1
觀天曆
Calendar for Observing Heaven
2
元祐《觀天曆》
The Yuanyou Calendar for Observing Heaven
3
演紀上元甲子，距元祐七年壬申，歲積五百九十四萬四千八百八算。 上考往古，每年減一； 下驗將來，每年加二。
From upper origin jiazi to the seventh year of Yuanyou (renzhen), the accumulated count is 5,944,808 years. When projecting backward into antiquity, subtract one per year; when verifying forward into the future, add two per year.
4
步氣朔
Procedures for Solar Terms and Lunations
5
統法：一萬二千三十。
Era divisor: 12,030
6
歲周：四百三十九萬三千八百八十。
Tropical year circuit: 4,393,880 parts
7
歲餘：六萬三千八十。
Year remainder: 63,080 parts
8
氣策：一十五、餘二千六百二十八、秒一十一。
Qi divisor: 15 days, remainder 2,628, seconds 11
9
朔實：三十五萬五千二百五十三。
Synodic month dividend: 355,253 parts
10
朔策：二十九、餘六千三百八十三。
New-moon divisor: 29 days, remainder 6,383
11
望策：一十四、餘九千二百六、秒一十八。
Full-moon divisor: 14 days, remainder 9,206, seconds 18
12
弦策：七、餘四千六百三、秒九。
Quarter-moon divisor: 7 days, remainder 4,603, seconds 9
13
歲閏：一十三萬八百四十四。
Year intercalation: 130,844 parts
14
中盈分：五千二百五十六、秒二十四。
Mid-term surplus: 5,256 parts, 24 seconds
15
朔虛分：五千六百四十七。
New-moon void: 5,647 parts
16
沒限分：九千四百二。
Extinction limit: 9,402 parts
17
閏限：三十四萬四千三百四十九、秒一十二。
Intercalation limit: 344,349 parts, 12 seconds
18
旬周：七十二萬一千八百。
Ten-day circuit: 721,800 parts
19
紀法：六十。
Sexagenary cycle divisor: 60
20
以上秒母同三十六。
For all the above constants, the second denominator is 36.
21
推天正冬至：置距所求積年，以歲周乘之，為氣積分； 滿旬周去之，不盡，以統法約之為大餘，不滿為小餘。 其大餘命甲子，算外，即所求年天正冬至日辰及餘。
To find the civil-year winter solstice: set the accumulated years from the epoch to the year sought, multiply by the tropical year circuit for the qi accumulation; divide out the ten-day circuit; the remainder, reduced by the era divisor, yields the greater remainder, and what is left is the lesser remainder. Count the greater remainder from jiazi; the day after the count gives the winter solstice date, double-hour, and remainder for the year sought.
22
求次氣：置天正冬至大、小餘，以氣策及餘秒累加之，秒盈秒法從小餘一，小餘盈統法從大餘一，大余盈紀法去之。
To find the next solar term: set the winter solstice greater and lesser remainders and add the qi divisor cumulatively; carry seconds into the lesser remainder, the lesser remainder into the greater, and remove full sexagenary cycles from the greater remainder.
23
命甲子，算外，即各得次氣日辰及餘秒。
Count from jiazi; the day after the count gives each successive solar term's date, double-hour, and remainder.
24
推天正經朔：置天正冬至氣積分，以朔實去之，不盡為閏餘； 以減天正冬至氣積分，餘為天正十一月經朔加時積分； 滿旬周去之，不盡，以統法約之為大餘，不滿為小餘。 其大餘命甲子，算外，即所求年天正十一月經朔日辰及餘。
To find the civil-year mean new moon: set the winter solstice qi accumulation and divide by the synodic month dividend; the remainder is the intercalation remainder; subtract this from the winter solstice qi accumulation; the remainder is the time-of-day accumulation for the eleventh month's mean new moon; divide out the ten-day circuit; the remainder, reduced by the era divisor, yields the greater remainder, and what is left is the lesser remainder. Count the greater remainder from jiazi; the day after the count gives the eleventh month's mean new moon date, double-hour, and remainder for the year sought.
25
求弦望及次朔經日：置天正十一月經朔大、小餘，以弦策累加之，去命如前，即各得弦、望及次朔經日及餘秒。
To find quarter, full, and successive mean new moons: set the eleventh-month mean new moon remainders, add the quarter-moon divisor cumulatively, and count as before to obtain each mean date and remainder.
26
求沒日：置有沒之氣小餘，以三百六十乘之，其秒進一位，從之，用減歲周，余滿歲餘除之為日，不滿為餘。 其日，命其氣初日日辰，算外，即為其氣沒日日辰。 凡氣小餘在沒限以上者，為有沒之氣。
To find the extinction day: set the lesser remainder of the qi subject to extinction, multiply by 360 with seconds carried one place, subtract from the tropical year circuit, and divide the remainder by the year remainder for the day count. Count that many days from the qi's opening date; the day after the count is the extinction day and double-hour. Any qi whose lesser remainder reaches the extinction limit or above is subject to extinction.
27
求滅日：置有滅之朔小餘，以三十乘之，滿朔虛分除之為日，不滿為餘。 其日命其月經朔初日日辰，算外，即為其月滅日日辰。 凡經朔小餘不滿朔虛分者，為有滅之朔。
To find the annihilation day: set the lesser remainder of the affected new moon, multiply by 30, and divide by the new-moon void for the day count. Count from that month's mean new moon opening date; the day after the count is the annihilation day and double-hour. Any mean new moon whose lesser remainder falls short of the new-moon void is subject to annihilation.
28
步發斂
Procedures for Issuing and Gathering
29
候策：五、餘八百七十六、秒四。
Pentad divisor: 5 days, remainder 876, seconds 4
30
卦策：六、餘一千五十一、秒一十二。
Hexagram divisor: 6 days, remainder 1,051, seconds 12
31
土王策：三、餘五百二十五、秒二十四。
Earth-kingship divisor: 3 days, remainder 525, seconds 24
32
月閏：一萬九百三、秒二十四。
Monthly intercalation: 10,903 parts, 24 seconds
33
辰法：二千五。
Double-hour divisor: 2,500
34
半辰法：一千二半。
Half double-hour divisor: 1,250
35
刻法：一千三百三。
Quarter-hour divisor: 1,303
36
秒母：三十六。
Second denominator: 36
37
推七十二候：各因中節大、小餘命之，為初候； 以候策加之，為次候； 又加之，為末候。
To find the seventy-two climatic pentads: for each mid-term, count from its greater and lesser remainders for the first pentad; add the pentad divisor for the second pentad; add again for the third pentad.
38
求六十四卦：各因中氣大、小餘命之，為初卦用事日； 以卦策加之，為中卦用事日； 又加之，得終卦用事日。 以土王策加諸侯內卦，得十有二節之初外卦用事日； 又加之，得大夫卦用事日； 復以卦策加之，得卿卦用事日。
To find the sixty-four hexagrams: for each mid-qi, count from its remainders for the first hexagram's effective day; add the hexagram divisor for the middle hexagram's effective day; add again for the final hexagram's effective day. Add the earth-kingship divisor to the feudal lord's inner hexagram for the outer hexagram's effective day at each of the twelve terms; add again for the grandee hexagram's effective day; add the hexagram divisor again for the minister hexagram's effective day.
39
推五行用事：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，命甲子，算外，為其月土始用事日。
To find when the five phases take effect: count from the four establishment terms' remainders for the opening days of spring wood, summer fire, autumn metal, and winter water; Subtract the earth-kingship divisor from each season's mid-qi remainders, count from jiazi, and the day after the count is that month's first earth-kingship day.
40
求中氣去經朔：置天正冬至閏餘，以月閏累加之，滿統法約之為日，不盡為餘，即各得每月中氣去經朔日及餘秒。 其閏余滿閏限者，為月內有閏也； 仍定其朔內無中氣者為閏月。
To find each mid-qi's offset from the mean new moon: set the winter solstice intercalation remainder, add the monthly intercalation cumulatively, reduce by the era divisor for days, and obtain each month's mid-qi offset in days and seconds. When the intercalation remainder reaches the intercalation limit, the month contains an intercalary day; and designate as an intercalary month any month whose lunation contains no mid-qi.
41
求卦候去經朔：以卦、候策累加減中氣，去經朔日及餘，中氣前，減； 中氣後，加。
To find hexagram and pentad offsets from the mean new moon: cumulatively add or subtract the hexagram and pentad divisors from the mid-qi; subtract before the mid-qi; add after the mid-qi.
42
即各得卦、候去經朔日及餘秒。
Thus obtaining each hexagram and pentad's offset from the mean new moon in days and seconds.
43
求發斂加時：倍所求小餘，以辰法除之為辰數，不滿，五因之，滿刻法為刻，不滿為餘。 其辰數命子正，算外，即各得所求加時辰、刻及分。
To find the time-of-day: double the lesser remainder, divide by the double-hour divisor for the double-hour count, multiply the remainder by five, and divide by the quarter divisor for quarters. Count the double-hours from midnight zi; the count gives each sought time's double-hour, quarter, and fractional parts.
44
步日躔
Procedures for Daily Solar Motion
45
周天分：四百三十九萬四千三十四、秒五十七。
Circumference parts: 4,394,034, seconds 57
46
周天度：三百六十五、餘三千八十四、秒五十七。
Circumference in degrees: 365, remainder 3,084, seconds 57
47
歲差：一百五十四、秒五十七。
Precession: 154 parts, 57 seconds
48
二至限日：一百八十二、餘七千四百八十。
Solstice limit: 182 days, remainder 7,480
49
冬至後盈初夏至後縮末限日：八十八、余一萬九百五十八。
Limit from post-winter-solstice surplus opening to post-summer-solstice contraction closing: 88 days, remainder 10,958
50
夏至後縮初冬至後盈末限日：九十三、餘八千五百五十二。
Limit from post-summer-solstice contraction opening to post-winter-solstice surplus closing: 93 days, remainder 8,552
51
求每日盈縮分：置入二至後全日，各在初限已下為初限； 已上，用減二至限，餘為末限。 列初、末限日及分於上，倍初、末限日及約分於下，相減相乘。 求盈縮分者，在盈初、縮末，以三千二百九十四除之。 在盈末、縮初，以三千六百五十九除之，皆為度，不滿，退除為分秒。 求朏朒積者，各退二位，在盈初縮末，以三百六十六而一； 在盈末縮初，以四百七而一，各得所求。 以盈縮相減，餘為升降分； 盈初縮末為升，縮初盈末為降。
To find each day's equation parts: set the full days after a solstice; days within the opening limit count as the opening limit; days above that, subtract from the solstice limit for the closing limit. Place the opening and closing limit days and parts above, double them below, and subtract and multiply. For the equation parts, in the surplus opening and contraction closing, divide by 3,294. In the surplus closing and contraction opening, divide by 3,659; reduce to degrees, parts, and seconds. For the phase accumulation, shift two places; in the surplus opening and contraction closing, divide by 366; in the surplus closing and contraction opening, divide by 407, to obtain the result. Subtract the two equation values; the remainder is the ascension-descension parts; surplus opening and contraction closing mean ascension; contraction opening and surplus closing mean descension.
52
以朏朒積相減，餘為損益率。 在初為益，在末為損。
Subtract the phase accumulations; the remainder is the correction rate. At the opening it increases; at the closing it decreases.
53
求經朔弦望入盈縮限：置天正閏日及餘，減縮末限日及餘，為天正十一月經朔入縮末限日及餘； 以弦策累加之，滿盈縮限日去之，即各得弦望及次朔入盈縮限日及餘秒。
To find mean conjunction and quarter-moon entry into the equation limit: set the civil-year intercalary day and remainder, subtract the contraction-closing limit, for the eleventh month's mean new moon entry; add the quarter-moon divisor cumulatively, remove full equation limits, and obtain each quarter, full, and new moon's entry day and remainder.
54
求經朔弦望朏朒定數：各置所入盈縮限日小餘，以其日下損益率乘之，如統法而一，所得，損益其下朏朒積為定數。
To find the true phase numbers for mean conjunctions and quarters: set each entry day's lesser remainder, multiply by that day's correction rate, divide by the era divisor, and apply to the phase accumulation below for the fixed value.
55
求定氣：冬夏二至以常氣為定氣。 自後，以其氣限日下盈縮分盈加縮減常氣約餘，即為所求之氣定日及分秒。
To find true solar terms: the winter and summer solstices use the mean terms as true terms. Thereafter, add surplus or subtract contraction from each term's mean remainder using the equation parts below its limit day to obtain the true term's date and seconds.
56
赤道宿度
Equatorial Lodge Degrees
57
鬥：二十六
Dipper: 26 degrees
58
牛：八
Ox: 8 degrees
59
女：十二
Girl: 12 degrees
60
虛：十少秒六十四
Void: 10 and a fraction, 64 seconds
61
危：十七
Rooftop: 17 degrees
62
室：十六
Encampment: 16 degrees
63
壁：九
Wall: 9 degrees
64
北方七宿九十八度少、秒六十四。
The seven northern lodges: 98 and a fraction degrees, 64 seconds
65
奎：十六
Striding: 16 degrees
66
婁：十二
Bond: 12 degrees
67
胃：十四
Stomach: 14 degrees
68
昴：十一
Hairy Head: 11 degrees
69
畢：十七
Net: 17 degrees
70
觜：一
Turtle Beak: 1 degree
71
參：十
Three Stars: 10 degrees
72
西方七宿八十一度。
The seven western lodges: 81 degrees.
73
井：三十三
Well: 33 degrees
74
鬼：三
Ghost: 3 degrees
75
柳：十五
Willow: 15 degrees
76
星：七
Star: 7 degrees
77
張：十八
Extended Net: 18 degrees
78
翼：十八
Wings: 18 degrees
79
軫：十七
Chariot Shaft: 17 degrees
80
南方七宿一百一十一度。
The seven southern lodges: 111 degrees.
81
角：十二
Horn: 12 degrees
82
亢：九
Neck: 9 degrees
83
氐：十五。
Root: 15 degrees.
84
房：五
Room: 5 degrees
85
心：五
Heart: 5 degrees
86
尾：十八
Tail: 18 degrees
87
箕：十一
Winnowing Basket: 11 degrees
88
東方七宿七十五度。
The seven eastern lodges: 75 degrees.
89
前皆赤道宿度，與古不同。 自《大衍曆》依渾儀測為定，用紘帶天中，儀極攸憑，以格黃道。
The foregoing are all equatorial lodge degrees, which differ from the ancient values. Since the Dayan Calendar they were fixed by armillary-sphere measurement, using the celestial girdle at heaven's center and the instrument's pole to lay out the ecliptic.
90
推天正冬至加時赤道日度：以歲差乘所求積年，滿周天分去之，不盡，用減周天分，餘以統法除之為度，不滿為餘。 命起赤道虛宿四度外去之，至不滿宿，即為所求年天正冬至加時赤道日度及餘秒。
To find the equatorial solar degree at the civil-year winter solstice: multiply the years sought by precession, divide out the circumference parts, subtract the remainder from the circumference parts, and reduce the result by the era divisor for degrees and remainder. Count from four degrees outside equatorial Void and remove lodge spans; the remainder gives the winter solstice equatorial solar degree and seconds for the year sought.
91
求夏至赤道日度：置天正冬至加時赤道日度，以二至限及余加之，滿赤道宿次去之，即得夏至加時赤道日度及餘秒。 因求後昏後夜半赤道日度者，以二至小餘減統法，餘以加二至赤道日度之餘，即二至初日昏後夜半赤道日度，以每日累加一度，去命如前，各得所求。
To find the summer solstice equatorial solar degree: set the winter solstice equatorial degree, add the solstice limit and remainder, remove full lodge spans, and obtain the summer solstice equatorial degree and seconds. To find post-dusk and midnight equatorial degrees thereafter: subtract the solstice lesser remainder from the era divisor, add the remainder to the solstice equatorial remainder for the first day's post-dusk and midnight degree, add one degree daily, and count as before for each value sought.
92
求二十八宿赤道積度：置二至加時日躔赤道全度，以二至加時赤道日度及約分減之，餘為距後度。 以赤道宿次累加之，即得二十八宿赤道積度及分秒。
To find the twenty-eight lodges' accumulated equatorial degrees: set the full equatorial solar motion at the solstice with time-of-day, subtract the solstice equatorial degree and reduced parts; the remainder is the post-solstice offset. Add equatorial lodge spans cumulatively to obtain each lodge's accumulated equatorial degrees and seconds.
93
求二十八宿赤道積度入初末限：各置赤道積度及分秒，滿象限九十一度三十一分、秒九即去之，若在四十五度六十五分、秒五十四半已下為初限； 已上，用減象限，餘為末限。
To find entry into opening and closing limits: set each lodge's accumulated equatorial degrees; remove full image limits of 91°31′9″; degrees at or below 45°65′54.5″ count as the opening limit; degrees above that, subtract from the image limit for the closing limit.
94
求二十八宿黃道度：各置赤道宿入初、末限度及分，三之，為限分。 用減四百，餘以限分乘之，一萬二千而一為度，命曰黃赤道差。 至後以減、分後以加赤道宿積度，為黃道積度； 以前宿黃道積度減之，余為二十八宿黃道度及分。 其分就近約為太、半、少，若二至之宿不足減者，即加二至限，然後減之，餘依術算。
To find the twenty-eight lodges' ecliptic degrees: set each lodge's opening or closing limit degrees and parts and treble them for limit parts. Subtract from 400, multiply by the limit parts, divide by 12,000 for degrees, and call this the ecliptic-equator difference. After the solstice subtract and after the equinox add to the equatorial accumulation for the ecliptic accumulation; subtract the preceding lodge's ecliptic accumulation; the remainder is that lodge's ecliptic degrees and parts. Reduce fractional parts to great, half, or small; if a solstice lodge cannot be subtracted, add the solstice limit first, then subtract, and proceed by the formula.
95
黃道宿度
Ecliptic Lodge Degrees
96
鬥：二十三半
Dipper: 23 and a half degrees
97
牛：七半
Ox: 7 and a half degrees
98
女：十一半
Girl: 11 and a half degrees
99
虛：十少秒六十四。
Void: 10 and a small fraction, 64 seconds.
100
危：十七太
Rooftop: 17 and a great fraction
101
室：十七少
Encampment: 17 and a small fraction
102
壁：九太
Wall: 9 and a great fraction
103
北方七宿九十七度半、秒六十四。
The seven northern lodges: 97 and a half degrees, 64 seconds.
104
奎：十七太
Striding: 17 and a great fraction
105
婁：十二太
Bond: 12 and a great fraction
106
胃：十四半
Stomach: 14 and a half degrees
107
昴：十一太
Hairy Head: 11 and a great fraction
108
畢：十六
Net: 16 degrees
109
觜：一
Turtle Beak: 1 degree
110
參：九少
Three Stars: 9 and a small fraction
111
西方七宿八十二度。
The seven western lodges: 82 degrees.
112
井：三十
Well: 30 degrees
113
鬼：二太
Ghost: 2 and a great fraction
114
柳：十四少
Willow: 14 and a small fraction
115
星：七
Star: 7 degrees
116
張：十八太
Extended Net: 18 and a great fraction
117
翼：十九半
Wings: 19 and a half degrees
118
軫：十八太
Chariot Shaft: 18 and a great fraction
119
南方七宿一百一十一度。
The seven southern lodges: 111 degrees.
120
角：十三
Horn: 13 degrees
121
亢：九半
Neck: 9 and a half degrees
122
氐：十五半
Root: 15 and a half degrees
123
房：五
Room: 5 degrees
124
心：四太
Heart: 4 and a great fraction
125
尾：十七
Tail: 17 degrees
126
箕：十
Winnowing Basket: 10 degrees
127
東方七宿七十四度太。
The seven eastern lodges: 74 and a great fraction degrees.
128
前黃道宿度，乃依今曆歲差變定。 若上考往古，下驗將來，當據歲差，每移一度，依曆推變，然後可步七曜，知其所在。
The foregoing ecliptic lodge degrees were fixed by adjusting for the present calendar's precession. To project backward or verify forward, one must rely on precession: for each degree of shift, derive the change by the calendar before stepping the seven luminaries to know their positions.
129
求天正冬至加時黃道日度：置天正冬至加時赤道日度及約分，三之，為限分； 用減四百，餘以限分乘之，一萬二千而一為度，命曰黃赤道差； 用減天正冬至加時赤道日度及分，即為所求年天正冬至加時黃道日度及分。 夏至日度，准此求之。
To find the ecliptic solar degree at the civil-year winter solstice: set the equatorial solar degree and reduced parts and treble them for limit parts; subtract from 400, multiply by the limit parts, divide by 12,000 for degrees, and call this the ecliptic-equator difference; subtract this from the winter solstice equatorial degree and parts to obtain the sought year's winter solstice ecliptic degree and parts. Find the summer solstice solar degree by the same method.
130
求二至初日晨前夜半黃道日度：置一萬分，以其日升降分升加降減之，以乘二至小餘，如統法而一，所得，以減二至加時黃道日度，餘為二至初日晨前夜半黃道日度及分。
To find the ecliptic degree before dawn and at midnight on a solstice's first day: set 10,000 parts, apply that day's ascension-descension parts, multiply by the solstice lesser remainder and divide by the era divisor, subtract from the solstice ecliptic degree, and the remainder is the predawn and midnight ecliptic degree.
131
求每日晨前夜半黃道日度：置二至初日晨前夜半黃道日度及分，每日加一度，百約其日下升降分，升加降減之，滿黃道宿次去之，即各得二至後每日晨前夜半黃道日度及分。
To find each day's predawn and midnight ecliptic degree: set the solstice first-day value, add one degree daily with ascension added or descension subtracted per the day's rate, remove full ecliptic lodge spans, and obtain each post-solstice predawn and midnight degree.
132
求太陽過宮日時刻：置黃道過宮宿度，以其日晨前夜半黃道宿度及分減之，餘以統法乘之，如其太陽行分而一，為加時小餘； 如發斂求之，即得太陽過宮日、時、刻及分。
To find when the sun passes a lodge: set the passage lodge degree, subtract that day's predawn and midnight ecliptic lodge degree, multiply by the era divisor and divide by the sun's daily motion for the time-of-day lesser remainder; derive the time by the issuing-and-gathering procedure to obtain the sun's lodge-passage day, time, quarter, and parts.
133
黃道過宮太史局吳澤等補治有此一段，開封進士吳時舉、國學進士程憙、常州百姓張文進本並無之。
The section on ecliptic lodge passage was supplemented by Wu Ze and others of the Astronomical Bureau; the versions of Wu Shiju of Kaifeng, Cheng Xi of the National University, and Zhang Wenjin of Changzhou all lack it.
134
危宿十五度少，入衛之分，亥。
Rooftop: 15 and a small fraction degrees, entering Wei's allotment, hai.
135
奎宿三度半，入魯之分，戌。
Striding: 3 and a half degrees, entering Lu's allotment, xu.
136
胃宿五度半，入趙之分，酉。
Stomach: 5 and a half degrees, entering Zhao's allotment, you.
137
畢宿十度半，入晉之分，申。
Net: 10 and a half degrees, entering Jin's allotment, shen.
138
井宿十二度，入秦之分，未。
Well: 12 degrees, entering Qin's allotment, wei.
139
柳宿七度半，入周之分，午。
Willow: 7 and a half degrees, entering Zhou's allotment, wu.
140
張宿十七度少，入楚之分，巳。
Extended Net: 17 and a small fraction degrees, entering Chu's allotment, si.
141
軫宿十二度，入鄭之分，辰。
Chariot Shaft: 12 degrees, entering Zheng's allotment, chen.
142
氐宿三度少，入宋之分，卯。
Root: 3 and a small fraction degrees, entering Song's allotment, mao.
143
尾宿八度，入燕之分，寅。
Tail: 8 degrees, entering Yan's allotment, yin.
144
鬥宿九度，入吳之分，醜。
Dipper: 9 degrees, entering Wu's allotment, chou.
145
女宿六度少，入齊之分，子。
Girl: 6 and a small fraction degrees, entering Qi's allotment, zi.
146
步月離
Procedures for Lunar Motion
147
轉周分：三十三萬一千四百八十二、秒三百八十九。
Rotation circuit: 331,482 parts, 389 seconds
148
轉周日：二十七、餘六千六百七十二、秒三百八十九。
Rotation circuit: 27 days, remainder 6,672, seconds 389
149
朔差日：一、余一萬一千七百四十、秒九千六百一十一。
New-moon difference: 1 day, remainder 11,740, seconds 9,611
150
弦策：七、餘四千六百三、秒二千五百。
Quarter-moon divisor: 7 days, remainder 4,603, seconds 2,500
151
望策：一十四、餘九千二百六、秒五千。
Full-moon divisor: 14 days, remainder 9,206, seconds 5,000
152
以上秒母同一萬。
For all the above constants, the second denominator is 10,000.
153
七日：初數一萬六百九十，初約八十九； 末數一千三百四十，末約一十一。
Day seven: initial count 10,690; initial reduced, 89; Final count 1,340; final reduced, 11.
154
十四日：初數九千三百五十一，初約七十八； 末數二千六百七十九，末約二十二。
Day fourteen: initial count 9,351; initial reduced, 78; Final count 2,679; final reduced, 22.
155
二十一日：初數八千一十一，初約六十七； 末數四千一十九，末約三十三。
Day twenty-one: initial count 8,011; initial reduced, 67; Final count 4,019; final reduced, 33.
156
二十八日：初數六千六百七十二，初約五十五。
Day twenty-eight: initial count 6,672; initial reduced, 55.
157
上弦：九十一度三十一分、秒四十一。
First quarter: 91 degrees, 31 parts, 41 seconds.
158
望：一百八十二度六十二分、秒八十二。
Full moon: 182 degrees, 62 parts, 82 seconds.
159
下弦：二百七十三度九十四分、秒二十三。
Last quarter: 273 degrees, 94 parts, 23 seconds.
160
平行：一十三度三十六分、秒八十七半。
Mean motion: 13 degrees, 36 parts, 87 and a half seconds.
161
以上秒母同一百。
For all the above constants, the second denominator is 100.
162
求天正十一月經朔加時入轉：置天正十一月經朔加時積分，以轉周分秒去之，不盡，以統法約之為日，不滿為餘。 命日，算外，即得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，即次朔加時入轉日及餘秒。 各以其月經朔小餘減之，餘為其月經朔夜半入轉。
To find entry into rotation at the civil-year eleventh month's mean-new-moon time of occurrence: set the accumulated parts for that mean new moon, cast out rotation-circuit parts and seconds; reduce the remainder by the era divisor for days, with the unfilled portion as remainder. Count the days from the stem; outside the count yields the sought year's eleventh-month mean-new-moon time-of-occurrence entry into rotation, with its day and remainder-seconds. Add the new-moon difference day and remainder-seconds; cast out full rotation-circuit days and remainder-seconds to obtain the next new moon's time-of-occurrence entry into rotation. For each month subtract that month's mean-new-moon minor remainder; the remainder is that month's mean-new-moon midnight entry into rotation.
163
求弦望入轉：因天正十一月經朔加時入轉日及餘秒，以弦策累加之，去命如前，即得弦、望入轉日及餘秒。 求朔弦望入轉朏朒定數：置入轉餘，乘其日算外損益率，如統法而一，所得，以損益其下朏朒積為定數。 其在四七日下余如初數已下，初率乘之，初數而一，以損益其下朏朒積為定數。 若初數已上者，以初數減之，餘乘末率，末數而一，用減初率，餘加其日下朏朒積為定數。 其十四日下余若在初數已上者，初數減之，餘乘末率，末數而一，便為朏定數。
To find quarter- and full-moon entry into rotation: from the eleventh month's mean-new-moon entry, add the quarter-moon divisor cumulatively; cast out and name as before for each quarter and full moon. To find the true waxing-waning numbers for mean conjunction and quarters entering rotation: set the rotation remainder, multiply by that day's correction rate beyond the count, divide by the era divisor, and apply to the phase accumulation below. On the fourth and seventh days, if the remainder is at or below the initial count, multiply by the initial rate and divide by the initial count, then adjust the phase accumulation below. If at or above the initial count, subtract the initial count; multiply the remainder by the final rate and divide by the final count; subtract from the initial rate and add the remainder to that day's phase accumulation. On the fourteenth day, if the remainder is at or above the initial count, subtract the initial count; multiply the remainder by the final rate and divide by the final count—that is the waxing fixed number.
164
求朔弦望定日、各以入限、入轉朏朒定數，朏減朒加經朔、弦、望小余，滿若不足，進退大餘，命甲子，算外。 各得定日及餘。 若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，秋分後在統法四分之三已上者，進一日； 若春分後定朔晨昏差如春分之日者，三約之，用減四分之三； 定朔小餘在此數已上者，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； 望若有交，虧初在日出分已前者，其定望小余雖滿日出分，亦退一日。 又有月行九道遲疾，曆有三大二小者； 依盈縮累增損之，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 true new-moon, quarter, and full-moon days: apply the equation-limit and rotation waxing-waning fixed numbers—subtract waxing and add waning from each mean minor remainder; on fullness or shortfall advance or retreat the major remainder and name from jiazi. Each yields the true day and remainder. If the true new moon's stem matches the next new moon's stem, the month is long; if they differ, it is short; a month with no mid-term inside it is intercalary. When publishing the calendar, inspect the true-new-moon minor remainder: after the autumn equinox, if it reaches three-quarters of the era divisor or more, advance one day. After the spring equinox, if the true new moon's dawn-dusk difference equals that of the equinox day, divide by three and subtract from three-quarters of the era divisor. If the true-new-moon minor remainder reaches this threshold or above, advance one day as well. Or if at a crossing the eclipse begins before sunset, do not advance the new moon. For quarters and full moon, if the true minor remainder falls short of the sunrise fraction, retreat one day. At full moon, if there is a crossing and the eclipse begins before the sunrise fraction, retreat one day even when the true full-moon minor remainder exceeds the sunrise fraction. Further, the moon's nine-path motion has slow and fast phases, and the calendar may show three long and two short months. If one accumulates adjustments by surplus and contraction, four long and three short months can appear—such is the arithmetic. If one merely follows ordinary practice, inspect whether the time of occurrence is early or late and advance or retreat toward the nearer case, keeping within three long and two short months.
165
求定朔弦望加時日度：置定朔、弦、望約分，副之，以乘其日升降分，一萬約之，所得，升加降減其副，以加其日夜半日度，命如前，各得定朔、弦、望加時日躔黃道宿度及分秒。
To find the solar degree at the true new moon, quarter, and full moon times of occurrence: set each reduced remainder aside, multiply by that day's ascension-descension parts and reduce by 10,000; ascension adds and descension subtracts from the duplicate; add to that day's midnight solar degree and name as before to obtain each true time-of-occurrence solar position on the ecliptic in lodge degrees and seconds.
166
求月行九道：凡合朔初交，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，出黃道東； 立冬、立夏後，青道半交在立春之宿，出黃道東南：至所沖之宿亦如之。
To find the moon's nine-path motion: at the initial crossing of conjunction, when winter enters the yin sequence and summer the yang sequence, the moon runs the green path. After winter and summer solstice, the green path's half-crossing stands at the spring-equinox lodges, east of the ecliptic. After Start of Winter and Start of Summer, the green path's half-crossing lies at the Start-of-Spring lodges, southeast of the ecliptic; the same holds at the opposite lodges.
167
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，出黃道西； 立冬、立夏後，白道半交在立秋之宿，出黃道西北； 至所沖之宿亦如之。
When winter enters the yang sequence and summer the yin sequence, the moon runs the white path. After winter and summer solstice, the white path's half-crossing stands at the autumn-equinox lodges, west of the ecliptic. After Start of Winter and Start of Summer, the white path's half-crossing lies at the Start-of-Autumn lodges, northwest of the ecliptic. The same holds at the opposite lodges.
168
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，出黃道南； 立夏、立秋後，朱道半交在立夏之宿，出黃道西南：至所沖之宿亦如之。
When spring enters the yang sequence and autumn the yin sequence, the moon runs the vermillion path. After the spring and autumn equinoxes, the vermillion path's half-crossing stands at the summer-solstice lodges, south of the ecliptic. After Start of Summer and Start of Autumn, the vermillion path's half-crossing lies at the Start-of-Summer lodges, southwest of the ecliptic; the same holds at the opposite lodges.
169
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，出黃道北； 立春、立秋後，黑道半交在立冬之宿，出黃道東北：至所沖之宿亦如之。
When spring enters the yin sequence and autumn the yang sequence, the moon runs the black path. After the spring and autumn equinoxes, the black path's half-crossing stands at the winter-solstice lodges, north of the ecliptic. After Start of Spring and Start of Autumn, the black path's half-crossing lies at the Start-of-Winter lodges, northeast of the ecliptic; the same holds at the opposite lodges.
170
四序離為八節，至陰陽之所交，皆與黃道相會，故月行有九道。 各視月行所入正交積度，滿交象去之，入交積度及交象度，並在交會術中。
The four seasons divide into eight nodes; where yin and yang meet they all intersect the ecliptic—hence the moon's motion comprises nine paths. For each case observe the accumulated degrees of the moon's entry into the true crossing; cast out full crossing images; entry-crossing accumulation and crossing-image degrees are both treated in the conjunction procedures.
171
若在半交象已下為初限； 已上，覆減交象，餘為末限。 置初、末限度及分，三之，為限分； 用減四百，餘以限分乘之，二萬四千而一為度，命曰月道與黃道差數。 距正交後、半交前，以差數加； 距半交後、正交前，以差數減。 此加減出入黃道六度，單與黃道相校之數，若校赤道，則隨氣遷變不常。
At or below the half-crossing image is the opening limit. Above that, invert-subtract from the crossing image; the remainder is the closing limit. Set the opening and closing limit degrees and parts, treble them for limit parts. Subtract from 400; multiply the remainder by the limit parts and divide by 24,000 for degrees—this is the lunar-path and ecliptic difference number. From after the true crossing to before the half-crossing, add the difference number. From after the half-crossing to before the true crossing, subtract the difference number. This addition and subtraction in and out of the ecliptic totals six degrees—the figure compared with the ecliptic alone; compared with the equator it shifts with the solar terms and is not constant.
172
仍計去冬、夏二至已來度數，乘差數，如九十而一，為月道與赤道差數。 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； 入春分交後行陽曆，秋分交後行陰曆，皆為異名。
Still compute the degrees elapsed since the preceding winter and summer solstices, multiply by the difference number, and divide by 90 for the lunar-path and equator difference number. In general the sun treats what lies inside the equator as yin and outside as yang. The moon treats what lies inside the ecliptic as yin and outside as yang. Hence when the moon's lodge motion after the spring-equinox crossing runs the yin sequence, and after the autumn-equinox crossing runs the yang sequence, both are same-name. After the spring-equinox crossing running the yang sequence, and after the autumn-equinox crossing the yin sequence, both are different-name.
173
其在同名者，以差數加者加之，減者減之； 其在異名者，以差數加者減之，減者加之。 二差皆增益黃道宿積度，為九道宿積度； 以前宿九道積度減之，為其宿九道度及分秒。 其分就近約之為太、半、少。
When same-name, add where the difference adds and subtract where it subtracts. When different-name, subtract where the difference adds and add where it subtracts. Both differences together augment the ecliptic lodge accumulated degrees to become nine-path lodge accumulated degrees. Subtract the previous lodge's nine-path accumulation to obtain that lodge's nine-path degree and parts-seconds. Round the parts nearby to great, half, and small.
174
求月行九道平交入氣：各以其月閏日及餘，加經朔加時入交泛日及餘秒，盈交終日及餘秒去之，乃減交終日及餘秒。 即各得平交入其月中氣日及餘秒； 若滿氣策即去之，余為平交入後月節氣日及餘秒。 若求朏朒定數，如求朔、望朏朒術入之，即得所求。
To find mean-crossing entry into qi for the nine paths: for each month add that month's intercalary day and remainder to the mean-new-moon generalized entry-crossing day and remainder-seconds; cast out full crossing-terminus days and remainder-seconds, then subtract the crossing terminus. Each yields mean-crossing entry into that month's mid-term day and remainder-seconds. If it fills the qi stride, remove it; the remainder is mean-crossing entry into the following month's solar node. To find the true waxing-waning number, apply the mean-new-moon and full-moon waxing-waning procedure to obtain the result.
175
求平交入轉朏朒定數：置所入氣餘，加其日夜半入轉餘，乘其日算外損益率，如統法而一，所得，以損益其下朏朒積，乃以交率乘之，交數而一，為定數。
To find the mean-crossing rotation waxing-waning fixed number: set the entered qi remainder, add that day's midnight rotation remainder, multiply by that day's correction rate beyond the count and divide by the era divisor; adjust the phase accumulation below, then multiply by the crossing rate and divide by the crossing number.
176
求正交入氣：以平交入氣、入轉朏朒定數，朏減朒加平交入氣余，滿若不足，進退其日，即正交入氣日及餘秒。
To find true-crossing entry into qi: apply the mean-crossing entry and rotation waxing-waning fixed number—subtract waxing and add waning from the mean-crossing qi remainder; advance or retreat the day on fullness or shortfall.
177
求正交加時黃道日度：置正交入氣餘，副之，以乘其日升降分，一萬約之，升加降減其副，乃以一百乘之，如統法而一，以加其日夜半日度，即正交加時黃道日度及分秒。
To find the ecliptic solar degree at the true-crossing time of occurrence: set the true-crossing qi remainder aside, multiply by that day's ascension-descension parts and reduce by 10,000; ascension adds and descension subtracts; multiply by 100, divide by the era divisor, and add to that day's midnight solar degree.
178
求正交加時月離九道宿度：置正交度加時黃道日及分，三之，為限分。 用減四百，餘以限分乘之，二萬四千而一，命曰月道與黃道差數。 以加黃道宿度，仍計去冬、夏二至已來度數，以乘差數，如九十而一，為月道與赤道差數。 同名以加，異名以減，二差皆增損正交度，即正交加時月離九道宿度及分秒。
To find the moon's nine-path lodge at the true-crossing time of occurrence: set the true-crossing added-time ecliptic solar day and parts, treble for limit parts. Subtract from 400; multiply the remainder by the limit parts and divide by 24,000—this is the lunar-path and ecliptic difference number. Add to the ecliptic lodge degree; still compute degrees elapsed since the preceding winter and summer solstices, multiply by the difference number, and divide by 90 for the lunar-path and equator difference. Same-name adds and different-name subtracts; both differences adjust the true-crossing degree to yield the moon's nine-path lodge at the true-crossing time of occurrence.
179
求定朔弦望加時月離黃道宿度：置定朔、弦、望加時日躔黃道宿度及分，凡合朔加時，月行潛在日下，與太陽同度，是為加時月度。 各以弦、望度加其所當日度，滿黃道宿次去之，即各得定朔、弦、望加時月離黃道宿度及分秒。
To find the moon's ecliptic lodge at the true new moon, quarter, and full moon times of occurrence: set each solar ecliptic position; at conjunction the moon moves hidden beneath the sun at the same degree—that is the time-of-occurrence lunar degree. For each, add the quarter or full-moon arc to the solar degree for that day and remove by ecliptic lodge sequence to obtain each true time-of-occurrence lunar ecliptic position.
180
求定朔弦望加時月離九道宿度：置定朔、弦、望加時月離黃道宿度及分秒，加前宿正交後黃道積度，如前求九道術入之，以前定宿正交後九道積度減之，餘為定朔、弦、望加時月離九道宿度及分秒。 凡合朔加時，若非正交，即日在黃道、月在九道所入宿度。 雖多少不同，考其去極，若應繩准，故曰加時九道。
To find the moon's nine-path lodge at the true new moon, quarter, and full moon times of occurrence: set each true ecliptic lunar position, add the ecliptic accumulation after the true crossing for the preceding lodge, apply the nine-path procedure as before, and subtract the preceding fixed lodge's nine-path accumulation. At conjunction, if it is not a true crossing, the sun lies on the ecliptic and the moon on the nine paths at the lodges entered—though the amounts differ. Though the figures differ, when their distance from the pole is tested they align like plumb line and square—hence the name time-of-occurrence nine paths.
181
求定朔午中入轉：各視經朔夜半入轉日及餘秒，以半法加之，若定朔及餘有進退者，亦進退轉日，否則因經為定。 因求次日，累加一日，滿轉周日及餘秒去之，即每日午中入轉。
To find true-new-moon noon entry into rotation: for each month take the mean-new-moon midnight rotation entry and add the half-divisor; if the true new moon advances or retreats, adjust the rotation day likewise; otherwise follow the mean. For the following days add one day cumulatively; cast out full rotation-circuit days and remainder-seconds for each day's noon entry into rotation.
182
求晨昏月度：以晨分乘其日算外轉定分，如統法而一，為晨轉分； 用減轉定分，餘為昏轉分； 乃以朔、弦、望小餘乘其日算外轉定分，如統法而一，為加時分； 以減晨昏轉分，餘為前； 不足減者，覆減之，餘為後； 以前加後減定朔、弦、望月度，即晨、昏月所在度。
To find dawn and dusk lunar degrees: multiply the dawn fraction by that day's rotation fixed parts beyond the count and divide by the era divisor for dawn rotation parts. Subtract from the rotation fixed parts; the remainder is dusk rotation parts. Then multiply the mean new moon, quarter, or full moon minor remainder by that day's rotation fixed parts beyond the count and divide by the era divisor for time-of-occurrence parts. Subtract from the dawn-dusk rotation parts; the remainder is before. If insufficient to subtract, invert-subtract; the remainder is after. Add before and subtract after from the true new moon, quarter, and full moon lunar degree to obtain the dawn and dusk lunar positions.
183
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 the true dawn-dusk intervals for new moon, quarter, and full moon: subtract the first-quarter dusk moon from that month's new-moon dusk moon for the post-new-moon dusk interval. Subtract the full-moon dusk moon from the first-quarter dusk moon for the post-first-quarter dusk interval. Subtract the last-quarter dawn moon from the full-moon dawn moon for the post-full-moon dawn interval. Subtract the following new moon's dawn moon from the last-quarter dawn moon for the post-last-quarter dawn interval.
184
求每日轉定度數：累計每程相距日轉定分，以減定程，餘為盈； 不足減者，覆減之，餘為縮； 以相距日除之，所得，盈加縮減每日轉定分，為每日轉定度及分秒。
To find the daily rotation fixed degrees: cumulatively tally the daily rotation fixed parts across the interval days and subtract from the fixed interval; the remainder is surplus. If insufficient to subtract, invert-subtract; the remainder is contraction. Divide by the intervening days; surplus adds and contraction subtracts from the daily rotation fixed parts to obtain the daily rotation fixed degree and parts-seconds.
185
求每日晨昏月：置朔、弦、望晨昏月，以每日轉定度及分加之，滿宿次去之，為每日晨昏月。 凡注曆，自朔日注昏月，望後一日注晨月。
To find the daily dawn and dusk moon: set each new moon, quarter, and full moon dawn and dusk position, add the daily rotation fixed degree and parts, and remove by lodge sequence. When publishing the calendar, annotate the dusk moon from the new-moon day and the dawn moon from the day after full moon.
186
已前月度並依九道所推，以究算術之精微，如求速要，即依後術求之。
All lunar positions above are derived by the nine paths to probe the arithmetic in full; for a quick approximation, use the later procedures.
187
求天正十一月經朔加時平行月：置歲周，以天正閏餘減之，餘以統法約之為度，不滿，退除為分秒，即天正十一月經朔加時平行月積度及分秒。
To find the parallel-moon position at the civil-year eleventh month's mean-new-moon time of occurrence: set the year circuit, subtract the civil-year intercalation remainder, and reduce the remainder by the era divisor for degrees, with the unfilled portion as parts and seconds, yielding the accumulated parallel-moon degree and parts-seconds at that mean new moon's time of occurrence.
188
求天正十一月定朔夜半平行月：置天正經朔小余，以平行月度分秒乘之，如統法而一為度，不滿，退除為分秒，以減天正十一月經朔加時平行月積度，即天正十一月經朔晨前夜半平行月。 其定朔大餘有進退者，亦進退平行度，否則因經為定，即天正十一月定朔晨前夜半平行月積度及分秒。
To find the true-new-moon midnight parallel moon for the eleventh month: set the civil-year mean-new-moon minor remainder, multiply by the parallel moon's monthly degree parts and seconds, divide by the era divisor for degrees, reduce the remainder to parts and seconds, and subtract from the mean-new-moon time-of-occurrence accumulated parallel moon to obtain the predawn midnight parallel moon on the mean new moon. If the true new moon's major remainder advances or retreats, adjust the parallel degree likewise; otherwise follow the mean as fixed, yielding the accumulated parallel-moon degree and parts-seconds before dawn at midnight on the true new moon.
189
求次定朔夜半平行月：置天正十一月定朔晨前夜半平行月積度及分秒，大月加三十五度八十分、秒六十一，小月加二十二度四十三分、秒七十三半，滿周天度及約分、秒去之，即得次定朔晨前夜半平行月積度及分秒。
To find the next true new moon's midnight parallel moon: set the eleventh month's predawn midnight accumulated parallel moon, add 35° 80′ 61″ for a long month or 22° 43′ 73.5″ for a short month, and cast out full circuit-of-heaven degrees and reduced parts to obtain the next true new moon's predawn midnight accumulated parallel moon.
190
求弦望定日夜半平行月：各計朔、弦、望相距之日，乘平行度及分秒，以加其月定朔晨前夜半平行月積度及分秒，即其月弦望定日晨前夜半平行月積度及分秒。
To find the quarter- and full-moon true-day midnight parallel moon: for each month count the days from new moon to quarter or full moon, multiply the parallel degree and parts-seconds, and add to that month's true-new-moon predawn midnight accumulated parallel moon to obtain the quarter or full moon's predawn midnight accumulated parallel moon.
191
求定朔晨前夜半入轉：置其月經朔晨前夜半入轉日及餘秒，若定朔大餘有進退者，亦進退轉日，否則因經為定，其餘如統法退除為分秒，即得其月定朔晨前夜半入轉日及分秒。 因求次日，累加一日，滿轉週二十七日五十五分、秒四十六去之，即每日晨前夜半入轉。
To find true-new-moon predawn midnight entry into rotation: set that month's mean-new-moon predawn midnight rotation entry day and remainder-seconds; if the true new moon's major remainder advances or retreats, adjust the rotation day likewise; otherwise follow the mean; reduce the remainder by the era divisor for parts and seconds to obtain that month's true-new-moon predawn midnight rotation entry. For the following days add one day cumulatively, cast out full rotation-circuit days of 27 days 55 minutes 46 seconds, and obtain each day's predawn midnight entry into rotation.
192
求定朔弦望晨前夜半定月：置定朔、弦、望晨前夜半入轉分，乘其日算外增減差，百約為分，分滿百為度，增減其下遲疾度，為遲疾定度； 遲減疾加定朔、弦、望晨前夜半平行月積度及分秒，以天正冬至加時黃道日度加而命之，即各得定朔、弦、望晨前夜半月離宿度及分秒。 如求每日晨、昏月，依前術入之，即得所求。
To find the true new moon, quarter, and full moon predawn midnight fixed moon: set the predawn midnight rotation-entry parts, multiply that day's correction rate beyond the count, scale by 100 for parts and by 100 parts for degrees, and apply to the slow-fast degree below to obtain the slow-fast fixed degree; Subtract slow and add fast to each true new moon, quarter, and full moon predawn midnight accumulated parallel moon, add the civil-year winter-solstice time-of-occurrence ecliptic solar degree, and name the lodges to obtain each true conjunction's predawn midnight lunar lodge departure in degrees, parts, and seconds. To find the dawn and dusk moon for each day, apply the preceding procedure and obtain the result sought.
193
步晷漏
Procedures for Gnomon Shadow and Clepsydra
194
二至限：一百八十二日六十二分。
Solstice limit: 182 days, 62 parts
195
一象：九十一日三十一分。
One quadrant: 91 days, 31 parts
196
消息法：九千七百三。
Message divisor: 9,703
197
半法：六千一十五。
Half-divisor: 6,015
198
辰法：二十五。
Double-hour divisor: 25
199
半辰法：一十二半。
Half double-hour divisor: 12.5
200
刻法：一千二百二。
Clepsydra graduation divisor: 1,202
201
辰刻：八、餘四百一。
Double-hour graduations: 8, remainder 401
202
昏明分：三百太。
Dusk-dawn parts: 300 and three-quarters
203
昏明刻：二、餘六百一半。
Dusk-dawn graduations: 2, remainder 601.5
204
冬至嶽台晷影常數：一丈二尺八寸五分。
Winter-solstice Yue Terrace gnomon-shadow constant: 1 zhang 2 chi 8 cun 5 fen
205
夏至嶽台晷影常數：一尺五寸七分。
Summer-solstice Yue Terrace gnomon-shadow constant: 1 chi 5 cun 7 fen
206
冬至後初限夏至後末限：四十五日、六十二分。
Opening limit after winter solstice and closing limit after summer solstice: 45 days, 62 parts
207
冬至後末限夏至後初限：一百三十七日、空分。
Closing limit after winter solstice and opening limit after summer solstice: 137 days, zero parts
208
求嶽台晷影入二至後日數：計入二至以來日數，以二至約分減之，乃加半日之分五十，即入二至後來午中日數及分。
To find the post-solstice day-count for Yue Terrace gnomon shadow: tally days since a solstice, subtract the solstice fractional parts, add fifty half-day parts, and obtain the noon day-count and parts after entering a solstice.
209
求嶽台午中晷影定數：置入二至後日及分，如初限已下者為初； 已上，覆減二至限，餘為末。 其在冬至後初限、夏至後末限者，以入限日入分減一千九百三十七半，為泛差。 仍以入限日及分乘其日盈縮積，其盈縮積者，以入盈縮限日及分與二百相減相乘，為盈縮積也。
To find the fixed noon gnomon shadow at Yue Terrace: set the post-solstice day and parts; days within the opening limit count as opening; days above that, invert-subtract from the solstice limit; the remainder is closing. Within the post-winter-solstice opening limit and post-summer-solstice closing limit, subtract the entry-limit day and parts from 1,937.5 for the provisional difference. Then multiply by that day's equation accumulation, which is obtained by subtracting the entry into the equation limit from 200 and multiplying by the entry-limit day and parts.
210
五因百約，用減泛差，為定差； 乃以入限日及分自相乘，以定差乘之，滿一百萬為尺，不滿為寸、分，以減冬至嶽台晷影常數，餘為其日午中晷影定數。 其在冬至後末限、夏至後初限者，以三約入限日及分，減四百八十五少，為泛差； 仍以盈縮差度減去極度，余者春分後、秋分前，四約，以加泛差，為定差。 春分前、秋分後，以去二分日數乘之，六百而一，以減泛差，為定差。 乃以入限日及分自相乘，以定差乘之，滿一百萬為尺，不滿為寸分，以加夏至嶽台晷影常數，為其日午中晷影定數。
Multiply by five and scale by 100, subtract from the provisional difference, and obtain the fixed difference; Square the entry-limit day and parts, multiply by the fixed difference, take full millions as chi and the remainder as cun and fen, subtract from the winter-solstice Yue Terrace gnomon-shadow constant, and the remainder is that day's fixed noon gnomon shadow. Within the post-winter-solstice closing limit and post-summer-solstice opening limit, divide the entry-limit day and parts by three, subtract from 485 and a minor quarter for the provisional difference; Subtract the equation difference from the polar degree; for the remainder after the spring equinox and before the autumn equinox, quarter it and add to the provisional difference to obtain the fixed difference. Before the spring equinox and after the autumn equinox, multiply the days from the equinoxes, divide by 600, subtract from the provisional difference, and obtain the fixed difference. Square the entry-limit day and parts, multiply by the fixed difference, take full millions as chi and the remainder as cun and fen, add to the summer-solstice Yue Terrace gnomon-shadow constant, and obtain that day's fixed noon gnomon shadow.
211
求每日午中定積日：置其日午中入二至後來日數及分，以其日盈縮分盈加縮減之，即每日午中定積日及分。
To find each day's fixed accumulated day at noon: set that day's noon post-solstice day-count and parts, add equation parts for surplus and subtract for contraction, and obtain each day's fixed accumulated day and parts at noon.
212
求每日午中消息定數：置定積日及分，在一象已下自相乘，已上，用減二至限，餘亦自相乘，七因，進二位元，以消息法除之，為消息常數； 副置之，用減六百一半，餘以乘其副，以二千六百七十除之，以加常數，為消息定數。 冬至後為息，夏至後為消。
To find each day's fixed noon message number: set the fixed accumulated day and parts; if within one quadrant, square it; if above, subtract from the solstice limit and square the remainder; multiply by seven, advance two places, divide by the message divisor, and obtain the message constant; Set aside a duplicate, subtract 601.5, multiply the duplicate by the remainder, divide by 2,670, add to the constant, and obtain the fixed message number. After the winter solstice is wax; after the summer solstice is wane.
213
求每日黃道去極度：置其日消息定數，十六乘之，滿四百一除之為度，不滿，退除為分，春分後加六十七度三十一分，秋分後減一百一十五度三十一分，即每日午中黃道去極度及分。
To find each day's ecliptic polar distance: set that day's fixed message number, multiply by sixteen, divide by 401 for degrees with the remainder as parts, add 67° 31′ after the spring equinox or subtract 115° 31′ after the autumn equinox, and obtain each day's noon ecliptic polar distance and parts.
214
求每日太陽去赤道內外度：置其日黃道去極度及分，與一象度相減，餘為太陽去赤道內、外度及分。 去極多為日在赤道外，去極少為日在赤道內。
To find each day's solar declination: set that day's ecliptic polar distance and parts, subtract from one-quadrant degree, and obtain the Sun's distance north or south of the equator in degrees and parts. Greater polar distance means the Sun is south of the equator; lesser polar distance means the Sun is north of the equator.
215
求每日晨昏分及日出入分半晝分。 置其日消息定數，春分後加二千一百少，秋分後減三千三百八少，各為其日晨分； 用減統法，餘為昏分。 以昏明分加晨分，為日出分； 減昏分，為日入分； 以日出分減半法，餘為半晝分。
To find each day's dawn and dusk parts, sunrise and sunset parts, and half-day parts. Set that day's fixed message number, add 2,100 and a minor quarter after the spring equinox or subtract 3,308 and a minor quarter after the autumn equinox, and obtain that day's dawn part; Subtract from the era divisor; the remainder is the dusk part. Add the dawn part to the dusk-dawn parts to obtain the sunrise part; Subtract from the dusk part to obtain the sunset part; Subtract the sunrise part from the half-divisor; the remainder is the half-day part.
216
求每日距中度：置其日晨分，進位，十四因之，以四千六百一十一除之為度，不滿，退除為分，即距子度。 用減半周天，餘為距中度； 五而一，為每更差數。
To find each day's mid-heaven distance: set that day's dawn part, advance one place, multiply by fourteen, divide by 4,611 for degrees with the remainder as parts, and obtain the distance from zi. Subtract from half the circuit of heaven; the remainder is the mid-heaven distance; Divide by five to obtain each watch's difference.
217
求每日夜半定漏：置晨分，進一位，如刻法而一為刻，不滿為刻分，即每日夜半定漏。
To find each day's fixed midnight clepsydra: set the dawn part, advance one place, divide by the clepsydra graduation divisor for graduations with the remainder as graduation parts, and obtain each day's fixed midnight clepsydra.
218
求每日晝夜刻及日出入辰刻：置夜半定漏，倍之，加五刻，為夜刻。 減百刻，為晝刻。 以昏明刻加夜半定漏，命子正，算外，得日出辰刻。 以晝刻加之，命如前，即日入辰刻。 其辰數依發斂術求之。
To find each day's day and night clepsydra graduations and sunrise-sunset double-hour graduations: set the fixed midnight clepsydra, double it, and add five graduations for night graduations. Subtract from 100 graduations to obtain day graduations. Add the dusk-dawn graduations to the fixed midnight clepsydra, count from zi midnight, and outside the count obtain the sunrise double-hour graduations. Add the day graduations and name as before to obtain the sunset double-hour graduations. The double-hour count is derived by the issuing-and-gathering procedure.
219
求更點辰刻：置其日夜半定漏，倍之，二十五而一為籌差； 半之，進位，為更差。 以昏明刻加日入辰刻，即甲夜辰刻； 以更籌差累加之，滿辰刻及分去之，各得每更籌所在辰刻及分。 若用司辰漏者，倍夜半定漏，減去待旦十刻，餘依術算，即得內中更籌也。
To find watch-point double-hour graduations: set that day's fixed midnight clepsydra, double it, and divide by twenty-five to obtain the tally difference; Halve it, advance one place, and obtain the watch difference. Add the dusk-dawn graduations to the sunset double-hour graduations to obtain the first night watch's double-hour graduations; Add the watch tally difference cumulatively, cast out full double-hour graduations and parts, and obtain each watch tally's double-hour graduations and parts. If using the Office-of-Time clepsydra, double the fixed midnight clepsydra, subtract the ten graduations awaiting dawn, and compute by the procedure to obtain the inner watch tallies.
220
求每日昏曉中星及五更中星：置距中度，以其日昏後夜半赤道日度加而命之，即得其日昏中星所格宿次，命之曰初更中星。 以每更差度加而命之，即乙夜中星。 以更差度累加之，去命如前，即五更及曉中星。 若依司辰星漏倍距子度，減去待旦三十六度五十二分半，餘依術求更點差度，即內中昏曉五更及攢點中星也。
To find each day's dusk, dawn, and five-watch midnight stars: set the mid-heaven distance, add that day's post-dusk midnight equatorial solar degree and name the lodges to obtain the dusk midnight star's lodge, which is named as the first-watch midnight star. Add each watch's difference in degrees and name the lodges to obtain the second night watch's midnight star. Add the watch difference cumulatively, cast out and name as before, and obtain the five watches and dawn midnight stars. If following the Office-of-Time star clepsydra, double the distance from zi, subtract thirty-six degrees fifty-two minutes and a half awaiting dawn, derive the watch-point difference by the procedure, and obtain the inner dusk, dawn, five watches, and gathered-point midnight stars.
221
求九服距差日：各於所在立表候之，若地在嶽臺北，測冬至後與嶽台冬至晷影同者，累冬至後至其日，為距差日。 若地在岳台南，測夏至後與嶽台晷影同者，累夏至後至其日，為距差日。
To find the nine regions' distance-difference day: at each location erect a gnomon table and observe; if the place lies north of Yue Terrace and after the winter solstice its noon shadow matches Yue Terrace's winter-solstice shadow, accumulate the days from the winter solstice to that day as the distance-difference day. If the place lies south of Yue Terrace and after the summer solstice its noon shadow matches Yue Terrace's, accumulate the days from the summer solstice to that day as the distance-difference day.
222
求九服晷影：若地在嶽臺北冬至前後者，以冬至前後日數減距差日，為餘日。 以餘日減一千九百三十七半，為泛差。 依前術求之，以加嶽台冬至晷影常數，為其地其日午中晷影定數。 冬至前後日多於距差日者，乃減去距差日，餘依法求之，即得其地其日午中晷影定數。 若地在岳台南夏至前後者，以夏至前後日數減距差日，為餘日。 乃三約之，以減四百八十五少，為泛差。 依前術求之，以減嶽台夏至晷影常數，即其地其日午中晷影定數。 如夏至前後日數多於距差日，乃減去距差日，餘依法求之，即得其地其日午中晷影定數，即晷在表南也。
To find gnomon shadows for the nine regions: if north of Yue Terrace around the winter solstice, subtract the distance-difference day from the winter-solstice day-count to obtain the remainder days. Subtract the remainder days from 1,937.5 for the provisional difference. Apply the preceding procedure and add to the Yue Terrace winter-solstice gnomon-shadow constant to obtain that place and day's fixed noon gnomon shadow. If the winter-solstice day-count exceeds the distance-difference day, subtract the distance-difference day and compute by the method to obtain that place and day's fixed noon gnomon shadow. If south of Yue Terrace around the summer solstice, subtract the distance-difference day from the summer-solstice day-count to obtain the remainder days. Then divide by three and subtract from 485 and a minor quarter for the provisional difference. Apply the preceding procedure and subtract from the Yue Terrace summer-solstice gnomon-shadow constant to obtain that place and day's fixed noon gnomon shadow. If the summer-solstice day-count exceeds the distance-difference day, subtract the distance-difference day and compute by the method to obtain that place and day's fixed noon gnomon shadow, with the shadow falling south of the table.
223
求九服所在晝夜漏刻：各於所在下水漏，以定二至夜刻，乃相減，餘為二至差刻。 乃置嶽台其日消息定數，以其處二至差刻乘之，如嶽台二至差刻二十除之，所得為其地其日消息定數。 乃倍消息定數，進位元，滿刻法約之為刻，不滿為分，以加減其處二至夜刻，春分後、秋分前，以加夏至夜刻； 秋分後、春分前，以減冬至夜刻。
To find day and night clepsydra graduations for each of the nine regions: at each location run the clepsydra below, fix the two-solstice night graduations, subtract them, and the remainder is the two-solstice difference in graduations. Set Yue Terrace's message fixed number for that day, multiply by that place's two-solstice difference graduations, divide by Yue Terrace's two-solstice difference of twenty graduations, and obtain that place and day's message fixed number. Then double the message fixed number, advance one place, reduce by the clepsydra graduation divisor for graduations with the remainder as parts, and add or subtract that place's two-solstice night graduations—after the spring equinox and before the autumn equinox add to the summer-solstice night graduations; After the autumn equinox and before the spring equinox subtract from the winter-solstice night graduations.
224
為其地其日夜刻； 以減百刻，餘為晝刻。 求日出入差刻及五更中星，並依嶽台法求之。
That yields that place and day's night graduations; Subtract from 100 graduations; the remainder is the day graduations. To find sunrise-sunset difference graduations and five-watch midnight stars, apply the Yue Terrace method throughout.