1
○庚午元曆下△步交會術交終分，一十四萬二千三百一十九，秒九千三百六，微二十。 交終日，二十七，余一千一百九，秒九千三百六，微二十。
Gengwu Yuan Calendar, lower part — Step Convergence Procedure: convergence termination parts 142,319; seconds 9,306; micro-units 20. Convergence termination day: 27 days, remainder 1,109 parts, seconds 9,306, micro-units 20.
2
交中日，一十三，余三千一百六十九，秒四千六百五十三，微一十。 交朔日，二，余一千六百六十五，秒六百九十三，微八十。 交望日，一十四，余四千二，秒五千。
Convergence mid-cycle day: 13 days, remainder 3,169 parts, seconds 4,653, micro-units 10. Convergence new-moon day: 2 days, remainder 1,665 parts, seconds 693, micro-units 80. Convergence full-moon day: 14 days, remainder 4,002 parts, seconds 5,000.
3
秒母，一萬。 微母，一百。 交終度，三百六十三，分七十九，秒三十六。 交中度，一百八十一，分八十九，秒六十八。
Seconds mother: 10,000. Micro mother: 100. Convergence termination degree: 363°, 79 parts, 36 seconds. Convergence middle degree: 181°, 89 parts, 68 seconds.
4
交象度，九十，分九十四，秒八十四。 半交象度，四十五，分四十七，秒四十二。 日食既前限，二千四百。 定法，二百四十八。 日食既後限，三千一百。 定法，三百二十。
Convergence image degree: 90°, 94 parts, 84 seconds. Half convergence image degree: 45°, 47 parts, 42 seconds. Solar eclipse totality anterior limit: 2,400. Fixed method: 248. Solar eclipse totality posterior limit: 3,100. Fixed method: 320.
5
月食限，五千一百。 月食既限，一千七百。 定法，三百四十。 分秒母，皆一百。
Lunar eclipse limit: 5,100. Lunar eclipse totality limit: 1,700. Fixed method: 340. Parts-and-seconds mother: each 100.
6
求朔望入交 〈（先置裏差，半之，如九而一，所得依其加減天正朔積分，然後求之。）〉
To find new and full moon entry into convergence. (First set the li difference, halve it, take one-ninth of that value, and add or subtract from the civil new-year new-moon accumulated parts accordingly; only then proceed.)
7
置天正朔積分，以交終分去之，不盡，如日法而一，為日，不滿為余，即得天正十一月中朔入交泛日及余秒。 〈（便為中朔加時入交泛日及余。）〉 交朔加之，得次朔； 交望加之，得望； 再加交望，亦得次朔； 各為朔望入交泛日及余秒。 〈（凡稱余秒者，微亦從之，余仿此。）〉
Set the civil new-year new-moon accumulated parts, discard full convergence termination parts, divide the remainder by the day divisor for days and parts, and obtain the eleventh-month central new-moon general convergence day, remainder, and seconds. (This is the central new-moon hour-of-addition general convergence day and remainder.) Add the convergence new-moon increment to obtain the next new moon; Add the convergence full-moon increment to obtain the full moon; Add the convergence full-moon increment again to obtain the next new moon; Each yields the general convergence day, remainder, and seconds for every new and full moon. (Whenever remainder seconds are named, micro-units follow likewise; treat other remainders the same way.)
8
〈【求定朔及每日夜半入交】〉
〈To find the fixed new moon and daily midnight entry into convergence〉
9
各置入交泛日及余秒，減去中朔望小余，即為定朔望夜半入交泛日及余秒。 若定朔望有進退者，亦進退交日，否則因中為定，大月加二日，小月加一日，餘皆加四千一百二十，秒六百九十三，微八十，即次朔夜半入交； 累加一日，滿交終日及余秒，去之，即每日夜半入交泛日及余秒。
Set each general convergence day and remainder seconds, subtract the central new- or full-moon minor remainder, and obtain the fixed new or full moon midnight general convergence day and remainder seconds. If a fixed new or full moon advances or retreats, adjust the convergence day likewise; otherwise take the central value as fixed. Add two days in a long month and one in a short month; always add 4,120 to the remainder with seconds 693 and micro-units 80 for the next new moon's midnight convergence entry; Add one day cumulatively, cast out full convergence termination days and remainder seconds, and obtain each day's midnight general convergence day and remainder seconds.
10
〈【求定朔望加時入交】〉
〈To find fixed new and full moon hour-of-addition entry into convergence〉
11
置中朔望加時入交泛日及余秒，以入氣入轉朓朒定數朓減朒加之，即得定朔望加時入交泛日及余秒。 〈【求定朔望加時入交積度及陰陽歷】〉
Set the central new- and full-moon hour-of-addition general convergence day and remainder seconds, apply the entry-into-qi and entry-into-rotation tiao–chuo fixed numbers (tiao subtract, chuo add), and obtain the fixed new- and full-moon hour-of-addition general convergence day and remainder seconds. 〈To find fixed new- and full-moon hour-of-addition convergence accumulated degrees and yang/yin calendar〉
12
置定朔望加時入交泛日，以日法通之，內余進二位，如三萬九千一百二十一而一，為度，不滿，退除為分秒，即得定朔望加時月行入交積度； 以定朔望加時入轉遲疾度遲減疾加之，即為月行入定交積度； 如交中度以下，為入陽歷積度，以上，去之，為入陰歷積度。 〈（每日夜半準此求之。）〉
Set the fixed new- or full-moon hour-of-addition general convergence day, convert through the day divisor with the remainder, shift two places, divide by 39,121 for degrees, reduce the remainder to parts and seconds, and obtain the hour-of-addition moon's entry into convergence accumulated degree; Apply the fixed syzygy hour's entry-into-rotation slow-fast degree (slow subtract, fast add) to obtain the moon's fixed entry into convergence accumulated degree; At or below convergence middle counts as yang-calendar accumulated degree; above, subtract it to obtain yin-calendar accumulated degree. (For each midnight, follow this procedure.)
13
〈【求月去黃道度】〉
〈To find the moon's ecliptic latitude (departure from the yellow path)〉
14
視月入陰陽歷積度及分，交象以下，為少象； 以上，覆減交中，余為老象。 置所入老少象度於上位，列交象度於下，相減，相乘，倍之，退位為分，分滿百為度，用減所入老少象度及分； 余，又與交中度相減、相乘，八因之，以一百一十除之，為分，分滿百為度，即得月去黃道度及分。
Inspect the entered yin- or yang-calendar accumulated degree and parts; at or below the convergence image counts as the young image; above, cover and subtract convergence middle; the remainder is the old image. Set the entered young or old image degree above and the convergence image degree below, subtract and multiply, double and shift one place for parts, convert hundreds to degrees, and subtract from the entered young or old image degree and parts; then with the remainder subtract and multiply against convergence middle, multiply by eight, divide by 110 for parts, convert hundreds to degrees, and obtain the moon's departure from the yellow path in degrees and parts.
15
〈【求朔望加時入交常日及定日】〉
〈To find new- and full-moon hour-of-addition general and fixed convergence days〉
16
置朔望入交泛日，以入氣朓朒定數朓減朒加，為入交常日。 又置入轉朓朒定數，進一位，以一百二十七而一，所得，朓減朒加交常日，為入交定日及余秒。
Set the new- or full-moon general convergence day, apply the entry-into-qi tiao–chuo fixed number (tiao subtract, chuo add), and obtain the general convergence day. Set the entry-into-rotation tiao–chuo fixed number, shift one place, divide by 127, apply tiao subtract and chuo add to the general convergence day, and obtain the fixed convergence day and remainder seconds.
17
〈【求入交陰陽歷交前後分】〉
〈To find entry-into-convergence yang/yin calendar before and after parts〉
18
視入交定日，如交中以下，為陽歷； 以上，去之，為陰歷。 如一日上下，以日法通日內分，內余為交後分； 十三日上下，覆減交中日，余為交前分。
Inspect the fixed entry-into-convergence day; at or below convergence middle counts as yang calendar; above, subtract it — yin calendar. If within about one day, convert the day through the day divisor into parts with the remainder included — that is the after-convergence part; If within about thirteen days, cover and subtract the convergence mid-cycle day to obtain the before-convergence part.
19
〈【求日月食甚定余】〉
〈To find the fixed remainder for solar and lunar eclipse greatest〉
20
置朔望入氣入轉朓朒定數，同名相從，異名相消，以一千三百三十七乘之，以定朔望加時入轉算外轉定分除之，所得，以朓減朒加中朔望小余，為泛余。 日食，視泛余，如半法以下，為中前，半法以上，去之，為中後。 置中前後分，與半法相減、相乘，倍之，萬約為分，曰時差。 中前以時差減泛余，為定余； 覆減半法，余為午前分； 中後以時差加泛余，為定余； 減去半法，余為午後分。 月食，視泛余，在日入後夜半前，如日法四分之三以下，減去半法，為酉前分； 四分之三以上，覆減日法，余為酉後分。 又視泛余，在夜半後日出前者，如日法四分之一以下，為卯前分； 四分之一以上，覆減半法，余為卯後分。 其卯酉前後分，自相乘，四因，退位，萬約為分，以加泛余，為定余。 各置定余，以發斂加時法求之，即得日月食甚辰刻及分秒。
Set the new- and full-moon entry-into-qi and entry-into-rotation tiao–chuo fixed numbers, combine same names and cancel different names, multiply by 1,337, divide by the fixed syzygy hour's entry-into-rotation count-outside rotation fixed parts, apply tiao subtract and chuo add to the central new- and full-moon minor remainder, and obtain the general remainder. Solar eclipse: if the general remainder is at or below the half method, it is before middle; at or above the half method, subtract the half method for after middle. Set the before- and after-middle parts, subtract and multiply against the half method, double, and reduce by 10,000 to obtain the time difference. Before middle: subtract the time difference from the general remainder to obtain the fixed remainder; cover and subtract the half method — the remainder is the before-noon part; After middle: add the time difference to the general remainder to obtain the fixed remainder; Subtract the half method to obtain the after-noon part. Lunar eclipse: if the general remainder falls after sunset and before midnight, at or below three-quarters of the day divisor subtract the half method for the before-you part; Above three-quarters, cover and subtract the day divisor for the after-you part. If after midnight and before sunrise, at or below one-quarter of the day divisor counts as before-mao; Above one-quarter, cover and subtract the half method for the after-mao part. Self-multiply each of the before- and after-mao and before- and after-you parts, multiply by four, shift one place back, reduce by 10,000 for parts, add to the general remainder, and obtain the fixed remainder. Set each fixed remainder and apply the aggregation-and-release hour-of-addition method to obtain the double-hour and ke of eclipse greatest.
21
〈【求日月食甚日行積度】〉
〈To find the solar motion accumulated degree at eclipse greatest〉
22
置定朔望食甚大小余，與中朔望大小余相減之，余以加減中朔望入氣日余， 〈（以中朔望少加多減。）〉 即為食甚入氣； 以加其氣中積，為食甚中積。 又置食甚入氣余，以所入氣日損益率 〈（盈縮之損益。）〉 乘之，如日法而一，以損益其日盈縮積，盈加縮減食甚中積，即為食甚日行積度及分。 先以食甚中積經分為約分，然後加減之，余類此者，依而求之。
Set the fixed syzygy eclipse-greatest major and minor remainders, subtract the central syzygy major and minor remainders, and with the difference adjust the central syzygy entry-into-qi day remainder, (When the central syzygy day count is less, add more and subtract less.) That is the entry into qi at eclipse greatest; Add that to the qi mid-accumulation to obtain the mid-accumulation at eclipse greatest. Set the entry-into-qi remainder at eclipse greatest and multiply by that qi day's gain-and-loss rate (the excess-and-deficit gain-and-loss.) Multiply, divide by the day divisor, adjust that day's excess-and-deficit accumulation, then excess adds and deficit subtracts from the eclipse-greatest mid-accumulation to obtain the eclipse-greatest daily solar motion accumulated degree and parts. First reduce the eclipse-greatest mid-accumulation canonical parts, then add and subtract; for similar remainders, follow the same procedure.
23
〈【求氣差】〉
〈To find qi difference〉
24
置日食食甚日行積度及分，滿中限去之，余在象限以下，為初限； 以上，覆減中限，為末限； 皆自相乘，進二位，以四百七十八而一，所得，用減一千七百四十四，余為氣差恒數； 以午前後分乘之，半晝分除之，所得，以減恒數，為定數。 〈（如不及減者，覆減為定數，應加者減之，應減者加之。）〉 春分後，陽歷減陰歷加； 秋分後，陽歷加陰歷減。 〈（春分前秋分後，各二日二千一百分為定氣，於此宜加減之。）〉
Set the solar eclipse-greatest daily solar motion accumulated degree and parts, cast out the central limit, and if the remainder is at or below the image limit it is the initial limit; if above, cover and subtract the central limit for the final limit; self-multiply in each case, shift two places, divide by 478, subtract from 1,744, and obtain the qi-difference constant; Multiply by the before- or after-noon part, divide by half the day-length part, subtract from the constant, and obtain the fixed number. (If subtraction fails, cover and subtract instead; where addition is indicated, subtract; where subtraction is indicated, add.) After the spring equinox, yang calendar subtracts and yin calendar adds; After the autumn equinox, yang calendar adds and yin calendar subtracts. (Before the spring equinox and after the autumn equinox, treat two days and 2,100 parts each as fixed qi and apply these additions and subtractions within that interval.)
25
〈【求刻差】〉
〈To find ke difference〉
26
置日食食甚日行積度及分，滿中限去之，余與中限相減、相乘，進二位，如四百七十八而一，所得，為刻差恒數； 以午前後分乘之，日法四分之一除，所得，為定數。 〈（若在恒數以上者，倍恒數，以所得之數減之，為定數，依其加減。）〉 冬至後，午前陽加陰減，午後陽減陰加； 夏至後，午前陽減陰加，午後陽加陰減。
Set the solar-eclipse-greatest solar daily accumulated degree and parts, discard full middle limits, multiply the remainder by its distance from the middle limit, advance two places, divide by 478, and obtain the ke-difference constant; Multiply by the before- and after-noon parts and divide by one quarter of the day divisor to obtain the fixed number. (If above the constant, double the constant, subtract the obtained number for the fixed number, and apply the indicated additions or subtractions.) After winter solstice, before noon add in the yang case and subtract in the yin case; after noon subtract in the yang case and add in the yin case; After summer solstice, before noon subtract in the yang case and add in the yin case; after noon add in the yang case and subtract in the yin case.
27
〈【求日食去交前後定分】〉
〈To find the solar eclipse fixed distance from crossing before and after〉
28
置氣刻二差定數，同名相從，異名相消，為食差； 依其加減去交前後分，為去交前後定分。 視其前後定分，如在陽歷，即不食； 如在陰歷，即有食之。 如交前陰歷不及減，反減之， 〈（反減食差。）〉 為交後陽歷； 交後陰歷不及減，反減之，為交前陽歷； 即不食。 交前陽歷不及減，反減之，為交後陰歷； 交後陽歷不及減，反減之，為交前陰歷； 即日有食之。
Set the qi- and quarter-mark-difference fixed numbers, combine those of the same name and cancel those of different name, and obtain the eclipse difference; Apply the additions and subtractions to the crossing before-and-after parts to obtain the fixed crossing before-and-after parts. Inspect the before-and-after fixed parts: in the yang calendar there is no eclipse; in the yin calendar there is an eclipse. If before crossing in the yin calendar subtraction is insufficient, reverse the subtraction, (reverse-subtract the eclipse difference.) that becomes after-crossing yang calendar; if after crossing in the yin calendar subtraction is insufficient, reverse the subtraction to obtain before-crossing yang calendar; then there is no eclipse. if before crossing in the yang calendar subtraction is insufficient, reverse the subtraction to obtain after-crossing yin calendar; if after crossing in the yang calendar subtraction is insufficient, reverse the subtraction to obtain before-crossing yin calendar; then that day there is a solar eclipse.
29
〈【求日食分】〉
〈To find the solar eclipse parts〉
30
視去交前後定分，如二千四百以下，為既前分； 以二百四十八除，為大分； 二千四百以上，覆減五千五百， 〈（不足減者不食。）〉 為既後分； 以三百二十除，為大分，不盡，退除為秒。 〈（其一分以下者，涉交太淺，太陽光盛，或不見食。）〉
Inspect the fixed crossing before-and-after parts: at 2,400 or below take before-totality parts; divide by 248 for great parts; If 2,400 or above, cover and subtract 5,500, (if subtraction is insufficient, there is no eclipse.) these are after-totality parts; divide by 320 for great parts and convert any remainder into seconds. (At one part or below the crossing is too shallow, the sun's light too strong, and the eclipse may not be visible.)
31
〈【求月食分】〉
〈To find the lunar eclipse parts〉
32
視去交前後分， 〈（不用氣刻差者。）〉 一千七百以下者，食既； 以上，覆減五千一百， 〈（不足減者不食。）〉 余以三百四十除之，為大分； 不盡，退除為秒，即月食之分秒。 去交分在既限以下，覆減既限，亦以三百四十除之，為既內之大分。
Inspect the crossing before-and-after parts, (without using the qi and quarter-mark differences.) at 1,700 or below the eclipse is total; if above, cover and subtract 5,100, (if subtraction is insufficient, there is no eclipse.) Divide the remainder by 340 for great parts; Convert any remainder into seconds to obtain the lunar eclipse parts and seconds. If the distance-from-crossing parts are at or below the totality limit, cover and subtract the totality limit and likewise divide by 340 for the great parts within totality.
33
〈【求日食定用分】〉
〈To find the solar eclipse fixed usage parts〉
34
置日食之大分，與二十分相減、相乘，又以二千四百五十乘之，如定朔入轉算外轉定分而一，所得，為定用分； 減定余，為初虧分； 加定余，為復圓分； 各以發斂加時法求之，即得日食三限辰刻也。
Set the solar eclipse great parts, subtract and multiply with 20 parts, multiply by 2,450, divide by the fixed new moon's outside-rotation fixed parts, and obtain the fixed usage parts; Subtract from the fixed remainder to obtain the first-diminishment parts; Add to the fixed remainder to obtain the restoration-of-roundness parts; Apply the expansion-contraction hour-addition method to each to obtain the solar eclipse's three-limit chen and ke.
35
〈【求月食定用分】〉
〈To find the lunar eclipse fixed usage parts〉
36
置月食之大分，與三十五分相減、相乘，又以二千一百乘之，如定望入轉算外轉定分而一，所得，為定用分； 加減定余，為初虧復圓分。 各如發斂加時法求之，即得月食三限辰刻。
Set the lunar eclipse great parts, subtract and multiply with 35 parts, multiply by 2,100, divide by the fixed full moon's outside-rotation fixed parts, and obtain the fixed usage parts; Add and subtract the fixed remainder to obtain the first-diminishment and restoration-of-roundness parts. Apply the expansion-contraction hour-addition method to each to obtain the lunar eclipse's three-limit chen and ke.
37
月食既者，以既內大分，以一十五分相減相乘，又以四千二百乘之，如定望入轉算外轉定分而一，所得為既內分； 用減定用分，為既外分。 置月食定余，減定用分，為初虧分； 因加既外分，為食既分； 又加既內分，為食甚分； 〈（即定余分是也。）〉 再加既內分，為生光分； 復加既外分，為復圓分。 各以發斂加時法求之，即得月食五限辰刻及分。 〈（如月食既者，以十分並既內大分，如其法而求其定用分也。）〉
For a total lunar eclipse, take the within-totality great parts, subtract and multiply with 15 parts, multiply by 4,200, divide by the fixed full moon's outside-rotation fixed parts, and obtain the within-totality parts; Subtract these from the fixed usage parts to obtain the outside-totality parts. Set the lunar eclipse fixed remainder, subtract the fixed usage parts, and obtain the first-diminishment parts; Then add the outside-totality parts to obtain the eclipse-totality parts; Again add the within-totality parts to obtain the greatest-eclipse parts; (This is the fixed remainder parts.) Again add the within-totality parts to obtain the light-generation parts; Again add the outside-totality parts to obtain the restoration-of-roundness parts. Apply the expansion-contraction hour-addition method to each to obtain the lunar eclipse's five-limit chen, ke, and parts. (For a total lunar eclipse, add ten parts to the within-totality great parts and by that method obtain the fixed usage parts.)
38
〈【求月食所入更點】〉
〈To find the lunar eclipse's entry into watch and point〉
39
置食甚所入日晨分，倍之，五約之，為更法； 又五約之，為點法。 乃置月食初末諸分，昏分以上者，減昏分； 晨分以下者，加晨分； 如不滿更法，為初更； 不滿點法，為一點。 依法以次求之，即得更點之數。
Set the day's dawn parts at greatest eclipse, double them, and five-reduce to obtain the watch method; Again five-reduce to obtain the point method. Then set the lunar eclipse's beginning-and-end parts: above dusk parts subtract dusk parts; below dawn parts add dawn parts; if insufficient for the watch method, that is the first watch; Any remainder short of the point divisor counts as one point. Apply the method in order to each limit to obtain the watch and point counts.
40
〈【求日食所起】〉
〈Procedure to find the solar eclipse's direction of first contact〉
41
食在既前，初起西南，甚於正南，復於東南。 食在既後，初起西北，甚於正北，復於東北。 其食八分以上者，皆起正西，復正東。 〈（此據正午地而論之。）〉
Before totality, first contact lies southwest, maximum due south, and recovery southeast. After totality, first contact lies northwest, maximum due north, and recovery northeast. For eclipses of eight-tenths or greater, first contact is always due west and recovery due east. (This assumes an observer on the local meridian (at noon).)
42
〈【求月食所起】〉
〈Procedure to find the lunar eclipse's direction of first contact〉
43
月在陽歷，初起東北，甚於正北，復於西北。 月在陰歷，初起東南，甚於正南，復於西南。 其食八分以上，皆起正東，復正西。 〈（此亦據正午地而論之。）〉
With the moon in the yang calendar, first contact is northeast, maximum due north, and recovery northwest. With the moon in the yin calendar, first contact is southeast, maximum due south, and recovery southwest. For eclipses of eight-tenths or greater, first contact is always due east and recovery due west. (This likewise assumes an observer on the local meridian (at noon).)
44
求日月出入帶食所見分數
Procedure to find the fraction of a solar or lunar eclipse visible when caught at sunrise or sunset
45
各以食甚小余，與日出入分相減，余為帶食差； 以乘所食之分，滿定用分而一， 〈（月食既者，以既內分減帶食差，余乘所食分，如既外分而一，不及減者，為帶食既出入。）〉 以減所食分，即日月出入帶食所見之分。 〈（其食甚在晝，晨為漸進，昏為已退； 食甚在夜，晨為已退，昏為漸進也。）〉
For each case, subtract the sun's rise-or-set parts from the eclipse-maximum minor remainder to obtain the horizon-partial difference; multiply by the eclipsed fraction and divide by the fixed usage parts; (For a total lunar eclipse, subtract the within-totality parts from the horizon-partial difference, multiply the remainder by the eclipsed fraction and divide by the outside-totality parts; if subtraction cannot be completed, the eclipse is partially total at rise or set.) Subtract the result from the eclipsed fraction to obtain the visible fraction at rise or set for sun or moon with horizon partial eclipse. (If greatest eclipse is in daylight, at dawn the eclipse is still advancing and at dusk it has already retreated; (if greatest eclipse is at night, at dawn it has already retreated and at dusk it is still advancing.)
46
〈【求日月食甚宿次】〉
〈Procedure to find the lodge at solar or lunar eclipse maximum〉
47
置日月食甚日行積度， 〈（望即更加望度。）〉 以天正冬至加時黃道日度加而命之，依黃道宿次去之，即各得日月食甚宿度及分秒。
Set the daily-motion accumulated degrees at eclipse maximum for sun or moon, (at full moon add the full-moon degree.) Add to the winter solstice ecliptic solar degree at hour-of-addition, name the lodges, and remove by the ecliptic lodge sequence to obtain each eclipse-maximum lodge degree, parts, and seconds for sun or moon.
48
○步五星術△木星周率，二百八萬六千一百四十二，秒九。 曆率，二千二百六十五萬五百五十七。
Procedures for the Five Planets — Jupiter cycle rate: 2,086,142; seconds 9. Calendar rate: 22,650,557.
49
歷度法，六萬二千一十四。 周日，三百九十八日八十八分。 歷度，三百六十五度二十四分九十秒。 歷中，一百八十二度六十二分四十五秒。
Circuit degree divisor: 62,014. Circuit day: 398 days, 88 parts. Circuit degree: 365 degrees, 24 parts, 90 seconds. Circuit mid: 182 degrees, 62 parts, 45 seconds.
50
歷策，一十五度二十一分八十七秒。 伏見，一十三度。
Circuit stride: 15 degrees, 21 parts, 87 seconds. Heliacal setting visibility: 13 degrees.
51
〈（以下表格略）〉 △火星周率，四百七萬九千四十二，秒一十四半。 曆率，三百五十九萬二千七百五十七，秒四十四少。
(The tables below are omitted.) Mars cycle rate: 4,079,042; seconds 14 and a half. Calendar rate: 3,592,757; seconds 44 slight.
52
歷度法，九千八百三十六半。 周日，七百七十九日九十三分一十六秒。 歷度，三百六十五度二十四分七十五秒。 歷中，一百八十二度六十二分三十七秒半。
Circuit degree divisor: 9,836½. Circuit day: 779 days, 93 parts, 16 seconds. Circuit degree: 365 degrees, 24 parts, 75 seconds. Circuit mid: 182°62′37.5″.
53
歷策，一十五度二十一分八十六秒。 伏見，一十九度。
Circuit stride: 15°21′86″. Heliacal setting visibility: 19 degrees.
54
〈（以下表格略）〉 △土星周率，一百九十七萬七千四百一十一，秒六十九。 曆率，五千六百二十二萬三千二百四十八半。
(The tables below are omitted.) Saturn cycle rate: 1,977,411; seconds 69. Calendar rate: 56,223,248½.
55
歷度法，一十五萬三千九百二十八。 周日，三百七十八日九分二秒。 歷度，三百六十五度二十五分六十八秒。 歷中，一百八十二度六十二分八十四秒。
Circuit degree divisor: 153,928. Circuit day: 378 days, 9 parts, 2 seconds. Circuit degree: 365°25′68″. Circuit mid: 182°62′84″.
56
歷策，一十五度二十一分九十秒。 伏見，一十七度。
Circuit stride: 15°21′90″. Heliacal setting visibility: 17 degrees.
57
〈（以下表格略）〉 △金星周率，三百五萬三千八百四，秒六十三太。 曆率，一百九十一萬二百四十，秒七十六半。
(The tables below are omitted.) Venus cycle rate: 3,053,804; seconds 63 great. Calendar rate: 1,910,240; seconds 76½.
58
歷度法，五千二百三十。 周日，五百八十三日九十分一十四秒。 合日，二百九十一日九十五分七秒。 歷度，三百六十五度二十四分六十八秒。
Circuit degree divisor: 5,230. Circuit day: 583 days, 90 parts, 14 seconds. Conjunction day: 291 days, 95 parts, 7 seconds. Circuit degree: 365°24′68″.
59
歷中，一百八十二度六十二分三十四秒。 歷策，一十五度二十一分八十六秒。 伏見，一十度半。
Circuit mid: 182°62′34″. Circuit stride: 15°21′86″. Heliacal setting visibility: 10½ degrees.
60
〈（以下表格略）〉 △水星周率，六十萬六千三十一，秒七十七半。 曆率，一百九十一萬二百四十二，秒一十三半。
(The tables below are omitted.) Mercury cycle rate: 606,031; seconds 77½. Calendar rate: 1,910,242; seconds 13½.
61
歷度法，五千二百三十。 周日，一百一十五日八十七分六十秒。 合日，五十七日九十三分八十秒。 歷度，三百六十五度二十四分七十秒。
Circuit degree divisor: 5,230. Circuit day: 115 days, 87 parts, 60 seconds. Conjunction day: 57 days, 93 parts, 80 seconds. Circuit degree: 365°24′70″.
62
歷中，一百八十二度六十二分三十五秒。 歷策，一十五度二十一分八十五秒。 晨伏夕見，一十四度。 夕伏晨見，一十九度。
Circuit mid: 182°62′35″. Circuit stride: 15°21′85″. Morning hidden, evening visible: 14 degrees. Evening hidden, morning visible: 19 degrees.
63
〈（以下表格略）〉 〈【求五星天正冬至後平合及諸段中積中星】〉
(The tables below are omitted.) 〈To find each planet's mean conjunction after heavenly-origin winter solstice and the central accumulation and central star for every segment〉
64
置通積分， 〈（先以裏差加減之。）〉 各以其星周率去之，不盡，為前合分； 覆減周率，余為後合分； 如日法而一，不滿，退除為分秒，即得其星天正冬至後平合中積中星。 〈（命為日，曰中積； 命為度，曰中星。）〉 以段日累加中積，即為諸段中積； 以平度累加中星，經退則減之，即為諸段中星。
Set the comprehensive accumulated parts, (First apply the inner difference.) For each star, cast out full cycle rates; the remainder is the anterior conjunction parts; Subtract the cycle rate again; the remainder is the posterior conjunction parts; Divide by the day divisor, reduce the remainder to parts and seconds, and obtain that star's mean-conjunction central accumulation and central star after heavenly-origin winter solstice. (Name the result as days—this is the central accumulation; (Name the result as degrees—this is the central star.) Add segment days cumulatively to the central accumulation to obtain each segment's central accumulation; Add mean degrees cumulatively to the central star, casting back when full, to obtain each segment's central star.
65
〈【求五星平合及諸段入歷】〉
〈To find each planet's mean conjunction and every segment's entry into the circuit cycle〉
66
置通積分，各加其星後合分，以曆率去之，不盡，各以其歷度法除為度，不滿，退除為分秒，即為其星平合入歷度及分秒； 以諸段限度累加之，即得諸段入歷度及分秒。
Set the comprehensive accumulated parts, add each star's posterior conjunction parts, cast out full calendar rates, divide the remainder by that star's circuit degree divisor for degrees, reduce the remainder to parts and seconds, and obtain the mean-conjunction entry-into-circuit degree and parts and seconds; Add each segment's limit degree cumulatively to obtain every segment's entry-into-circuit degree and parts and seconds.
67
〈【求五星平合及諸段盈縮定差】〉
〈To find each planet's mean conjunction and every segment's excess-and-deficit fixed difference〉
68
各置其星段入歷度及分秒，如在歷中以下，為盈； 以上，減去歷中，余為縮。 以其星曆策除之，為策數； 不盡，為入策度及分。 命策數算外，以其策損益率乘之，余歷策而一，為分，以損益其下盈縮積度，即為其星段盈縮定差。
Set each star and segment's entry-into-circuit degree and parts and seconds; at or below the circuit mid counts as excess; if above, subtract the circuit mid; the remainder is in deficit. Divide by that star's calendar stride to obtain the stride count; the remainder is the entry-into-stride degree and parts. Name from outside the stride count, multiply the tabulated loss-and-gain rate, divide the remainder by the circuit stride for parts, apply it to the underlying excess-and-deficit accumulation, and obtain that star and segment's excess-and-deficit fixed difference.
69
〈【求五星平合及諸段定積】〉
〈To find each planet's mean conjunction and every segment's fixed accumulation〉
70
各置其星段中積，以其段盈縮定差盈加縮減之，即得其段定積日及分； 加天正冬至大余及約分，滿紀法，去之，不滿，命壬戌算外，即得日辰也。 〈【求五星平合及諸段所在月日】〉
Set each star and segment's central accumulation and add or subtract the segment's excess-and-deficit fixed difference according to excess or deficit to obtain the segment's fixed-accumulation days and parts; Add to the heavenly-origin winter solstice greater remainder and approximate parts, cast out full era rules, name from renxu outside the count, and obtain the day and double-hour. 〈To find each planet's mean conjunction and every segment's month and day〉
71
各置其段定積，以加天正閏日及約分，以朔策及約分除之，為月數； 不盡，為入月以來日數及分。 其月數，命天正十一月算外，即得其段入月中朔日數及分； 乃以日辰相距，為所在定朔月日。
Set each segment's fixed accumulation, add heavenly-origin intercalary days and approximate parts, and divide by the new-moon policy and approximate parts for month count; the remainder is the days and parts elapsed within the month. Name the month count from heavenly-origin month eleven outside the count to obtain the segment's days and parts since mean new moon; take the day-and-double-hour interval as the fixed new-moon month and day.
72
〈【求五星平合及諸段加時定星】〉
〈To find each planet's mean conjunction and every segment's hour-of-addition fixed star〉
73
各置中星，以盈縮定差盈加縮減， 〈（金星倍之，水星三之，然後加減。）〉 即為五星諸段定星； 以加天正冬至加時黃道日度，依宿次命之，即其星其段加時所在宿度及分秒。
Set each central star and apply the excess-and-deficit fixed difference (add in excess, subtract in deficit), (Double for Venus and triple for Mercury, then add or subtract.) The result is each planet's fixed star for every segment; Add to the heavenly-origin winter solstice ecliptic solar degree at hour of addition, name the lodge accordingly, and obtain that star and segment's lodge degree and parts and seconds at hour of addition.
74
〈【求五星諸段初日晨前夜半定星】〉
〈To find each planet's fixed star at dawn before midnight on the first day of each segment〉
75
各以其段初行率，乘其段定積日下加時分，百約之，乃順減退加其日加時定星，即其段初日晨前夜半定星所在宿度及分秒。 〈【求諸段日率度率】〉
Multiply each segment's initial motion rate by the hour-of-addition parts under the segment's fixed-accumulation day, reduce by 100, subtract in forward motion and add in retrograde to that day's hour-of-addition fixed star, and obtain the lodge degree and parts and seconds of the segment-first-day dawn-before-midnight fixed star. 〈To find each segment's day rate and degree rate〉
76
各以其段日辰，距後段日辰為日率。 以其段夜半宿次，與後段夜半宿次相減，余為度率。 〈【求諸段平行分】〉
Take each segment's day-and-double-hour interval to the next segment as the day rate. Subtract this segment's midnight lodge sequence from the next segment's; the remainder is the degree rate. 〈To find each segment's parallel motion parts〉
77
各置其段度率及分秒，以其段日率除之，即得其段平行度日及分秒。 〈【求諸段總差及日差】〉
Set each segment's degree rate and parts and seconds, divide by that segment's day rate, and obtain its parallel motion degree and parts and seconds. 〈To find each segment's total difference and daily difference〉
78
本段前後平行分相減，為其段泛差； 〈（假令求木星次疾泛差，乃以順疾順遲平行分相減，余為次疾泛差，他皆仿此。）〉 倍而退位，為增減差； 加減其段平行分，為初末日行分； 〈（前多後少者，加為初，減為末； 前少後多者，減為初，加為末。）〉 倍增減差，為總差； 以日率減一除之，為日差。
Subtract this segment's prior and subsequent parallel motion parts; the remainder is its general difference; (To find Jupiter's second-fast general difference, subtract forward-swift from forward-slow parallel motion parts; the remainder is the second-fast general difference; all other cases follow this pattern.) Double the general difference and shift back one place for the increase-decrease difference; Add or subtract it from parallel motion parts to obtain initial and final day motion parts; (If the prior exceeds the subsequent, add for the initial day and subtract for the final day; (If the prior is less and the subsequent more, subtract for the initial day and add for the final day.) Double the increase-decrease difference for the total difference; Divide by the day rate minus one to obtain the daily difference.
79
〈【求前後伏遲退段增減差】〉
〈To find the increase-decrease differences for prior and subsequent concealment, slow, and retrograde segments〉
80
前伏者，置後段初日行分，加其日差之半，為末日行分； 後伏者，置前段末日行分，加其日差之半，為初日行分； 以減伏段平行分，余為增減差。 前遲者，置前段末日行分，倍其日差減之，為初日行分； 後遲者，置後段初日行分，倍其日差減之，為末日行分； 以遲段平行分減之，余為增減差。 〈（前後近留遲段。）〉 木火土三星，退行者，六因平行分，退一位，為增減差。 金星，前後伏退者，三因平行分，半而退位，為增減差。 前退者，置後段初日之行分，以其日差減之，為末日行分。 後退者，置前段末日之行分，以其日差減之，為初日行分； 以本段平行分減之，余為增減差。 水星，平行分為增減差，皆以增減差加減平行分，為初末日行分。 〈（前多後少，加初減末； 前少後多，減初加末。）〉 又倍增減差為總差，以日率減一，除之，為日差。
For prior concealment, set the next segment's initial-day motion parts, add half its daily difference, and obtain final-day motion parts; For subsequent concealment, set the prior segment's final-day motion parts, add half its daily difference, and obtain initial-day motion parts; Subtract from the concealment segment's parallel motion parts; the remainder is the increase-decrease difference. For prior slow motion, set the prior segment's final-day motion parts, subtract double its daily difference, and obtain initial-day motion parts; For subsequent slow motion, set the next segment's initial-day motion parts, subtract double its daily difference, and obtain final-day motion parts; Subtract from the slow segment's parallel motion parts; the remainder is the increase-decrease difference. (Slow segments before and after that lie near station.) For Jupiter, Mars, and Saturn in retrograde, multiply parallel motion parts by six, shift back one place, and obtain the increase-decrease difference. For Venus at prior and subsequent concealment and retrograde, multiply parallel motion parts by three, halve and shift back one place, and obtain the increase-decrease difference. For prior retrograde, set the next segment's initial-day motion parts, subtract its daily difference, and obtain final-day motion parts. For subsequent retrograde, set the prior segment's final-day motion parts, subtract its daily difference, and obtain initial-day motion parts; Subtract from this segment's parallel motion parts; the remainder is the increase-decrease difference. For Mercury, take the parallel motion parts as the increase-decrease difference, then add or subtract it from parallel motion parts to obtain initial and final day motion parts. (If the prior exceeds the subsequent, add for the initial day and subtract for the final day; (if the prior is less and the subsequent more, subtract for the initial day and add for the final day.) Again double the increase-decrease difference for the total difference and divide by the day rate minus one to obtain the daily difference.
81
〈【求每日晨前夜半星行宿次】〉
〈To find each day's star lodge sequence at dawn before midnight〉
82
各置其段初日行分，以日差累損益之， 〈（後少則損之，後多則益之。）〉 為每日行度及分秒； 乃順加退減之，滿宿次去之，即得每日晨前夜半星行宿次。 〈（視前段末日後段初日行分相較之數，不過一二日差為妙； 或多日差數倍，或顛倒不倫，當類同前後增減差稍損益之，使其有倫，然後用之。 或前後平行分俱多俱少，則平註之； 或總差之秒不盈一分，亦平註之； 若有不倫而平註得倫者，亦平註之。）〉
Set each segment's initial-day motion parts and apply the daily difference cumulatively, (Subtract when the subsequent value is less and add when it is more.) The result is each day's motion in degrees, parts, and seconds; Then add in forward motion and subtract in retrograde, cast out full lodge sequences, and obtain each day's star lodge sequence at dawn before midnight. (Compare the prior segment's final-day and next segment's initial-day motion parts; a gap of no more than one or two daily differences is ideal; If the daily difference is many times too large or the sequence is inverted and incoherent, reconcile it with the surrounding increase-decrease differences, adjust slightly until coherent, and then apply it. If prior and subsequent parallel motion parts are both too large or both too small, distribute the correction evenly; If the total difference's seconds do not reach one part, distribute evenly as well; (If the values are incoherent but even distribution restores coherence, distribute evenly.)
83
〈【求五星平合及見伏入氣】〉
〈To find each planet's mean conjunction and appearance-and-concealment entry into qi〉
84
置定積，以氣策及約分除之，為氣數； 不滿，為入氣日及分秒； 命天正冬至算外，即得所求平合及見伏入氣日及分秒。 〈【求五星平合及見伏行差】〉
Set the fixed accumulation, divide by the qi stride and approximate parts for the qi count, take the remainder as days, parts, and seconds entered into qi; name from heavenly-origin winter solstice outside the count, and obtain the mean conjunction and appearance-and-concealment entry-into-qi days, parts, and seconds. 〈To find each planet's mean conjunction and appearance-and-concealment motion difference〉
85
各以其段初日星行分與太陽行分相減，余為行差。 若金在退行、水在退合者，相並為行差。 如水星夕伏晨見者，直以太陽行分為行差。 〈【求五星定合及見伏泛積】〉
Subtract the sun's motion parts from each segment's initial-day star motion parts; the remainder is the motion difference. If Venus is retrograde or Mercury at retrograde conjunction, add the two for the motion difference. When Mercury is evening concealed and morning visible, take the sun's motion parts directly as the motion difference. 〈To find each planet's fixed conjunction, appearance, and concealment general accumulation〉
86
木火土三星，各以平合晨疾夕伏定積，為定合定見定伏泛積。 金水二星，置其段盈縮定差， 〈（水星倍之。）〉 各以行差除之，為日，不滿，退除為分秒； 若在平合夕見晨伏者，盈減縮加； 如在退合夕伏晨見，盈加縮減； 皆以加減定積為定合定見定伏泛積。
For Jupiter, Mars, and Saturn, the mean-conjunction morning-swift evening-concealed fixed accumulation is the fixed conjunction, appearance, and concealment general accumulation. For Venus and Mercury, set that segment's excess-and-deficit fixed difference, (Double it for Mercury.) divide each by the motion difference for days, and reduce any remainder to parts and seconds; at mean conjunction with evening visibility and morning concealment, subtract in excess and add in deficit; at retrograde conjunction with evening concealment and morning visibility, add in excess and subtract in deficit; Apply the indicated additions and subtractions to the fixed accumulation to obtain the fixed conjunction, appearance, and concealment general accumulation.
87
〈【求五星定合定積定星】〉
〈To find each planet's fixed conjunction fixed accumulation and fixed star〉
88
木火土三星，各以平合行差除其日太陽盈縮差，為距合差日； 以太陽盈縮差減之，為距合差度； 日在盈縮，以差日差度減之； 在縮歷，加之； 加減其星定合泛積，為定合定積定星。 金水二星，順合退合，各以平合退合行差，除其日太陽盈縮差，為距合差日； 順加退減太陽盈縮差，為距合差度； 順在盈歷，以差日差度加之； 在縮歷，減之； 退在盈歷，以差日減之，差度加之； 在縮歷，以差日加之，差度減之； 皆以加減其定星定合再定合泛積，為定合再定合定積定星； 以冬至大余及約分加定積，滿紀法，去之，命得定合日辰； 以冬至加時黃道日度加定星，滿宿次，去之，即得定合所在宿次。 〈（其順退所在盈縮，即太陽盈縮。）〉
For Jupiter, Mars, and Saturn, divide each day's solar excess-and-deficit difference by the mean-conjunction motion difference to obtain the conjunction-distance difference day; Subtract from the solar excess-and-deficit difference to obtain the conjunction-distance difference degree; When the sun is in excess, subtract the difference day and difference degree; When in the deficit cycle, add them; Add or subtract on that star's fixed-conjunction general accumulation to obtain the fixed conjunction fixed accumulation and fixed star. For Venus and Mercury in direct or retrograde conjunction, divide that day's solar excess-and-deficit difference by the corresponding mean-conjunction motion difference to obtain the conjunction-distance difference day; Add the solar excess-and-deficit difference in direct motion and subtract it in retrograde motion to obtain the conjunction-distance difference degree; In direct motion when in the excess cycle, add the difference day and difference degree; When in the deficit cycle, subtract them; In retrograde motion when in the excess cycle, subtract the difference day and add the difference degree; When in the deficit cycle, add the difference day and subtract the difference degree; Apply the indicated additions and subtractions to that star's fixed star, fixed conjunction, and second fixed-conjunction general accumulation to obtain the fixed conjunction, second fixed conjunction, fixed accumulation, and fixed star; Add the fixed accumulation to the winter solstice greater remainder and approximate parts, cast out full era rules, name the count, and obtain the fixed conjunction day and double-hour; Add the fixed star to the winter solstice ecliptic solar degree at hour of addition, remove full lodge sequences, and obtain the lodge of the fixed conjunction. (Whether the motion is direct or retrograde, the excess or deficit used is the sun's excess and deficit.)
89
〈【求木火土三星定見伏定日】〉
〈To find the fixed appearance, concealment, and fixed day for Jupiter, Mars, and Saturn〉
90
各置其星定見伏泛積，晨加夕減象限日及分秒； 〈（半中限為象限。）〉 如中限以下，自相乘； 以上，覆減歲周日及分秒，余亦自相乘； 滿七十五而一，所得，以其星伏見度乘之，一十五除之，為差。 其差，如其段行差而一，為日，不滿，退除為分秒； 見加伏減泛積，為定積； 加命如前，即得日辰。
Set each star's fixed appearance-and-concealment general accumulation; at dawn add and at dusk subtract the image-limit day and parts and seconds; (Half the middle limit serves as the image limit.) If at or below the middle limit, square it; If above, overlay-subtract from the year circuit day and parts and seconds and square the remainder likewise; Divide by 75, multiply by that star's heliacal appearance-and-concealment degree, and divide by 15 to obtain the difference. Divide the difference by that segment's motion difference for days; cast back the remainder into parts and seconds; Add at appearance and subtract at concealment on the general accumulation to obtain the fixed accumulation; Add and name as before to obtain the day and double-hour.
91
〈【求金水二星定見伏定日】〉
〈To find the fixed appearance, concealment, and fixed day for Venus and Mercury〉
92
各以伏見日行差，除其日太陽盈縮差，為日。 若晨伏夕見，日在盈歷，加之； 在縮歷，減之； 如夕伏晨見，日在盈縮，減之，在縮歷，加之； 加減其星泛積，為常積。 視常積，如中限以下，為冬至後； 以上，去之，余為夏至後。 其二至後，如象限以下，自相乘； 以上，覆減中限，余亦自相乘； 各如法而一為分， 〈（冬至後晨，夏至後夕，以一十八為法； 冬至後夕，夏至後晨，以七十五為法。）〉 以伏見度乘之，一十五除之，為差。 其差，滿行差而一，為日，不滿，退除為分秒； 加減常積，為定積； 〈（冬至後，晨見夕伏，加之； 夕見晨伏，減之。 夏至後，晨見夕伏，減之； 夕見晨伏，加之。）〉 加命如前，即得定見伏日辰。
For each star, divide that day's solar excess-and-deficit difference by the appearance-concealment daily motion difference to obtain days. For morning concealment and evening appearance, add when the sun is in the excess cycle; when in the deficit cycle, subtract; For evening concealment and morning appearance, subtract when the sun is in excess and add when in the deficit cycle; Add or subtract on that star's general accumulation to obtain the regular accumulation. Inspect the regular accumulation: at or below the middle limit is after winter solstice; above it, cast out the middle limit and take the remainder as after summer solstice. After each solstice, square values at or below the image limit; above it, overlay-subtract from the middle limit and square the remainder likewise; divide each by the appropriate divisor to obtain parts, (After winter solstice at dawn and after summer solstice at dusk, use 18 as the divisor; (after winter solstice at dusk and after summer solstice at dawn, use 75 as the divisor.) Multiply by the heliacal appearance-and-concealment degree and divide by 15 to obtain the difference. When the difference fills the motion difference, take one for days; cast back the remainder into parts and seconds; Add or subtract on the regular accumulation to obtain the fixed accumulation; (After winter solstice, for morning appearance and evening concealment, add; for evening appearance and morning concealment, subtract. After summer solstice, for morning appearance and evening concealment, subtract; (for evening appearance and morning concealment, add.) Add and name as before to obtain the fixed appearance-and-concealment day and double-hour.
93
其水星，夕疾在大暑氣初日至立冬氣九日三十五分以下者，不見； 晨留在大寒氣初日至立夏氣九日三十五分以下者，不見。 春不晨見，秋不夕見者，亦舊歷有之。
For Mercury in evening swift motion from the start of Great Heat through day 9, part 35 of Start of Winter, it is invisible; When in morning station from the start of Great Cold through day 9, part 35 of Start of Summer, it is invisible. That it is not seen at dawn in spring or at dusk in autumn was also recorded in older calendars.