1
步日躔
Procedure: Solar Daily Motion along the Ecliptic.
2
周天分：三百八十六萬八千六十五、秒二。
Circuit-of-heaven parts: 3,868,065 parts, 2 seconds.
3
周天度：三百六十五度。 虛分二千七百一十五、秒二，約分二十五、秒六十四。
Circuit-of-heaven degrees: 365°. Void fractional parts: 2,715 parts, 2 seconds; simplified parts: 25 parts, 64 seconds.
4
歲差：一百二十五、秒二。
Precession: 125 parts, 2 seconds.
5
乘法：三十二。
Multiplication factor: 32.
6
除法：四百八十七。
Division factor: 487.
7
秒法：一百。
Second divisor: 100.
8
求每日盈縮定數：以乘法乘所入氣升降分，如除法而一，為其氣中平率； 與後氣中平率相減，為差率； 半差率，加減其氣中平率，為其氣初、末泛率。 至後加為初，減為末； 分後減為初，加為末。
To find the daily fixed excess–deficit value: multiply the ascending–descending parts of the qi-period entered by the multiplication factor and divide by the division factor to obtain that qi-period's mid-level rate; subtract from the next qi-period's mid-level rate to obtain the difference rate; take half the difference rate and add or subtract it from the qi-period mid-level rate to obtain the initial and final general rates for that qi-period. After a solstice: add to the initial, subtract from the final; After an equinox: subtract from the initial, add to the final.
9
又以乘法乘差率，除法而一，為日差； 半之，加減初、末泛率，為初、末定率。 至後減初加末，分後加初減末。
Again multiply the difference rate by the multiplication factor and divide by the division factor to obtain the daily difference; halve it and add or subtract from the initial and final general rates to obtain the initial and final fixed rates. After a solstice: subtract from the initial and add to the final; after an equinox: add to the initial and subtract from the final.
10
以日差累加減氣之定率，為每日升降定率； 至後減，分後加。
Accumulate the daily difference against the qi-period's fixed rate to obtain the daily ascending–descending fixed rate; After a solstice subtract; after an equinox add.
11
以每日升降定率，冬至後升加降減，夏至後升減降加，其氣初日盈縮分，為每日盈縮定數； 其分、至前一氣先後率相減，以前末泛率為其氣初泛率，以半日差，至前加之，分前減之。
Using the daily ascending–descending fixed rate, after winter solstice add ascent and subtract descent, after summer solstice subtract ascent and add descent; the first-day excess–deficit parts of the qi-period become the daily fixed excess–deficit value; At an equinox or solstice, subtract the prior and subsequent rates of the preceding qi-period; use the preceding final general rate as this qi-period's initial general rate; with half the daily difference, add before a solstice and subtract before an equinox.
12
為其氣初日定率。 餘依本術。 求朏朒准此。
This gives the first-day fixed rate for that qi-period. For the remainder, follow the original procedure. To find the waxing and waning corrections, apply the same method.
13
求經朔弦望入氣：置天正閏日及餘，如氣策及餘秒以下者，以減氣策及餘秒，為入大雪氣； 已上者去之，餘以減氣策及餘秒，為入小雪氣：即得天正十一月經朔入大、小雪氣日及餘秒。 求弦、望及後朔入氣，以弦策累加之，滿氣策及餘秒去之，即得。
To find which qi-period a month-beginning, quarter-moon, or full moon enters: set the civil new-year intercalary day and remainder; if it falls at or below the qi-divisor and remainder in seconds, subtract from the qi-divisor and remainder in seconds to obtain entry into Major Snow; if above that threshold, discard it and subtract the remainder from the qi-divisor and remainder in seconds to obtain entry into Minor Snow; this yields the day and remainder in seconds for the eleventh-month beginning entering Major or Minor Snow at civil new year. To find quarter-moon, full moon, and later month-beginnings entering qi-periods, add the quarter-moon divisor repeatedly and discard full qi-divisors and remainder in seconds.
14
求定氣日：冬、夏二至以常氣為定。 餘即以其氣下盈縮分縮加盈減常氣約餘為定氣，滿若不足，進退大餘，命甲子，算外，即定氣日及分。
To find the fixed qi-period day: the winter and summer solstices are fixed at the regular qi. For the others, use that qi-period's lower excess–deficit parts to contract, add excess, or subtract from the regular qi's simplified remainder to obtain the fixed qi; if full or short, advance or retreat the major remainder, count from jiazi outward, and the result is the fixed qi-period day and parts.
15
求經朔弦望入氣朏朒定數：各以所入氣小餘乘其日損益率，如樞法而一，即得。
To find the fixed waxing-and-waning value for month-beginnings, quarter-moons, and full moons entering qi-periods: multiply the minor remainder of the qi-period entered by its daily gain-and-loss rate and divide by the pivot divisor.
16
求赤道宿度
Procedure to Find Equatorial Lodge Degrees.
17
鬥：二十六度牛：八度女：十二度虛：十度及分
Dipper: 26°; Ox: 8°; Maiden: 12°; Void: 10° and fractional parts.
18
危：十七度室：十六度壁：九度
Rooftop: 17°; House: 16°; Wall: 9°.
19
北方七宿九十八度虛分二千七百一十五、秒二，約分二十五、秒六十四。
The seven northern lodges total 98°, with Void fractional parts of 2,715 parts and 2 seconds, or simplified parts of 25 parts and 64 seconds.
20
奎：十六度婁：十二度胃：十四度昴：十一度
Legs: 16°; Bond: 12°; Stomach: 14°; Hairy Head: 11°.
21
畢：十七度觜：一度參：十度
Net: 17°; Turtle Beak: 1°; Orion: 10°.
22
西方七宿八十一度。
The seven western lodges total 81°.
23
井：三十三度鬼：三度柳：十五度星：七度
Well: 33°; Ghost: 3°; Willow: 15°; Star: 7°.
24
張：十八度翼：十八度軫：十七度
Extended Net: 18°; Wings: 18°; Chariot Base: 17°.
25
南方七宿一百一十一度。
The seven southern lodges total 111°.
26
角：十二度亢：九度氐：十七度房：五度
Horn: 12°; Neck: 9°; Root: 17°; Chamber: 5°.
27
心：五度尾：十八度箕：十一度
Heart: 5°; Tail: 18°; Winnowing Basket: 11°.
28
東方七宿七十五度。
The seven eastern lodges total 75°.
29
前皆赤道度，其畢、觜、參及輿鬼四宿度數與古度不同，自《大衍曆》依渾天儀以測定，為用紘帶天中，儀極是憑，以格黃道。
The figures above are all equatorial degrees. The degrees for Net, Turtle Beak, Orion, and Ghost differ from the ancient values; since the 《Dayan Calendar》 they have been measured with the armillary sphere, using the celestial girdle at the center of heaven and the instrument's pole as reference to delimit the ecliptic.
30
推天正冬至赤道日度：以歲差乘距所求積年，滿周天分去之，不盡，用減周天分，餘以樞法除之為度，不盡為餘秒。 其度，命以赤道虛宿七度外起算，依宿次去之，不滿者，即得天正冬至加時赤道日躔所距宿度及餘秒。 其餘以樞法退除為分及秒，各以一百為度。
To derive the equatorial solar degree at civil new-year winter solstice: multiply precession by the accumulated years from the epoch to the year sought, discard full circuit-of-heaven parts, subtract the remainder from the circuit-of-heaven parts, divide the result by the pivot divisor for degrees, and treat the remainder as parts and seconds. Count the degrees from beyond Void lodge 7° on the equator, discard full lodges in sequence, and the lodge not yet filled gives the equatorial solar lodge degree and remainder in seconds at hour-of-addition for civil new-year winter solstice. Convert the remainder by retreating through the pivot divisor into parts and seconds, using one hundred per degree.
31
求二十四氣赤道日度：置天正冬至加時赤道日度及餘秒，以氣策及餘秒累加之。 先以三十六乘赤道秒，以一百乘氣策秒，然後加之，即秒母皆同三千六百。
To find the equatorial solar degrees for the twenty-four qi-periods: set the winter solstice equatorial degree and remainder in seconds at hour-of-addition and add the qi-divisor and remainder in seconds repeatedly. First multiply equatorial seconds by 36 and qi-divisor seconds by 100, then add them so that all second denominators become 3,600.
32
滿赤道宿次去之，即各得二十四氣加時赤道日躔宿度及餘秒。
Discard full equatorial lodge cycles to obtain the equatorial solar lodge degree and remainder in seconds at hour-of-addition for each of the twenty-four qi-periods.
33
求二十四氣昏後夜半赤道日度：各以其氣小餘減樞法，其秒亦以一百乘，然乃減之。
To find post-dusk and midnight equatorial solar degrees for the twenty-four qi-periods: subtract each qi-period's minor remainder from the pivot divisor, likewise multiplying its seconds by 100 before subtracting.
34
余加其氣加時赤道日躔宿度及餘秒，即其氣初日昏後夜半赤道日度及餘秒。 求次日累加一度，滿宿次去之，各得所求。
Add the remainder to that qi-period's equatorial solar lodge degree and remainder in seconds at hour-of-addition to obtain the post-dusk and midnight equatorial solar degree and remainder in seconds on its first day. For each following day add one degree cumulatively, discarding full lodge cycles, to obtain the values sought.
35
求赤道宿積度：置冬至加時日躔赤道宿全度，以冬至加時日躔赤道宿度及約分秒減之，餘為距後度及分秒； 以赤道宿度累加距後度，即得各赤道宿積度及分秒。
To find accumulated equatorial lodge degrees: set the full equatorial lodge degree at winter solstice hour-of-addition, subtract the lodge degree and simplified parts and seconds at that moment, and the remainder is the degrees-after value in degrees, parts, and seconds; add equatorial lodge degrees cumulatively to the degrees-after to obtain the accumulated equatorial degree and parts and seconds for each lodge.
36
求赤赤道宿積度入初末限：各置赤道宿積度及分秒，滿九十一度三十一分、秒一十一去之，餘四十五度六十六分以下為入初之限； 已上者，用減九十一度三十一分，餘為入末限度及分秒。
To find whether accumulated equatorial lodge degrees fall within the initial or final limit: set each accumulated degree with parts and seconds, discard full 91° 31′ 11″, and a remainder of 45° 66′ or below counts as within the initial limit; if above that, subtract from 91° 31′ to obtain the final-limit remainder in degrees, parts, and seconds.
37
求二十八宿黃道度：各置赤道宿入初、末限度及分，用減一百二十五，余以初、末限度及分乘之，十二除為分，分滿百為度，命為黃、赤道差度及分； 至後分前以減、分後至前以加赤道宿積度，為其宿黃道積度； 以前宿黃道積度減其宿黃道積度，為其宿黃道度及分。 其分就近約為太、半、少。
To find ecliptic degrees for the twenty-eight lodges: set each equatorial lodge's initial or final limit in degrees and parts, subtract from 125, multiply the remainder by that limit, divide by 12 for parts and convert parts filling 100 into degrees to obtain the ecliptic–equatorial difference in degrees and parts; before an equinox after a solstice subtract, after an equinox before a solstice add to the accumulated equatorial degree to obtain the lodge's accumulated ecliptic degree; subtract the preceding lodge's accumulated ecliptic degree to obtain that lodge's ecliptic degree and parts. Round the parts to the nearest excess, half, or deficient fraction.
38
黃道宿度
Ecliptic Lodge Degrees.
39
鬥：二十三太
Dipper: 23¾°.
40
牛：七半
Ox: 7½°.
41
女：十一半
Maiden: 11½°.
42
虛：十秒六十四
Void: 10° 64″.
43
危：十七太
Rooftop: 17¾°.
44
室：十七
House: 17°.
45
壁：九少
Wall: 9¼°.
46
北方七宿九十七度。 半、秒六十四
The seven northern lodges total 97°. ½°, 64″.
47
奎：十七半
Legs: 17½°.
48
妻：十二太
Bond: 12¾°.
49
胃：十四太
Stomach: 14¾°.
50
昴：十一
Hairy Head: 11°.
51
畢：十六
Net: 16°.
52
觜：一
Turtle Beak: 1°.
53
參：九少
Orion: 9¼°.
54
西方七宿八十二度。
The seven western lodges total 82°.
55
井：三十
Well: 30°.
56
鬼：二
Ghost: 2°.
57
柳：十四
Willow: 14°.
58
星：七
Star: 7°.
59
張：十八太
Extended Net: 18¾°.
60
翼：十九少
Wings: 19¼°.
61
軫：十八
Chariot Base: 18°.
62
南方七宿一百一十度。
The seven southern lodges total 110°.
63
角：十三
Horn: 13°.
64
亢：九半
Neck: 9½°.
65
氐：十五半
Root: 15½°.
66
房：五
Chamber: 5°.
67
心：四
Heart: 4°.
68
尾：十七
Tail: 17°.
69
箕：十
Winnowing Basket: 10°.
70
東方七宿七十四度。
The seven eastern lodges total 74°.
71
求冬至加時黃道日躔宿次：以冬至加時赤道日躔宿度，用減一百二十五，餘以冬至加時赤道度及分乘之，十二除為分，分滿百為度，用減九十一度赤道日度及分，即冬至加時黃道日躔宿度及分。
To find the ecliptic solar lodge at winter solstice hour-of-addition: take the equatorial solar lodge degree at that moment, subtract from 125, multiply the remainder by the equatorial degree and parts at winter solstice hour-of-addition, divide by 12 for parts and convert parts filling 100 into degrees, subtract from 91° of equatorial solar degree and parts, and the result is the ecliptic solar lodge degree and parts at winter solstice hour-of-addition.
72
求二十四氣初日加時黃道日躔宿次：置所求年冬至日躔黃道赤道差，以次年黃赤道差減之，餘以所氣數乘之，二十四而一，所得，以加其氣下中積及約分，又以其氣初日盈縮分盈加縮減之，用加冬時黃道日度，依宿次命之，即各得其氣初日加時黃道日躔所在宿度及分。 若其年冬至加時赤道日躔度空，分、秒在歲差已下者，即如前宿全度，乃求黃赤道差，以次年冬至加時黃赤道差減之，餘依本術，各得所求。 此術以究算理之微，亟求其當，止以盈縮分加減中積，以天正冬至加時黃道日度加而命之。
To find the ecliptic solar lodge at hour-of-addition on the first day of each qi-period: set the winter solstice ecliptic–equatorial difference for the year sought, subtract the next year's difference, multiply the remainder by the qi-period number and divide by 24, add the result to that qi-period's lower mid-accumulation and simplified parts, then add excess or subtract deficit using its first-day excess–deficit parts, add to the winter ecliptic solar degree, and count through the lodges to obtain the ecliptic solar lodge degree and parts at hour-of-addition on each qi-period's first day. If the equatorial solar degree at winter solstice hour-of-addition for that year is void, with parts and seconds below the precession value, use the preceding lodge's full degree to find the ecliptic–equatorial difference, subtract the next year's winter solstice ecliptic–equatorial difference, and apply the remainder by the original procedure to obtain each value sought. This procedure probes the subtlety of computational theory and seeks exactitude; it only adds and subtracts excess–deficit parts from the mid-accumulation, then adds to the ecliptic solar degree at civil new-year winter solstice hour-of-addition and counts through the lodges.
73
求二十四氣初日晨前夜半黃道日躔宿次：置一百分，分以一百約其氣初日升降分，升加降減之，一日所行之分乘其初日約分，所得滿百為分，分滿百為度，不滿百分為秒，以減其初日黃道加時日躔宿次，即其日晨前夜半黃道日躔宿次。
To find the ecliptic solar lodge pre-dawn and at midnight on the first day of each qi-period: set 100 parts, simplify that qi-period's first-day ascending–descending parts per 100 and add for ascent or subtract for descent, multiply the daily motion in parts by the first-day simplified parts, convert results filling 100 into parts and then degrees with remainder as seconds, subtract from the ecliptic solar lodge at hour-of-addition on the first day, and the result is the ecliptic solar lodge pre-dawn and at midnight.
74
求每日晨前夜半黃道日躔宿次：各因二十四氣初日晨前夜半黃道日躔宿次，日加一度，以一百約每日升降為分秒，升加降減之，以黃道宿次命之，即每日晨前夜半黃道日躔所距宿度及分。
To find the ecliptic solar lodge pre-dawn and at midnight for each day: start from each qi-period's first-day values, add one degree per day, simplify daily ascending–descending per 100 into parts and seconds with ascent added and descent subtracted, count through ecliptic lodges, and obtain the ecliptic solar lodge degree and parts pre-dawn and at midnight for each day.
75
步月離
Procedure: Lunar Motion and Separation.
76
轉周分：二十九萬一千八百三、秒五百九十四。
Rotation circuit parts: 291,803 parts, 594 seconds.
77
轉周日：二十七、餘五千八百七十三、秒五百九十四。
Rotation circuit days: 27 days, remainder 5,873 parts, 594 seconds.
78
朔差日：一、余一萬三百三十五、秒九千四百六。
Month-beginning difference days: 1 day, remainder 10,335 parts, 9,406 seconds.
79
望差：一十四、餘八千一百四、秒五千。
Full-moon difference: 14 days, remainder 8,104 parts, 5,000 seconds.
80
弦策：七、餘四千五十二、秒二千五百。
Quarter-moon divisor: 7 days, remainder 4,052 parts, 2,500 seconds.
81
七日：初數九千四百四十一，初約分八十九； 末數一千一百七十九，末約分一十一。
Day 7: initial number 9,441, initial simplified parts 89; final number 1,179, final simplified parts 11.
82
十四日：初數八千二百三十二，初約分七十八； 末數二千三百五十八，末約分二十二。
Day 14: initial number 8,232, initial simplified parts 78; final number 2,358, final simplified parts 22.
83
二十一日：初數七千五十二，初約分六十九； 末數三千五百三十八，末約分二十三。
Day 21: initial number 7,052, initial simplified parts 69; final number 3,538, final simplified parts 23.
84
二十八日：初數五千八百七十三，初約分五十六。
Day 28: initial number 5,873, initial simplified parts 56.
85
已上秒法一萬。
For the above, the second divisor is 10,000.
86
上弦：九十一度三十一分、秒四十一。
First quarter: 91° 31′ 41″.
87
望：一百八十二度六十二分、秒八十二。
Full moon: 182° 62′ 82″.
88
下弦：二百七十三度九十四分、秒二十三。
Last quarter: 273° 94′ 23″.
89
平行：一十三度三十六分、秒八十七半。
Mean motion: 13° 36′ 87.5″.
90
已上秒母一百。
For the above, the second denominator is 100.
91
推天正十一月經朔入轉：置天正十一月經朔積分，以轉周分秒去之，不盡，以樞法除之為日，不滿為餘秒，命日，算外，即所求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，即次日加時入轉。
To derive entry into rotation for the eleventh-month beginning at civil new year: set the accumulated parts for that month-beginning, discard full rotation circuit parts and seconds, divide the remainder by the pivot divisor for days with remainder as parts and seconds, count days outward, and obtain the rotation day and remainder in seconds at hour-of-addition for the eleventh-month beginning. Add the month-beginning difference in days and remainder seconds, discard full rotation circuit days and remainder seconds, and obtain entry into rotation at hour-of-addition for the following day.
92
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，即上弦、望及下弦加時入轉日及餘秒。 若以經朔、弦、望小餘減之，各得其日夜半入轉日及餘秒。
To find entry into rotation for quarter-moons and full moon: start from the eleventh-month beginning values at hour-of-addition, add the quarter-moon divisor repeatedly, and count as before to obtain rotation day and remainder in seconds at hour-of-addition for first quarter, full moon, and last quarter. Subtract the minor remainder for month-beginning, quarter-moon, or full moon to obtain entry into rotation day and remainder in seconds at midnight for each.
93
求朔弦望入轉朏朒
Procedure to Find Waxing-and-Waning Corrections for Month-Beginnings, Quarter-Moons, and Full Moons Entering Rotation.
94
定數：置所入轉餘，乘其日損益率，樞法而一，所得，以損益其下朏朒積為定數。 其四七日下余如初數下，以初率乘之，初數而一，以損益朏朒為定數。 若初數已上者，以初數減之，餘乘末率，末數而一，用減初率，餘加朏朒，各為定數。 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，為朏定數。
Fixed value: set the rotation remainder entered, multiply by its daily gain-and-loss rate, divide by the pivot divisor, and apply gain or loss to the lower tuoke accumulation below to obtain the fixed value. If the remainder within the four sevens falls below the initial number, multiply by the initial rate, divide by the initial number, and apply gain or loss to tuoke for the fixed value. If above the initial number, subtract the initial number, multiply the remainder by the final rate and divide by the final number, subtract from the initial rate, add the remainder to tuoke, and each yields a fixed value. If within day 14 the remainder is above the initial number, subtract the initial number, multiply the remainder by the final rate and divide by the final number to obtain the tuoke fixed value.
95
求朔望定日：各以入氣、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。 若定朔幹名與後朔同名者大，不同者小，其月無中氣者為閏月。 凡注曆，觀朔小餘，如日入分已上者，進一日，朔或當定，有食應見者，其朔不進。 弦、望定小餘不滿日出分，退一日，其望定小余雖滿此數，若有交食虧初起在日出已前者，亦如之。 有月行九道遲疾，曆有三大二小； 若行盈縮累增損之，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，不過三大二小。 若正朔有加交，時虧在晦、二正見者，消息前後一兩月，以定大小。
To find fixed days for month-beginnings and full moons: for each apply the fixed waxing-and-waning values from qi-period and rotation entry—subtract tuoke and add tuoke to the minor remainder for month-beginning, quarter-moon, or full moon, advance or retreat the major remainder if full or short, count from jiazi outward, and obtain the fixed day and remainder. If the fixed month-beginning stem-branch matches the following month-beginning's name the month is long; if different it is short; a month without a mid-qi is intercalary. When annotating the calendar, observe the month-beginning minor remainder; if the sun's entry parts are already reached, advance one day, but if the month-beginning is fixed and an eclipse should be visible, do not advance that month-beginning. If fixed minor remainders for quarter-moon or full moon do not reach the sun-out parts, retreat one day; even if the full-moon fixed minor remainder reaches that number, retreat likewise if eclipse dimming at conjunction begins before sunrise. Because the moon follows the nine paths with varying speed, the calendar has three long and two short months; if excess and deficit motion is accumulated with adjustments, there will be four long and three short months, as the numbers require; if one follows the usual practice, inspect whether the hour-of-addition is early or late and advance or retreat accordingly, not exceeding three long and two short. If the proper month-beginning has added conjunction and the eclipse dimming falls on the month's last day or is visible across two proper months, adjust the preceding or following one or two months to determine long or short.
96
求定朔弦望加時日所在度：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，各得其日加時日躔黃道宿次。
To find the solar degree at hour-of-addition for fixed month-beginnings, quarter-moons, and full moons: set the simplified parts as auxiliary, multiply by that day's ascending–descending parts and simplify per 10,000, add for ascent or subtract for descent from the auxiliary, add to midnight solar degree for that day, count through lodges as before, and obtain the ecliptic solar lodge at hour-of-addition.
97
推月行九道：凡合朔所交，冬在陰曆，夏在陽曆，月行青道； 冬、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 derive the moon's nine-path motion: at conjunction, when winter falls in the yin calendar and summer in the yang calendar, the moon follows the green path; after winter and summer solstices, the green path's half-crossing lies at the spring equinox lodge, east on the ecliptic; after Start of Winter and Start of Summer, the green path's half-crossing lies at the Start of Spring lodge, southeast on the ecliptic; the opposite lodge is treated likewise.
98
冬在陽曆，夏在陰曆，月行白道； 冬、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。
If winter falls in the yang calendar and summer in the yin calendar, the moon follows the white path; after winter and summer solstices, the white path's half-crossing lies at the autumn equinox lodge, west on the ecliptic; after Start of Winter and Start of Summer, the white path's half-crossing lies at the Start of Autumn lodge, northwest on the ecliptic; the opposite lodge is treated likewise.
99
春在陽曆，秋在陰曆，月行朱道； 春、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。
If spring falls in the yang calendar and autumn in the yin calendar, the moon follows the vermilion path; after spring and autumn equinoxes, the vermilion path's half-crossing lies at the summer solstice lodge, south on the ecliptic; after Start of Spring and Start of Autumn, the vermilion path's half-crossing lies at the Start of Summer lodge, southwest on the ecliptic; the opposite lodge is treated likewise.
100
春在陰曆，秋在陽曆，月行黑道。 春、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。
If spring falls in the yin calendar and autumn in the yang calendar, the moon follows the black path. After spring and autumn equinoxes, the black path's half-crossing lies at the winter solstice lodge, north on the ecliptic; after Start of Spring and Start of Autumn, the black path's half-crossing lies at the Start of Winter lodge, northeast on the ecliptic; the opposite lodge is treated likewise.
101
四序月離雖為八節，至陰陽之所交，皆與黃道相會，故月行有九道。 各視月所入正交積度，滿象度及分去之，入交積度及象度並在交會術中。
Although the moon's seasonal motion forms eight nodes, wherever yin and yang meet it intersects the ecliptic, so the moon's motion has nine paths. For each, observe the moon's accumulated degree at proper crossing, discard full image degrees and parts, and treat entry-crossing accumulated degrees and image degrees within the conjunction procedure.
102
若在半象以下者為入初限； 已上者，復減象度，餘為入末限； 用減一百二十五，餘以所入初、末限度及分乘之，滿二十四而一為分，分滿百為度，所得，為月行與黃道差數。 距半交後、正交前，以差數為減； 距正交後、半交前，以差數為加。 此加減出入六度，單與黃道相較之數，若較赤道，則隨氣遷變不常。
If below half an image, it falls within the initial limit; if above that, subtract image degrees again and the remainder enters the final limit; subtract from 125, multiply the remainder by the initial or final limit in degrees and parts entered, divide by 24 for parts and convert parts filling 100 into degrees to obtain the difference between lunar motion and the ecliptic. From after half-crossing to before proper crossing, subtract the difference number; from after proper crossing to before half-crossing, add the difference number. This addition and subtraction varies within six degrees—the value compared solely with the ecliptic; compared with the equator it shifts with the qi-periods and is not constant.
103
計去冬、夏至以來度數，乘黃道所差，九十而一，為月行與赤道差數。 凡日以赤道內為陰，外為陽； 月以黃道內為陰，外為陽。 故月行宿度，入春分交後行陰曆，秋分交後行陽曆，皆為同名； 春分交後行陽曆，秋分交後行陰曆，皆為異名。 其在同名，以差數加者加之，減者減之； 其在異名，以差數加者減之，減者加之。 皆以增損黃道宿積度，為九道宿積度； 以前宿九道積度減之，為其九道宿度及分。 其分就近約為少、半、太之數。
Calculate degrees elapsed since winter or summer solstice, multiply the ecliptic difference, divide by 90, and obtain the difference between lunar motion and the equator. In general the sun treats the region inside the equator as yin and outside as yang; the moon treats the region inside the ecliptic as yin and outside as yang. Thus for lunar lodge degrees, after the spring equinox crossing it travels the yin calendar and after the autumn equinox crossing the yang calendar—both are same-name; after the spring equinox crossing it travels the yang calendar and after the autumn equinox crossing the yin calendar—both are different-name. 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. Apply gain and loss to accumulated ecliptic lodge degrees to obtain accumulated nine-path lodge degrees; subtract the preceding lodge's accumulated nine-path degree to obtain that lodge's nine-path degree and parts. Round the parts to the nearest deficient, half, or excess fraction.
104
推月行九道平交入氣：各以其月閏日及餘，加經朔加時入交泛日及餘秒，盈交終日去之，乃減交終日及餘秒，即各平交入其月中氣日及餘秒。 滿氣策及餘秒去之，余即平交入後月節氣日及餘秒。 因求次交者，以交終日及餘秒加之，滿氣策及餘秒去之，余為平交入其氣日及餘秒，若求其氣朏朒定數，如求朔、弦、望經日術入之，各得所求也。
To derive mean-crossing entry into qi-periods for the moon's nine-path motion: for each month add intercalary day and remainder to month-beginning entry into crossing at hour-of-addition, discard full crossing-end days, subtract crossing-end day and remainder seconds, and obtain mean crossing entry into mid-qi day and remainder seconds for that month. Discard full qi-divisors and remainder in seconds; the remainder gives mean crossing entry into the following month's solar node day and remainder in seconds. To find the next crossing, add crossing-end day and remainder seconds, discard full qi-divisors and remainder in seconds, and the remainder is mean crossing entry into that qi-period's day and remainder in seconds; to find fixed waxing-and-waning for the qi-period, apply the month-beginning procedure and obtain each value sought.
105
求平交入轉朏朒定數：置所入氣餘，加其日夜半入轉餘，以乘其日損益率，樞法而一，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 fixed waxing-and-waning value for mean crossing entering rotation: set the qi-period remainder entered, add midnight rotation remainder for that day, multiply by daily gain-and-loss rate and divide by the pivot divisor, apply gain or loss to the lower tuoke accumulation, then multiply by crossing rate and divide by crossing number to obtain the fixed value.
106
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 proper-crossing entry into qi-periods: apply fixed waxing-and-waning from mean crossing and rotation entry—subtract tuoke and add tuoke to mean-crossing qi-period remainder, advance or retreat the day if full or short, and obtain proper-crossing entry day and remainder in seconds.
107
求正交加時黃道宿度：置正交入氣餘，副之，以乘其日升降分，一百約之，升加降減其副，乃一百乘之，樞法而一，以加其日夜半日度，即正交加時黃道日度及分秒。
To find ecliptic solar degree at proper-crossing hour-of-addition: set proper-crossing qi-period remainder as auxiliary, multiply by that day's ascending–descending parts and simplify per 100, add for ascent or subtract for descent, multiply by 100 and divide by the pivot divisor, add to midnight solar degree for that day, and obtain ecliptic solar degree and parts and seconds at proper-crossing hour-of-addition.
108
求正交加時月離九道宿度：以正交度及分減一百二十五，餘以正交度及分乘之，滿二十四，餘為定差。 以差加黃道宿度，仍計去冬、夏至以來度數乘差，九十而一，所得，依名同異而加減之，滿若不足，進退其度，命如前，即正交加時月離九道宿度及分。
To find nine-path lunar lodge at proper-crossing hour-of-addition: subtract proper-crossing degree and parts from 125, multiply the remainder by proper-crossing degree and parts, and the remainder after dividing by 24 is the fixed difference. Add the difference to the ecliptic lodge degree, calculate degrees since winter or summer solstice and multiply the difference, divide by 90, add or subtract according to same-name or different-name rules, advance or retreat the degree if full or short, count through lodges as before, and obtain nine-path lunar lodge degree and parts at proper-crossing hour-of-addition.
109
推定朔、弦、望加時月離所在度：各置其日加時日躔所在，變從九道，循次相當。 凡合朔加時，月行潛在日下，與太陽同度，是為加時月離宿次； 先置朔、弦、望加時黃道宿度，以正交加時黃道宿度減之，餘以加其正交加時九道宿度，命起正交宿度，算外，即朔、弦、望加時所當九道宿度。 其合朔加時若非正交，則日在黃道、月在九道各入宿度，雖多少不同，考其去極，若應繩准，故云月行潛在日下，與太陽同度。
To derive the lunar degree at hour-of-addition for fixed month-beginnings, quarter-moons, and full moons: for each set where the sun lies at hour-of-addition that day, convert to the nine paths, and proceed in corresponding sequence. At conjunction hour-of-addition the moon travels hidden beneath the sun at the same degree—that is the lunar lodge at hour-of-addition; Set ecliptic lodge at hour-of-addition for month-beginning, quarter-moon, or full moon, subtract ecliptic lodge at proper-crossing hour-of-addition, add the remainder to nine-path lodge at proper-crossing hour-of-addition, count from proper-crossing lodge outward, and obtain the nine-path lodge at hour-of-addition. If conjunction hour-of-addition is not at proper crossing, the sun lies on the ecliptic and the moon on the nine paths at their respective lodge degrees; though they differ in amount, their distance from the pole aligns like a plumb line—hence the moon is said to travel hidden beneath the sun at the same degree.
110
各以弦、望度及分秒加其所當九道宿度，滿宿次去之，命如前，即各得加時九道月離宿次。
For each add quarter-moon or full-moon degree with parts and seconds to the corresponding nine-path lodge degree, discard full lodge cycles, count as before, and obtain the nine-path lunar lodge at hour-of-addition.
111
求定朔夜半入轉：各視經朔夜半入轉，若定朔大餘有進退者，亦加減轉日，不則因經為定。
To find fixed month-beginning entry into rotation at midnight: observe mean month-beginning entry at midnight; if the fixed month-beginning major remainder advances or retreats, adjust rotation days accordingly, otherwise use the mean value as fixed.
112
求次定朔夜半入轉：因定朔夜半入轉，大月加二，小月加一，餘皆四千七百一十六、秒九千四百六，滿轉周日及餘秒去之，即次定朔夜半入轉； 累加一日，去命如前，各得次日夜半轉日及餘秒。
To find the next fixed month-beginning entry into rotation at midnight: start from the current fixed value, add 2 for long months or 1 for short months with remainder 4,716 parts and 9,406 seconds, discard full rotation circuit days and remainder in seconds, and obtain the next fixed month-beginning entry at midnight; add one day cumulatively, discard and count as before, and obtain rotation day and remainder in seconds at midnight for each following day.
113
求月晨昏度：以晨昏乘其日轉定分，樞法而一，為晨轉分； 減轉定分，餘為昏轉分； 乃以朔、弦、望定小餘乘轉定分，樞法而一，為加時分； 以減晨昏轉分，餘為前； 不足覆減，餘為後； 仍前加後減加時月，即晨、昏月所在度。
To find lunar degrees at dawn and dusk: multiply dawn and dusk by that day's fixed rotation parts and divide by the pivot divisor to obtain dawn rotation parts; subtract from fixed rotation parts and the remainder is dusk rotation parts; then multiply fixed minor remainder for month-beginning, quarter-moon, or full moon by fixed rotation parts and divide by the pivot divisor to obtain hour-of-addition parts; subtract from dawn and dusk rotation parts and the remainder is the before portion; if insufficient, borrow and subtract and the remainder is the after portion; add the before portion and subtract the after portion from the moon at hour-of-addition to obtain the degrees where the moon stands at dawn and dusk.
114
求朔、弦、望晨昏定程：各以其朔昏定月減上弦昏定月，為朔後定程； 以上弦昏定月減望日昏定月，為上弦後定程； 以望日晨定月減下弦晨定月，為望後定程； 以下弦晨定月減後朔晨定月，為下弦後定程。
To find fixed dawn-and-dusk intervals for month-beginnings, quarter-moons, and full moons: subtract first-quarter dusk moon from month-beginning dusk moon to obtain the fixed interval after month-beginning; subtract full-moon dusk moon from first-quarter dusk moon for the fixed interval after first quarter; subtract last-quarter dawn moon from full-moon dawn moon for the fixed interval after full moon; subtract the following month-beginning dawn moon from last-quarter dawn moon for the fixed interval after last quarter.
115
求每日轉定度：累計每程相距日轉定分，以減定程為盈； 不足，覆減為縮； 以相距日均其盈縮，盈加縮減每日轉定分，為每日轉定度及分。
To find each day's fixed rotation degree: accumulate fixed rotation parts for the days spanning each interval, subtract from the fixed interval for excess; if insufficient, borrow and subtract for deficit; average excess and deficit over the separated days, add excess and subtract deficit from each day's fixed rotation parts, and obtain each day's fixed rotation degree and parts.
116
求每日晨昏月：因朔、弦、望晨昏月，加每日轉定度及分，盈縮次去之，為每日晨昏月。 凡注曆，自朔日注昏，望後次日注晨。
To find dawn and dusk moon for each day: start from month-beginning, quarter-moon, and full-moon values, add each day's fixed rotation degree and parts, discard by lodge sequence, and obtain dawn and dusk moon for each day. When annotating the calendar, record dusk from month-beginning day and dawn from the day after full moon.
117
已前月度並依九道所推，以究算理之精微。 如求其速要，即依後術求之。
The lunar degrees above all follow derivation through the nine paths, probing the subtlety of computational theory. For a rapid summary, follow the later procedure instead.
118
推天正經朔加時平行月：置歲周，以天正閏餘減之，餘以樞法除之為度，不盡，退除為分秒，即天正經朔加時平行月積度。
To derive mean moon at hour-of-addition for the civil new-year month-beginning: set the year circuit, subtract the civil new-year intercalary remainder, divide the remainder by the pivot divisor for degrees, convert the remainder into parts and seconds, and obtain the accumulated mean moon degree at hour-of-addition.
119
求天正十一月定朔夜半平行月：置天正經朔小余，以平行分乘之，樞法而一為度，不盡，退除為分秒，所得，為加時度； 用減天正經朔加時平行月，即經朔晨前夜半平行月，其定朔有進退者，即以平行度分加減之。
To find mean moon at midnight for the eleventh-month fixed month-beginning at civil new year: set the month-beginning minor remainder, multiply by mean-motion parts and divide by the pivot divisor for degrees, convert the remainder into parts and seconds, and obtain the hour-of-addition degree; subtract from mean moon at civil new-year month-beginning hour-of-addition to obtain mean moon pre-dawn and at midnight; if the fixed month-beginning advances or retreats, add or subtract mean-motion degrees and parts accordingly.
120
即天正十一月定朔晨前夜半平行月積度。
This gives the accumulated mean moon degree pre-dawn and at midnight for the eleventh-month fixed month-beginning at civil new year.
121
求次定朔夜半平行月：置天正定朔夜半平行月，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，滿周天度分去之，即每月定朔晨前夜半平行月積度及分。
To find mean moon at midnight for the next fixed month-beginning: set the civil new-year value, add 35° 80′ 61″ for long months or 22° 43′ 73.5″ for short months, discard full circuit-of-heaven degrees, and obtain accumulated mean moon degree and parts pre-dawn and at midnight for each month.
122
求定望夜半平行月：計定朔距定望日數，以乘平行度及分秒，所得，加其定朔夜半平行月積度及分，即定望夜半平行月積度及分。
To find mean moon at midnight for fixed full moon: count days from fixed month-beginning to fixed full moon, multiply by mean-motion degree with parts and seconds, add to the fixed month-beginning midnight accumulated degree and parts, and obtain the fixed full-moon midnight accumulated degree and parts.
123
求天正定朔夜半入轉：因天正經朔夜半入轉，若定朔大餘有進退者，亦進退之，不則因經而定，即所求年天正定朔晨前夜半入轉及其餘； 以樞法退除為約分及秒，皆一百為母。
To find entry into rotation at midnight for the civil new-year fixed month-beginning: start from mean month-beginning entry at midnight, advance or retreat if the fixed month-beginning major remainder does so, otherwise use the mean value as fixed, and obtain entry into rotation and remainder pre-dawn and at midnight; convert by retreating through the pivot divisor into simplified parts and seconds, using one hundred as the denominator throughout.
124
求定望及次定朔夜半入轉：因天正定朔夜半入轉及分秒，以朔望相距日累加之，滿轉周日二十七及分五十五、秒四十六去之，即各得定望及次定朔晨前夜半入轉日及分秒。
To find entry into rotation at midnight for fixed full moon and the next fixed month-beginning: start from the civil new-year fixed value, add days between month-beginning and full moon cumulatively, discard full rotation circuit of 27 days 55 parts 46 seconds, and obtain rotation day and parts and seconds pre-dawn and at midnight for each.
125
求定朔望夜半定月：置定朔、望夜半入轉分，乘其日增減差，一百約之為分，分滿百為度，增減其下遲疾度，為遲疾定度，遲減疾加夜半平行月，為朔望夜半定月； 以冬至加時黃道日度加而命之，即朔望夜半月離宿次。 其入轉若在四七日下，如求朏朒術入之，即得所求。
To find fixed moon at midnight for month-beginnings and full moons: set midnight rotation parts, multiply by daily increase-and-decrease difference and simplify per 100, convert parts filling 100 into degrees, adjust the lower slow–fast degree to obtain fixed slow–fast degree, subtract slow and add fast to midnight mean moon, and obtain fixed moon at midnight; add to the ecliptic solar degree at winter solstice hour-of-addition and count through lodges to obtain the lunar lodge at midnight for month-beginnings and full moons. If entry into rotation falls within the four sevens, apply the waxing-and-waning procedure to obtain the value sought.
126
求朔望定程：以朔定月減望定月，為朔後定程； 以望定月減次朔定月，即望後定程。
To find fixed intervals for month-beginnings and full moons: subtract full-moon fixed moon from month-beginning fixed moon to obtain the fixed interval after month-beginning; subtract the following month-beginning fixed moon from full-moon fixed moon to obtain the fixed interval after full moon.
127
求朔望轉積：計朔至望轉定分，為朔後轉積； 自望至次朔亦如之，為望後轉積。
To find rotation accumulation for month-beginnings and full moons: calculate fixed rotation parts from month-beginning to full moon for the accumulation after month-beginning; from full moon to the next month-beginning is treated likewise for the accumulation after full moon.
128
求每日夜半月離宿次：各以其朔、望定程與轉積相減，余為程差； 以距後程日數除之，為日差； 加歲轉定分，為每日行度及分； 定程多，加之； 定程少，減之。
To find lunar lodge at midnight for each day: subtract rotation accumulation from the fixed interval for month-beginning or full moon to obtain interval difference; divide by the days remaining in the interval to obtain daily difference; add the year's fixed rotation parts to obtain each day's motion in degrees and parts; if the fixed interval is larger, add; if the fixed interval is smaller, subtract.
129
以每日行度及分累加朔、望夜半宿次，命之，即每日晨前夜半月離宿次。 若求晨昏月，以其日晨昏分乘其日轉定度及分，樞法而一，以加夜半月，即晨昏月所在度及分。 若以四象為程，兼求弦日平行積餘，各依次入之。 若以九終轉定分累加之，依宿次命之，亦得所求。
Add each day's motion in degrees and parts cumulatively to the midnight lodge at month-beginning or full moon and count through lodges to obtain the lunar lodge pre-dawn and at midnight for each day. To find dawn and dusk moon, multiply that day's dawn and dusk parts by fixed rotation degree and parts and divide by the pivot divisor, add to midnight moon, and obtain the degree and parts where the moon stands at dawn and dusk. If using the four phases as intervals, also find mean-motion accumulated remainder for quarter-moon days and enter each in sequence. Alternatively, add fixed rotation parts for the nine-cycle cumulatively and count through lodges to obtain the values sought.
130
步晷漏
Procedure: Gnomon Shadow and Clepsydra.
131
二至限：一百八十二、六十二分。
Solstice limit: 182 days, 62 parts.
132
一象：九十一、三十二分。
One image: 91 days, 32 parts.
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消息法：七千八百七十三。
Message divisor: 7,873.
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辰法：八百八十二半，八刻三百五十三。
Chronogram divisor: 882.5 parts, eight ke 353 parts.
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昏明刻：一百二十九半。
Dusk and dawn ke: 129.5.
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昏明餘數：二百六十四太。
Dusk and dawn remainder number: 264¾.
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冬至陽城晷景：一丈二尺七寸一分半； 初限六十二，末限一百二十六、十二分。
Winter solstice gnomon shadow at Yangcheng: 1 zhang 2 chi 7 cun 1½ fen; initial limit 62, final limit 126 days 12 parts.
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夏至陽城晷景：一尺四寸七分，小分八十； 初限一百二十六、十二分，末限六十二。
Summer solstice gnomon shadow at Yangcheng: 1 chi 4 cun 7 fen, 80 minor parts; initial limit 126 days 12 parts, final limit 62.
139
求陽城晷景入二至後日數：各計入二至後日數，乃如半日之分五十，又以二至約分減之，即入二至後來午中日數及分。
To find days after a solstice for Yangcheng gnomon shadow: calculate days after each solstice, treat as half-day parts of 50, subtract solstice simplified parts, and obtain days and parts from solstice to noon.
140
求陽城晷景入初末限定日及分：置其日中入二至後求日數及分，以其日午中入氣盈縮分盈加縮減之，各如初限已下為在初限； 已上，覆減二至限，餘為入末限定日及分。 求盈縮分，置入二至後來午中日數及分，以氣策及約分除之為氣數，不盡，為入氣以來日數及分； 加其氣數，命以冬、夏至，算外，即其日午中所入氣日及分。 置所入氣日約分，如出朏朒術入之，即得所求。
To find fixed day and parts within initial or final limit for Yangcheng gnomon shadow: set days and parts from solstice to noon for that day, add excess or subtract deficit using noon excess–deficit parts for the qi-period entered, and values at or below the initial limit fall within the initial limit; if above that, borrow and subtract the solstice limit and the remainder gives fixed day and parts within the final limit. To find excess–deficit parts, set days and parts from solstice to noon, divide by qi-divisor and simplified parts for qi-period number, and treat the remainder as days and parts since entering the qi-period; add the qi-period number, count from winter or summer solstice outward, and obtain the qi-period day and parts at noon for that day. Set simplified parts for the qi-period day entered and apply the waxing-and-waning procedure to obtain the value sought.
141
求陽城每日中晷定數：置入二至初、末限定日及分，如冬至後初限、夏至後末限者，以初、末限日及分減一百四十六，餘退一等，為定差； 又以初、末限日及分自相乘，以乘定差，滿六千六百四十五為尺，不滿，退除為寸分，命曰晷差； 以晷差減冬至晷數，即其日陽城午中晷景定數。 如冬至後末限、夏至後初限者，以初、末限日及分減一千二百一十七，餘再退，為定差； 亦以初末限日及分自相乘，以乘定差，滿二萬四千九百三十，餘為尺，不滿，退除為寸分，命曰晷差； 以晷差加夏至晷數，即其日陽城中晷定數。 若以中積求之，即得每日晷影常數。
To find fixed noon gnomon value for each day at Yangcheng: set fixed day and parts within solstice initial or final limit; for winter solstice initial limit or summer solstice final limit, subtract limit day and parts from 146, retreat the remainder one rank for fixed difference; square initial or final limit day and parts, multiply by fixed difference, convert filling 6,645 into chi with remainder as cun and fen, and name the result gnomon difference; subtract gnomon difference from the winter solstice gnomon value to obtain the fixed noon gnomon shadow at Yangcheng for that day. For winter solstice final limit or summer solstice initial limit, subtract limit day and parts from 1,217 and retreat the remainder again for fixed difference; likewise square initial and final limit day and parts, multiply by fixed difference, convert filling 24,930 into chi with remainder as cun and fen, and name the result gnomon difference; add gnomon difference to the summer solstice gnomon value to obtain the fixed noon gnomon value at Yangcheng for that day. If sought by mid-accumulation, one obtains the constant gnomon shadow value for each day.
142
求每日消息定數：以所入氣日及加其氣下中積，一象已下，自相乘； 已上者，用減二至限，餘亦自相乘，皆五因之，進二位元，以消息法除之，為消息常數； 副置常數，用減五百二十九半，餘乘其副，以二千三百五十除之，加於常數，為消息定數。 冬至後為消，夏至後為息。
To find fixed message value for each day: take the qi-period day entered plus its lower mid-accumulation; if below one image, square the value; if above that, subtract the solstice limit and square the remainder; multiply all by five, advance two decimal places, divide by the message divisor, and obtain the message constant; Set the constant as auxiliary, subtract 529.5, multiply the remainder by the auxiliary, divide by 2,350, add to the constant, and obtain the fixed message value. After winter solstice is waning; after summer solstice is waxing.
143
求每日黃道去極度及赤道內外度：置其日消息數，十六乘之，以三百五十三除為度，不滿，退除為分，所得，在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，即每日黃道去極度分度。 又以每日黃道去極度及分，與一象度相減，餘為赤道內、外度。 若去極度少，為日在赤道內； 去極度多，為日在赤道外，即各得所求。 其赤道內外度，為黃、赤道相去度分。
To find daily ecliptic distance from the pole and equator inner–outer degrees: set that day's message number, multiply by 16, divide by 353 for degrees with remainder as parts, add 67° 31′ after spring equinox or subtract 115° 31′ after autumn equinox, and obtain daily ecliptic distance from the pole in degrees and parts. Subtract each day's ecliptic distance from the pole and parts from one image degree to obtain equator inner and outer degrees. If distance from the pole is less, the sun lies inside the equator; if distance from the pole is greater, the sun lies outside the equator, and each value sought is obtained. The equator inner and outer degrees are the degrees and parts separating ecliptic and equator.
144
求每日晨昏分日出入分及半晝分：以每日消息定數，春分後加一千八百五十三少，秋分後減二千九百一十二少，各為每日晨分； 用減樞法，為昏分。 以昏明餘數加晨分，為日出分； 減昏分，為日入分； 以日出分減半法，為晝分。
To find daily dawn and dusk parts, sun-out and sun-in parts, and half-day parts: take each day's fixed message value, add 1,853¼ after spring equinox or subtract 2,912¼ after autumn equinox to obtain dawn parts for each day; subtract from the pivot divisor to obtain dusk parts. Add dusk-and-dawn remainder to dawn parts to obtain sun-out parts; subtract dusk parts to obtain sun-in parts; subtract sun-out parts from the half divisor to obtain day parts.
145
求每日距中度：置每日晨分，三因，進二位，以八千六百九十八除為度，不滿，退除為分，即距子度； 用減半周天，餘為距中度； 又倍距子度，五除，為每更差度及分。
To find daily distance-from-center degree: set dawn parts, multiply by three and advance two decimal places, divide by 8,698 for degrees with remainder as parts, and obtain distance-from-zi degree; subtract from half the circuit of heaven and the remainder is distance-from-center degree; double distance-from-zi degree and divide by five to obtain difference degree and parts for each watch.
146
求夜半定漏：置晨分，進一位，以刻法除為刻，不滿為分，即每日夜半定漏。
To find fixed clepsydra at midnight: set dawn parts, advance one decimal place, divide by ke divisor for ke with remainder as parts, and obtain fixed clepsydra at midnight for each day.
147
求晝夜刻及日出入辰刻：倍夜半定漏，加五刻，為夜刻； 減一百刻，餘為晝刻。 以昏明刻加夜半定漏，命子正，算外，即日出辰刻； 以晝刻加之，命如前，即日入辰刻。
To find day and night ke and sun-out and sun-in chronogram ke: double fixed clepsydra at midnight and add 5 ke for night ke; subtract from 100 ke and the remainder is day ke. Add dusk-and-dawn ke to fixed clepsydra at midnight, count from zi proper outward, and obtain sun-out chronogram ke; add day ke and count as before to obtain sun-in chronogram ke.
148
求更籌辰刻：倍夜半定漏，二十五而一，為籌差刻； 五乘之，為更差刻。 以昏明刻加日入辰刻，即甲夜辰刻； 以更籌差刻累加之，滿辰刻及分去之，各得每更籌所入辰刻及分。
To find watch and tally chronogram ke: double fixed clepsydra at midnight and divide by 25 to obtain tally-difference ke; multiply by five to obtain watch-difference ke. Add dusk-and-dawn ke to sun-in chronogram ke to obtain first-night chronogram ke; add watch and tally difference ke cumulatively, discard full chronogram ke and parts, and obtain chronogram ke and parts entered for each watch and tally.
149
求每日昏明度：置距中度，以其日昏後夜半赤道日度加而命之，即昏中星所格宿次； 又倍距子度，加昏中星命之，即曉中星所格宿次。
To find dusk and dawn degrees for each day: set distance-from-center degree, add post-dusk and midnight equatorial solar degree for that day and count through lodges to obtain the lodge where the dusk culmination star stands; Double distance-from-zi degree, add to the dusk culmination star and count through lodges to obtain the lodge where the dawn culmination star stands.
150
求五更中星：皆以昏中星為初更中星，以每更差加而命之，即乙夜所格宿次； 累加之，各得五更中星所格宿次。
To find culmination stars for the five watches: use the dusk culmination star as the first-watch star, add each watch's difference and count through lodges to obtain the second night's lodge sequence; add cumulatively to obtain the lodge sequence for each of the five watches' culmination stars.
151
求九服距差日：各於所在立表候之，若地在陽城北，測冬至後與陽城冬至晷景同者，累冬至後至其日，為距差日； 若地在陽城南，測夏至後與陽城夏至晷景同者，累夏至後至其日，為距差日。
To find distance-difference days for the nine domains: at each location set up a gnomon and observe; if the place lies north of Yangcheng, observe after winter solstice when the shadow matches Yangcheng's winter solstice shadow and accumulate days from winter solstice to that day as distance-difference days; if the place lies south of Yangcheng, observe after summer solstice when the shadow matches Yangcheng's summer solstice shadow and accumulate days from summer solstice to that day as distance-difference days.
152
求九服晷景； 若地在陽城北冬至前後者，置冬至前後日數，用減距差日，為餘日； 以餘日減一百四十六，餘退一等，為定差； 以餘日自相乘而乘之，滿六千六百四十五除之為尺，不滿，退除為寸分，加陽城冬至晷景，為其地其日中晷常數。 若冬至前後日多於距差日，即減去距差日，余依陽城法求之，各其地其日中晷常數。 若地在陽城南夏至前後者，以夏至前後日數減距差日，為餘日，以減一千二百一十七，餘再退，為定差； 以餘日自相乘而乘之，滿二萬四千九百三十為尺，不滿，退除為寸分，以減陽城夏至晷數，即其地其日中晷常數； 如不及減，乃減去陽城夏至日晷景，餘即晷在表南也。 若夏至前後日多於距差日，即減去距差日，余依陽城法求之，各其地其日中晷常數。 若求中晷定數，先以盈縮分加減之，乃用法求之，即各得其地其日中晷定數。
To find gnomon shadows for the nine domains; if the place lies north of Yangcheng around winter solstice, set days before and after winter solstice, subtract distance-difference days to obtain remainder days; subtract remainder days from 146 and retreat the remainder one rank for fixed difference; square remainder days and multiply, divide by 6,645 for chi with remainder as cun and fen, add Yangcheng winter solstice gnomon shadow, and obtain the constant noon gnomon value for that place. If days before and after winter solstice exceed distance-difference days, subtract distance-difference days and apply the Yangcheng procedure to obtain each place's constant noon gnomon value. If the place lies south of Yangcheng around summer solstice, subtract distance-difference days from days before and after summer solstice for remainder days, subtract from 1,217, and retreat the remainder again for fixed difference; square remainder days and multiply, convert filling 24,930 into chi with remainder as cun and fen, subtract from Yangcheng summer solstice gnomon value, and obtain the constant noon gnomon value for that place; If insufficient to subtract, subtract Yangcheng summer solstice gnomon shadow instead; the remainder indicates the shadow falls south of the gnomon. If days before and after summer solstice exceed distance-difference days, subtract distance-difference days and apply the Yangcheng procedure to obtain each place's constant noon gnomon value. To find fixed noon gnomon values, first apply excess–deficit parts, then use the procedure to obtain the fixed noon gnomon value for each place.
153
求九服所在晝夜漏刻：冬、夏至各於所在下水漏，以定其處二至夜刻數，相減為冬、夏至差刻。 乃置陽城其日消息定數，以其處二至差刻乘之，如陽城二至差刻二十而一，所得，為其地其日消息定數。 乃倍消息定數，進一位，滿刻法約之為刻，不滿為分，乃加減其處二至夜刻，秋分後、春分前，減冬至夜刻； 春分後、秋分前，加夏至夜刻。
To find day and night clepsydra ke for the nine domains: at winter and summer solstices run the water clepsydra at each location to fix solstice night ke and subtract to obtain solstice difference ke. Set Yangcheng's fixed message value for that day, multiply by that place's solstice difference ke proportional to Yangcheng's solstice difference ke of 20, and obtain the fixed message value for that place and day. Double the fixed message value, advance one decimal place, simplify per ke divisor into ke with remainder as parts, then add or subtract that place's solstice night ke—subtract winter solstice night ke after autumn equinox and before spring equinox; after spring equinox and before autumn equinox, add summer solstice night ke.
154
為其地其日夜刻； 用減一百刻，餘為晝刻。 求日出入辰刻及距中度五更中星，皆依陽城法。
This gives day and night ke for that place and day; subtract from 100 ke and the remainder is day ke. To find sun-out and sun-in chronogram ke, distance-from-center degree, and five watches' culmination stars, all follow the Yangcheng procedure.