事日。 以土王策減四季中氣大、小餘，即其季土始用事之日。 各命己卯，算外，即得所求日辰。
To find the dominion of the five phases (wuxing yong shi): for each, take the major and minor remainders of the four establishment nodes (si li) and designate them to obtain the first dominion days of spring wood, summer fire, autumn metal, and winter water. subtract the earth ruler stride from each season's mid-qi major and minor remainders to obtain the day when earth first takes dominion in that season. for each, count from jimao (outside the count) to obtain the requested day and double-hour.
41
七十二候及卦目與前曆同。
The seventy-two hou and hexagram list are the same as in the preceding calendar.
42
求中氣去經朔：置天正閏餘，以月閏累加之，滿日法為閏日，不滿為餘，即其月中氣去經朔日算。 因求卦候者，各以卦、候策依次累加減之，中氣前減，中氣後加。
To find the mid-qi's distance from the canonical new moon: set the celestial-standard intercalary remainder (run yu) and add the month intercalation (yue run) cumulatively; when the total fills the day divisor (ri fa) it makes an intercalary day, and the unfilled portion is the remainder—that is the count of days from the canonical new moon to the mid-qi in that month. For those also seeking hexagrams and hou, add and subtract the hexagram and hou strides in sequence for each; subtract before the mid-qi and add after the mid-qi.
43
各得其月卦、候去經朔日算。
Thus obtain for each month the count of days from the canonical new moon to the hexagram and hou.
44
求發斂加時：置所求小餘，倍之，如辰法而一為辰數，不滿，五因之，如刻法而一為刻，不盡為分。 命辰數起子正，算外，即各得加時所在辰、刻及分。 如半辰數，即命起子初。
To find the added time of expansion and contraction (fa lian jia shi): set the requested lesser remainder, double it, and divide by the chen divisor (chen fa) to obtain the chen number; if any remainder remains, multiply by five and divide by the ke divisor (ke fa) to obtain ke, with the unfilled portion as fen. count the chen number outward from midnight of zi (zi zheng) (outside the count) to obtain the double-hour, ke, and fen of the added time. if the half-chen number is included, count from the beginning of the zi ke (zi chu).
45
步日躔
Procedures for solar motion along the ecliptic (bu ri chan).
46
周天分：二億一千三百一萬八千一十七。
Circuit-of-heaven parts (zhou tian fen): 213,018,017.
47
歲差：七千九百三十七。
Precession (sui cha): 7,937.
48
周天度：三百六十五、約分二十五、秒七十二。
Circuit of heaven (zhou tian du): 365° 25′ 72″.
49
象限：九十一、約分三十一、秒九。
Quadrant limit (xiang xian): 91° 31′ 9″.
50
乘法：一百一十九。
Multiplier (cheng fa): 119.
51
除法：一千八百一十一。
Divisor (chu fa): 1,811.
52
秒法：一百。
Second divisor (miao fa): 100.
53
求每日盈縮分先後數：置所求盈縮分，以乘法乘之，如除法而一，為其氣中平率； 與後氣中平率相減，為合差； 半合差，加減其氣中平率，為初、末泛率。 至後加為初、減為末，分後減為初、加為末。
To find the daily excess–deficit parts (ying suo fen) and sequence numbers (xian hou shu): set the requested excess–deficit parts, multiply by the multiplier (cheng fa), and divide by the divisor (chu fa) to obtain the mid-qi mean rate (qi zhong ping lu) for that qi; subtract it from the next qi's mid-qi mean rate to obtain the combined difference (he cha); add or subtract half the combined difference from the mid-qi mean rate to obtain the initial and final general rates (fan lu). After a solstice (zhi), add to the initial and subtract from the final; after an equinox (fen), subtract from the initial and add to the final.
54
又以乘法乘合差，如除法而一，為日差； 半日差，加減初、末泛率，為初、末定率。 至後減初加末，分後加初減末。 以日差累加減其氣初定率，為每日盈縮分； 至後減，分後加。
set the combined difference, multiply by the multiplier, and divide by the divisor to obtain the daily difference (ri cha); add or subtract half the daily difference from the initial and final general rates to obtain the initial and final fixed rates (ding lu). After a solstice, subtract from the initial and add to the final; after an equinox, add to the initial and subtract from the final. by cumulative daily difference add and subtract the qi's initial fixed rate to obtain the daily excess–deficit parts; after a solstice subtract; after an equinox add.
55
各以每日盈縮分加減氣下先後數。 冬至後，積盈為先，在縮減之； 夏至後，積縮為後，在盈減之。 其分、至前一氣，無後氣相減，皆因前氣合差為其氣合差。 餘依前術，求朏朒仿此。
apply each day's excess–deficit parts to add and subtract the sequence numbers (xian hou shu) below each qi. After the winter solstice, accumulated surplus counts as early (xian)—where there is contraction (suo), subtract; after the summer solstice, accumulated contraction counts as late (hou)—where there is surplus (ying), subtract. For the qi before an equinox or solstice, when there is no following qi from which to subtract, always take the preceding qi's combined difference as that qi's combined difference. For the remainder, follow the preceding procedures; find deficit–surfeit (fei nei) by the same method.
56
求經朔弦望入氣：置天正閏日及餘，如氣策以下者，以減氣策，為入大雪氣； 以上者去之，餘以減氣策，為入小雪氣：即天正十一月經朔入氣日及餘。 求弦、望及後朔入氣，以弦策累加之，滿氣策去之，即各得弦、望及次朔入氣日及餘。
To find qi entry for canonical new moon, quarter-moon, and full moon: set the celestial-standard intercalary day and remainder; if below the qi stride (qi ce), subtract from the qi stride to obtain entry into Major Snow (da xue) qi; if above, cast it out; subtract the qi stride from the remainder again to obtain entry into Minor Snow (xiao xue) qi: that is the day and remainder of the celestial-standard month-11 canonical new moon's qi entry. To find qi entry for quarter-moon, full moon, and later new moons, add the quarter-moon stride (xian ce) cumulatively; cast out a full qi stride each time to obtain the day and remainder of qi entry for each quarter-moon, full moon, and successive new moon.
57
求經朔弦望入氣朏朒定數：各以所入氣小餘乘其日損益率，如日法而一，所得，以損益其日下朏朒積，各為定數。
To find the fixed deficit–surfeit number for canonical new moon, quarter-moon, and full moon qi entry: for each, multiply the lesser remainder of the entered qi by its daily gain-and-loss rate (sun-yi-lu), divide by the day divisor (ri fa), and apply the result to add or subtract the deficit–surfeit accumulation (fei nei ji) below that day—each yields a fixed number.
58
赤道宿度
Equatorial lodge degrees (chida xiu du).
59
鬥：二十五
Dipper (Dou): 25°.
60
牛：七少
Ox (Niu): 7° minor.
61
女：十一少
Maid (Nü): 11° minor.
62
虛：九少秒七十二
Void (Xu): 9° minor, 72″.
63
危：十五半
Rooftop (Wei): 15° half.
64
室：十七
Encampment (Shi): 17°.
65
壁：八太。
Wall: 8° major.
66
北方七宿九十四度秒七十二。
Northern seven lodges: 94° 72″.
67
奎：十六半
Striding Man: 16° half.
68
婁：十二
Bond: 12°.
69
胃：十五
Stomach: 15°.
70
昴：十一少
Hairy Head: 11° minor.
71
畢：十七少。
Net: 17° minor.
72
觜：半。
Beak: 0.5°.
73
參：十半。
Three Stars: 10° half.
74
西方七宿八十三度。
Western seven lodges: 83°.
75
井：三十三少
Well: 33° minor.
76
鬼：二半
Ghost: 2° half.
77
柳：十三太。
Willow: 13° major.
78
星：六太
Star: 6° major.
79
張：十七少
Extended Net: 17° minor.
80
翼：十八太
Wings: 18° major.
81
軫：十七
Chariot Axis: 17°.
82
南方七宿一百九度少。
Southern seven lodges: 109° minor.
83
角：十二
Horn: 12°.
84
亢：九少
Gullet: 9° minor.
85
氐：十六
Base: 16°.
86
房：五太
Chamber: 5° major.
87
心：六少
Heart: 6° minor.
88
尾：十九少
Tail: 19° minor.
89
箕：十半
Winnowing Basket: 10° half.
90
東方七宿七十九度。
Eastern seven lodges: 79°.
91
按諸曆赤道宿次，就立全度，頗失真數。 今依宋朝渾儀校測距度，分定太、半、少，用為常數，校之天道，最為密近。 如考唐，用唐所測； 考古，用古所測：即各得當時宿度。
In various calendars, equatorial lodge positions were set as whole degrees, which departs significantly from true values. The present calendar instead uses arc degrees measured with the Song armillary sphere, fixing major, half, and minor fractions as constants—a fit to the heavens that is closest and most precise. To verify against the Tang, use Tang-era measurements; to verify antiquity, use ancient measurements—each then yields the lodge degrees of that era.
92
求冬至赤道日度：以歲差乘所求積年，滿周天分去之，不滿，覆減周天分，餘如五千八百三十二而一為分，不盡，退除為秒。 其分，滿百為度，命起赤道虛宿七度外去之，至不滿宿，即所求年天正冬至加時日躔赤道宿度及分秒。
Procedure to find the winter solstice equatorial solar degree: multiply precession by the accumulated years sought, discard full circuit-of-heaven parts, and if a remainder remains subtract the circuit-of-heaven parts again; divide by 5,832 for minutes and convert the remainder into seconds. Carry hundreds into degrees, count from beyond Void lodge 7° on the equator and discard full lodges, and the lodge not yet filled gives the winter solstice hour-of-addition equatorial solar lodge degree and minutes and seconds for the civil new year of the year sought.
93
求春分、夏至、秋分赤道日度：置天正冬至加時赤道日度，累加象限，滿赤道宿次去之，即各得春分、夏至、秋分加時日在宿度及分秒。
Procedure to find the equatorial solar degrees at spring equinox, summer solstice, and autumn equinox: set the winter solstice equatorial degree at hour-of-addition, add the image quadrant repeatedly, discard full equatorial cycles, and obtain the equatorial lodge degrees and minutes and seconds at hour-of-addition for each cardinal.
94
求四正後赤道宿積度：置四正赤道宿全度，以四正赤道日度及分減之，餘為距後度； 以赤道宿度累加之，各得四正後赤道宿積度及分。
Procedure to find accumulated equatorial degrees after the four cardinals: set the full width of each cardinal equatorial lodge, subtract the cardinal equatorial solar degree and minutes, and the remainder is the post-cardinal distance; add equatorial lodge widths cumulatively to obtain the accumulated equatorial degree and minutes after each cardinal.
95
求赤道宿積度入初末限：視四正後赤道宿積度及分，在四十五度六十五分、秒五十四半已下為入初限； 已上，用減象限，餘為入末限。
Procedure to find whether an equatorial accumulated degree falls in the initial or terminal limit: if the post-cardinal equatorial accumulation is 45° 65′ 54.5″ or less, it is in the initial limit; otherwise subtract it from the image quadrant and the remainder is in the terminal limit.
96
求二十八宿黃道度：以四正後赤道宿入初、末限度及分，減一百一度，余以初、末限度及分乘之，進位，滿百為分，分滿百為度，至後以減、分後以加赤道宿積度，為其宿黃道積度； 以前宿黃道積度減之，其四正之宿，先加象限，然後以前宿減之。
Procedure to find the ecliptic degrees of the twenty-eight lodges: subtract the post-cardinal equatorial limit from 101°, multiply by the limit’s degrees and minutes, carry as needed, convert hundreds into minutes and then degrees, subtract after winter solstice and add after the equinoxes to convert equatorial accumulation into ecliptic accumulation for each lodge; subtract the prior lodge’s ecliptic accumulation; for the four cardinal lodges, add the image quadrant first, then subtract the prior lodge.
97
為其宿黃道度分。 其分就近約為太、半、少。
The result is that lodge’s ecliptic degree and minutes. Round the minutes to major, half, and minor fractions.
98
黃道宿度
Ecliptic lodge degrees.
99
鬥：二十三
Dipper: 23°.
100
牛：七
Ox: 7°.
101
女：十一
Maid: 11°.
102
虛：九少秒七十二
Void: 9° minor, 72″.
103
危：十六。
Rooftop: 16°.
104
室：十八。
Encampment: 18°.
105
壁：九半。
Wall: 9° half.
106
北方七宿九十三度太秒七十二。
Northern seven lodges: 93° major, 72″.
107
奎：十八
Striding Man: 18°.
108
婁：十二太
Bond: 12° major.
109
胃：十五半
Stomach: 15° half.
110
昴：十一
Hairy Head: 11°.
111
畢：十六半
Net: 16° half.
112
觜； 半
Beak: 0.5°.
113
參：九太
Three Stars: 9° major.
114
西方七宿八十四度。
Western seven lodges: 84°.
115
井：三十半
Well: 30° half.
116
鬼：二半
Ghost: 2° half.
117
柳：十三少
Willow: 13° minor.
118
星：六太
Star: 6° major.
119
張：十七太
Extended Net: 17° major.
120
翼：二十
Wings: 20°.
121
軫：十八半
Chariot Axis: 18° half.
122
南方七宿一百九度。
Southern seven lodges: 109°.
123
角：十二太
Horn: 12° major.
124
亢：九太
Gullet: 9° major.
125
氐：十六少
Base: 16° minor.
126
房：五太
Chamber: 5° major.
127
心：六
Heart: 6°.
128
尾：十八少
Tail: 18° minor.
129
箕：九半
Winnowing Basket: 9° half.
130
東方七宿七十八度少。
Eastern seven lodges: 78° minor.
131
前黃道宿度，依今曆歲差所在算定。 如上考往古，下驗將來，當據歲差，每移一度，依術推變當時宿度，然後可步七曜，知其所在。 如徑求七曜所在，置所在積度，以前黃道宿積度減之，為所在黃道宿度及分。
The ecliptic lodge degrees above are determined by the present calendar’s precessional position. To check the past and test the future, rely on precession: whenever it shifts a degree, recalculate contemporary lodge degrees by the procedure—only then can the seven luminaries be stepped and their places known. To find a luminary’s position directly, set its accumulated longitude, subtract the prior ecliptic lodge accumulation, and obtain its ecliptic lodge degree and minutes.
132
求天正冬至加時黃道日度：以冬至加時赤道日度及分秒，減一百一度，餘以冬至加時赤道日度及分秒乘之，進位，滿百為分，分滿百為度，命曰黃赤道差； 用減冬至赤道日度及分秒，即所求年天正冬至加時黃道日度及分秒。
Procedure to find the ecliptic solar degree at civil new-year winter solstice hour-of-addition: subtract 101° from the winter solstice equatorial degree at hour-of-addition, multiply the remainder by that equatorial degree and its minutes and seconds, carry, convert hundreds into minutes and then degrees, and call the result the ecliptic–equator difference; subtract that difference from the winter solstice equatorial degree at hour-of-addition to obtain the ecliptic solar degree and minutes and seconds at civil new-year winter solstice for the year sought.
133
求二十四氣加時黃道日度：置所求年冬至日躔黃赤道差，以次年黃赤道差減之，餘以所求氣數乘之，二十四而一，所得以加其氣中積及約分，又以其氣初日先後數先加後減之，用加冬至加時黃道日度，依宿次命之，即各得其氣加時黃道日躔宿度及分秒。 如其年冬至加時赤道宿度空，分秒在歲差已下者，即加前宿全度。 然求黃赤道差，餘依術算。
Procedure to find the ecliptic solar degrees at hour-of-addition for all twenty-four qi: take the winter solstice ecliptic–equator difference for the year sought, subtract the next year’s difference, multiply the remainder by the qi number and divide by twenty-four, add to the qi’s central accumulation and approximate parts, adjust by the first-day lead-and-lag counts (add first, subtract after), add to the winter solstice ecliptic degree at hour-of-addition, and discard full lodge cycles to obtain each qi’s ecliptic solar lodge at hour-of-addition. If the winter solstice equatorial lodge at hour-of-addition has fractional minutes and seconds below the precessional threshold, add the prior lodge’s full width. Then compute the ecliptic–equator difference; otherwise follow the procedure as above.
134
求二十四氣晨前夜半黃道日度：置日法，以其氣小餘減之，餘副置之； 以其氣初日盈縮分乘之，如萬約之，所得，盈加縮減其副，滿日法為度，不滿，退除為分秒，以加其氣加時黃道日度，即各得其氣一日晨前夜半黃道日度及分秒； 每日加一度，以百約每日盈縮分為分秒，盈加縮減之，滿黃道宿次去之，即每日晨前夜半黃道日躔宿度及分秒。 其二十四氣初日晨前夜半黃道日度，系屬前氣，自前氣攤算，即各得所求。
Procedure to find the ecliptic solar degrees before dawn and at midnight for each of the twenty-four qi: set the day divisor, subtract that qi’s minor remainder, and set the remainder aside; multiply by that qi’s first-day expansion–contraction parts, reduce by ten thousand, add for expansion and subtract for contraction to the subsidiary, convert a full day divisor into degrees and the remainder into minutes and seconds, add to that qi’s ecliptic degree at hour-of-addition, and obtain that qi’s first-day ecliptic solar degree before dawn and at midnight with minutes and seconds; add one degree each day, reduce the daily expansion–contraction parts by one hundred to obtain minutes and seconds, add for expansion and subtract for contraction, discard full ecliptic cycles, and obtain the daily ecliptic solar lodge degree and minutes and seconds before dawn and at midnight. Each qi’s first-day ecliptic degrees before dawn and at midnight belong to the preceding qi; spread the calculation from that prior qi to obtain each value sought.
135
求每日午中黃道日度：置一萬分，以所入氣日盈縮分盈加縮減而半之，滿百為分，不滿為秒，以加其日晨前夜半黃道日度，即其日午中日躔黃道宿度及分。
Procedure to find the daily noon ecliptic solar degree: set 10,000 parts, apply the entered qi’s daily expansion–contraction parts (add for expansion, subtract for contraction) and halve, convert hundreds to minutes and the remainder to seconds, add to that day’s ecliptic degree before dawn and at midnight, and obtain the noon ecliptic solar lodge degree and minutes.
136
求夏至加時黃道日度：置天正冬至加時黃道日度及分秒，以二至限及分秒加之，滿黃道宿次去之，不滿，為夏至加時黃道日度及分秒。
Procedure to find the summer solstice ecliptic solar degree at hour-of-addition: set the winter solstice ecliptic degree at hour-of-addition with minutes and seconds, add the two-solstice limit with minutes and seconds, discard full ecliptic cycles, and the remainder is the summer solstice ecliptic degree at hour-of-addition with minutes and seconds.
137
求每日午中黃道積度：以二至加時黃道日度距至所求日午中黃道日度，為入二至後黃道積度及分。
Procedure to find the daily noon accumulated ecliptic degree: measure from the solstice ecliptic degree at hour-of-addition to the sought day’s noon ecliptic degree to obtain the accumulated ecliptic degree after the solstice with minutes.
138
求每日午中黃道入初末限：視二至後黃道積度，在四十三度一十二分、秒八十七以下為初限； 以上，用減象限，餘為入末限。 其積度滿象限去之，為二分後黃道積度，在四十八度一十八分、秒二十二以下為初限； 以上，用減象限，餘為入末限。
Procedure to find daily noon ecliptic entry into initial and terminal limits: for accumulated ecliptic degree after a solstice, 43° 12′ 87″ or below is the initial limit; above that, subtract the image quadrant and the remainder is in the terminal limit. when the accumulation fills the image quadrant, discard it to obtain the post-equinox accumulated ecliptic degree; 48° 18′ 22″ or below is the initial limit; above that, subtract the image quadrant and the remainder is in the terminal limit.
139
求每日午中赤道日度：以所求日午中黃道積度，入至後初限、分後末限度及分秒，進三位，加二十萬二千五十少，開平方除之，所得，減去四百四十九半，余在初限者，直以二至赤道日度加而命之； 在末限者，以減象限，餘以二分赤道日度加而命之：即每日午中赤道日度。 以所求日午中黃道積度，入至後末限、分後初限度及分秒，進三位，用減三十萬三千五十少，開平方除之，所得，以減五百五十半，余在初限者，直以二分赤道日度加而命之； 在末限者，以減象限，餘以二至赤道日度加而命之：即每日午中赤道日度。
Procedure to find the daily noon equatorial solar degree: take the sought day’s noon accumulated ecliptic degree—after a solstice in the initial limit, after an equinox in the terminal limit—with degrees, minutes, and seconds; advance three places, add 202,050 and a fraction less, take the square root and divide; subtract 449.5; if in the initial limit, add the two-solstice equatorial solar degree directly and assign the lodges; if in the terminal limit, subtract the image quadrant, add the two-equinox equatorial solar degree to the remainder, and assign the lodges: that yields the daily noon equatorial solar degree. Take the sought day’s noon accumulated ecliptic degree—after a solstice in the terminal limit, after an equinox in the initial limit—with degrees, minutes, and seconds; advance three places, subtract 303,050 and a fraction less, take the square root and divide; subtract 550.5; if in the initial limit, add the two-equinox equatorial solar degree directly and assign the lodges; if in the terminal limit, subtract the image quadrant, add the two-solstice equatorial solar degree to the remainder, and assign the lodges: that yields the daily noon equatorial solar degree.
140
求太陽入宮日時刻及分：各置入宮宿度及分秒，以其日晨前夜半日度減之，餘以二十四乘，為時實； 以其日太陽行度及分秒為法實，如法而一，為半時數； 不滿，進二位，為刻實； 以二十四乘，前法除之為刻，不滿，退除為分。 其半時命起子正，算外，即得太陽入宮初正時、刻及分。 其逐刻日、時及分，舊曆均其日數，從其簡略，未盡其詳。 今但依入宮正術求之，即允協天道。
Procedure to find the day, hour, quarter, and minutes of solar palace entry: set each palace-entry lodge degree with minutes and seconds, subtract that day’s ecliptic solar degree before dawn and at midnight, and multiply the remainder by twenty-four for the hour dividend; take that day’s solar motion degree with minutes and seconds as the divisor dividend, divide by the divisor for the half-hour count; If it does not fill, shift two places for the quarter dividend; Multiply by 24 and divide by the previous divisor for quarters; the remainder, stepped back, gives minutes. At half the double-hour count, count from zi zheng (outside the count) to obtain the exact time, quarter, and minutes when the Sun enters the palace. For day, time, and minute at each quarter, the old calendar averaged day-counts—a simplified treatment that did not fully elaborate the procedure. We now derive it solely by the correct palace-entry method, which aligns with celestial principles.
141
步晷漏
Stepping Gnomon and Clepsydra.
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二至限：一百八十二、分六十二、秒一十八。
Solstice limit (er zhi xian): 182°, 62′, 18″.
143
象限：九十一、分二十一、秒九。
Quadrant limit (xiang xian): 91°, 21′, 9″.
144
一象度：九十一、分二十一、秒四十三。
One-quadrant arc: 91°, 21′, 43″.
145
冬至後初限夏至後末限：六十二日、分二十。
Initial limit after winter solstice and final limit after summer solstice: 62 days, 20′.
146
夏至後初限冬至後末限：一百二十日、分四十二。
Initial limit after summer solstice and final limit after winter solstice: 120 days, 42′.
147
已上分秒母各同一百。
For the above values, minutes and seconds share the divisor 100.
148
冬至嶽台晷影常數：一丈二尺八寸三分。
Standard Yuetai noon gnomon shadow at winter solstice: 1 zhang 2 chi 8 cun 3 fen.
149
夏至嶽台晷影常數：一尺五寸六分。
Standard Yuetai noon gnomon shadow at summer solstice: 1 chi 5 cun 6 fen.
150
昏明分：一百八十二少。
Twilight parts: 182 shao (182 and a fractional shortfall).
151
昏明刻：二分三百六十四半。
Twilight quarters: 2 ke, 364.5 fen.
152
辰刻：八分二百四十三。
Double-hour quarters: 8 ke, 243 fen.
153
半辰刻：四分一百二十一半。
Half double-hour quarters: 4 ke, 121.5 fen.
154
刻法：七百二十九。
Quarter divisor (ke fa): 729.
155
求午中入氣：置所求日大餘及半法，以所入氣大、小餘減之，為其日午中入氣日及餘。
To find midday solar-term entry: take the sought day's greater remainder plus the half divisor, subtract the entered qi's greater and lesser remainders—this gives the day's midday entry into the qi in days and remainder.
156
求午中中積：置其氣中積，以午中入氣日及余加之，其餘以日法退除為分秒。
To find the midday mid-term accumulation: start from the qi's mid-term accumulation, add midday entry days and remainder, then convert excess remainder through the day divisor into minutes and seconds.
157
為所求日午中中積及分秒。
The result is the sought day's midday mid-term accumulation in days, minutes, and seconds.
158
求午中入二至後初末限； 置午中中積及分，為入冬至後； 滿二至限去之，為入夏至後。 其二至後，如在初限已下為入初限； 已上，覆減二至限，餘為入末限。
To find midday entry into the solstice initial or final limit: Take the midday mid-term accumulation and minutes as entry after winter solstice; When it exceeds the solstice limit, subtract that limit to obtain entry after summer solstice. Within each post-solstice interval: if at or below the initial limit, count as initial-limit entry; If at or above that threshold, subtract the solstice limit from it; the remainder is final-limit entry.
159
求嶽台晷影午中定數：冬至後初限、夏至後末限，以百通日，內分，自相乘為實，置之； 以七百二十五除之，所得，加一十萬六百一十七，併入限分，折半為法，實如法而一為分，不滿，退除為小分，其分滿十為寸，寸滿十為尺，用減冬至嶽台晷影常數，即得所求午中晷影定數。 夏至後初限、冬至後末限，以百通日，內分，自相乘，為實，乃置入限分，九因，再折，加一十九萬八千七十五為法，其夏至前後，日如在半限以上者，減去半限，餘置於上，列半限於下，以上減下，餘以乘上，進二位，七十七除之，所得加法為定法，然後除之。
To find the fixed Yuetai noon gnomon shadow: in the winter-solstice initial / summer-solstice final interval, express days in hundredths including minutes, square the value for the dividend; Divide by 725; add the quotient to 100,617, combine with limit minutes and halve for the divisor; divide the dividend by the divisor for minutes (remainder stepped back for fractional minutes); convert ten minutes to one cun and ten cun to one chi; subtract from the standard winter-solstice Yuetai shadow—the result is the fixed noon gnomon shadow sought. In the summer-solstice initial / winter-solstice final interval: express days in hundredths including minutes and square for the dividend; set limit minutes, multiply by 9, halve twice, add 198,075 as divisor; if the day lies at or beyond half the limit around summer solstice, subtract half the limit, place remainder over half-limit below, subtract and multiply; carry two places, divide by 77, add quotient to divisor for the fixed divisor—then perform the division.
160
實如法而一為分，不滿，退除為小分，其分滿十為寸，寸滿十為尺，以加夏至嶽台晷影常數，即得所求日午中晷影定數。
Divide dividend by divisor for minutes (remainder stepped back for fractional minutes); convert by tens to cun and chi; add to the standard summer-solstice Yuetai shadow—the result is that day's fixed noon gnomon shadow.
161
求每日日行積度：以午中入氣餘乘其日盈縮分，日法而一，冬至後盈加縮減、夏至後縮加盈減先後數，以先加後減中積日及分秒，滿與不足，進退其日，為所求日行積度及分秒。
To find daily accumulated solar motion: multiply midday qi-entry remainder by the day's surplus-deficit part and divide by the day divisor; after winter solstice add surplus and subtract deficit, after summer solstice the reverse; apply advance-retard to the mid-term day accumulation (add early, subtract late), adjusting days as needed—the result is accumulated daily solar motion in degrees, minutes, and seconds.
162
求每日赤道內外度：置所求日午中日行積度及分，如不滿二至限，在象限已下為冬至後度； 象限已上，用減二至限，為夏至前度。 如滿二至限去之，餘在象限以下為夏至後度； 象限以上，用減二至限，為冬至前度。 並置之於上，列象限於下，以上減下，餘以乘上，冬至前後五百一十七而一，夏至前後四百而一為度，不滿，退除為分，以加二至前後度，所得，用減象限，餘置於上，列二至限於下，以上減下，餘以乘上，其度、分、秒皆以百通，然後乘之。
To find daily equatorial declination: take the day's midday accumulated solar degrees and minutes; if below the solstice limit and under the quadrant limit, count as post–winter-solstice degrees; If at or above the quadrant limit, subtract from the solstice limit for pre–summer-solstice degrees. If it exceeds the solstice limit, subtract that limit; if the remainder is below the quadrant limit, count as post–summer-solstice degrees; If at or above the quadrant limit, subtract from the solstice limit for pre–winter-solstice degrees. Place the computed value over the quadrant limit below, subtract and multiply; divide by 517 before winter solstice or 400 before summer solstice for degrees (remainder stepped back for minutes); add to pre-solstice degrees; subtract from the quadrant limit, repeat the subtract-multiply with the solstice limit below, converting degrees, minutes, and seconds to hundredths before multiplication.
163
退一位，如三十四萬八千八百五十六而一為秒，滿百為分，分滿百為度，即所求日黃道去赤道內外度及分。 冬至前後為外，夏至前後為內。
Shift one place and divide by 348,856 for seconds; 100 seconds make one minute, 100 minutes one degree—the result is the day's ecliptic declination in degrees and minutes. South of the equator before and after winter solstice; north of the equator before and after summer solstice.
164
求每日午中太陽去極度； 以每日午中黃道去赤道內、外度及分，內減外加一象度及分，為每日午中太陽去極度及分。
To find daily midday solar polar distance: From the day's midday ecliptic declination, subtract if north of the equator or add if south, plus one quadrant arc—the result is midday solar polar distance.
165
求每日日出入分晨昏分半晝分：置所求日黃道去赤道內外度及分，以三百六十三乘之，進一位，如二百三十九而一，所得，以加減一千八百二十二半，赤道內以減，赤道外以加。
To find sunrise, sunset, dawn, dusk, and half-day parts: take the day's ecliptic declination, multiply by 363, shift one place, divide by 239; from 1,822.5 subtract if north of the equator or add if south.
166
為所求日日出分； 用減日法，為日入分。 以昏明分減日出分，為晨分； 加日入分，為昏分； 以日出分減半法，為半晝分。
This gives the day's sunrise parts; Subtract from the day divisor for sunset parts. Twilight parts minus sunrise parts give dawn parts; Add sunset parts for dusk parts; Half divisor minus sunrise parts gives half-day parts.
167
求每日晝夜刻日出入辰刻：置日出分，倍之，進一位，滿刻法為刻，不滿為分，即所求日夜刻； 以減百刻，餘為晝刻； 半夜刻，滿辰刻為辰數； 命子正，算外，即日出辰刻； 以半辰刻加之，即命起時初。
To find day/night quarters and sunrise/sunset double-hour quarters: double sunrise parts, shift one place; divide by the quarter divisor for night quarters (remainder = minutes); Subtract from 100 quarters for day quarters; Halve night quarters; divide by double-hour quarters for the double-hour count; Count from zi zheng (outside the count) for sunrise double-hour and quarters; Add half double-hour quarters to name from the start of the double-hour.
168
以晝刻加之，滿辰刻為辰數； 命日出，算外，即日入辰刻及分。
Add day quarters; divide by double-hour quarters for the count; Count from sunrise (outside the count) for sunset double-hour and quarters.
169
求每更點差刻及逐更點辰刻：置夜刻，減去十五刻，五而一，為更差； 又五而一，為點差。 以昏明刻加日入辰刻，即初更辰刻； 以更點差刻累加之，滿辰刻及分去之，各得更點所入辰刻及分。
To find watch and clepsydra-mark differential quarters: take night quarters, subtract 15, divide by 5 for watch difference; Divide by 5 again for mark difference. Add twilight quarters to sunset double-hour quarters for the first watch; Accumulate watch and mark differentials; reduce by full double-hour quarters and minutes at each step to obtain double-hour and quarter for each watch and mark.
170
求每日距中度及每更差度：置所求日黃道去赤道內、外度及分，以四千四百三十五乘之，如五千八百一十二而一為度，不滿，退除為分，以內加外減一百度七十二分、秒七為距中度。 用減一百六十四度八十一分、秒五十七，餘四因，退一位，為每更差度。
To find daily culmination distance and per-watch differential degrees: multiply ecliptic declination by 4,435 and divide by 5,812 for degrees (remainder stepped back for minutes); add if north of the equator or subtract if south from 100° 72′ 7″ for culmination distance. Subtract from 164° 81′ 57.5″; multiply the remainder by 4 and shift one place for per-watch difference degrees.
171
求昏曉五更及攢點中星：置距中度，以其日午中赤道日度加而命之，即昏中星所格宿次，命為初更中星； 以每更差度加而命之，即二更中星； 以每更差度累加之，滿赤道宿度去之，即逐更及攢點中星； 加三十六度六十二分、秒五十七，滿赤道宿度去之，即曉中星。
To find culminating stars at dusk, dawn, five watches, and clepsydra marks: add culmination distance to the day's midday equatorial solar longitude and name the lodge—the star culminating at dusk; this is the first watch culmination star; Add per-watch difference and name the lodge for the second watch culmination star; Accumulate per-watch differences, reducing by full equatorial lodge arcs at each step for each watch and mark culmination; Add 36° 62′ 57.5″ and reduce by lodge arcs for the dawn culmination star.
172
求九服晷景：各于所在測冬夏二至晷數，乃相減之，餘為二至差數。 如地在岳台南測夏至晷景在表南者，並冬夏二至晷數為二至差數。 其所求日在冬至後初限、夏至後末限者，置嶽台冬至晷景常數，以所求日嶽台午中晷景定數減之，餘以其處二至差數乘之，如嶽台二至差數一丈一尺二寸七分而一，所得，以減其處冬至晷數，即其地其日中晷定數。 所求日在夏至後初限、冬至後末限者，置所求日嶽台午中晷景定數，以嶽台夏至晷景常數減之，餘以其處二至差數乘之，如嶽台二至差數而一，所得，以加其處夏至晷數，即其地其日中晷定數。 如其處夏至景在表南者，以所得之數減其處夏至晷數，餘為其地其日中晷定數，亦在表南也。 其所得之數多於其處夏至晷數，即減去夏至晷數，餘為其地其日中晷定數，在表北也。
To find gnomon shadows across the nine domains: at each site measure solstice shadows, subtract, and take the remainder as the solstice differential. Where a site south of Yuetai shows the summer-solstice shadow south of the gnomon, add the two solstice shadows to get the solstice differential. In the winter-solstice initial / summer-solstice final interval: take the standard Yuetai winter-solstice shadow, subtract that day's fixed Yuetai noon shadow, multiply by the local solstice differential, divide by Yuetai's differential (1 zhang 1 chi 2 cun 7 fen), and subtract from the local winter-solstice shadow for that site's fixed noon shadow. In the summer-solstice initial / winter-solstice final interval: take the day's fixed Yuetai noon shadow, subtract the standard summer-solstice shadow, scale by local solstice differential over Yuetai's differential, and add to the local summer-solstice shadow. If the local summer shadow falls south of the gnomon, subtract the quotient from the local summer-solstice shadow—the remainder is the fixed noon shadow, likewise south of the gnomon. If the quotient exceeds the local summer-solstice shadow, subtract that shadow from the quotient—the remainder is the fixed noon shadow north of the gnomon.
173
求九服所在晝夜漏刻：各於所在下水漏，以定其處冬夏二至夜刻，但得一至可矣，不必須要冬夏二至。
To find day/night clepsydra quarters by region: set a clepsydra at each site and measure solstice night quarters—one solstice measurement suffices; both are not mandatory.
174
乃與五十刻相減，餘為至差刻。 置所求日黃道去赤道內外度及分，以至差刻乘之，進一位，如二百三十九而一為刻，不盡，以刻法乘之，復八而一為分，內減外加五十刻，即所求日夜刻； 減百刻，餘為晝刻。 其日日出入辰刻及更點差刻、每更點辰刻，並依嶽台術求之。
Subtract from 50 quarters; the remainder is the solstice differential in quarters. Multiply the day's ecliptic declination by solstice differential quarters, shift one place, divide by 239 for quarters (convert remainder via the quarter divisor and divide by 8 for minutes); subtract if north of the equator or add if south from 50 quarters for night quarters; Subtract from 100 quarters for day quarters. Sunrise/sunset quarters, watch and mark differentials, and per-watch quarters at each site all follow the Yuetai method.
175
步月離
Stepping Lunar Motion.
176
轉周分：二十萬八百七十三、秒九百九十。
Rotation circuit parts (zhuan zhou fen): 200,873, seconds 990.
177
轉周日：二十七、餘四千四十三、秒九百九十。
Rotation circuit days: 27 days, remainder 4,043, seconds 990.
178
朔差日：一、餘七千一百一十四、秒九千一十。
New-moon difference days: 1 day, remainder 7,114, seconds 9,010.
179
望策：一十四、餘五千五百七十九。
Full-moon stride (wang ce): 14, remainder 5,579.
180
弦策：七、餘二千七百八十九半。
Quadrature stride (xian ce): 7, remainder 2,789.5.
181
已上秒母一萬。
For the above values, the second divisor is 10,000.
182
七日：初數六千四百七十八，初約分八十九； 末數八百一十二，末約分一十一。
Seven days: initial number 6,478, initial approximate minutes 89; final number 812, final approximate minutes 11.
183
十四日：初數五千六百六十六，初約分七十八； 末數一千六百二十四，末約分二十二。
Fourteen days: initial number 5,666, initial approximate minutes 78; final number 1,624, final approximate minutes 22.
184
二十一日：初數四千八百五十四，初約分六十七； 末數二千四百三十六，末約分三十三。
Twenty-one days: initial number 4,854, initial approximate minutes 67; final number 2,436, final approximate minutes 33.
185
二十八日：初數四千四十三，初約分五十五。
Twenty-eight days: initial number 4,043, initial approximate minutes 55.
186
上弦：九十一度、分三十一、秒四十三。
First quadrature: 91°, 31′, 43″.
187
望：一百八十二度、分六十二、秒八十六。
Full moon: 182°, 62′, 86″.
188
下弦：二百七十三度、分九十四、秒二十九。
Last quadrature: 273°, 94′, 29″.
189
月平行：十三度、分三十六、秒八十七太。
Mean lunar daily motion: 13°, 36′, 87″ tai (with fractional excess).
190
已上分、秒母皆同一百。
For the above, minutes and seconds share the divisor 100.
191
求天正十一月經朔入轉：置天正十一月經朔加時積分，以轉周分及秒去之，不盡，滿日法除之為日，不滿為餘秒，命日，算外，即所求年天正十一月經朔加時入轉日及餘秒。 若以朔差日及餘秒加之，滿轉周日及餘秒去之，即次朔加時入轉日。
To find entry into rotation at the civil eleventh month's standard new moon: from accumulated added-time parts, remove full rotation-circuit parts and seconds; divide remainder by day divisor for days, the rest is remainder-seconds; count from the day (outside the count) for the year's eleventh-month new-moon rotation entry. Add new-moon difference days and remainder-seconds; subtract full rotation-circuit days and remainder-seconds for the next new moon's rotation entry day.
192
求弦望入轉：各因其月經朔加時入轉日及餘秒，以弦策累加之，去命如前，即上弦、望及下弦經日加時入轉日及餘秒。
For quadrature and full moon: from each month's new-moon rotation entry, accumulate the quadrature stride and reduce as before to obtain first quadrature, full moon, and last quadrature rotation entry days and remainder-seconds.
193
求朔弦望入轉朏朒定數：置入轉餘，以其日算外損益率乘之，如日法而一，所得，以損益其下朏朒積為定數。 其四七日下余如初數已下者，初率乘之，初數而一，以損益朏朒為定數。 如初數已上者，以初數減之，餘乘末率，末數而一，用減初率，餘加朏朒為定數。 其十四日下余如初數已上者，初數減之，餘乘末率，末數而一，為朏朒定數。
To find fixed phase/defect (fei/si) numbers at new moon, quadrature, and full moon: from rotation remainder, multiply the excess-deficit rate outside that day's count, divide by day divisor, and apply to the phase/defect accumulation for the fixed value. Within the first seven-day interval, if remainder is at or below the initial number, multiply by the initial rate, divide by the initial number, and adjust phase/defect for the fixed number. When remainder exceeds the initial number, subtract the initial number, multiply the rest by the final rate and divide by the final number; subtract that quotient from the initial rate and add the remainder to phase/defect for the fixed number. For days below fourteen, if the remainder is at or above the initial count, subtract the initial count, multiply the remainder by the final rate, divide by the final count, and take the result as the anomalistic fixed number.
194
求朔弦望定日：各置經朔、弦、望小餘，以入氣、入轉朏朒定數朏減朒加之，滿與不足，進退大餘，命己卯，算外，各得定日日辰及餘。 定朔幹名與後朔幹名同者月大，不同者月小，其月內無中氣者為閏月。 凡注曆，觀定朔小余，秋分後在日法四分之三已上者，進一日； 春分後定朔日出分差如春分之日者，三約之，用減四分之三； 定朔小餘及此數已上者，亦進一日； 或當交虧初在日入已前者，其朔不進。 弦、望定小餘不滿日出分者，退一日； 望若有食虧初在日出已前者，定望小余進滿日出分，亦進一日。 又月行九道遲疾，有三大二小； 日行盈縮累增損之，則有四大三小，理數然也。 若俯循常儀，當察加時早晚，隨其所近而進退之，使不過三大二小。
To find fixed syzygy dates: set each mean syzygy minor remainder; apply the fixed anomalistic numbers for entered qi and rotation—subtract for fast, add for slow; carry or borrow the major remainder; count from ji-mao to obtain each fixed date, time, and remainder. When the fixed new-moon stem name matches the next new moon, the month is long; when they differ, it is short; a month containing no central qi is intercalary. In calendar annotation, if the fixed new-moon minor remainder after the Autumn Equinox is three-fourths of the day divisor or more, advance the date one day; After the Spring Equinox, if the fixed new-moon sunrise-sunset difference equals that of the equinox day, divide by three and subtract from three-fourths; if the fixed new-moon minor remainder reaches this threshold or above, also advance one day; or when eclipse onset at conjunction occurs before sunset, do not advance the new moon. If the fixed minor remainder at quarter or full moon does not reach the sunrise fraction, set the date back one day; If a full-moon eclipse begins before sunrise and advancing the fixed full-moon minor remainder fills the sunrise fraction, also advance one day. 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 two short months.
195
求定朔弦望加時日所在度：置定朔、弦、望約餘，副之，以乘其日盈縮分，萬約之，所得，盈加縮減其副，滿百為分，分滿百為度，以加其日夜半日度，命之，各得其日加時日躔黃道宿次。
To find the ecliptic lodge at fixed syzygy hour-addition: set each fixed syzygy reduced remainder and keep a duplicate; multiply by that day’s equation of time, divide by ten thousand, and add or subtract from the duplicate according to surplus or deficit; carry parts into degrees and degrees into lodge positions; add to that night’s midnight solar longitude and name the lodge—each yields the ecliptic lodge at syzygy hour-addition.
196
求平交日辰：置交終日及餘秒，以其月經朔加時入交泛日及餘秒減之，余為平交入其月經朔加時後日算及餘秒，以加減其月經朔大、小餘，其大餘命己卯，算外，即平交日辰及餘秒。 求次交者，以交終日及餘秒加之，大余滿紀法去之，命如前，即次平交日辰及餘秒。
To find mean node-crossing date and time: set the nodal-period days and remainder-seconds; subtract the month’s mean new-moon hour-added general conjunction entry; the remainder is days and seconds after mean new-moon hour-addition; adjust the month’s mean new-moon major and minor remainders; count from ji-mao to obtain mean node-crossing date, time, and remainder-seconds. For the next crossing, add the nodal-period days and remainder-seconds; when the major remainder fills the era divisor, remove it and name as before to obtain the next mean node-crossing date, time, and remainder-seconds.
197
求平交入轉朏朒定數：置平交小餘，加其日夜半入轉餘，以乘其日損益率，日法而一，所得，以損益其下朏朒積為定數。
To find the mean crossing’s rotation anomalistic fixed number: set the mean crossing minor remainder; add the day’s midnight rotation entry remainder; multiply by that day’s increase-decrease rate, divide by the day divisor, and adjust the anomalistic accumulation below to obtain the fixed number.
198
求正交日辰：置平交小余，以平交入轉朏朒定數朏減朒加之，滿與不足，進退日辰，即正交日辰及餘秒； 與定朔日辰相距，即所在月日。
To find true node-crossing date and time: set the mean crossing minor remainder; apply the rotation anomalistic fixed number—subtract for fast, add for slow; carry or borrow days and seconds to obtain true node-crossing date, time, and remainder-seconds; compare its offset from the fixed new-moon date and time to determine the month and day of occurrence.
199
求經朔加時中積：各以其月經朔加時入氣日及餘，加其氣中積及餘，其日命為度，其餘以日法退除為分秒，即其月經朔加時中積度及分秒。
To find central accumulated longitude at mean new-moon hour-addition: add each month’s mean new-moon entered-qi days and remainder to its qi central accumulation; convert days to degrees and the remainder to parts and seconds by the day divisor to obtain central accumulated longitude at mean new-moon hour-addition.
200
求正交加時黃道月度：置平交入經朔加時後日算及約餘秒，以日法通日，內餘，進一位，如五千四百五十三而一為度，不滿，退除為分秒，以加其月經朔加時中積，然後以冬至加時黃道日度加而命之，即得其月正加時月離黃道宿度及分秒。 如求次交者，以交終度及分秒加而命之，即得所求。
To find ecliptic lunar longitude at true-conjunction hour-addition: set days and reduced seconds after mean new-moon hour-addition at mean crossing; convert through the day divisor, advance one place, and divide by 5453 for degrees; convert the remainder to parts and seconds; add to mean new-moon hour-added central accumulation; then add and name from the winter-solstice hour-added ecliptic solar degree to obtain true-conjunction hour-added lunar ecliptic lodge longitude. For the next crossing, add the nodal-period degrees, parts, and seconds and name the lodge to obtain the result.
201
求黃道宿積度：置正交加時黃道宿全度，以正交加時月離黃道宿度及分秒減之，餘為距後度及分秒，以黃道宿度累加之，即各得正交後黃道宿積度及分秒。
To find ecliptic lodge accumulated longitude: set the full ecliptic lodge at true-conjunction hour-addition; subtract the lunar ecliptic lodge longitude there; the remainder is post-distance longitude; accumulate by adding lodge degrees to obtain post-true-conjunction ecliptic accumulated longitude.
202
求黃道宿積度入初末限：各置黃道宿積度及分秒，滿交象度及分去之，在半交象已下為初限； 已上者，以減交象度，餘為入末限。 入交積度、交象度並在交會術中。
To determine initial or final limit for ecliptic accumulated longitude: set accumulated degrees, parts, and seconds; remove full conjunction-image intervals—at or below half the conjunction image is the initial limit; if above, subtract from the conjunction-image interval; the remainder is entry into the final limit. Entered-conjunction accumulated degrees and conjunction-image degrees are both defined in the conjunction procedure.
203
求月行九道宿度：凡月行所交，冬入陰曆，夏入陽曆，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：至所沖之宿亦如之。
To find nine-path lodge degrees: at each crossing, in winter the moon enters yin months and in summer yang months, following 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.
204
冬入陽曆，夏入陰曆，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：至所沖之宿亦如之。
In winter it enters yang months and in summer yin months, and 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.
205
春入陽曆，秋入陰曆，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：至所沖之宿亦如之。
In spring it enters yang months and in autumn yin months, and the moon follows the vermillion path. After the Spring and Autumn Equinoxes, the vermillion path’s half-intersection lies at the summer-solstice lodge, south of the ecliptic; after Start of Spring and Start of Autumn, the vermillion path’s half-intersection lies at the Start-of-Summer lodge, southwest of the ecliptic—at the opposing lodge, the same applies.
206
春入陰曆，秋入陽曆，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：至所沖之宿亦如之。
In spring it enters yin months and in autumn yang months, and 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.
207
四序離為八節，至陰陽之所交，皆與黃道相會，故月行有九道。 各以所入初、末限度及分減一百一度，餘以所入初、末限度及分乘之，半而退位為分，分滿百為度，命為月道與黃道泛差。 凡日以赤道內為陰，外為陽； 月以黃道內為陰、外為陽。 故月行正交，入夏至後宿度內為同名，入冬至後宿度內為異名。 其在同名者，置月行與黃道泛差，九因八約之，為定差。 半交後、正交前以差減，正交後、半交前以差加。 此加減出入六度，正如黃、赤道相交同名之差。 若較之漸異，則隨交所在，遷變不常。
The four seasons yield eight nodes; at each yin–yang crossing the moon meets the ecliptic—hence nine paths of lunar motion. For each, subtract the initial or final limit from 101°; multiply the remainder by that limit; halve and shift one place for parts; carry parts into degrees—this is the general lunar–ecliptic 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 at true conjunction, entry after the Summer Solstice within the path is same-named; entry after the Winter Solstice within the path is differently named. When same-named, set the general lunar–ecliptic difference; multiply by nine and reduce by eight to obtain the fixed difference. Between half-intersection and true conjunction subtract the difference; between true conjunction and half-intersection add it. This adjustment shifts latitude by six degrees—matching the same-name difference at ecliptic–equator crossing. Compared incrementally they diverge; the adjustment varies with the conjunction position and is not constant.
208
仍以正交度距秋分度數乘定差，如象限而一，所得，為月道與赤道定差，前加者為減，減者為加。 其在異名者，置月行與黃道泛差，七因八約之，為定差； 半交後、正交前以差加，正交後、半交前以差減。 此加減出入六度，異如黃赤道相交異名之差，若較之漸同，則隨交所在，遷變不常。
Multiply the fixed difference by the true-conjunction degree’s distance from the autumn equinox and divide by a quadrant to obtain the fixed lunar–equator difference; reverse prior addition and subtraction. When differently named, set the general lunar–ecliptic difference; multiply by seven and reduce by eight for the fixed difference; between half-intersection and true conjunction add the difference; between true conjunction and half-intersection subtract it. This adjustment shifts latitude by six degrees—unlike the different-name ecliptic–equator difference; compared incrementally they converge, yet the adjustment still varies with conjunction position.
209
仍以正交度距春分度數乘定差，如象限而一，所得，為月行與赤道定差，前加者為減，減者為加； 皆加減黃道宿積度，為九道宿積度； 以前宿九道積度減之，為其宿九道度及分。 其分就近約為太、半、少。 論春、夏、秋、冬，以四時日所在宿度為正。
Multiply the fixed difference by the true-conjunction degree’s distance from the spring equinox and divide by a quadrant to obtain the fixed lunar–equator difference; reverse prior addition and subtraction; add or subtract both differences from ecliptic accumulated longitude to obtain nine-path accumulated longitude; subtract the preceding lodge’s nine-path accumulated longitude to obtain that lodge’s nine-path degree and parts. Round parts to the nearest major, half, or minor fraction. For the four seasons, take the solar lodge at each season’s position as the standard.
210
求正交加時月離九道宿度：以正交加時黃道日度及分減一百一度，餘以正交度及分乘之，半而退位為分，分滿百為度，命為月道與黃道泛差。 其在同名者，置月行與黃道泛差，九因八約之，為定差，以加； 仍以正交度距秋分度數乘定差，如象限而一，所得，為月道與赤道定差，以減。 其在異名者，置月行與黃道泛差，七因八約之，為定差，以減； 仍以正交度距春分度數乘定差，如象限而一，所得，為月道與赤道定差，以加。 置正交加時黃道月度及分，以二差加減之，即正交加時月離九道宿度及分。
To find nine-path lodge longitude at true-conjunction hour-addition: subtract 101° from the hour-added ecliptic solar degree; multiply the remainder by the true-conjunction degree; halve and shift for parts; carry into degrees—this is the general lunar–ecliptic difference. When same-named, set the general lunar–ecliptic difference; multiply by nine and reduce by eight for the fixed difference, and add it; multiply the fixed difference by the true-conjunction degree’s distance from the autumn equinox, divide by a quadrant for the fixed lunar–equator difference, and subtract it. When differently named, set the general lunar–ecliptic difference; multiply by seven and reduce by eight for the fixed difference, and subtract it; multiply the fixed difference by the true-conjunction degree’s distance from the spring equinox, divide by a quadrant for the fixed lunar–equator difference, and add it. Set the hour-added ecliptic lunar degree; apply both differences to obtain the nine-path lodge longitude at true-conjunction hour-addition.
211
求定朔弦望加時月所在度：置定朔加時日躔黃道宿次，凡合朔加時，月行潛在日下，與太陽同度，是為加時月離宿次； 各以弦、望度及分秒加其所當弦、望加時日躔黃道宿度，滿宿次去之，命如前，各得定朔、弦、望加時月所在黃道宿度及分秒。
To find lunar longitude at fixed syzygy hour-addition: set the ecliptic lodge at fixed new-moon hour-added solar transit; at conjunction hour-addition the moon lies hidden beneath the sun at the same longitude—this is the hour-added lunar lodge; add each quarter- or full-moon interval to the corresponding hour-added ecliptic solar longitude; carry lodge positions and name as before to obtain ecliptic lunar longitude at each fixed syzygy hour-addition.
212
求定朔弦望加時九道月度：各以定朔、弦、望加時月離黃道宿度及分秒，加前宿正交後黃道積度，為定朔、弦、望加時正交後黃道積度。 如前求九道積度，以前宿九道積度減之，餘為定朔、弦、望加時九道月離宿度及分秒。 其合朔加時若非正交，則日在黃道、月在九道。 所入宿度雖多少不同，考其兩極，若應繩准，故云月行潛在日下，與太陽同度。
To find nine-path lunar longitude at fixed syzygy hour-addition: add each hour-added ecliptic lunar longitude to the preceding lodge’s post-true-conjunction ecliptic accumulation. As in the nine-path accumulation procedure, subtract the preceding lodge’s nine-path accumulation to obtain nine-path lunar longitude at fixed syzygy hour-addition. If syzygy hour-addition is 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.
213
求定朔午中入轉：以經朔小餘與半法相減，餘以加減經朔加時入轉，經朔小餘少，如半法加之； 多，如半法減之。
To find noon rotation entry at fixed new moon: subtract the mean new-moon minor remainder from the half divisor and use the result to adjust mean new-moon hour-added rotation entry; if the minor remainder is smaller, add toward the half divisor; if larger, subtract toward the half divisor.
214
為經朔午中入轉。 若定朔大餘有進退，亦加減轉日，否則因經為定，命日，算外，即得所求。 次月仿此求之。
This is mean new-moon noon rotation entry. If the fixed new-moon major remainder advances or retreats, adjust the rotation day likewise; otherwise take the mean value as fixed, name the day, and count outward to obtain the result. For the following month, repeat the same procedure.
215
求每日午中入轉：因定朔午中入轉日及餘秒，每日累加一日，滿轉周日及餘秒去之，命如前，即得每日午中入轉日及餘秒。
To find daily noon rotation entry: from fixed new-moon noon rotation days and seconds, add one day per day; when the rotation-cycle interval is filled, remove it and name as before to obtain each day’s noon rotation entry.
216
求晨昏月度：置其日晨分，乘其日算外轉定分，日法而一，為晨轉分； 用減轉定分，餘為昏轉分； 又以朔、弦、望定小餘乘轉定分，日法而一，為加時分； 以減晨昏轉分，為前； 不足，覆減之，餘為後； 乃前加後減加時月度，即晨、昏月所在宿度及分秒。
To find morning and evening lunar longitude: set that day’s morning fraction; multiply the day’s rotation fixed parts by it and divide by the day divisor for morning rotation parts; subtract from the rotation fixed parts to obtain evening rotation parts; multiply the fixed syzygy minor remainder by the rotation fixed parts and divide by the day divisor for hour-added parts; subtract from the morning and evening rotation parts for the before portion; if insufficient, wrap around and subtract; the remainder is the after portion; add the before portion and subtract the after portion from hour-added lunar longitude to obtain morning and evening lunar lodge positions.
217
求朔弦望晨昏定程：各以其朔昏定月減上弦昏定月，餘為朔後昏定程； 以上弦昏定月減望昏定月，餘為上弦後昏定程； 以望晨定月減下弦晨定月，餘為望後晨定程； 以下弦晨定月減後朔晨定月，餘為下弦後晨定程。
To find fixed morning-and-evening intervals at syzygies: subtract upper-quarter evening longitude from new-moon evening longitude for the post-new-moon evening interval; subtract full-moon evening longitude from upper-quarter evening longitude for the post-upper-quarter evening interval; subtract last-quarter morning longitude from full-moon morning longitude for the post-full-moon morning interval; subtract the next new-moon morning longitude from last-quarter morning longitude for the post-last-quarter morning interval.
218
求每日轉定度：累計每程相距日轉定分，與晨昏定程相減，餘以相距日數除之，為日差； 定程多為加，定程少為減。
To find daily rotation fixed degrees: accumulate daily rotation fixed parts over the interval and subtract from the fixed morning-and-evening interval; divide the remainder by intervening days for the daily difference; add when the fixed interval is larger and subtract when it is smaller.
219
以加減每日轉定分，為每日轉定度及分秒。
Apply the daily difference to each day’s rotation fixed parts to obtain daily rotation fixed degrees, parts, and seconds.
220
求每日晨昏月：因朔、弦、望晨昏月，加每日轉定度及分秒，滿宿次去之，為每日晨昏月。 凡注曆，目朔日注昏月，望後次日注晨月。
To find daily morning and evening lunar positions: from syzygy morning and evening longitudes, add daily rotation fixed degrees; carry lodge positions to obtain each day’s morning and evening moon. In calendar annotation, record the evening moon on new-moon day and the morning moon on the day after full moon.
221
已前月度以究算術之精微，如求其速要，即依後術徑求。
The preceding lunar-longitude methods exhaust computational refinement; for a quick result, use the shortcut procedure below.
222
求經朔加時平行月：各以其月經朔入氣日及余秒，其餘以日法退除為分秒。 加其氣中積日及約分，命日為度，即為經朔加時平行月積度及分秒。
To find mean lunar longitude at mean new-moon hour-addition: set each month’s mean new-moon entered-qi days and remainder-seconds; convert the remainder to parts and seconds by the day divisor. Add qi central accumulated days and reduced parts; convert days to degrees to obtain mean lunar accumulated longitude at mean new-moon hour-addition.
223
求所求日加時平行月：置所求日大余及加時小餘，以其月經朔大、小餘減之，餘為入經朔加時後日數及餘； 以其日乘月平行度及分秒，列於上位，又以其餘乘月平行度及分秒，滿日法除之為度，不滿，退除為分秒，並上位，用加經朔加時平行月，滿周天度及分秒去之，即得所求日加時平行月積度及分秒。
To find mean lunar longitude at the sought day’s hour-addition: set the sought day’s major remainder and hour-added minor remainder; subtract the month’s mean new-moon remainders to obtain days and remainder after mean new-moon hour-addition; multiply mean lunar motion by the day-count in the upper position and by the remainder; divide by the day divisor for degrees and convert the remainder to parts and seconds; add to mean new-moon hour-added mean longitude and remove full circuit intervals to obtain mean lunar accumulated longitude at the sought hour-addition.
224
求所求日加時入轉：以所求日加時入經朔加時後日數及餘，加經朔加時入轉日及余秒，滿轉周日及餘秒去之，命日，算外，即得所求。 其餘先以日法退除為分秒。
To find rotation entry at the sought hour-addition: add days and remainder after mean new-moon hour-addition to mean new-moon hour-added rotation entry; remove full rotation-cycle intervals; name the day and count outward to obtain the result. First convert the remainder to parts and seconds by the day divisor.
225
求所求日加時定月：置所求日加時入轉分，以其日算外加減差乘之，百約為分，分滿百為度，加減其下遲疾度，為遲疾定度； 乃以遲減疾加所求日加時平行月，為定月。 各以天正冬至加時黃道日度加而命之，即得所求日加時月離黃道宿度及分秒。 其入轉若在四、七日者，如求朏朒術入之。
To find true lunar longitude at the sought hour-addition: set hour-added rotation parts; multiply by the day’s increase-decrease difference outside the day-count; divide by one hundred for parts and carry into degrees; adjust the fast-slow degree below to obtain the fixed fast-slow degree; then subtract for slow and add for fast to the hour-added mean moon to obtain the true moon. Add and name from the winter-solstice hour-added ecliptic solar degree to obtain ecliptic lunar lodge longitude at the sought hour-addition. If rotation entry falls on days four or seven, apply the anomalistic procedure as above.