1
步晷漏術
Procedure for Gnomon Shadow and Clepsydra.
2
二至限：一百八十一日六十二分。
Two-solstice limit: 181 days, 62 parts.
3
一象度：九十一度三十一分。
One-quadrant arc: 91°31′.
4
消息法：一萬六百八十九。
Ebb-and-flow divisor: 10,689.
5
辰法：三千二百五十。
Double-hour divisor: 3,250.
6
刻法：三百九十。
Clepsydra-graduation divisor: 390.
7
半辰法：一千六百二十五。
Half double-hour divisor: 1,625.
8
昏明刻分：九百七十五。
Dusk-and-dawn clepsydra parts: 975.
9
昏明：二刻一百九十五分。
Dusk and dawn: 2 ke and 195 parts.
10
冬至岳台晷景常數：一丈二尺八寸五分。
Winter-solstice Yue Terrace noon-shadow constant: 12 chi 8 cun 5 fen (about 8 ft).
11
夏至岳台晷景常數：一尺五寸七分。
Summer-solstice Yue Terrace noon-shadow constant: 1 chi 5 cun 7 fen.
12
冬至後初限，夏至後末限：四十五日六十二分。
After winter solstice (initial) and after summer solstice (final): 45 days, 62 parts.
13
夏至後初限，冬至後末限：一百三十七日。
After summer solstice (initial) and after winter solstice (final): 137 days.
14
求岳台晷景入二至後日數：計入二至後來日數，以二至約餘減之，仍加半日之分，即為入二至後來日午中積數及分。
To find days after a solstice for Yue Terrace shadow: take post-solstice elapsed days, subtract the solstice reduced remainder, add half a day's parts—the noon accumulated days and parts since the solstice.
15
求岳台晷景午中定數：置所求午中積數，如初限以下者為在初； 已上者，覆減二至限，餘為在末其在冬至後初限，夏至後末限者，以入限日減一千九百三十七半，為泛差； 仍以入限日分乘其日盈縮積，盈縮積在日度術中。
To find the fixed noon Yue Terrace shadow: set the noon accumulated number; if at or below the initial limit, it falls in the initial period; if above, subtract from the two-solstice limit; the remainder is in the final period. In the post–winter-solstice initial or post–summer-solstice final limit, subtract days within the limit from 1,937½ for the general difference; then multiply by the days-within-limit parts and the daily excess-and-deficiency accumulation (from the solar-degree procedure).
16
五因百約之，用減泛差，為定差； 乃以入限日分自相乘，以乘定差，滿一百萬為尺，不滿為寸，為分及小分，以減冬至常晷，餘為其日午中晷景定數若所求入冬至後末限，夏至後初限者，乃三約入限日分，以減四百八十五少，餘為泛差。 仍以盈縮差減極數，餘者若在春分後，秋分前者，直以四約之，以加泛差，為定差； 若春分前，秋分後者，以去二分日數及分乘之，滿六百而一，以減泛差，餘為定差； 乃以入限日分自相乘，以乘定差，滿一百萬為尺，不滿為寸，為分及小分，以加夏至常晷，即為其日午中晷景定數。
Multiply by five and reduce by a hundred to get the fixed difference; square the days-within-limit parts and multiply by the fixed difference; one million fills a chi, otherwise cun, fen, and minor fen; subtract from the winter-solstice constant—the day's fixed noon shadow. For post–winter-solstice final or post–summer-solstice initial limits, divide days-within-limit parts by three, subtract from 485 and a fraction less for the general difference. Subtract the excess-and-deficiency difference from the extreme number; if between spring and autumn equinox, divide by four and add to the general difference for the fixed difference; if before spring equinox or after autumn equinox, multiply days and parts from the equinox, divide by 600, subtract from the general difference for the fixed difference; square days-within-limit parts, multiply by fixed difference; one million = chi, else cun, fen, minor fen; add to summer-solstice constant for that day's fixed noon shadow.
17
求每日消息定數：置所求日中日度分，如在二至限以下者為在息； 以上者去之，餘為在消又視入消息度加一象以下者為在初； 以上者，覆減二至限，餘為在末。 其初，末度自相乘，以一萬乘而再折之，滿消息法除之，為常數。 乃副之，用減一千九百五十，餘以乘其副，滿八千六百五十除之，所得以加常數，為所求消息定數。
To find the daily fixed ebb-and-flow number: set noon solar degree parts; at or below the two-solstice limit is waxing (xi); if above, remove it; remainder is waning (xiao). If entered ebb-and-flow degree is at or below one quadrant plus one, it is initial; if above, subtract from the two-solstice limit; remainder is final. Square initial or final degrees, multiply by 10,000, fold twice, divide by ebb-and-flow divisor for the constant. Set aside; subtract from 1,950, multiply remainder by auxiliary, divide by 8,650; add to constant for the fixed ebb-and-flow number.
18
求每日黃道去極度及赤道內外度：置其日消息定數，以四因之，滿三百二十五除之為度，不滿，退除為分，所得，在春分後加六十七度三十一分，在秋分後減一百一十五度三十一分，即為所求日黃道去極度及分。 以黃道去極度與一象度相減，餘為赤道內，外度。 若去極度少，為日在赤道內； 若去極度多，為日在赤道外。
Set daily fixed ebb-and-flow number, ×4, ÷325 for degrees (remainder → parts); after spring equinox add 67°31′, after autumn equinox subtract 115°31′ for yellow-path polar distance. Subtract polar distance from one-quadrant arc; remainder is equatorial inner or outer degree. If polar distance is less, the sun is south of the equator; if greater, the sun is north of the equator.
19
求每日晨昏分及日出入分：以其日消息定數，春分後加六千八百二十五，秋分後減一萬七百二十五，餘為所求日晨分； 用減元法，餘為昏分以昏明分加晨分，為日出分； 減昏分，為日入分。
For dawn/dusk and sunrise/sunset parts: add 6,825 after spring equinox or subtract 10,725 after autumn equinox from fixed ebb-and-flow; remainder = dawn parts; subtract from origin method for dusk; dawn + dusk-and-dawn = sunrise; subtract dusk for sunset.
20
求每日距中距子度及每更差度：置其日晨分，以七百乘之，滿七萬四千七百四十二除為度，不滿，退除為分，命曰距子度； 用減半周天，餘為距中度若倍距子度，五除之，即為每更差度及分若依司辰星漏曆，則倍距子度，減去待旦三十六度五十二分半，餘以五約之，即每更差度。
For distance-from-center/zi and watch difference: dawn parts ×700 ÷74,742 = degrees (remainder → parts), named distance-from-zi; subtract from half circuit for distance-from-center; double distance-from-zi ÷5 = watch difference. Per Directorate calendar: double distance-from-zi, subtract awaiting-dawn 36°52′½, divide by five.
21
求每日夜半定漏：置其日晨分，以刻法除之為刻，不滿為分，即所求日夜半定漏。
Midnight fixed clepsydra: dawn parts ÷ clepsydra divisor = ke, remainder = parts.
22
求每日晝夜刻及日出入辰刻：倍夜半定漏，加五刻，為夜刻。 用減一百刻，餘為晝刻。 以昏明刻加夜半定漏，滿辰法除之為辰數，不滿，刻法除之為刻，又不滿，為刻分命辰數從子正，算外，即日出辰刻； 以晝刻加之，命如前，即日入辰刻若以。 半辰刻加之，即命從辰初也。
Day/night ke: double midnight clepsydra + 5 ke = night ke. 100 ke − night = day ke. Dusk-dawn + midnight clepsydra ÷ double-hour method → hours; remainder ÷ clepsydra → ke; count from zi for sunrise; add day ke for sunset; if using half double-hour ke, count from chen initial.
23
求更點辰刻：倍夜半定漏，二十五而一，為點差刻； 五因之，為更差刻。 以昏明刻加日入辰刻，即甲夜辰刻； 以更點差刻累加之，滿辰刻及分去之，各得更點所入辰刻及分。 若同司辰星漏曆者，倍夜半定漏，減去待旦一十刻，餘依術求之，即同內中更點。
Watch points: double midnight clepsydra ÷25 = point-difference ke; ×5 = watch-difference ke. Sunset + dusk-dawn = first watch (jia ye); accumulate watch-difference; remove full double-hours for each watch entry. Directorate calendar: double midnight clepsydra − awaiting-dawn 10 ke, same inner watch points.
24
求昏曉及五更中星：置距中度，以其日昏後夜半赤道日度加而命之，即其日昏中星所格宿次，其昏中星便為初更中星； 以每更差度加而命之，即乙夜所格中星； 累加之，得逐更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，若同司辰星漏曆中星，則倍距子度，減去待旦十刻之度三十六度五十二分半，餘約之為五更，即同內中更點中星。
Dusk/dawn/five-watch stars: distance-from-center + post-dusk midnight equatorial sun degree → dusk median lodge; dusk = first watch; add watch difference for second watch; accumulate per watch; double distance-from-zi + dusk = dawn; Directorate: double distance-from-zi − awaiting-dawn 36°52′½, ÷5 watches.
25
求九服距差曰：各於所在立表候之，若地在岳台北，測冬至後與岳台冬至晷景同者，累冬至後至其日，為距差日； 若地在岳台南，測夏至後與岳台晷景同者，累夏至後至其日，為距差日。
Nine-domain distance difference: erect gnomon locally; north of Yue Terrace, when post–winter-solstice shadow matches Yue Terrace winter shadow, accumulated days = distance-difference days; south of Yue Terrace, when post–summer-solstice shadow matches, accumulate from summer solstice = distance-difference days.
26
求九服晷景：若地在岳台北冬至前後者，以冬至前後日數減距差日，為餘日； 以餘日減一千九百三十七半，為泛差； 依前術求之，以加岳台冬至晷景常數，為其地其日中晷常數。 若冬至前後日多於距差日，乃減去距差日，餘依前術求之，即得其地其日中晷常數若地在岳台南夏至前後者，以夏至前後日數減距差日，為餘日； 乃三約之，以減四百八十五少，為泛差； 依前術求之，以減岳台夏至晷景常數，即其地其日中晷常數。 如夏至前後日數多於距差日，乃減岳台夏至常晷，餘即晷在表南也。 若夏至前後日多於距差日，即減去距差日，餘依前術求之，各得其地其日中晷常數。 若求定數，依立成以求午中晷景定數。
Nine-domain shadow: north of Yue Terrace at winter solstice, winter-solstice days − distance-difference = remainder; remainder days subtracted from 1,937½ = general difference; by prior procedure, add to Yue Terrace winter constant = local noon constant. If winter days > distance-difference, subtract and seek; south at summer solstice: summer days − distance-difference = remainder; ÷3, subtract from 485 and a fraction less for general difference; by prior procedure, subtract from Yue Terrace summer constant. If summer days > distance-difference, subtract summer constant—shadow south of table. If summer days exceed distance-difference, subtract and seek each local noon constant. For fixed values, use the ready reckoner for noon fixed shadow.
27
求九服所在晝夜漏刻：冬，夏二至各於所在下水漏，以定其地二至夜刻，乃相減，餘為冬，夏至差刻置岳台其日消息定數，以其地二至差刻乘之，如岳台二至差刻二十而一，所得，為其地其日消息定數。 乃倍消息定數，滿刻法約之為刻，不滿為分，乃加減其地二至夜刻，秋分後，春分前，減冬至夜刻； 春分後，秋分前，加夏至夜刻。
Local day/night clepsydra: fix solstice night ke by water clepsydra; solstice difference. Yue Terrace fixed ebb-and-flow × local solstice difference ÷ Yue Terrace difference ÷20 = local fixed ebb-and-flow. Double fixed ebb-and-flow ÷ clepsydra → ke; adjust local solstice night: after autumn/before spring subtract winter night; after spring/before autumn add summer night.
28
為其地其日夜刻； 用減一百刻，餘為晝刻。 其日出入辰刻及距中度五更中星，並依前術求之。
That yields the night ke for that place on that day; 100 − night = day ke. Sunrise/sunset, distance-from-center, five-watch stars—by prior procedure.
29
步月離術
Procedure for Lunar Motion.
30
轉度母：八千一百一十二萬。
Rotation degree mother: 81,120,000.
31
轉終分：二百九十八億八千二百二十四萬二千二百五十一。
Rotation terminal parts: 298,824,222,251.
32
朔差：二十一億四千二百八十八萬七千。
Synodic difference: 2,142,887,000.
33
朔差：二十六度餘三千三百七十六萬七千，約餘四千一百六十二半。
Synodic difference: 26° remainder 33,767,000, reduced remainder 4,162½.
34
轉法：一十億八千四百四十七萬三千。
Rotation method: 1,084,473,000.
35
會周：三百二十億二千五百一十二萬九千二百五十一。
Conjunction cycle: 320,251,292,251.
36
轉終：三百六十八度餘三十八萬二千二百五十一，約餘三千七百八。
Rotation terminal: 368° remainder 382,251, reduced remainder 3,708.
37
轉終：二十七日餘六億一百四十七萬一千二百五十一，約餘五千五百四十六。
Rotation terminal: 27 days remainder 614,712,251, reduced remainder 5,546.
38
中度：一百八十四度餘一千五百四萬一千一百二十五半，約餘一千八百五十四。
Mid-degree: 184° remainder 15,041,125½, reduced remainder 1,854.
39
象度：九十二度餘七百五十二萬五百六十二太，約分九百二十七。
Quadrant degree: 92° remainder 7,520,562 and a fraction more, reduced parts 927.
40
月平行：十三度餘二千九百九十一萬三千，約分三千六百八十七半。
Moon mean motion: 13° remainder 29,913,000, reduced parts 3,687½.
41
望差：一百九十七度餘三千一百九十二萬四千六百二十五半，約分三千九百三十四。
Full-moon difference: 197° remainder 31,924,625½, reduced parts 3,934.
42
弦差：九十八度餘五千六百五十二萬二千三百一十二太，約分六千九百六十七。
Quarter difference: 98° remainder 56,522,312 and a fraction more, reduced parts 6,967.
43
日衰：一十八，小分九。
Daily decay: 18, minor parts 9.
44
求月行入轉度：以朔差乘所求積月，滿轉終分去之，不盡為轉餘。 滿轉度母除為度，不滿為餘，其餘若以一萬乘之，滿轉度母除之，即得約分； 若以轉法除轉餘，即為入轉日及餘。
Lunar entry rotation: accumulated months × synodic difference; remove rotation terminal; remainder = rotation remainder. ÷ rotation mother → degrees, remainder parts; ×10,000 ÷ mother → reduced parts; rotation remainder ÷ rotation method = entry rotation day and remainder.
45
即得所求月加時入轉度及餘。 若以弦度及餘累加之，即得上弦、望、下弦及後朔加時入轉度及分； 其度若滿轉終度及餘去之。
Thus the moon's hour-of-addition entry rotation degree and remainder. Accumulate quarter degree for first quarter, full, last quarter, next new moon entry; if degree fills rotation terminal, remove.
46
其入轉度如在中度以下為月行在疾曆； 如在中度以上者，乃減去中度及餘，為月入遲曆。
If entry rotation ≤ mid-degree, moon is in fast sequence; if above, subtract mid-degree; moon enters slow sequence.
47
求月行遲疾差度及定差：。 置所求月行入遲速度，如在象度以下為在初以上，覆減中度，餘為在末其度餘用約分百為母。
To find lunar fast-slow difference and fixed difference: Set entry slow-fast degree; ≤ quadrant = initial, above invert mid-degree = final; use reduced parts 100 as mother.
48
置初，末度於上，列二百一度九分於下，以上減下，餘以下乘上，為積數； 滿一千九百七十六除為度，不滿，退除為分，命曰遲疾差度。 在疾為減，在遲為加。
Initial/final above, 201°9 parts below; lower subtract upper, lower × upper = accumulation; ÷1,976 = degrees, remainder → parts = fast-slow difference. Fast: subtract; slow: add.
49
以一萬乘積數，滿六千七百七十三半除之，為遲疾定差。 疾加，遲減，若用立成者，以其度下損益率乘度餘，滿轉度母而一，所得，隨其損益，即得遲疾及定差。 其遲疾，初末損益分為二日者，各加其初，末以乘除。
Accumulation ×10,000 ÷6,773½ = fixed difference. Fast add/slow subtract; ready reckoner: rate × degree remainder ÷ rotation mother. If initial/final gain-loss spans two days, add each initial/final then multiply/divide.
50
求朔弦望所直度下月行定分：置遲疾所入初，末度分，進一位，滿七百三十九除之，用減一百二十七，餘為衰差。 乃以衰差疾初遲末減，遲初疾末加，皆加減平行度分，為其度所直月行定分。 其度以百命為分。
Fixed parts at degree: fast-slow initial/final parts advance one place ÷739; 127 − result = decay difference. Fast-initial slow-final subtract, slow-initial fast-final add, ± mean motion = fixed parts at degree. Its degree uses one hundred as the divisor for parts.
51
求朔弦望定日：各以日躔盈縮，月行遲疾定差加減經朔、弦、望小餘，滿若不足，進退大餘，命甲子，算外，各得定日日辰及餘。 若定朔干名與後朔干名同者月大，不同月小，月內無中氣者為閏月。 凡注曆，觀定朔小餘，秋分後四分之三已上者，進一日； 若春分後，其定朔晨分差如春分之日者，三約之，以減四分之三； 如定朔小餘及此數已上者，進一日； 朔或當交有食，初虧在日入已前者，其朔不進弦，望定小餘不滿日出分者，退一日； 其望或當交有食，初虧在日出已前，其定望小餘雖滿日出分者，亦退之又月行九道遲疾，歷有三大二小； 日行盈縮累增損之，則有四大三小，理數然也若。 循其常，則當察加時早晚，隨其所近而進退之，使月之大小不過連三。 舊說，正月朔有交，必須消息前後一兩月，移食在晦，二之日。 且日食當朔，月食當望蓋自然之理夫日之食，蓋天之垂誡，警悟時政，若道化得中，則變咎為祥，國家務以至公理天下，不可私移晦朔，宜順天誡。 故《春秋傳》書日食，乃糾正其朔，不可專移食於晦，二。 其正月朔有交，一從近典，不可移避。
To find fixed syzygy days: apply solar excess-deficiency and lunar fast-slow fixed differences to mean minor remainders; adjust major remainder, assign from jiazi for each fixed day and time. If fixed new-moon stem matches the next, the month is long; if not, short; no mid-qi within the month makes an intercalary month. In calendar annotation, if fixed new-moon minor remainder is three-fourths or more after autumn equinox, advance one day; after spring equinox, if fixed new-moon dawn difference equals the spring-equinox value, reduce three-fourths by one-third; if fixed new-moon minor remainder reaches that number or above, advance one day; if new moon coincides with eclipse with first contact before sunset, do not advance; for quarter/full, if fixed minor remainder is less than sunrise parts, retreat one day; if full moon has eclipse with first contact before sunrise, retreat even if minor remainder fills sunrise; lunar nine-path also yields three long and two short months; solar excess-and-deficiency cumulatively adjusts, four long and three short months arise by numerical principle. If follow the constant rule, inspect addition time and advance or retreat accordingly so long or short months do not exceed three in succession. Old theory held that a first-month new moon with conjunction required moving the eclipse to the last day or second day, one or two months around the ebb-and-flow period. Solar eclipses belong at new moon and lunar at full moon by natural principle. An eclipse is Heaven's warning to government; with proper governance calamity becomes blessing, and the state must govern fairly—not shift new moons privately but heed Heaven. The Zuo Commentary records eclipses to correct the new moon; one cannot simply move eclipses to the last or second day. If the first-month new moon has conjunction, follow the recent canon—it cannot be shifted away.
52
求定朔弦望加時日度：置朔，弦，望中日及約分，以日躔盈縮度及分盈加縮減之，又以元法退除遲疾定差，疾加遲減之，餘為其朔，弦，望加時定日。 以天正冬至加時黃道日度加而命之，即所求朔，弦，望加時定日所在宿次。 朔，望有交，則依後術。
For fixed syzygy addition-time solar degree: set mean syzygy midday parts; apply solar excess-deficiency; remove fast-slow fixed difference by origin method (fast add, slow subtract). Add to winter-solstice yellow-path degree and assign for the lodge of fixed syzygy addition time. If new or full moon has conjunction, use the later procedure.
53
求月行九道：凡合朔所交，冬在陰曆，夏在陽曆，月行青道。 冬至，夏至後，青道半交在春分之宿，當黃道東立夏，立冬後，青道半交在立春之宿，當黃道東南； 至所衝之宿亦如之。
For lunar nine-path motion: at conjunction, winter in yin and summer in yang calendars—the moon follows the green path. After solstices, green-path half-crossing lies at spring-equinox lodge, east on the yellow path; after summer and winter beginnings, at spring-beginning lodge, southeast; the opposite solstice lodge is the same.
54
冬在陽曆，夏在陰曆，月行白道冬至，夏至後，白道半交在秋分之宿，當黃道西； 立冬，立夏後，白道半交在立秋之宿，當黃道西北； 至所衝之宿亦如之。
Winter in yang and summer in yin—the moon follows the white path. After solstices, white-path half-crossing is at autumn-equinox lodge, west on the yellow path; after winter and summer beginnings, at autumn-beginning lodge, northwest on the yellow path; the opposite solstice lodge is likewise.
55
春在陽曆，秋在陰曆，月行朱道春分，秋分後，朱道交在夏至之宿，當黃道南； 立春，立秋後，朱道半交在立夏之宿，當黃道西南：至所衝之宿亦如之。
Spring in yang and autumn in yin—the vermilion path. After equinoxes, vermilion crossing at summer-solstice lodge, south on the yellow path; after spring and autumn beginnings, half-crossing at summer-beginning lodge, southwest; the opposite solstice lodge is likewise.
56
春在陰曆，秋在陽曆，月行黑道春分，秋分後，黑道半交在冬至之宿，當黃道正北立春，立秋後，黑道半交在立冬之宿，當黃道東北； 至所衝之宿亦如之。
Spring in yin and autumn in yang—the black path. After equinoxes, half-crossing at winter-solstice lodge, due north; after spring and autumn beginnings, at winter-beginning lodge, northeast; the opposite solstice lodge is likewise.
57
四序離為八節，至陰陽之所交，皆與黃道相會，故月行九道。 各視月所入正交積度，視正交九道宿度所入節候，即其道，其節所起。
Four seasons yield eight nodes; yin-yang crossings meet the yellow path—hence nine lunar paths. Observe the moon's orthodox-crossing accumulated degree and which node the nine-path lodge enters—that path and its starting node.
58
滿象度及分去之餘，入交積度及象度並在交會術中。
Remove full quadrant degrees; entered-crossing and quadrant degrees are in the conjunction procedure.
59
若在半象以下為在初限。 以上，覆減象度及分，為在末限。 用減一百一十一度三十七分，餘以所入初，末限度及分乘之，退位，半之，滿百為度，不滿為分，所得為月行與黃道差數距半交後，正交前，以差數減； 距正交後，半交前，以差數加。 此加減出入六度，單與黃道相較之數，若較之赤道，隨數遷變不常。
At or below half a quadrant is the initial limit. If above, invert-subtract quadrant degree and parts for the final limit. Subtract from 111°37′, multiply by limit degrees, halve, ÷100 for the lunar yellow-path difference; subtract after half-crossing before orthodox crossing; add after orthodox crossing before half-crossing. This adjustment varies within six degrees against the yellow path; against the equator the value shifts irregularly.
60
計去二至以來度數，乘黃道所差，九十而一，為月行與黃道差數。 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； 若入春分交後行陽曆，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，減者加之皆加減黃道宿積度，為九道宿積度； 以前宿九道宿積度減其宿九道宿積度，餘為其宿九道宿度及分。 其分就近約為太，半，少三數。
Degrees since solstice × yellow-path difference ÷ 90 = lunar yellow-path difference. For the sun, inside the equator is yin, outside yang; for the moon, inside the yellow path is yin, outside yang. Thus the moon after spring-equinox crossing travels yin calendar, after autumn yang—same-name; yang after spring or yin after autumn is different-name; in same-name add/subtract with the difference; in different-name reverse the operation—all yield nine-path accumulated degrees; Prior nine-path accumulated degree subtracted from current yields that lodge's nine-path degree and parts. Reduce parts to greater, half, or lesser fractions.
61
求月行九道入交度：置其朔加時定日度，以其朔交初度及分減之，餘為其朔加時月行入交度及餘。 其餘以一萬乘之，以元法退除之，即為約餘。
Entry-crossing degree: fixed new-moon solar degree minus initial orthodox-crossing degree and parts. Remainder ×10,000, remove by origin method = reduced remainder.
62
以天正冬至加時黃道日度加而命之，即正交月離所在黃道宿度。
Add to winter-solstice yellow-path degree and assign for orthodox-crossing lunar lodge.
63
求正交加時月離九道宿度：以正交度及分減一百一十一度三十七分，餘以正交度及分乘之，退一等，半之，滿百為度，不滿為分，所得，命曰定差。 以定差加黃道宿度，計去冬，夏至以來度數，乘定差，九十而一，所得，依同異名加減之，滿若不足，進退其度，命如前，即正交加時月離九道宿度及分。
Orthodox-crossing nine-path lodge: 111°37′ − crossing parts, × crossing parts, halve, ÷100 = fixed difference. Add fixed difference to yellow-path degree; solstice degrees × fixed difference ÷90; adjust by same/different name for orthodox-crossing nine-path lodge.
64
求定朔弦望加時月離所在宿度：各置其日加時日躔所在，變從九道，循次相加凡合朔加時，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，命起正交宿次，算外，即朔、弦、望加時所當九道宿度。 其合朔加時若非正近，則日在黃道，月在九道各入宿度，雖多少不同，考其去極，若應繩準。 故云月行潛在日下，與太陽同度。
Fixed syzygy lunar lodge: shift solar progression to nine-path; at conjunction the moon is beneath the sun at the same degree. Subtract orthodox-crossing yellow-path degree from syzygy yellow-path degree; add to nine-path degree and assign from orthodox-crossing lodge. If not exactly at crossing, sun and moon lodge degrees differ but polar distances align. Hence: the moon lies hidden beneath the sun at the same degree.
65
各以弦，望度及分加其所當九道宿度，滿宿次去之，各得加時九道月離宿次。
Add quarter/full degrees to nine-path lodge; remove full lodges for each syzygy addition-time lunar lodge.
66
求定朔夜半入轉：以所求經朔小餘減其朔加時入轉日餘，其經朔小餘，以二萬七千八百七乘之，即母轉法。
Midnight entry rotation: mean minor remainder minus addition-time entry-rotation remainder; ×27,807 = mother rotation method.
67
為其經朔夜半入轉。 若定朔大餘有進退者，亦進退轉日，無進退則因經為定。 其餘以轉法退收之，即為約分。
That yields mean new-moon midnight entry rotation. If fixed new-moon major remainder shifts, shift rotation day; else use the mean as fixed. Remove remainder by rotation method for reduced parts.
68
求次月定朔夜半入轉：因定朔夜半入轉，大月加二日，小月加一日，餘，分皆加四千四百五十四，滿轉終日及約分去之，即次月定朔夜半入轉； 累加一日，去命如前，各得逐日夜半入轉日及分。
Next month: add 2 days (long) or 1 (short) and 4,454 parts; remove rotation terminal. Accumulate daily for each midnight entry-rotation day and parts.
69
求定朔弦望夜半月度：各置加時小餘，若非朔，望有交者，有用定朔、弦、望小餘。
Midnight lunar degree: use addition-time or fixed syzygy minor remainder as applicable.
70
以其日月行定分乘之，滿元法而一為度，不滿，退除為分，命曰加時度。 以減其日加時月度，即各得所求夜半月度。
× lunar fixed parts ÷ origin = addition-time degrees. Subtract from addition-time lunar degree for midnight lunar degree.
71
求晨昏月：以晨分乘其日月行定分，元法而一，為晨度； 用減月行定分，餘為昏度。 各以晨昏度加夜半月度，即所求晨昏月所在宿度。
Dawn moon: dawn parts × lunar fixed parts ÷ origin = dawn degree; lunar fixed parts − dawn degree = dusk degree. Add dawn/dusk degrees to midnight lunar degree for dawn and dusk lodges.
72
求朔弦望晨昏定程：各以其朔昏定月減上弦昏定月，餘為朔後昏定程； 以上弦昏定月減望昏定月，餘為上弦後昏定程； 以望晨定月減下弦晨定月，餘為望後晨定程； 以下弦晨定月減次朔晨定月，餘為下弦後晨定程。
Fixed intervals: new dusk − first-quarter dusk = post-new dusk interval; first-quarter dusk − full dusk = post-first-quarter interval; full dawn − last-quarter dawn = post-full dawn interval; last-quarter dawn − next new dawn = post-last-quarter dawn interval.
73
求轉積度：計四七日月行定分，以日衰加減之，為逐日月行定程； 乃自所入日計求定之，為其程轉積度分其四七日月行定分者，初日益遲一千二百一十，七日漸疾一千三百四十一，十四日損疾一千四百六十一，二十一日漸遲一千三百二十八，乃觀其遲疾之極差而損益之，以百為分母。
Rotation accumulation: four-seven-day fixed parts ± daily decay per day; Compute from entry day; day 1 +1,210 slow, day 7 +1,341 fast, day 14 −1,461, day 21 +1,328 slow—adjust by fast-slow extremes, denominator 100.
74
求每日晨昏月：以轉積度與晨昏定程相減，餘以距後程日數除之，為日差定程多為加，定程少為減。
Daily dawn/dusk moon: rotation accumulation vs fixed interval ÷ days after = daily adjustment.
75
以加減每日月行定分，為每日轉定度及分。 以每日轉定度及分加朔、弦、望晨昏月，滿九道宿次去之，即為每日晨，昏月離所在宿度及分。 凡注曆，朔後注昏，望後注晨。
Adjust daily lunar fixed parts for daily rotation fixed degree. Add daily rotation to syzygy dawn/dusk moon; remove full lodges for daily dawn and dusk lunar positions. Calendar annotation: dusk after new moon, dawn after full moon.
76
已前月度，並依九道所推，以究算術之精微。 若注曆求其速要者，即依後術以推黃道月度。
Earlier lunar degrees follow nine-path computation for full precision. For brevity, use the later procedure for yellow-path lunar degrees.
77
求天正十一月定朔夜半平行月：以天正經朔小餘乘平行度分，元法而一為度，不滿，退除為分秒，所得，為經朔加時度。 用減其朔中日，即經朔晨前夜半平行月積度。 若定朔有進退，以平行度分加減之。
Eleventh-month midnight mean moon: mean minor remainder × mean-motion parts ÷ origin. Subtract from new-moon midday for dawn-before midnight mean-motion degree. If fixed new moon shifts, adjust mean-motion parts.
78
即為天正十一月定朔之日晨前夜半平行月積度及分。
That is the eleventh-month fixed new-moon dawn-before midnight mean-motion degree.
79
求次月定朔之日夜半平行月：置天正定朔之日夜半平行月，大月加三十五度八十分六十一秒，小月加二十二度四十三分七十三秒半，滿周天度分即去之，即每月定朔之晨前夜半平行月積度及分秒。
Next month: long +35°80′61″, short +22°43′73.5″ from prior midnight mean moon.
80
求定弦望夜半平行月、計弦、望距定朔日數，以乘平行度及分秒，以加其定朔夜半平行月積度及分秒，即定弦，望之日夜半平行月積度及分秒。 亦可直求朔望，不复求度，從簡易也。
Quarter/full midnight mean moon: days × mean-motion parts + fixed new-moon midnight mean moon. Or seek new and full directly without separate degree steps—for simplicity.
81
求天正定朔夜半入轉度：置天正經朔小餘，以平行月度及分乘之，滿元法除為度，不滿，退除為分秒，命為加時度； 以減天正十一月經朔加時入轉度及約分，餘為天正十一月經朔夜半入轉度及分。 若定朔大餘有進退者，亦進退平行度分，即為天正十一月定朔之日晨前夜半入轉度及分秒。
First fixed new-moon midnight entry rotation: mean minor remainder × mean lunar parts ÷ origin; subtract from mean addition-time entry rotation for eleventh-month midnight entry rotation. Adjust mean-motion parts if fixed new-moon major remainder shifts.
82
求次月定朔及弦望夜半入轉度：因天正十一月定朔夜半入轉度分，大月加三十二度六十九分一十七秒，小月加十九度三十二分二十九秒半，即各得次月定朔夜半入轉度及分。 各以朔、弦、望相距日數乘平行度分以加之，滿轉終度及秒即去之，如在中度以下者為在疾； 以上者去之，餘為入遲曆，即各得次朔、弦、望定日晨前夜半入轉度及分若以平行月度及分收之，即為定朔、弦、望入轉日。
Next month entry rotation: long +32°69′17″, short +19°32′29.5″. add syzygy-interval days × mean-motion parts; ≤ mid-degree = fast; above mid-degree → slow; collect by mean lunar parts for entry-rotation day.
83
求定朔弦望夜半定月：以定朔、弦、望夜半入轉度分乘其度損益衰，以一萬約之為分，百約之為秒，損益其度下遲疾度，為遲疾定度。 乃以遲加疾減夜半平行月，為朔，弦，望夜半定月積度。 以冬至加時黃道日度加而命之，即定朔、弦、望夜半月離所在。 宿次若有求晨昏月，以其日晨昏分乘其日月行定分，元法而一，所得為晨昏度； 以加其夜半定月，即得朔、弦、望晨昏月度。
Midnight fixed moon: entry rotation × gain-loss decay; adjust fast-slow degree. Slow add, fast subtract from midnight mean moon for fixed moon. Add to winter-solstice yellow-path degree for midnight lunar lodge. Dawn/dusk moon: dawn/dusk parts × lunar fixed parts ÷ origin; add to midnight fixed moon for dawn and dusk lunar degrees.
84
求朔弦望定程：各以朔、弦、望定月相減，餘為定程若求晨昏定程，則用晨昏定月相減，朔後用昏，望後用晨。
Fixed interval: difference of fixed moons; dawn/dusk versions use dawn/dusk moons.
85
求朔弦望轉積度分：計四七日月行定分，以日衰加減之，為逐日月行定分； 乃自所入日計之，為其程轉積度分。 其四七日月行定分者，初日益遲一千二百一十，七日漸疾一千三百四十一，十四日損疾一千四百六十一，二十一日漸遲一千三百二十八，乃視其遲疾之極差而損益之，分以百為母。
Rotation accumulation: four-seven-day parts ± daily decay; compute from entry day for rotation accumulation. Day 1 +1,210 slow through day 21 +1,328 slow—adjust by extremes, denominator 100.
86
求每日月離宿次：。 各以其朔、弦、望定程與轉積度相減，餘為程差以距後程日數除之，為日差定程多為益差，定程少為損差。
To find each day's lunar departure lodge: Subtract rotation accumulation from each syzygy fixed interval; ÷ days after interval = daily difference (increase if interval greater, decrease if less).
87
以日差加減月行定分。 為每日月行定分； 以每日月行定分累加定朔、弦、望夜半月在宿次，命之，即每日晨前夜半月離宿次。 如晨昏宿次，即得每日晨昏月度。
Adjust lunar fixed parts by daily difference. each day's lunar motion fixed parts; Accumulate daily fixed parts from syzygy midnight moon to assign each dawn-before midnight lunar lodge. From dawn and dusk lodges obtain daily dawn and dusk lunar degrees.
88
步交會術
Procedure for Conjunction and Eclipse.
89
交度母：六百二十四萬。
Crossing degree mother: 6,240,000.
90
周天分：二十二億七千九百二十萬四百四十七。
Circuit-of-heaven parts: 2,279,200,447.
91
朔差：九百九十萬一千一百五十九。
Synodic difference: 9,901,159.
92
朔差：一度，餘三百六十六萬一千一百五十九。
Synodic difference: 1° remainder 3,661,159.
93
望差：空度，餘四百九十五萬五百七十九半。
Full-moon difference: 0° remainder 4,950,579½.
94
半周天：一百八十二度餘三百九十二萬二百二十三半，約分六千二百八十二。
Half circuit: 182° remainder 3,920,223½, reduced parts 6,282.
95
日食限：一千四百六十四。
Solar eclipse limit: 1,464.
96
月食限：一千三百三十八。
Lunar eclipse limit: 1,338.
97
盈初限縮末限：六十度八十七分半。
Excess-initial and deficiency-final limit: 60°87′½.
98
縮初限盈末限：一百二十一度七十五分。
Deficiency-initial and excess-final limit: 121°75′.
99
求交初度：置所求積月，以朔差乘之，滿周天分去之，不盡，覆減周天分，滿交度母除之為度，不滿為餘，即得所求月交初度及餘； 以半周天加之，滿周天去之，餘為交中度及餘若以望差減之，即得其月望交初度及餘； 以朔差減之，即得次月交初度及餘； 以交度母退除，即得餘分若以天正黃道日度加而命之，即各得交初，中所在宿度及分。
Initial crossing degree: accumulated months × synodic difference; remove circuit of heaven; ÷ crossing mother = degrees and remainder; add half circuit, remove full circuit for mid-crossing; subtract full-moon difference for full-moon initial crossing; subtract synodic difference for next month's initial crossing; remove by crossing mother; add winter-solstice yellow-path degree and assign for crossing lodges.
100
求日月食甚小餘及加時辰刻：以其朔，望月行遲疾定差疾加遲減經朔望小餘，若不足減者，退大餘一，加元法以減之； 若加之滿元法者，但積其數。
Greatest-eclipse minor remainder: apply lunar fast-slow fixed difference to mean syzygy minor remainder; retreat major remainder if needed; if addition fills origin method, accumulate the number.
101
以一千三百三十七乘之，滿其度所直月行定分除之，為月行差數； 乃以日躔盈定差盈加縮減之，餘為其朔，望食甚小餘。 凡加減滿若不足，進退其日，此朔望加時以究月行遲疾之數，若非有交會，直以經定小餘為定。
×1,337 ÷ lunar fixed parts at that degree = lunar motion difference; then ± solar excess-deficiency fixed difference for greatest-eclipse minor remainder. Advance or retreat day if needed; without conjunction use mean fixed minor remainder.
102
置之，如前發斂加時術入之，即各得日，月食甚所在晨刻視食甚小餘，如半法以下者，覆減半法，餘為午前分； 半法已上者，減去半法，餘為午後分。
Assign by convergence procedure; ≤ half method → before-noon parts; ≥ half method → after-noon parts.
103
求朔望加時日月度：以其朔，望加時小餘與經朔望小餘相減，餘以元法退收之，以加減其朔，望中日及約分，經朔望少，加； 經朔望多，減。
Addition-time degrees: difference of minor remainders ÷ origin; adjust midday sun (add if mean less, subtract if more); if mean more, subtract.
104
為其朔，望加時中日乃以所入日升降分乘所入日約分，以一萬約之，所得，隨以損益其日下盈縮積，為盈縮定度； 以盈加縮減加時中日，為其朔，望加時定日； 望則更加半周天，為加時定月； 以天正冬至加時黃道日度加而命之，即得所求朔，望加時日月所在宿度及分。
Multiply ascent-descent parts × reduced parts ÷10,000; adjust daily excess-deficiency accumulation; apply to addition-time midday for fixed sun; for full moon add half circuit for fixed moon; add to winter-solstice yellow-path degree and assign for syzygy addition-time lodges.
105
求朔望日月加時去交度分：置朔望日月加時定度與交初，交中度相減，餘為去交度分就近者相減之，其度以百通之為分。
Distance from crossing: addition-time fixed degree − initial or mid crossing; subtract nearer value; degrees ×100 for parts.
106
加時度多為後，少為前，即得其朔望去交前，後分交初後，交中前，為月行外道陽曆； 交中後，交初前，為月行內道陰曆。
Greater = after, less = before crossing. After initial, before mid = outer path yang calendar; after mid, before initial = inner path yin calendar.
107
求日食四正食差定數：置其朔加時定日，如半周天以下者為在盈。 以上者去之，餘為在縮視之，如在初限以下者為在初以上。 者，覆減二至限，餘為在末，置初，末限度及分，盈初限，縮末限者倍之。
Four-quadrant solar eclipse difference: if fixed sun ≤ half circuit = excess. if above, remove; = deficiency; ≤ initial limit = initial; if above invert two-solstice limit for final; double limits for excess-initial/deficiency-final.
108
置於上位，列二百四十三度半於下，以上減下，餘以下乘上，以一百六乘之，滿三千九十三除之，為東西食差泛數。 用減五百八，餘為南北食差泛數。 其求南北食差定數者，乃視午前，後分，如四分法之一以下者覆減之，餘以乘泛數。 若以上者即去之，餘以乘泛數，皆滿九千七百五十除之，為南北食差定數盈初縮末限者，食甚在卯酉以南，內減外加； 食甚在卯酉以北，內加外減。
Array 243°½ below, upper − lower, lower × upper ×106 ÷3,093 = east-west eclipse general difference. 508 − result = north-south eclipse general difference. North-south fixed difference: before/after-noon parts ≤¼ method → invert, × general difference. if above, remove then ×; ÷9,750; excess-initial/deficiency-final: south of mao-you inner subtract outer add; north of mao-you: inner add outer subtract.
109
縮初盈末限者，食甚在卯酉以南，內加外減； 食甚在卯酉以北，內減外加。
deficiency-initial/excess-final south of mao-you: inner add outer subtract; north of mao-you: inner subtract outer add.
110
其求東西食差定數者，乃視午前，後分，如四分法之一以下者以乘泛數； 以上者，覆減半法，餘乘泛數，皆滿九千七百五十除之前，為東西食差定數。 盈初縮末限者，食甚在子午以東，內減外加； 食甚在子午以西，內加外減。 外減； 食甚在子午以西，內減外加。
East-west fixed: ≤¼ method × general difference; if above, invert half method, × general difference, ÷9,750. excess-initial/deficiency-final east of zi-wu: inner subtract outer add; west of zi-wu: inner add outer subtract. outer subtract; west of zi-wu: inner subtract outer add.
111
即得其朔四正食差加減定數。
Thus the four-quadrant eclipse adjustment fixed numbers.
112
求日月食去交定分：視其朔四正食差，加減定數，同名相從，異名相消，餘為食差加減總數； 以加減去交分，餘為日食去交定分其去交定分不足減，乃覆減食差總數，若陽曆覆減入陰曆，為入食限； 若陰曆覆減入陽曆，為不入食限凡加之滿食限以上者，亦不入食限。
Eclipse fixed parts: combine quadrant adjustments (same add, different subtract); adjust distance from crossing; if yang inverts to yin = enters eclipse limit; yin to yang = no eclipse; filling limit also excludes eclipse.
113
其望食者，以其望去交分便為其望月食去交定分。
Full-moon eclipse uses full-moon distance-from-crossing directly.
114
求日月食分：日食者，視去交定分，如食限三之一以下者倍之，類同陽曆食分。 以上者，覆減食限，餘為陰曆食分皆進一位，滿九百七十六除為大分，不滿，退除為小分，命十為限，即日食之大，小分。 月食者，視去交定分，如食限三之一以下者，食既； 以上者，覆減食限餘進一位，滿八百九十二除之為大分，不滿，退除為小分，命十為限，即月食之大，小分其食不滿大分者，雖交而數淺，或不見食也。
Solar eclipse parts: ≤⅓ limit doubled = yang eclipse parts. above: invert limit for yin parts; advance one place ÷976 for major/minor solar eclipse parts. Lunar eclipse: ≤⅓ limit = total eclipse; else invert, ÷892; shallow numbers may hide eclipse despite crossing.
115
求日食泛用刻分：置陰、陽曆食分於上，列一千九百五十二於下，以上減下，餘以乘上，滿二百七十一除之，為日食泛用刻、分。
Solar general usable ke: yin/yang parts vs 1,952, upper − lower, × upper ÷271.
116
求月食泛用刻分：置去交定分，自相乘，交初以四百五十九除，交中以五百四十除之，所得，交初以減三千九百，交中以減三千三百一十五，餘為月食泛用刻、分。
Lunar general usable ke: distance²; initial ÷459, mid ÷540; subtract from 3,900 or 3,315.
117
求日月食定用刻分：置日月食泛用刻、分，以一千三百三十七乘之，以所直度下月行定分除之，所得為日月食定用刻、分。
Fixed usable ke: general ×1,337 ÷ lunar fixed parts at degree.
118
求日月食虧初复滿時刻：以定用刻分減食甚小餘，為虧初小餘； 加食甚，為复滿小餘； 各滿辰法為辰數，不盡，滿刻法除之為刻數，不滿為分。 命辰數從子正，算外，即得虧初、复末辰、刻及分。 若以半辰數加之，即命從時初也。
First loss: greatest-eclipse minor remainder − fixed usable ke; greatest eclipse + fixed usable = full return; ÷ double-hour method for hours, clepsydra for ke and parts. Count from zi midnight for first loss and last return times. Add half double-hour to count from hour initial.
119
求日月食初虧复滿方位：其日食在陽曆者，初食西南，甚於正南，复於東南； 日在陰曆者，初食西北，甚於正北，复於東北其食過八分者，皆初食正西，复於正東其月食者，月在陰曆，初食東南，甚於正南，复於西南； 月在陽曆，初食東北，甚於正北，复於西北。 其食八分已上者，皆初食正東，复於正西。 此皆審其食甚所向，據午正而論之，其食餘方察其斜正，則初虧，复滿乃可知矣。
Solar yang path: first SW, greatest south, return SE; yin path: NW, N, NE; >8 parts: W/E. Lunar yin: SE, S, SW; yang moon: NE, N, NW. ≥8 parts: first east, return west. Orient by greatest-eclipse direction from noon south; obliquity reveals first loss and return.
120
求月食更點定法：倍其望晨分，五而一，為更法； 又五而一，為點法若依司辰星注曆，同內中更點，則倍晨分，減去待旦十刻之分，餘，五而一，為更法； 又五而一，為點法。
Lunar eclipse watches: double full-moon dawn parts ÷5 = watch method; ÷5 again for points; Directorate version subtract awaiting-dawn 10 ke first; ÷5 again for point method.
121
求月食入更點：各置初虧，食甚，复滿小餘，如在晨分以下者加晨分，如在昏分以上者減去昏分，餘以更法除之為更數，不滿，以點法除之為點數。 其更數命初更，算外，即各得所入更，點。
Eclipse watch entry: adjust by dawn/dusk; ÷ watch and point methods. Count from first watch for entered watch and point.
122
求月食既內外刻分：置月食去交分，覆減食限三之一，不及減者為食不既。 餘列於上位； 乃列三之二於下，以上減下，餘下下乘上，以一百七十除之，所得，以定用刻分乘之，滿泛用刻分除之，為月食既內刻分； 用減定用刻分，餘為既外刻，分。
Totality inner/outer: distance from crossing invert − ⅓ limit; cannot subtract = not total. Set remainder above; array ⅔ below, compute, × fixed usable ÷ general = totality inner ke; fixed usable − inner = outer ke and parts.
123
求日月帶食出入所見分數：視食甚小餘在日出分以下者，為月見食甚，日不見食甚； 以日出分減复滿小餘，若食甚小餘在日出分已上者，為日見食甚，月不見食甚； 以初虧小餘減日出分，各為帶食差； 若月食既者，以既內刻分減帶食差，餘乘所食分，既外刻分而一，不及減者，即帶食既出入也。
Grazing eclipse visibility: if greatest eclipse < sunrise parts, moon sees, sun does not; if above sunrise, sun sees greatest; subtract first loss from sunrise for grazing difference; each yields grazing difference; total lunar eclipse: adjust grazing by totality inner/outer ke.
124
以乘所食之分，滿定用刻分而一，即各為日帶食出，月帶食入所見之分。 凡虧初小餘多如日出分為在晝，复滿小餘多如日出分為在夜，不帶食出入也。
× eclipse parts ÷ fixed usable = visible grazing parts. First loss > sunrise = daytime; full return > sunrise = night—no grazing.
125
若食甚小餘在日入分以下者，為日見食甚，月不見食甚； 以日入分減复滿小餘，若食甚小餘在日入分已上者，為月見食甚，日不見食甚； 以初虧小餘減日入分，各為帶食差； 若月食既者，以既內刻分減帶食差，餘乘所差分，既外刻分而一，不及減者，即帶食既出入也。
If greatest eclipse < sunset parts, sun sees, moon does not; if above sunset, moon sees; subtract first loss from sunset for grazing difference; each yields grazing difference; total lunar: adjust grazing by totality inner/outer ke.
126
以乘所食之分，滿定用刻分而一，即各為日帶食入，月帶食出所見之分。 凡虧初小餘多如日入分為在夜，复滿小餘少如日入分為在晝，並不帶食出入也。
× eclipse parts ÷ fixed usable = grazing entry/exit parts. First loss > sunset = night; full return < sunset = day—no grazing.
127
步五星術
Procedure for the Five Planets.
128
木星終率：一千五百五十五萬六千五百四。
Jupiter terminal rate: 15,556,504.
129
終曰：三百九十八日餘三萬四千五百四，約分八千八百四十七。
Terminal period: 398 days remainder 34,504, reduced parts 8,847.
130
曆差：六萬一千七百五十。
Sequence difference: 61,750.
131
見伏常度：一十四度。
Appearance and hiding constant arc: 14°.
132
火星終率：三千四十一萬七千五百三十六。
Mars terminal rate: 30,417,536.
133
終曰：七百七十九日餘三萬六千五百三十六，約分九千三百六十八。
Mars terminal period: 779 days remainder 36,536, reduced parts 9,368.
134
曆差：六萬一千二百四十。
Sequence difference: 61,240.
135
見伏常度：一十八度。
Appearance constant arc: 18°.
136
土星終率：一千四百七十四萬五千四百四十六。
Saturn terminal rate: 14,745,446.
137
終曰：三百七十八餘三千四百四十六，約分八百八十三。
Saturn terminal period: 378 days remainder 3,446, reduced parts 883.
138
曆差：六萬一千三百五十。
Sequence difference: 61,350.
139
見伏常度：一十八度半。
Appearance constant arc: 18°30′.
140
金星終率：二千二百七十七萬二千一百九十六。
Venus terminal rate: 22,772,196.
141
終曰：五百八十三日餘三萬五千一百九十六，約分九千二十四。
Venus terminal period: 583 days remainder 35,196, reduced parts 9,024.
142
見伏常度：一十一度少。
Appearance constant arc: 11° slightly less.
143
水星終率：四百五十一萬九千一百八十四改九千一百九十四。
Mercury terminal rate: 4,519,184 (revised 9,194).
144
終曰：一百一十五日餘三萬四千一百八十四，約分八千七百六十五。
Mercury terminal period: 115 days remainder 34,184, reduced parts 8,765.
145
見伏常度：一十八度。
Appearance constant arc: 18°.
146
求五星天正冬至後諸段中積中星：置氣積分，各以其星終率去之，不盡，覆減終率，餘滿元法為日，不滿，退除為分，即天正冬至後其星平合中積。 重列之為中星，因命為前一段之初，以諸段變日，變度累加減之，即為諸段中星，變日加減中積，變度加減中星。
Post–winter-solstice mean conjunction: qi parts mod terminal rate; ÷ origin = days. Re-array as center star; accumulate segment day/degree changes.
147
求木火土三星入曆：以其星曆差乘積年，滿周天分去之，不盡，以度母除之為度，不滿，退除為分，命曰差度； 以減其星平合中星，即為平合入曆度分； 以其星其段曆度加之，滿周天度分即去之，各得其星其段入曆度分金，水附日而行，更不求曆差。 其木，火，土三星前變為晨，後變為夕。 金，水二星前變為夕，後變為晨。
Wood/Mars/Saturn sequence entry: years × sequence difference mod circuit; ÷ degree mother. subtract from mean-conjunction center star for sequence entry; add segment sequence degree; Venus/Mercury follow the sun. Wood/Mars/Saturn: prior change = morning, posterior = evening. Venus/Mercury: prior = evening, posterior = morning.
148
求木火土三星諸段盈縮定差：木，土二星，置其星其段入曆度分，如半周天以下者為在盈以上者，減去半周天，餘為在縮置。 盈縮度分，如在一象以下者為在初限。 以上者，覆減半周天，餘為在末限。 置初，末限度及分於上，列半周天於下，以上減下，以下乘上，木進一位，土九因之。
Segment excess-deficiency: ≤ half circuit = excess; else deficiency. ≤ one quadrant = initial limit. Above: invert half circuit for final limit. Initial/final vs half circuit; wood ×10, earth ×9.
149
皆滿百為分，分滿百為度，命曰盈縮定差。 其火星，置盈縮度分，如在初限以下者為在初。 以上者，覆減半周天，餘為在末。 以四十五度六十五分半為盈初，縮末限度，以一百三十六度九十六分半為縮初，盈末限度分。
÷100 twice = excess-deficiency fixed difference. Mars: ≤ initial limit = initial. Above: invert half circuit for final. Mars limits: 45°65′½ excess-initial/deficiency-final; 136°96′½ deficiency-initial/excess-final.
150
置初，末限度於上，盈初，縮末三因之。
Triple initial/final for excess-initial/deficiency-final.
151
列二百七十三度九十三分於下，以上減下，餘以下乘上，以一十二乘之，滿百為度，不滿，百約為分，命曰盈縮定差若。 用立成法，以其度下損益率乘度下約分，滿百者，以損益其度下盈縮差度為盈縮定差，若在留退段者，即在盈縮泛差。
273°93′ procedure ×12 = Mars fixed difference. If Ready reckoner: rate × parts; station-retreat uses general difference.
152
求木火土三星留退差：置後退，後留盈縮泛差，各列其星盈縮極度於下，木極度，八度三十三分； 火極度，二十二度五十一分； 土極度，七度五十分。
Station-retreat difference: posterior retreat/station general vs extremes—wood 8°33′; Mars extreme 22°51′; Saturn extreme 7°50′.
153
以上減下，餘以下乘上，水，土三因之，火倍之。
Upper − lower; lower × upper; earth/water ×3, Mars ×2.
154
皆滿百為度，命曰留退差。 後退初半之，後留全用。
÷100 = station-retreat difference. Posterior retreat: half; posterior station: full.
155
其留退差，在盈益減損加，在縮損減益加其段盈縮泛差，為後退，後留定差。 因為後遲初段定差，各須類會前留定差，觀其盈縮，察其降差也。
Gain/loss rules by excess/deficiency for station fixed difference. Posterior slow initial fixed difference matches prior station.
156
求五星諸段定積：各置其星其段中積，以其段盈縮定差盈加縮減之，即其星其段定積及分； 以天正冬至大餘及約分加之，滿紀法去之，不盡，命甲子，算外，即得日辰其五星合見，伏，即為推算段定日； 後求見，伏合定日，即曆注其日。
Segment fixed accumulation: center ± segment fixed difference; + winter solstice remainder; assign from jiazi for segment fixed day/time. then appearance/hiding conjunction fixed day for annotation.
157
求五星諸段所在月曰：各置諸段定積，以天正閏日及約分加之，滿朔策及分去之，為月數； 不滿，為入月以來日數及分其月數命從天正十一月，算外，即其星其段入其月經朔日數及分。 定朔有進退者，亦進退其日，以日辰為定。 若以氣策及約分去定積，命從冬至，算外，即得其段入氣日及分。
Segment month/day: fixed accumulation + intercalary days; ÷ new-moon policy. assign months from eleventh month for segment entry new-moon days. Adjust day if fixed new moon shifts. Or assign from winter solstice by qi policy.
158
求五星諸段加時定星：各置其星其段中星，以其段盈縮定差盈加縮減之，即五星諸段定星若以天正冬至加時黃道日度加而命之，即其段加時定星所在宿次，五星皆以前留為前退初定星，後留為後順初定星。
Segment fixed star: center ± fixed difference; assign from winter solstice yellow path.
159
求五星諸段初日晨前夜半定星：木，火，土三星，以其星其段盈縮定差與次度下盈縮定差相減，餘為其度損益差； 以乘其段初行率，一百約之，所得，以加減其段初行率，在盈，益加損減； 在縮，益減損加。
First-day midnight star: degree gain-loss × initial motion rate; ± initial rate by excess/deficiency; in deficiency reverse.
160
以一百乘之，為初行積分； 又置一百分，亦依其數加減之，以除初行積分，為初日定行分以乘其段初日約分，以一百約之，順減退加其段定星，為其段初日晨前夜半定星； 以天正冬至加時黃道日度加而命之，即得所求金，水二星，直以初行率便為初日定行分。
×100 = initial accumulated parts; compute first-day fixed motion; adjust segment fixed star. Assign from winter solstice; Venus/Mercury use initial rate directly.
161
求太陽盈縮度：各置其段定積，如二至限以下為在盈； 以上者去之，餘為在縮又視入盈縮度，如一象以下者為在初； 以上者，覆減二至限，餘為在末。 置初，末限度及分，如前日度術求之，即得所求。 若用立成者，直以其度下損益分乘度餘，百約之，所得，損益其度下盈縮差，亦得所求。
Solar excess-deficiency: ≤ two-solstice limit = excess; above = deficiency; ≤ quadrant = initial; above: invert two-solstice limit for final. Use solar-degree procedure for initial/final limits. Ready reckoner: gain-loss × remainder ÷100.
162
求諸段日度率：以二段日晨相距為日率，又以二段夜半定星相減，餘為其段度率及分。
Day rate = dawn separation; degree rate = midnight star difference.
163
求諸段平行分：各置其段度率及分，以其段日率除之，為其段平行分。
Mean motion = degree rate ÷ day rate.
164
求諸段泛差：各以其段平行分與後段平行分相減，餘為泛差； 並前段泛差，四因之，退一等，為其段總差五星前留前，後留後一段，皆以六因平行分，退一等，為其段總差，水星為半總差。 其在退行者，木，火，土以十二乘其段平行分，退一等，為其段總差。 金星退行者，以其段泛差為總差，後變則反用初，末。 水星退行者，以其段平行分為總差，若在前後順第一段者，乃半次段總差，為其段總差。
General difference = prior mean − posterior mean; accumulate ×4 for total difference; station segments ×6; Mercury half. Retrograde: mean motion ×12 ÷10 = total difference. Venus retrograde: general difference as total; reverse initial/final. Mercury retrograde: mean motion as total; first direct uses half next total.
165
求諸段初末日行分：各半其段總差，加減其段平行分，為其段初，末日行分前變加為初，減為末； 後變減為初，加為末。 其在退段者，前則減為初，加為末； 後則加為初，減為末若前後段行分多少不倫者，乃平注之； 或總差不備大分者，亦平注之：皆類會前後初，末，不可失其衰殺。
First/last motion: mean ± half total difference; rules by segment type. Posterior change: subtract initial, add final. Retrograde prior: subtract initial, add final; posterior reverse; level-annotate if unequal. level-annotate total difference if needed.
166
求諸段日差：減其段日率一，以除其段總差，為其段日差後行分少為損，後行分多為益。
Daily difference = total difference ÷ (day rate − 1).
167
求每日晨前夜半星行宿次：置其段初日行分，以日差累損益之，為每日行分以每日行分累加減其段初日晨前夜半宿次，命之，即每日星行宿次。
Daily lodge: first-day motion ± daily difference accumulated.
168
徑求其日宿次：置所求日，減一，以乘日差，以加減初日行分，後少，減之； 後多，加之。
Direct lodge: (day−1) × daily difference ± first-day motion. if posterior more, add.
169
為所求日行分； 乃加初日行分而半之，以所求日數乘之，為徑求積度； 以加減其段初日宿次，命之，即徑求其日星宿次。
sought day's motion parts; halve (first + sought motion) × days = direct accumulation; ± segment first-day lodge for direct result.
170
求五星定合定曰：木，火，土三星，以其段初日行分減一百分，餘以除其日太陽盈縮餘為日，不滿，退除為分，命曰距合差日及分。 以差日及分減太陽盈縮分，餘為距合差度。 以差日，差度盈減縮加。 金，水二星平合者，以百分減初日行分，餘以除其日太陽盈縮餘為日，不滿，退除為分，命曰距合差日及分。 以減太陽盈縮分，餘為距合差度。 以差日，差度盈加縮減。 金，水星再合者，以初日行分加一百分，以除其日太陽盈縮分為日，不滿，退除為分，命曰再合差日； 以減太陽盈縮分，餘為再合差度。 以差日，差度盈加縮減。 差度則反其加減。
Fixed conjunction: (initial motion −100) divides solar excess-deficiency. difference day/parts from solar excess-deficiency → difference degree. excess subtract, deficiency add. Venus/Mercury mean: (initial −100) divides solar excess-deficiency. subtract for difference degree. excess add, deficiency subtract. Second conjunction: (initial +100) divides solar excess-deficiency. subtract for second-conjunction difference degree. excess add, deficiency subtract. difference degree reverses gain/loss.
171
皆以加減定積，為再合定日。 以天正冬至大餘及約分加而命之，即得定合日辰。
± fixed accumulation for second-conjunction fixed day. Assign from winter solstice for fixed conjunction time.
172
求五星定見伏：木，火，土三星，各以其段初日行分減一百分，餘以除其日太陽盈縮分為日，不滿，退除為分，以盈減縮加金，水二星夕見，晨伏者，以一百分減初日行分，餘以除其日太陽盈縮分為日，不滿，退除為分，以盈加縮減。 其在晨見，夕伏者，以一百分加其段初日行分，以除其日太陽盈縮分為日，不滿，退除為分，以盈減縮加。 皆加減其段定積，為見，伏定日。 以加冬至大餘及約分，滿紀法去之，命從甲子，算外，即得五星見，伏定日日辰。
To find fixed planetary appearance and hiding: for Jupiter, Mars, and Saturn, subtract 100 from the segment's initial daily motion, divide the day's solar excess-deficiency remainder for days and parts, and apply excess by subtracting and deficiency by adding. For Venus and Mercury in evening appearance and morning hiding, subtract 100 from the initial daily motion, divide likewise, and apply excess by adding and deficiency by subtracting. Morning appearance/evening hiding: add 100 to initial motion; excess subtract, deficiency add. ± segment fixed accumulation for appearance/hiding day. Assign from jiazi for appearance/hiding fixed time.
173
琮又論曆曰：「古今之曆，必有術過於前人，而可以為萬世之法者，乃為勝也若一行為《大衍曆》，議及略例，校正曆世，以求曆法強弱，為曆家體要，得中平之數。 劉焯悟日行有盈縮之差。 舊曆推日行平行一度，至此方悟日行有盈縮，冬至前後定日八十八日八十九分，夏至前後定日九十三日七十四分，冬至前後日行一度有餘，夏至前後日行不及一度。
Zhou Cong wrote: A calendar surpasses predecessors only when its methods can serve all generations. Yixing's Dayan Calendar established the calendrist's essentials and a balanced mean. Liu Chuo realized solar motion varies in excess and deficiency. Older calendars used uniform solar motion; Liu Chuo showed winter days ~88.89 parts and summer ~93.74, with solar speed above and below one degree accordingly.
174
李淳風悟定朔之法，並氣朔，閏餘，皆同一術。 舊曆定朔平注一大一小，至此以日行盈縮，月行遲疾加減朔餘，餘為定朔，望加時，以定大小，不過三數。 自此後日食在朔，月食在望，更無晦，二之差。 舊曆皆須用章歲，章月之數，使閏餘有差，淳風造《麟德曆》，以氣朔，閏餘同歸一母。
Li Chunfeng unified fixed new moon, qi, and intercalary remainder in one procedure. Earlier calendars alternated long and short months mechanically; Li Chunfeng used solar and lunar corrections for fixed syzygies, limiting consecutive long or short months to three. Thereafter solar eclipses align with new moon and lunar with full moon, without shifting to the last or second day. Older calendars used separate rule-year and rule-month figures for intercalary remainder; Li Chunfeng's Lindé Calendar unified qi, syzygy, and intercalary remainder under one divisor.
175
張子信悟月行有交道表裏，五星有入氣加減。 北齊學士張子信因葛榮亂，隱居海島三十餘年，專以圓儀揆測天道，始悟月行有交道表裡，在表為外道陽曆，在裏為內道陰曆。 月行在內道，則日有食之，月行在外道則無食。 若月外之人北戶向日之地，則反觀有食。 又舊曆五星率無盈縮，至是始悟五星皆有盈縮，加減之數。
Zhang Zixin discovered the moon's nodal crossing (inner and outer paths) and qi-entry adjustments for the five planets. Zhang Zixin of Northern Qi spent thirty years on an island observing with an armillary sphere during Ge Rong's rebellion, discovering lunar nodes: outer path = yang calendar, inner path = yin calendar. Eclipses occur when the moon is on the inner path; none when on the outer path. Observers in far northern latitudes facing the sun would see an eclipse where others do not. Earlier five-planet rates ignored variation; Zhang Zixin showed all five planets have excess-deficiency adjustments.
176
宋何承天始悟測景以定氣序景極長，冬至； 景極短，夏至始立八尺之表，連測十餘年，即知舊《景初曆》冬至常遲天三日。 乃造《元嘉曆》，冬至加時比舊退減三日。
He Chengtian of Song first used gnomon shadows to order the qi: longest shadow marks winter solstice; shortest shadow marks summer solstice. After ten years with an eight-chi gnomon he found the Jingchu Calendar's winter solstice was three days late. His Yuanjia Calendar moved winter-solstice addition time back three days from the old reckoning.
177
晉姜岌始悟以月食所衝之宿，為日所在之度。 日所在不知宿度，至此以月食之宿所衝，為日所在宿度。
Jiang Ji of Jin first inferred the sun's lodge from the lodge opposite a lunar eclipse. Where the sun's lodge was unknown, he took the lodge opposite the eclipsed moon as the sun's position.
178
後漢劉洪作《乾象曆》，始悟月行有遲疾數。 舊曆，月平行十三度十九分度之七，至是始悟月行有遲疾之差，極遲則日行十二度強，極疾則日行十四度太，其遲疾極差五度有餘。
Liu Hong of Later Han, in the Qianxiang Calendar, first modeled lunar fast and slow motion. Older calendars used uniform lunar motion of 13°7/19 per day; Liu Hong showed extremes of about 12°+ slow and 14°+ fast, differing by more than five degrees.
179
宋祖衝之始悟歲差。 《書·堯典》曰：「日短星昴，以正仲冬； 宵中星虛，以殷仲秋。」 至今三千餘年，中星所差三十餘度，則知每歲有漸差之數，造《大明曆》，率四十五年九月而退差一度。
Zu Chongzhi of Liu Song first accounted for precession. The Yao Canon says: "When days are short and the stars are in Mao, mid-winter is fixed; at midnight the stars are in Xu, mid-autumn is fixed." From then until now median stars have shifted more than thirty degrees, proving annual precession; the Daming Calendar used 45 years 9 months per degree of precession.
180
唐徐升作《宣明曆》，悟日食有氣，刻差數。 舊曆推日食皆平求食分，多不允合，至是推日食，以氣刻差數增損之，測日食分數，稍近天驗。
Xu Sheng of Tang, in the Xuānmíng Calendar, added qi and clepsydra corrections to solar eclipses. Earlier eclipse magnitudes were computed uniformly and poorly matched observation; Xu Sheng's qi and ke adjustments brought eclipse fractions closer to the sky.
181
《明天曆》悟日月會合為朔，所立日法，積年有自然之數，及立法推求晷景，知氣節加時所在。 自《元嘉曆》後所立日法，以四十九分之二十六為強率，以十七分之九為弱率，並強弱之數為日法，朔餘，自後諸曆效之。 殊不知日月會合為朔，並朔餘虛分為日法，蓋自然之理。 其氣節加時，晉，漢以來約而要取，有差半日，今立法推求，得盡其數。
The Tomorrow Calendar treated syzygy as new moon, used natural divisors for the day method, and computed gnomon shadows to fix qi addition times. After Yuanjia, calendars used 26/49 and 9/17 to build the day method and syzygy remainder—a pattern later calendars copied. They missed that syzygy defines new moon and that the day method should unite syzygy remainder and void parts by natural principle. Qi addition times had been approximated since Jin and Han with half-day errors; the new procedures recover the full values.
182
後之造曆者，莫不遵用焉。 其疏謬之甚者，即苗守信之《乾元曆》，馬重績之《調元曆》，郭紹之《五紀曆》也。 大概無出於此矣。 然造曆者，皆須會日月之行，以為晦朔之數，驗「春秋」日食，以明強弱。 其於氣序，則取驗於「傳」之南至。 其日行盈縮，月行遲疾，五星加減，二曜食差，日宿月離，中星晷景，立數立法，悉本之於前語。 然後較驗，上自夏仲康五年九月「辰弗集於房」，以至於今，其星辰氣朔，日月交食等，使三千年間若應準繩。 而有前有後，有親有疏者，即為中平之數，乃可施於後世。 其較驗則依一行，孫思恭，取數多而不以少，得為親密。 較日月交食，若一分二刻以下為親，二分四刻以下為近，三分五刻以上為遠。 以曆注有食而天驗無食，或天驗有食而曆注無食者為失。 其較星度，則以差天二度以下為親，三度以下為近，四以上為遠； 其較晷景尺寸，以二分以下為親，三分以下為近，四分以上為遠若較古而得數多，又近於今，兼立法，立數，得其理而通於本者為最也。」 琮自謂善曆，嘗曰：「世之知曆者甚少，近世獨孫思恭為妙」。 而思恭又嘗推劉羲叟為知曆焉。
Later calendar makers all followed these principles. Among the worst were Miao Shouxin's Qianyuan, Ma Chongji's Tiaoyuan, and Guo Shao's Wuji calendars. In general, nothing goes beyond this framework. Calendar makers must reconcile solar and lunar motion for syzygy reckoning and test against Spring and Autumn eclipses for magnitude. For qi order, verify against the Zuo Commentary's winter-solstice record. Solar excess-deficiency, lunar fast-slow, planetary adjustments, eclipse differences, lodges, median stars, and gnomon shadows—all derive from the principles above. Then test from the Xia record of stars not gathering in Fang to the present so three millennia of stars, qi, syzygies, and eclipses align like a plumb line. Where results run early or late, close or loose, the balanced mean can serve later generations. Verification follows Yixing and Sun Sigong: prefer more data points over fewer for a close fit. Eclipse comparison: within 1 part 2 ke = close; within 2 parts 4 ke = near; 3 parts 5 ke or more = distant. Failure is when the calendar predicts an eclipse the sky lacks, or the sky shows one the calendar omits. Star comparison: within 2° = close, within 3° = near, 4° or more = distant; Shadow comparison: within 2 fen close, 3 fen near, 4 fen or more distant. Best is many ancient matches, nearness to today, and laws that grasp the underlying principle. Zhou Cong claimed calendrical skill and said few truly understand calendars—in his day only Sun Sigong excelled. Sun Sigong in turn praised Liu Yishou as a master of calendars.