1
昔者聖人因天道以授人時，厘百工以熙庶政，步推之法，其來尚矣。 自漢太初迄于前宋，治曆者奚啻七十餘家，大概或百年或數十年，率一易焉。 蓋日月五星盈縮進退，與夫天運，至不齊也，人方制器以求之，以俾其齊，積寡至多，不能無爽故爾。
Long ago the sages took Heaven's Way to give mankind the seasons, set the hundred offices in order to brighten common affairs—and the arts of stepwise reckoning reach back to a remote antiquity. From the Han Taichu era down to the former Song, calendar-makers numbered more than seventy houses; as a rule, every century or few decades brought a new system. Sun, moon, and five stars swell and shrink, advance and retreat; Heaven's motion is never uniform. Men fashion reckonings to force alignment—but piling small errors into large ones, some discrepancy is inevitable.
2
金有天下百餘年，曆惟一易。 天會五年，司天楊級紹造《大明曆》，十五年春正月朔，始頒行之。 其法，以三億八千三百七十六萬八千六百五十七為曆元，五千二百三十為日法。 然其所本，不能詳究，或曰因宋《紀元曆》而增損之也。 正隆戊寅三月辛酉朔，司天言日當食，而不食。 大定癸巳五月壬辰朔，日食，甲午十一月甲申朔，日食，加時皆先天。 丁酉九月丁酉朔，食乃後天。 由是占候漸差，乃命司天監趙知微重修《大明曆》，十一年曆成。 時翰林應奉耶律履亦造《乙未曆》。 二十一年十一月望，太陰虧食，遂命尚書省委禮部員外郎任忠傑與司天曆官驗所食時刻分秒，比校知微、履及見行曆之親疏，以知微曆為親，遂用之。 明昌初，司天又改進新曆，禮部郎中張行簡言：「請俟他日月食，覆校無差，然後用之。」 事遂寢。 是以終金之世，惟用知微曆，我朝初亦用之，後始改《授時曆》焉。 今其書存乎太史，采而錄之，以為《曆志》。
Jin held the realm for a little over a century and changed its calendar only once. In Tianhui year 5 the Director of Heaven Yang Ji completed the Grand Clarity Calendar; in spring of year 15, on the new moon of the first month, it was first promulgated. Its constants take 383,768,657 as the calendar origin and 5,230 as the day divisor. Its sources cannot be traced in full; some say it was adapted from the Song Era Origin Calendar with additions and cuts. Zhenglong, wuyin year, third month, on xinyou, the new moon: the calendar office predicted an eclipse, but none occurred. Dading guisi, fifth month, on renchen, the new moon—a solar eclipse; jiawu, eleventh month, on jiashen, the new moon—another eclipse; in both cases the predicted time came before the event. Dingyou, ninth month, on dingyou, the new moon—the eclipse came after the predicted hour. Prognostication and observation drifted apart; the Directorate ordered Zhao Zhiwei to revise the Grand Clarity Calendar, and in eleven years the revision was done. At the same time Hanlin Attendant Yelü Lü also devised the Yiwei Calendar. In year 21, at the full moon of the eleventh month, the moon was eclipsed; the Ministry ordered Ren Zhongjie of the Rites Bureau and the calendar officers to verify the hour, minute, and second of the eclipse and compare Zhao Zhiwei's, Yelü Lü's, and the current calendar—Zhiwei's proved closer, and his calendar was adopted. Early in Mingchang the calendar office proposed yet another revision; Zhang Xingjian of the Rites Bureau said, "Wait for another solar eclipse, re-check until there is no discrepancy, and only then adopt it." The proposal was shelved. Through the whole Jin age only Zhao Zhiwei's calendar was used; our dynasty used it at first as well, then replaced it with the Season-Granting Calendar. Its text still rests with the Grand Astrologer; we have gathered and recorded it here as the Calendar Treatise.
3
步氣朔第一
Procedures for qi and new moon, part one.
4
演紀：上元甲子距今大定庚子，八千八百六十三萬九千六百五十六年。
Era computation: from upper origin jiazi to the present Dading gengzi, 88,639,656 years.
5
日法：五千二百三十分。
Day divisor: 5,230 parts.
6
歲實：一百九十一萬二百二十四分。
Tropical year dividend: 1,910,224 parts.
7
通餘：二萬七千四百二十四分。
Common remainder: 27,424 parts.
8
朔實：一十五萬四千四百四十五分。
New-moon dividend: 154,445 parts.
9
通閏：五萬六千八百八十四分。
Common intercalation: 56,884 parts.
10
歲策：三百六十五日，餘一千二百七十四分。
Year stride: 365 days, remainder 1,274 parts.
11
朔策：二十九日，餘二千七百七十五分。
New-moon stride: 29 days, remainder 2,775 parts.
12
氣策：一十五日，餘一千一百四十二分，六十秒。
Qi stride: 15 days, remainder 1,142 parts, 60 seconds.
13
望策：一十四日，餘四千二分，四十五秒。
Full-moon stride: 14 days, remainder 4,002 parts, 45 seconds.
14
象策：七日，餘二千一分，二十二秒半。
Phase stride: 7 days, remainder 2,001 parts, 22 and a half seconds.
15
沒限：四千八十七分，三十秒。
Submergence limit: 4,087 parts, 30 seconds.
16
朔虛分：二千四百五十五分。
New-moon void fraction: 2,455 parts.
17
旬周：三十一萬三千八百分。
Ten-day circuit: 313,800 parts.
18
紀法：六十。
Era divisor: 60.
19
秒母：九十。
Second parent: 90.
20
求天正冬至
To find the winter solstice of the celestial standard year.
21
置上元甲子以來積年，歲實乘之，為通積分。 滿旬周去之，不盡以日法約之為日，不盈為餘，命甲子算外，即所求天正冬至日大小餘。
Set the accumulated years since upper origin jiazi, multiply by the tropical year dividend to obtain the general accumulated parts. Cast out full ten-day circuits; divide the remainder by the day divisor for days, with the unfilled portion as the remainder; count outside jiazi—that yields the greater and lesser remainders for the celestial-standard winter solstice.
22
求次氣
To find successive qi.
23
置天正冬至大小餘，以氣策累加之，秒盈秒母從分，分滿日法從日，即得次氣日及餘秒。
Set the winter solstice greater and lesser remainders and add the qi stride cumulatively; when seconds fill the second parent they carry into parts, when parts fill the day divisor they carry into days—thus each successive qi day with its remainder and seconds.
24
求天正經朔
To find the canonical new moon of the celestial standard year.
25
以朔實去通積分，不盡為閏餘，以減通積分為朔積分。 滿旬周去之，不盡如日法而一為日，不盈為餘，即所求天正經大小餘也。
Divide the general accumulated parts by the new-moon dividend; the remainder is the intercalary remainder; subtract it from the general accumulated parts to obtain the new-moon accumulated parts. Cast out full ten-day circuits; divide the remainder by the day divisor for days, with the unfilled portion as the remainder—that yields the greater and lesser remainders for the celestial-standard canonical new moon.
26
求弦望及次朔
To find quarter-moon, full-moon, and successive new moons.
27
置天正經朔大小餘，以象策累加之，即各得弦、望及次朔經日及餘秒也。
Set the canonical new moon greater and lesser remainders and add the phase stride cumulatively—each addition yields the quarter-moon, full-moon, and successive canonical days with their remainders and seconds.
28
求沒日
To find submergence days.
29
置有沒之恆氣小餘，如沒限以上，為有沒之氣。 以秒母乘之，內其秒，用減四十七萬七千五百五十六，余滿六千八百五十六而一，所得並恆氣大餘，命為沒日。
Take the lesser remainder of the constant qi that bears submergence; if it reaches the submergence limit or above, that qi has a submergence day. Multiply by the second parent and include the seconds; subtract from 477,556; divide the remainder by 6,856 and add to the constant qi greater remainder—name the result as the submergence day.
30
求滅日
To find extinction days.
31
置有滅之朔小餘，經朔小餘不滿朔虛分者。 六因之，如四百九十一而一，所得並經朔大餘，命為滅日。
Take the lesser remainder of the new moon that bears extinction—when the canonical new moon lesser remainder falls short of the new-moon void fraction. Multiply by six, divide by 491, add to the canonical new moon greater remainder—name the result as the extinction day.
32
步卦候第二
Procedure 2: Hexagrams and Period-Markers.
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候策：五，餘三百八十，秒八十。
Hou rule: 5 days, remainder 380 parts, 80 seconds.
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卦策：六，餘四百五十七，秒六。
Hexagram rule: 6 days, remainder 457 parts, 6 seconds.
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貞策：三，餘二百二十八，秒四十六。
Zhen rule: 3 days, remainder 228 parts, 46 seconds.
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秒母：九十。
Second denominator: 90.
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辰法：二千六百一十五。
Chen divisor: 2,615.
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半辰法：一千三百七半。
Half-chen divisor: 1,307½.
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刻法：三百一十三，秒八十。
Ke divisor: 313 parts, 80 seconds.
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辰刻：八，十百四分，秒六十。
Chen-ke: 8 parts, 104 fen, 60 seconds.
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半辰刻：四，五十二分，秒三十。
Half-chen ke: 4 parts, 52 fen, 30 seconds.
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秒母：一百。
Second denominator: 100.
43
求七十二候
To find the seventy-two hou.
44
置中氣大小餘，命之為初候，以候策累加之，即次候及末候也。
Take the mid-qi major and minor remainders as the first hou; by cumulative addition of the hou rule you reach each subsequent hou and the final hou.
45
求六十四卦
To find the sixty-four hexagrams.
46
置中氣大小餘，命之為公卦； 以卦策累加之，得辟卦； 又加之，得侯內卦。 以貞策加之，得節氣之初，為侯外卦； 又以貞策加之，得大夫卦。 又以卦策加之，為卿卦。
Take the mid-qi major and minor remainders and designate them as the duke hexagram; by cumulative addition of the hexagram rule you obtain the lord hexagram; add again to obtain the marquis inner hexagram. add the zhen rule to reach the start of the qi-period, the marquis outer hexagram; add the zhen rule again to obtain the grandee hexagram. add the hexagram rule again for the minister hexagram.
47
求土王用事
To find the Earth ruler's dominion.
48
以貞策減四季中氣大小餘，即土王用事日也。
Subtract the zhen rule from each season's mid-qi major and minor remainders to obtain the Earth ruler's dominion day.
49
求發斂
To find expansion and contraction.
50
置小餘，以六因之，如辰法而一為辰。 如不盡，以刻法除之為刻。 命子下算外，即得加時所在辰刻及分。 如加半辰法，即命子刻初。
Set the minor remainder, multiply by six, and divide by the chen divisor to obtain the chen number. If any remainder remains, divide it by the ke divisor to obtain ke. Count outward from zi below the stem sequence to obtain the double-hour, ke, and fen of the added time. If the half-chen divisor is included, count from the beginning of the zi ke.
51
步日躔第三
Procedure 3: Solar Motion along the Ecliptic.
52
周天分：一百九十一萬二百九十三分，五百三十秒。
Circuit-of-heaven parts: 1,910,293 parts, 530 seconds.
53
歲差：六十九，五百三十秒。 秒母一萬。
Precession: 69 parts, 530 seconds. Second denominator: 10,000.
54
周天諶：三百六十五度，二十五分，六十八秒。
Calibrated circuit of heaven: 365° 25′ 68″.
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象限：九十一，三十一分，九秒。
Quadrant limit: 91° 31′ 9″.
56
二十四氣日積度及盈縮
Daily accumulated degrees and excess–deficit for the twenty-four qi-periods.
57
求每日盈縮朓朒
To find the daily excess–deficit and tuoke.
58
各置其氣損益率，求盈縮之損益，求朒磕用朒磕之損益。 六因，如象限而一，為氣中率。 與後氣中率相減，為合差。 半合差加減其氣中率，為初末泛率。 至後：加初，減末。 分後：減初，加末。 又置合差，六因，如象限而一，為日，半之，加減初未泛率，為初末定率。 至後：減初，加末。 分後：加初，減末。 以日差累加減其氣初末定率，為每日損益分。 至後減，分後加。 各以每日損益分加減氣下盈縮、朒磕，為每日盈縮、朒磕二分前一氣元後率相減為合差者，皆用前氣合差。
Set each qi-period's gain-and-loss rate to derive excess–deficit gain and loss, and use the tuoke gain-and-loss rate for tuoke. Multiply by six and divide by the quadrant limit to obtain the mid-qi rate. Subtract from the next qi-period's mid-qi rate to obtain the combined difference. Add or subtract half the combined difference from the mid-qi rate to obtain the initial and final general rates. After a solstice: add to the initial, subtract from the final. After an equinox: subtract from the initial, add to the final. Set the combined difference, multiply by six, divide by the quadrant limit for days; halve 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, add to the final. After an equinox: add to the initial, subtract from the final. By cumulative daily difference add and subtract the qi-period's initial and final fixed rates to obtain the daily gain-and-loss parts. After a solstice subtract; after an equinox add. Apply each day's gain-and-loss parts to adjust the qi-period's excess–deficit and tuoke accumulations for the daily values; whenever the combined difference derives from subtracting the preceding qi's original and following rates, use that preceding qi's combined difference.
59
求經朔弦望入氣
To find qi entry for major new moon, first quarter, full moon, and last quarter.
60
置天正閏餘，以日法除為日，不滿為餘，如氣策以下，以減氣策，為入大雪氣。 以上去之，餘亦減氣策，為入小雪氣。 即得天正經朔入氣日及餘也。 以象策累加之，滿氣策去之，即得弦、望入次氣日及餘。 因加，後朔入氣日及餘也。
Set the Heavenly Origin intercalary remainder; divide by the day divisor for days and keep the remainder; if below the qi rule, subtract from the qi rule to obtain entry into Major Snow. Remove what is above; subtract the qi rule from the remainder again to obtain entry into Minor Snow. This yields the day and remainder of the civil new year’s mean conjunction within its qi phase. Add the image strategy repeatedly; discard a full qi strategy each time to obtain the day and remainder within the next qi for first quarter and full moon. Continue adding likewise to obtain each later conjunction’s day and remainder within its qi phase.
61
求每日損益、盈縮、朒磕
Procedure to find each day’s deficit and surplus, expansion and contraction, and deficit–surfeit.
62
以日差益加減損加減其氣初損益率，為每日損益率。 馴積損益其氣盈縮、朒磕積，為每日盈、朒磕積。
Apply the daily difference to add or subtract the initial qi-phase deficit–surplus rate and obtain the daily deficit–surplus rate. Accumulate stepwise to adjust the qi-phase expansion–contraction and deficit–surfeit totals into the daily expansion and deficit–surfeit accumulations.
63
求經朔弦望入氣朒磕定數
Procedure to find the fixed deficit–surfeit number for mean conjunction and waxing and full moon within qi.
64
各以所入恆氣小餘，以乘其日損益率，如日法而一，以所損益其下朒磕積為定數。
Multiply the entered constant qi’s minor remainder by the daily deficit–surplus rate, divide by the day divisor, and apply the result to the underlying deficit–surfeit accumulation to obtain the fixed number.
65
赤道宿度
Equatorial lodge degrees.
66
斗二十五度牛七度少女十一度少虛九度少秒六十八危十五度半室十七度壁八度太
Dipper 25°; Ox 7° minor; Maid 11° minor; Void 9° minor, 68″; Rooftop 15° half; Encampment 17°; Wall 8° major.
67
右北方七宿九十四度秒六十八
At right: the northern seven lodges total 94° 68″.
68
奎十六度半婁十二度胃十五度昴十一度少畢十七度少觜半度參十度半
Striding Man 16° half; Bond 12°; Stomach 15°; Hairy Head 11° minor; Net 17° minor; Beak 0.5°; Three Stars 10° half.
69
右西方七宿八十三度井三十三度少鬼二度半柳十三度太星六度太張十七度少翼十八度太軫十七度
At right: the western seven lodges 83°; Well 33° minor; Ghost 2° half; Willow 13° major; Star 6° major; Extended Net 17° minor; Wings 18° major; Chariot Axis 17°.
70
右南方七宿一百九度少角十二度亢九度少氐一十六度房五度太心六度少尾十九度少箕十度半
At right: the southern seven lodges 109° minor; Horn 12°; Gullet 9° minor; Base 16°; Chamber 5° major; Heart 6° minor; Tail 19° minor; Winnowing Basket 10° half.
71
右東方七宿七十九度
At right: the eastern seven lodges total 79°.
72
求冬至赤道日度
Procedure to find the winter solstice equatorial solar degree.
73
置通積分，以周天分去之，餘日法而一為度，不滿退除為分秒。 以百為母。 命起赤道虛宿七度外去之，至不滿宿，即所求年天正冬至加時日躔赤道度及分秒。
Set the general accumulated parts, subtract the circuit-of-heaven parts, divide the remainder by the day divisor for degrees, and convert the remainder into minutes and seconds. Use one hundred as the denominator. Count from beyond Void lodge 7° on the equator and discard full lodges; the lodge not yet filled gives the winter solstice hour-of-addition equatorial solar degree and minutes and seconds for the civil new year of the year sought.
74
求春分夏至秋分赤道日度
Procedure to find the equatorial solar degrees at spring equinox, summer solstice, and autumn equinox.
75
置天正冬至加時赤道日度，累加象限，滿赤道宿次去之，即各得春分、夏至、秋分加時日在宿度及分秒。
Set the winter solstice equatorial degree at hour-of-addition, add the image quadrant repeatedly, and discard full equatorial cycles to obtain the equatorial lodge degrees and minutes and seconds at hour-of-addition for spring equinox, summer solstice, and autumn equinox.
76
求四正赤道宿積度
Procedure to find the accumulated equatorial degrees after the four cardinals.
77
置四正赤道宿全度，以四正赤道日度及分減之，餘為距後度。 以赤道宿度累加之，各得四正後赤道宿積度及分。
Set the full width of each cardinal equatorial lodge, subtract the cardinal equatorial solar degree and minutes, and the remainder is the post-cardinal distance. Add equatorial lodge widths cumulatively to obtain the accumulated equatorial degree and minutes after each cardinal.
78
求赤道宿積度入初末限
Procedure to find whether an equatorial accumulated degree falls in the initial or terminal limit.
79
視四正後赤道宿積度及分，在四十五度六十五分秒五十四半以下為入初限，以上者用減象限，餘為入末限。
If the post-cardinal equatorial accumulation is 45° 65′ 54.5″ or less, it is in the initial limit; otherwise subtract it from the image quadrant and the remainder is in the terminal limit.
80
求二十宿黃道度
Procedure to find the ecliptic degrees of the twenty-eight lodges.
81
以四正後赤道宿入初末限度及分，減一百一度，余以初末限度及分乘之，進位，滿百為分，分滿百為度。 至後以減、分後以加赤道宿積度，為其宿黃道積度。 以前宿黃道積度減之其四正之宿，先加象限，然後前宿減之。 為其宿黃道度及分。 其分就約為太、半、少。
Subtract the post-cardinal equatorial limit from 101°, multiply by the limit’s degrees and minutes, carry as needed, and convert hundreds into minutes and then into degrees. Subtract after winter solstice and add after the equinoxes to convert equatorial accumulation into ecliptic accumulation for each lodge. Subtract the prior lodge’s ecliptic accumulation; for the four cardinal lodges, add the image quadrant first, then subtract the prior lodge. The result is that lodge’s ecliptic degree and minutes. Round the minutes to major, half, and minor fractions.
82
黃道宿度
Ecliptic lodge degrees.
83
斗二十三度牛七度女十一度虛九度少秒六十八危十六度室十八度少壁九度半
Dipper 23°; Ox 7°; Maid 11°; Void 9° minor, 68″; Rooftop 16°; Encampment 18° minor; Wall 9° half.
84
右北方七宿九十四度六十八秒
At right: the northern seven lodges total 94° 68″.
85
奎十七度太婁十二度太胃十五度半昴十一度畢十六度半觜半度參九度太
Striding Man 17° major; Bond 12° major; Stomach 15° half; Hairy Head 11°; Net 16° half; Beak 0.5°; Three Stars 9° major.
86
右西方七宿八十三度太一百七十七、七十五、六十八
At right: the western seven lodges 83° major; cumulative 177°, 75′, 68″.
87
井三十度半鬼二度半柳十三度少星六度太張十七度太翼二十度軫十八度半
Well 30° half; Ghost 2° half; Willow 13° minor; Star 6° major; Extended Net 17° major; Wings 20°; Chariot Axis 18° half.
88
右南方七宿一百九度少二百八十七、六十八
At right: the southern seven lodges 109° minor; cumulative 287°, 68″.
89
角十二度太亢九度太氐十六度少房五度太心六度尾十八度少箕九度半
Horn 12° major; Gullet 9° major; Base 16° minor; Chamber 5° major; Heart 6°; Tail 18° minor; Winnowing Basket 9° half.
90
右東方七宿七十八度少三百六十五、二十五、六十八
At right: the eastern seven lodges 78° minor; cumulative 365°, 25′, 68″.
91
前黃道宿度，依今曆歲差所在算定。 如上考往古，下驗將來，當據歲差，每移一度，依術推變當時宿度，然後可步七曜，知其所在。
The ecliptic lodge degrees above are determined by the present calendar’s precessional position. To check the past and test the future, rely on precession: whenever it shifts a degree, recalculate contemporary lodge degrees by the procedure—only then can the seven luminaries be stepped and their places known.
92
求天正冬至加時黃道日度
Procedure to find the ecliptic solar degree at civil new-year winter solstice hour-of-addition.
93
以冬至加時赤道日度及分秒，減一百一度，餘以冬至赤度及分秒乘之，進位，滿百為分，分滿百為度。 命曰黃赤道差。 用減冬至加時赤道日度及分秒，即所求年天正冬至加時黃道日度及分秒。
Subtract 101° from the winter solstice equatorial degree at hour-of-addition, multiply the remainder by that equatorial degree and its minutes and seconds, carry, and convert hundreds into minutes and then degrees. This is called the ecliptic–equator difference. Subtract that difference from the winter solstice equatorial degree at hour-of-addition to obtain the ecliptic solar degree and minutes and seconds at civil new-year winter solstice for the year sought.
94
求二十四氣加時黃道日度
Procedure to find the ecliptic solar degrees at hour-of-addition for all twenty-four qi.
95
置所求年冬至日躔黃赤道差，以次年黃赤道差減之，餘以所求氣數乘之，二十四而一，所得以加其氣中積及約分，又以其氣初日盈縮數盈加縮減之，用加冬至加時黃道日度，依宿次去之，即各得其氣加時黃道日躔宿度及分秒。 如其年冬至加時赤道宿度空分秒在歲差以下者，即加前宿全度，然後求黃赤道差，餘依術算。
Take the winter solstice ecliptic–equator difference for the year sought, subtract the next year’s difference, multiply the remainder by the qi number and divide by twenty-four, add to the qi’s central accumulation and approximate parts, adjust by the first-day expansion–contraction (surplus add, contraction subtract), add to the winter solstice ecliptic degree at hour-of-addition, and discard full lodge cycles to obtain each qi’s ecliptic solar lodge at hour-of-addition. If the winter solstice equatorial lodge at hour-of-addition has fractional minutes and seconds below the precessional threshold, add the prior lodge’s full width before computing the ecliptic–equator difference; otherwise follow the procedure as above.
96
求二十四氣每日晨前夜半黃道日度
To find the daily ecliptic solar degree before dawn and at midnight for each of the twenty-four qi.
97
副置其氣小餘，以其氣初日損益率乘之，盈縮之損益。 萬約之為分，應益者盈加縮減，應損者盈減縮加其副，日法除之為度，不滿退除為分秒，以減其氣加時黃道日度，即各得其氣初日晨前夜半黃道日度。 每日加一度，以百約每日損益率，盈縮之損益。 應益者盈加縮減，應損者盈減縮加，為每日晨前夜半黃道日度及分秒。
Set aside the qi's minor remainder, multiply by that qi's first-day decrease-increase rate, and apply excess-and-deficit to the decrease and increase. Reduce by ten thousand to obtain fen; for increase, excess-add and deficit-subtract; for decrease, excess-subtract and deficit-add the subsidiary; divide by the day factor for degrees, and any remainder steps back into fen and miao; subtract from that qi's added-hour ecliptic solar degree to obtain each qi's first-day ecliptic solar degree before dawn and at midnight. Add one degree each day, reduce the daily decrease-increase rate by one hundred, and apply excess-and-deficit to the decrease and increase. For increase, excess-add and deficit-subtract; for decrease, excess-subtract and deficit-add — yielding the daily ecliptic solar degree before dawn and at midnight with fen and miao.
98
求每日午中黃道日度
To find the daily noon ecliptic solar degree.
99
置一萬分，以所入氣日盈縮損益率，應益者盈加縮減，應損者盈減縮加，皆加減損益率，餘半之，滿百為分，不滿為秒，以加其日清晨前夜半黃道日度，即其日午中日躔黃道宿度及分秒。
Set ten thousand fen, take the entered qi's daily excess-deficit decrease-increase rate, and for increase excess-add and deficit-subtract, for decrease excess-subtract and deficit-add; add and subtract the decrease-increase rate throughout, halve the remainder, convert hundreds to fen and the remainder to miao, and add to that day's early-morning before-dawn ecliptic solar degree to obtain the noon solar progression ecliptic lodge degree with fen and miao.
100
求每日午中黃道積度
To find the daily noon accumulated ecliptic degree.
101
以二至加時黃道日度，距至所求日午中黃道日度，為入二至後黃道積度及分秒。
From the solstice added-hour ecliptic solar degree, measure to the sought day's noon ecliptic solar degree to obtain the accumulated ecliptic degree after the solstice, with fen and miao.
102
每日午中黃道入初末限
Daily noon ecliptic entry into initial and final limits.
103
視二至後黃道積度，在四十三度一十二分秒八十七以下為初限，以上用減象限，餘為入末限。 其積度滿象限去之，為二分後黃道積度，在四十八度一十八分秒二十二以下為初限，以上用減象限，餘為入末限。
For accumulated ecliptic degree after a solstice: at 43°12′87″ or below, it is the initial limit; above that, subtract the quadrant limit and the remainder is entry into the final limit. When the accumulated degree fills the quadrant limit, remove it to obtain the post-equinox accumulated ecliptic degree; at 48°18′22″ or below it is the initial limit; above that, subtract the quadrant limit and the remainder is entry into the final limit.
104
求每日午中赤道日度
To find the daily noon equatorial solar degree.
105
以所求日午中黃道積度，入至後初限，分後末限，度及分秒，進三位，加二十萬二千五十少，開平方除之，所得減去四百四十九半，余在初限者，直以二至赤道日度加而命之。 在末限者，以減象限，餘以二分赤道日度加而命之。 即每日午中赤道日度。 以所求日午中黃道積度，入至後末限，分後初限，度及分秒，進三位，用減三十萬三千五十少，開平方除之，所得，以減五百五十半，其在初限者，以所減之餘，直以二分赤道日度加而命之。 在末限者，以減象限，餘以二至赤道日度加而命之。 即每日午中赤道日度。
Take the sought day's noon accumulated ecliptic degree — after a solstice in the initial limit, after an equinox in the final limit — with degrees, fen, and miao; advance three places, add 202,050 and a fraction less, take the square root and divide; subtract 449.5; if in the initial limit, add the two-solstice equatorial solar degree directly and assign the mansions. If in the final limit, subtract the quadrant limit, add the two-equinox equatorial solar degree to the remainder, and assign the mansions. That yields the daily noon equatorial solar degree. Take the sought day's noon accumulated ecliptic degree — after a solstice in the final limit, after an equinox in the initial limit — with degrees, fen, and miao; advance three places, subtract 303,050 and a fraction less, take the square root and divide; subtract 550.5; if in the initial limit, add the two-equinox equatorial solar degree to the remainder and assign the mansions. If in the final limit, subtract the quadrant limit, add the two-solstice equatorial solar degree to the remainder, and assign the mansions. That yields the daily noon equatorial solar degree.
106
太陽黃道十二次入宮宿度，雨水，危十三度三十九分五十秒外，入衛分，陬訾之次，辰在亥。 春分，奎二度三十五分八十五秒外，入魯分，降婁之次，辰在戌。 穀雨，胃四度二十四分三十三秒外，入趙分，大樑之次，辰在酉。 小滿，畢七度九十六分六秒外，入晉分，實沈之次，辰在申。 夏至，井九度四十七分一十秒外，入秦分，鶉首之次，辰在未。 大暑，柳四度九十五分一十六秒外，入周分，鶉火之次，辰在午。 處暑，張十五度五十六分三十五秒外，入楚分，鶉尾之次，辰在巳。 秋分，軫十度四十四分五秒外，入鄭分，壽星之次，辰在辰。 霜降，氐一度七十七分七十七秒外，入宋分，大火之次，辰在卯。 小雪，尾三度九十七分九十二秒外，入燕分，析木之次，辰在寅。 冬至，斗四度三十六分六十六秒外，入吳越分，星紀之次，辰在醜。 大寒，女二度九十一分九十一秒外，入齊分，玄枵之次，辰在子。
Solar ecliptic twelve-lodge palace-entry degrees: Rain Water — beyond 13°39′50″ of Wei, enters the Wei allotment, Zouzi station, sign at hai. Spring Equinox — beyond 2°35′85″ of Kui, enters the Lu allotment, Jianglou station, sign at xu. Grain Rain — beyond 4°24′33″ of Stomach, enters the Zhao allotment, Daliang station, sign at you. Lesser Fullness — beyond 7°96′6″ of Net, enters the Jin allotment, Shishen station, sign at shen. Summer Solstice — beyond 9°47′10″ of Well, enters the Qin allotment, Chunshou station, sign at wei. Greater Heat — beyond 4°95′16″ of Willow, enters the Zhou allotment, Chunhuo station, sign at wu. End of Heat — beyond 15°56′35″ of Extended Net, enters the Chu allotment, Chunwei station, sign at si. Autumn Equinox — beyond 10°44′5″ of Chariot, enters the Zheng allotment, Shouxing station, sign at chen. Frost Descent — beyond 1°77′77″ of Root, enters the Song allotment, Dahu station, sign at mao. Lesser Snow — beyond 3°97′92″ of Tail, enters the Yan allotment, Ximu station, sign at yin. Winter Solstice — beyond 4°36′66″ of Dipper, enters the Wu-Yue allotment, Xingji station, sign at chou. Greater Cold — beyond 2°91′91″ of Woman, enters the Qi allotment, Xuanxiao station, sign at zi.
107
求入宮時刻
To find the time of entering the palace.
108
各置入宮宿度及分秒，以其日晨前夜半日度減之，相近一度之間者求之。 餘以日法乘其分，其秒從於下，亦通乘之，為實； 以其日太陽行分為法，實如法而一，所得，依發斂加時求之，即得其日太陽入宮時刻及分秒。
Set each palace-entry lodge degree with fen and miao, subtract that day's before-dawn solar degree, and find where the remainder falls within one degree. Multiply the remainder's fen by the day factor, carry the miao below and multiply through likewise to form the dividend; use that day's solar motion fen as divisor; divide; then seek by emission-and-absorption timing to obtain that day's solar palace-entry time with fen and miao.
109
○步晷漏第四
○ Procedure for Gnomon and Clepsydra, Fourth.
110
中限：一百八十二日，六十二分，一十八秒。
Central limit: 182 days, 62 fen, 18 miao.
111
冬至初限，夏至末限：六十二日，二十分。
Winter-solstice initial limit, summer-solstice final limit: 62 days, 20 fen.
112
夏至初限，冬至末限：一百二十日，四十二分。
Summer-solstice initial limit, winter-solstice final limit: 120 days, 42 fen.
113
冬至地中晷影常數：一丈二尺八寸三分。
Winter-solstice geocentric gnomon-shadow constant: 1 zhang 2 chi 8 cun 3 fen.
114
夏至地中晷影常數：一尺五寸六分。
Summer-solstice geocentric gnomon-shadow constant: 1 chi 5 cun 6 fen.
115
周法：一千四百二十八。
Circumference factor: 1,428.
116
內外法：一萬八百九十六。
Inner-outer factor: 10,896.
117
半法：二千六百一十五。
Half factor: 2,615.
118
日法：四分之三：三千九百二十二半。
Day factor, three-fourths: 3,922.5.
119
日法：四分之一：一千三百七半。
Day factor, one-fourth: 1,307.5.
120
昏明分：一百三十分，七十五秒。
Dusk-and-dawn fen: 130 fen, 75 miao.
121
昏明刻：二刻，一百五十六分，九十秒。
Dusk-and-dawn ke: 2 ke, 156 fen, 90 miao.
122
刻法：三百一十三分，八十秒。
Ke factor: 313 fen, 80 miao.
123
秒母：一百。
Miao mother: 100.
124
求午中入氣中積
To find the noon mid-accumulation within qi.
125
置所求日大餘及半法，以所入之氣大小餘減之，為其日午中入氣。 以加其氣中積，為其日午中中積。 小餘以日法除為約分。
Set the sought day's greater remainder and half factor, subtract the entered qi's greater and lesser remainders, and obtain that day's noon entry into qi. Add that to the qi mid-accumulation to obtain the day's noon mid-accumulation. Divide the lesser remainder by the day factor to obtain approximate fen.
126
求二至後午中入初末限
To find noon entry into initial and final limits after the two solstices.
127
置午中中積及分，如中限以下，為冬至後。 以上去中限，為夏至後。 其二至後，如在初限以下，為初限。 以上覆減中限，餘為入末限也。
Set the noon mid-accumulation and fen; at or below the central limit counts as after winter solstice. If above, subtract the central limit — after summer solstice. Within each solstice period, at or below the initial limit counts as the initial limit. If above, cover and subtract the central limit; the remainder is entry into the final limit.
128
求午中晷影定數
To find the noon gnomon-shadow fixed number.
129
視冬至後初限、夏至後末限，百通日，內分，自相乘，副置之。 以一千四百五十除之，所得加五萬三百八十，折半限分並之； 除其副為分。 分滿十為寸，寸滿十為尺，用減冬至地中晷影常數，為所求晷影定數。 視夏至後初限、冬至後末限，百通日，內分，自相乘為上位。 下置入限分，以二百二十五乘，百約之，加一十九萬八千七十五為法。 夏至前後半限以上者，減去半限，列於上位。 下位置半限。 各百通日，內分，先相減，後相乘。 以七千七百除之，所得以加其法反除上位，為分，分滿十為寸，寸滿十為尺，用加夏至地中晷影常數，為所求晷影定數。
After winter solstice in the initial limit or after summer solstice in the final limit: multiply the day count by one hundred, include the fen, square the product, and set it aside. Divide by 1,450, add the quotient to 50,380, and combine with half the limit fen; divide the subsidiary by that sum to obtain fen. Ten fen make one cun and ten cun one chi; subtract from the winter-solstice geocentric gnomon-shadow constant to obtain the sought gnomon-shadow fixed number. After summer solstice in the initial limit or after winter solstice in the final limit: multiply the day count by one hundred, include the fen, and square the product for the upper position. Set the entered limit fen below, multiply by 225, reduce by one hundred, and add 198,075 to form the divisor. When before or after summer solstice the value reaches half the limit or above, subtract half the limit and place the result in the upper position. Set half the limit below. In each case multiply the day count by one hundred with the fen included, subtract first, then multiply. Divide by 7,700, add the quotient to the divisor, and divide the upper position to obtain fen; ten fen make one cun and ten cun one chi; add to the summer-solstice geocentric gnomon-shadow constant to obtain the sought gnomon-shadow fixed number.
130
求四方所在晷影
To find the gnomon shadow at each of the four directions.
131
各於其處測冬夏二至晷影，乃相減之餘，為其處二至晷差。 亦以地中二至晷數相減，為地中二至晷差。 其所求日在冬至後初限、夏至後末限者，如在半限以下，倍之； 半限以上，覆減半限，餘亦倍之，併入限日，三因折半，以日為分，十為寸，以減地中二至晷差為法。 置地中冬至晷影常數，以所求日地中晷影定數減之，餘以其年二至晷差乘之為實。 實如法而一，所得，以減其處冬至晷數，即得其處其日晷影定數。 所求日在夏至後初限、冬至後末限者，如在半限以下，倍之； 半限以上，覆減半限，餘亦倍之，併入限日，三因四除，以日為分，十為寸，以加地中二至晷差為法。 置所求日地中晷影定數，以地中夏至晷影常數減之，餘以其處二至晷差乘之為實。 實如法而一，所得，以加其處夏至晷數，即得其處其日晷影定數。
At each place measure the winter and summer solstice gnomon shadows; their difference is that place's solstice gnomon difference. Subtract the geocentric solstice gnomon figures likewise to obtain the geocentric solstice gnomon difference. When the sought day falls after winter solstice in the initial limit or after summer solstice in the final limit: if at or below half the limit, double it; if above half the limit, cover and subtract half the limit and double the remainder, combine with the entered limit days, multiply by three and halve, take days as fen and ten as cun, and subtract from the geocentric solstice gnomon difference to form the divisor. Set the geocentric winter-solstice gnomon-shadow constant, subtract the sought day's geocentric gnomon-shadow fixed number, and multiply the remainder by that year's solstice gnomon difference for the dividend. Divide the dividend by the divisor and subtract the quotient from that place's winter-solstice gnomon figure to obtain that place's gnomon-shadow fixed number for the day. When the sought day falls after summer solstice in the initial limit or after winter solstice in the final limit: if at or below half the limit, double it; if above half the limit, cover and subtract half the limit and double the remainder, combine with the entered limit days, multiply by three and divide by four, take days as fen and ten as cun, and add to the geocentric solstice gnomon difference to form the divisor. Set the sought day's geocentric gnomon-shadow fixed number, subtract the geocentric summer-solstice gnomon-shadow constant, and multiply the remainder by that place's solstice gnomon difference for the dividend. Divide the dividend by the divisor and add the quotient to that place's summer-solstice gnomon figure to obtain that place's gnomon-shadow fixed number for the day.
132
二十四氣陟降及日出分
Twenty-four qi ascension–descension and sunrise fen.
133
二分前後陟降率
Ascension–descension rates before and after the two equinoxes.
134
春分前三日太陽入赤道內，秋分後三日太陽出赤道外，故其陟降與他日不倫，今各別立數而用之。
Three days before the vernal equinox the sun enters inside the equator, and three days after the autumnal equinox it emerges outside it — so its ascension and descension diverge from ordinary days; separate rates are therefore assigned for each period.
135
驚蟄，十二日，陟四六十七，一十六此為末率，于此用畢。 其減差亦止於此。 十三日，陟四四十一，六。 十四日，陟四三十六，九十。 十五日，陟四一。
Awakening of Insects, day twelve: ascend 467, 116 — the final rate; use ends here. The decrease-difference likewise ends here. Day thirteen: ascend 441, 6. Day fourteen: ascend 436, 90. Day fifteen: ascend 41.
136
秋分，初日，降四三十八。 一日，降四三十九。 二日，降四五十七。 三日降四六十八。 此為初率，始用之。 其加差亦始於此。
Autumnal Equinox, first day: descend 438. Day one: descend 439. Day two: descend 457. Day three: descend 468. This is the initial rate; use begins here. The increase-difference likewise begins here.
137
求每日出入晨昏半晝分
To find the daily exit and entry, dawn and dusk, and half-daylight fen.
138
各以陟降初率，陟減降加其氣初日日出分，為一日下日出分。 以增損差，仍加減加減差。 增損陟降率，馴積而加減之，即為每日日出分。 覆減日法，餘為日入分。 以日出分日入分而半之，為半晝分。 以昏明分減日出分為晨分，加日入分為昏分。
Take the ascension–descension initial rate for each qi, subtract ascension and add descension to that qi's first-day sunrise fen, and obtain the sunrise fen for the day below. Apply the increase-decrease difference and continue adding and subtracting the adjustment difference. Adjust the ascension–descension rate and stepwise accumulate the additions and subtractions to obtain the daily sunrise fen. Cover and subtract the day factor; the remainder is sunset fen. Halve the sunrise and sunset fen together to obtain half-daylight fen. Subtract dusk-and-dawn fen from sunrise fen for dawn fen; add it to sunset fen for dusk fen.
139
求日出入辰刻
To find the double-hours and ke of exit and entry.
140
置日出入分，以六因之，滿辰法而一，為辰數，不盡，刻法除之為刻數，不滿為分，命子正算外，即得所求。
Set the exit-and-entry fen, multiply by six, and divide by the chen divisor for the chen count; divide any remainder by the ke factor for ke, with the unfilled portion as fen; count outward from zi zheng to obtain the result.
141
求晝夜刻
To find day and night ke.
142
置日出分，十二乘之，刻法而一，為刻，不滿為分，即為夜刻。 覆減百刻，餘為晝刻。
Set sunrise fen, multiply by twelve, and divide by the ke factor for ke, with the unfilled portion as fen — that is night ke. Cover and subtract from one hundred ke; the remainder is day ke.
143
求更點率
To find the geng and point rates.
144
置晨分，四因，退位為更率。 二因更率，退位為點率。
Set dawn fen, multiply by four, and step back one place to obtain the geng rate. Multiply the geng rate by two and step back one place to obtain the point rate.
145
求更點所在辰刻
To find the double-hour and ke where a geng or point falls.
146
置更點率，以所求更點數因之，又六因，內加昏明分，滿辰法而一，為辰數。 不盡，滿刻法除之為刻數，不滿為分，命其辰刻算外，即得所求。
Set the geng or point rate, multiply by the sought number of geng or points and again by six, add dusk-and-dawn fen within, and divide by the chen divisor for the chen count. Divide any remainder by the ke factor for ke, with the unfilled portion as fen, and count outward from that chen and ke to obtain the result.
147
求四方所在漏刻
To find the clepsydra graduations at each of the four directions.
148
各於所在下水漏，以定其處冬至或夏至夜刻，乃與五十刻相減，餘為至差刻。 置所求日黃道去赤道內外度及分，以至差刻乘之，進一位，如二百三十九而一，為刻，不盡以刻法乘之，退除為分，內減外加五十刻，即所求日夜刻，以減百刻，餘為晝刻。 其日出入辰刻及更點差率算等，並依術求之。
At each place run the water clepsydra to fix the winter- or summer-solstice night ke, subtract it from fifty ke, and take the remainder as the solstice-difference ke. Set the sought day's ecliptic distance inside or outside the equator with fen, multiply by the solstice-difference ke and advance one place, divide by 239 for ke, convert any remainder through the ke factor into fen, subtract within and add without fifty ke for night ke, and subtract from one hundred ke for day ke. The day's exit-and-entry chen and ke, together with the geng and point differential rates and related reckonings, are all found by the same procedures.
149
求黃道內外度
To find ecliptic inner and outer degrees.
150
置日出分，如日法四分之一以上，去之，餘為外分。 如日法四分之一以下，覆減之，餘為內分。 置內外分，千乘之，如內外法而一，為度，不滿退除為分，即為黃道去赤道內外度。 內減外加象限，即得內道去極度。
Set sunrise fen; if it reaches one-fourth of the day factor or above, remove that portion — the remainder is outer fen. If it falls at one-fourth of the day factor or below, cover and subtract — the remainder is inner fen. Set inner and outer fen, multiply by one thousand, divide by the inner-outer factor for degrees, and convert any remainder into fen — the ecliptic distance inside or outside the equator. Subtract within and add without the quadrant limit to obtain the inner path's distance from the pole.
151
求距中度及更差度
To find the distance-from-center degree and the geng differential degree.
152
置半法，以晨分減之，餘為距中分，百乘之，如周法而一，為距中度。 用減一百八十三度一十二分八十四秒，餘四因退位，為每更差度。
Set the half factor and subtract dawn fen; multiply the remainder by one hundred and divide by the circumference factor for distance-from-center degrees. Subtract from 183°12′84″, multiply the remainder by four, and step back one place to obtain the differential degree for each geng.
153
求昏明五更中星
To find the culmination stars at dusk, dawn, and the five geng.
154
置距中度，以其日午中赤道日度加而命之，即昏中星所格宿次，因為初更中星。 以更差度累加之，命赤道宿次去之，即得逐更及明中星。
Set the distance-from-center degree, add that day's noon equatorial solar degree and assign the mansions — the lodge of dusk culmination, which is also the first-geng culmination star. Add the geng differential degree cumulatively, assign the equatorial mansions and cast out full circuits — thus obtain the culmination stars for each successive geng and at dawn.