1
明天曆
The Tianli Calendar
2
調日法朔余、周天分、鬥分、歲差、日度母附
Appendix: Adjusting the Day Divisor, New-Moon Remainder, Circuit-of-Heaven Parts, Dipper Fraction, Precession, and Day-Degree Mother
3
造曆之法，必先立元，元正然後定日法，法定然後度周天，以定分、至，三者有程，則曆可成矣。 日者，積余成之； 度者，積分成之。 蓋日月始離，初行生分，積分成日。 自《四分曆》洎古之六曆，皆以九百四十為日法。 率由日行一度，經三百六十五日四分之一，是為周天； 月行十三度十九分之七，經二十九日有餘，與日相會，是為朔策。 史官當會集日月之行，以求合朔。
To construct a calendar, one must first establish the epoch. Once the epoch is settled, the day divisor is fixed; once the divisor is fixed, the circuit of heaven is measured to determine the solstitial points. When all three have their proper measures, the calendar can be completed. Days arise from the accumulation of fractional remainders; and degrees arise from the accumulation of fractional parts. When sun and moon first separate, their initial motion generates fractional parts, and those accumulated parts become days. From the Quarter-Remainder Calendar through the six ancient calendars, the day divisor was uniformly 940. The rate assumes the sun moves one degree per day through 365¼ days, the full circuit of heaven; while the moon moves 13° 7/19 per synodic month (~29+ days) until it meets the sun—that is the new-moon cycle. Astronomers must combine the motions of sun and moon to find true conjunction.
4
自漢太初至於今，冬至差十日，如劉歆《三統》復強于古，故先儒謂之最疏。 後漢劉洪考驗《四分》，於天不合，乃減朔余，苟合時用。 自是已降，率意加減，以造日法。 宋世何承天更以四十九分之二十六為強率，十七分之九為弱率，於強弱之際以求日法。 承天日法七百五十二，得一十五強一弱。 自後治曆者，莫不因承天法、累強弱之數，皆不悟日月有自然合會之數。
From the Han Taichu era to the present, winter solstice has drifted by ten days. Liu Xin's Triple Concordance was even more forced than antiquity, so earlier scholars called it the least accurate. In Later Han, Liu Hong tested the Quarter-Remainder Calendar against the heavens, found it wanting, and reduced the new-moon remainder to fit contemporary practice. Thereafter calendar-makers arbitrarily adjusted figures to derive new day divisors. In the Liu Song, He Chengtian used 26/49 as the "strong" rate and 9/17 as the "weak" rate, seeking the day divisor between them. He Chengtian's day divisor was 752, equivalent to fifteen strong units and one weak. Later calendar-makers all followed He Chengtian's strong-and-weak method without realizing that sun and moon possess a natural number of conjunction.
5
今稍悟其失，定新曆以三萬九千為日法，六百二十四萬為度母，九千五百為鬥分，二萬六百九十三為朔餘，可以上稽于古，下驗於今，反覆推求，若應繩准。 又以二百三十萬一千為月行之餘，月行十三度之餘。
Having discerned this flaw, the new calendar sets the day divisor at 39,000, the degree mother at 6,240,000, the Dipper fraction at 9,500, and the new-moon remainder at 20,693—checking against antiquity above and the present below, repeated inquiry shows it true as a plumb line. Further, 2,301,000 is set as the remainder of the moon's motion—the fractional part of its 13° motion.
6
以一百六十萬四百四十七為日行之餘。 日行周天之餘。
1,600,447 is set as the remainder of the sun's motion. This is the fractional remainder of the sun's daily circuit.
7
乃會日月之行，以盈不足平之，並盈不足，是為一朔之法。 日法也，名元法。
Combining the sun's and moon's motions, excess and deficit are balanced and joined; this yields the method for one new moon. This is the day divisor, also termed the origin divisor.
8
今乃以大月乘不足之數，以小月乘盈行之分，平而並之，是為一朔之實。 周天分也。
The long month multiplies the deficit figure and the short month the excess parts; averaged and combined, this gives the true value of one new moon. This is the circuit-of-heaven fraction.
9
以法約實，得日月相會之數，皆以等數約之，悉得今有之數。 盈為朔虛，不足為朔餘。
Reducing the true value by the divisor yields the conjunction number; reducing by common divisors produces exactly the figures used today. Excess becomes the new-moon void fraction; deficit becomes the new-moon remainder.
10
又二法相乘為本母，各母互乘，以減周天，餘則歲差生焉，亦以等數約之，即得歲差、度母、周天實用之數。 此之一法，理極幽眇，所謂反覆相求，潛遁相通，數有冥符，法有偶會，古曆家皆所未達。 以等數約之，得三萬九千為元法，九千五百為鬥分，二萬六百九十三為朔餘，六百二十四萬為日度母，二十二億七千九百二十萬四百四十七為周天分，八萬四百四十七為歲差。
Multiplying the two divisors gives the fundamental mother; cross-multiplying and subtracting from the circuit of heaven yields the precession remainder, which common reduction converts to the precession, degree mother, and practical circuit-of-heaven values. This method is profoundly subtle: repeated mutual calculation, hidden interconnection, numbers in secret correspondence, methods in felicitous alignment—none of the ancient calendar-makers had attained it. Common reduction yields 39,000 as the origin divisor, 9,500 as the Dipper fraction, 20,693 as the new-moon remainder, 6,240,000 as the day-degree mother, 2,279,200,447 as the circuit-of-heaven parts, and 80,447 as the precession value.
11
歲餘：九千五百。 古曆曰鬥分。
Annual remainder: 9,500. Ancient calendars termed this the Dipper fraction.
12
古者以周天三百六十五度四分度之一，是為鬥分。 夫舉正於中，上稽往古，下驗當時，反覆參求，合符應准，然後施行於百代，為不易之術。 自後治曆者，測今冬至日晷，用校古法，過盈，以萬為母，課諸氣分，率二千五百以下、二千四百二十八已上為中平之率。 新曆鬥分九千五百，以萬平之，得二千四百二十五半盈，得中平之數也。 而三萬九千年冬至小余成九千五百日，滿朔實一百一十五萬一千六百九十三，年齊於日分，而氣朔相會。
Antiquity defined the circuit of heaven as 365¼ degrees—the Dipper fraction. Taking the mean as standard, checking past and present, comparing until all accord—only then may it serve for generations as an unchanging method. Later makers measured contemporary winter-solstice shadows, compared them to antique methods, and when excess appeared used 10,000 as denominator; qi-period tests showed rates between 2428 and 2500 as the mean. The new calendar's Dipper fraction of 9,500, normalized to 10,000, gives 2,425.5 excess—the mean figure. Over 39,000 years the winter-solstice small remainder accrues to 9,500 days; filling the new-moon true value of 1,151,693 aligns the year with the day fraction so qi and new moon coincide.
13
歲周：一千四百二十四萬四千五百。 以元法乘三百六十五度，內鬥分九千五百，得之，即為一歲之日分，故曰歲周。 若以二十四均之，得一十五日、餘八千五百二十、秒一十五，為一氣之策也。
Annual circuit: 14,244,500. Multiplying the origin divisor by 365° and adding the Dipper fraction 9,500 gives one year's day-parts—the annual circuit. Divided by 24, this yields 15 days, 8,520 parts, and 15 seconds per qi period.
14
朔實：一百一十五萬一千六百九十三。 本會日月之行，以盈不足平而得二萬六百九十三，是為朔餘，備在調日法術中。
New-moon true value: 1,151,693. Combining the sun's and moon's motions and balancing excess and deficit yields 20,693 as the new-moon remainder, as detailed in the day-divisor adjustment method.
15
是則四象全策之餘也。 今以元法乘四象全策二十九，總而並之，是為一朔之實也。 古曆以一百萬平朔餘之分，得五十三萬六百以下、五百七十已上，是為中平之率。 新曆以一百萬平之，得五十三萬五百八十九，得中平之數也。 若以四象均之，得七日，余一萬四千九百二十三、秒，是為弦策也。
This is the remainder of the full four-phase cycle. Multiplying the origin divisor by the four-phase full cycle of 29 and summing them gives the true value of one new moon. Ancient calendars normalized new-moon remainder parts to 1,000,000, taking figures below 530,600 and from 570 upward as the mean rate. The new calendar, normalized to 1,000,000, gives 530,589—the mean figure. Divided by four phases, this yields 7 days, 14,923 parts, and seconds—the quarter-moon cycle.
16
中盈、朔虛分：閏餘附
Mid-Qi Excess and New-Moon Void Fractions: Appendix on Intercalary Remainder
17
日月以會朔為正，氣序以鬥建為中，是故氣進而盈分存焉。 置中節兩氣之策，以一月之全策三十減之，每至中氣，即一萬七千四十、秒十二，是為中盈分。 朔退而虛分列焉，置一月之全策三十，以朔策及餘減之，余一萬八千三百七，是為朔虛分。 綜中盈、朔虛分，而閏余章焉。 閏余三萬五千三百四十五、秒一十三。
Sun and moon take conjunction at new moon as the standard; the qi sequence takes the Dipper's establishment as the center—thus as qi advances, excess parts accumulate. Taking the two qi cycles of the mid-season node and subtracting from the month's full cycle of 30, each mid-qi yields 17,040 parts and 12 seconds—the mid-qi excess fraction. As new moon retreats, void fractions appear: subtract the new-moon cycle and remainder from the month's full cycle of 30; the remainder 18,307 is the new-moon void fraction. Combining mid-qi excess and new-moon void fractions yields the intercalary remainder chapter. Intercalary remainder: 35,345, seconds 13.
18
從消息而自致，以盈虛名焉。
It arises naturally from waxing and waning, and is named for excess and void.
19
紀法：六十。 《易·乾》象之爻九，《坤》象之爻六，《震》、《坎》、《艮》象之爻皆七，《巽》、《離》、《兌》象之爻皆八。 綜八卦之數凡六十，又六旬之數也。 紀者，終也，數終八卦，故以紀名焉。
Cycle divisor: 60. In the Book of Changes, the Qian trigram has nine lines, Kun six; Zhen, Kan, and Gen each seven; Xun, Li, and Dui each eight. The eight trigrams' line-counts sum to 60—the same as a sexagenary cycle. A cycle (ji) means an end; the count completes the eight trigrams, hence the name.
20
天正冬至：大余五十七，小余一萬七千。 先測立冬晷景，次取測立春晷景，取近者通計，半之，為距至泛日； 乃以晷數相減，餘者以法乘之，滿其日晷差而一，為差刻； 乃以差刻求冬至，視其前晷多則為減，少則為加，求夏至者反之。
Heaven's first month, winter solstice: large remainder 57, small remainder 17,000. First measure the Beginning-of-Winter gnomon shadow, then the Beginning-of-Spring shadow; take the nearer, compute through, and halve it to get the approximate day-distance to the solstice; subtract the shadow lengths; multiply the remainder by the divisor and divide by the daily shadow difference to obtain the correction in clepsydra marks; then apply the correction marks to find winter solstice: if the prior shadow is longer, subtract; if shorter, add; reverse the procedure for summer solstice.
21
加減距至泛日，為定日； 仍加半日之刻，命從前距日辰，算外，即二至加時日辰及刻分所在。 如此推求，則加時與日晷相協。 今須積歲四百一年，治平元年甲辰歲，氣積年也。
Adding or subtracting from the approximate day-distance yields the fixed date; add half a day's clepsydra marks, count from the prior day's distance hour, and the result gives the day, hour, and marks of true solstice time. Calculated thus, true time agrees with the gnomon shadow. The accumulated count must be 401 years; the first year of Zhiping (1064), year jiachen, is the accumulated qi epoch.
22
則冬至大、小餘與今適會。
Then the winter solstice large and small remainders match the present exactly.
23
天正經朔：大余三十四，小余三萬一千。 閏余八十八萬三千九百九十。
Heaven's first month, standard new moon: large remainder 34, small remainder 31,000. Intercalary remainder: 883,990.
24
則經朔大、小餘與今有之數，偕閏余而相會。
Then the standard new moon's large and small remainders, together with the intercalary remainder, coincide with present figures.
25
日度歲差：八萬四百四十七。 《書》舉正南之星以正四方，蓋先王以明時授人，奉天育物。 然先儒所述，互有同異。 虞喜云：「堯時冬至日短星昴，今二千七百餘年，乃東壁中，則知每歲漸差之所至。」 又何承天云：「《堯典》：'日永星火，以正仲夏； 宵中星虛，以正仲秋。 '今以中星校之，所差二十七八度，即堯時冬至，日在須女十度。」 故祖沖之修《大明曆》，始立歲差，率四十五年九月卻一度。 虞鄺、劉孝孫等因之，各有增損，以創新法。 若從虞喜之驗，昴中則五十餘年日退一度； 若依承天之驗，火中又不及百年日退一度。 後《皇極》綜兩曆之率而要取其中，故七十五年而退一度，此乃通其意未盡其微。 今則別調新率，改立歲差，大率七十七年七月，日退一度，上元命於虛九，可以上覆往古，下逮於今。 自帝堯以來，迴圈考驗，新曆歲差，皆得其中，最為親近。
Day-degree precession: 80,447. The Documents cites the star due south to orient the four directions, for the ancient kings used the seasons to instruct the people, serving Heaven and nurturing all things. Yet earlier scholars' accounts differ among themselves. Yu Xi said: "At Yao's time, winter solstice fell when the day was short and Mao was culminating; after more than 2,700 years it is at the center of Eastern Wall—thus we know the extent of the annual drift." He Chengtian also said: "The Canon of Yao reads: 'The day is longest and the Fire star is bright, to mark midsummer; the night is centered and Xu is culminating, to mark mid-autumn. Checking by the culminating stars today, the difference is 27 or 28 degrees—at Yao's winter solstice the sun stood at Woman, 10°." Hence Zu Chongzhi, revising the Daming Calendar, first established precession at the rate of one degree every 45 years and 9 months. Yu Kui, Liu Xiaosun, and others followed suit, each adjusting the rate to devise new methods. By Yu Xi's verification, with Mao at culmination the sun retreats one degree in just over 50 years; by He Chengtian's verification, with Fire at culmination the sun retreats one degree in less than a century. Later the Huangji Calendar averaged the two rates, yielding one degree every 75 years—this captured the idea but not the full subtlety. Now a new rate is set and precession re-established at roughly one degree every 77 years and 7 months; the upper origin is fixed at Void 9°, covering antiquity above and the present below. Tested cyclically from Emperor Yao to the present, the new calendar's precession hits the mean most closely.
26
周天分：二十二億七千九百二十萬四百四十七。 本齊日月之行，會合朔而得之。 在調日法。
Circuit-of-heaven parts: 2,279,200,447. Derived by aligning the sun's and moon's motions at conjunction at new moon. See the day-divisor adjustment method.
27
使上考仲康房、宿之交，下驗薑岌月食之沖，三十年間，若應準繩，則新曆周天，有自然冥符之數，最為密近。
Checking above at Zhong Kang's Room-Lodging crossing and below at Jiang Kui's lunar eclipse, over thirty years if all matches like a plumb line, the new calendar's circuit of heaven has a naturally corresponding number—most precise of all.
28
日躔盈縮定差：張胄玄名損益率曰盈縮數，劉孝孫以盈縮數為朏朒積，《皇極》有陟降率、遲疾數，《麟德》曰先後、盈縮數，《大衍》曰損益、朏朒積，《崇天》曰損益、盈縮積。 所謂古曆平朔之日，而月或朝覿東方，夕見西方，則史官謂之朏朒。 今以日行之所盈縮、月行之所遲疾，皆損益之，或進退其日，以為定朔，則舒亟之度，乃勢數使然，非失政之致也。 新曆以七千一為盈縮之極，其數與月離相錯，而損益、盈縮為名，則文約而義見。
Solar motion excess-deficit fixed difference: Zhang Zhouxuan called the increase-decrease rate the excess-deficit number; Liu Xiaosun termed it the crescent-gibbous accumulation; the Huangji had ascent-descent rate and slow-fast number; the Lindé calendar used before-after and excess-deficit number; the Dayan used increase-decrease and crescent-gibbous accumulation; the Chongtian used increase-decrease and excess-deficit accumulation. Ancient mean-new-moon calendars could show the moon visible in the east at dawn and in the west at dusk on the same day—astronomers called this a crescent-gibbous anomaly. Now adjusting for the sun's excess-deficit and the moon's slow-fast motion, advancing or retreating the day to obtain true new moon—the varying pace follows numerical necessity, not governmental failure. The new calendar sets 7,001 as the maximum excess-deficit; its figure interlocks with lunar motion, and naming them increase-decrease and excess-deficit keeps the text concise yet clear.
29
升降分：《皇極》躔衰有陟降率，《麟德》以日景差、陟降率、日晷景消息為之，義通軌漏。 夫南至之後，日行漸升，去極近，故晷短而萬物皆盛； 北至之後，日行漸降，去極遠，故晷長而萬物浸衰。 自《大衍》以下，皆從《麟德》。 今曆消息日行之升降，積而為盈縮焉。
Ascent-descent fraction: the Huangji calendar's lodge decline used an ascent-descent rate; the Lindé used daily shadow difference, ascent-descent rate, and gnomon shadow waxing-waning—equivalent to clepsydra regulation. After the southern solstice, the sun's path gradually rises toward the pole, shadows shorten, and all things flourish; after the northern solstice, the sun's path gradually descends from the pole, shadows lengthen, and all things gradually decline. From the Dayan calendar onward, all followed the Lindé method. The present calendar tracks the sun's waxing and waning motion, accumulating it into excess-deficit.
30
赤道宿：漢百二年議造曆，乃定東西，立晷儀，下漏刻，以追二十八宿相距于四方，赤道宿度，則其法也。 其赤道，鬥二十六度及分，牛八度，女十二度，虛十度，危十七度，室十六度，壁九度，奎十六度，婁十二度，胃十四度，昴十一度，畢十六度，觜二度，參九度，井三十三度，鬼四度，柳十五度，星七度，張十八度，翼十八度，軫十七度，角十二度，亢九度，氐十五度，房五度，心五度，尾十八度，箕十一度，自後相承用之。 至唐初，李淳風造渾儀，亦無所改。 開元中，浮屠一行作《大衍曆》，詔梁令瓚作黃道遊儀，測知畢、觜、參及輿鬼四宿赤道宿度，與舊不同。 畢十七度，觜一度，參十度，鬼三度。
Equatorial lodges: in 99 BCE the court debated calendar reform, fixed east and west, erected the gnomon, set clepsydra marks, and measured the twenty-eight lodges' spacing in the four directions—equatorial lodge degrees were the method. Its equatorial degrees: Dipper 26°+, Ox 8°, Woman 12°, Void 10°, Rooftop 17°, Encampment 16°, Wall 9°, Stride 16°, Bond 12°, Stomach 14°, Hairy Head 11°, Net 16°, Turtle Beak 2°, Three Stars 9°, Well 33°, Ghost 4°, Willow 15°, Star 7°, Extended Net 18°, Wings 18°, Chariot Shaft 17°, Horn 12°, Gullet 9°, Base 15°, Room 5°, Heart 5°, Tail 18°, Winnowing Basket 11°—used in succession thereafter. In early Tang, Li Chunfeng built the armillary sphere without altering these figures. In the Kaiyuan era, the monk Yixing compiled the Dayan Calendar; Liang Lingzan was ordered to build the ecliptic touring instrument and found the equatorial degrees of Net, Turtle Beak, Three Stars, and Ghost differed from the old values. Net 17°, Turtle Beak 1°, Three Stars 10°, Ghost 3°.
31
自一行之後，因相沿襲，下更五代，無所增損。 至仁宗皇祐初，始有詔造黃道渾儀，鑄銅為之。 自後測驗赤道宿度，又一十四宿與一行所測不同。 鬥二十五度，牛七度，女十一度，危十六度，室十七度，胃十五度，畢十八度，井三十四度，鬼二度，柳十四度，氐十六度，心六度，尾十九度，箕十度。
After Yixing, the values were passed down unchanged through five dynasties. Early in Emperor Renzong's Huangyou reign, an edict ordered a bronze ecliptic armillary sphere to be cast. Later measurements found fourteen more lodges whose equatorial degrees differed from Yixing's values. Dipper 25°, Ox 7°, Woman 11°, Rooftop 16°, Encampment 17°, Stomach 15°, Net 18°, Well 34°, Ghost 2°, Willow 14°, Base 16°, Heart 6°, Tail 19°, Winnowing Basket 10°.
32
蓋古今之人，以八尺圓器，欲以盡天體，決知其難矣。 又況圖本所指距星，傳習有差，故今赤道宿度與古不同。 自漢太初後至唐開元治曆之初，凡八百年間，悉無更易。 今雖測驗與舊不同，亦歲月未久。 新曆兩備其數，如淳風從舊之意。
Ancient and modern observers alike have known that an eight-foot circular instrument cannot fully compass the heavens. Moreover, reference stars in inherited star charts vary, so present equatorial degrees differ from ancient ones. From the Han Taichu calendar through the start of Tang Kaiyuan calendar reform—some eight hundred years—there had been no revisions. Though recent measurements differ from older values, not much time has passed. The new calendar retains both sets of figures, in the spirit of Li Chunfeng's adherence to tradition.
33
月度轉分：《洪範傳》曰：「晦而月見西方謂之朏。 月未合朔，在日後； 今在日前，太疾也。 朏者，人君舒緩、臣下驕盈專權之象。 朔而月見東方謂之側匿。 合朔則月與日合，今在日後，太遲也。 側匿者，人君嚴急、臣下危殆恐懼之象。」 盈則進，縮則退，躔離九道，周合三旬，考其變行，自有常數。 《傳》稱，人君有舒疾之變，未達月有遲速之常也。 後漢劉洪粗通其旨。 爾後治曆者多循舊法，皆考遲疾之分，增損平會之朔，得月後定追及日之際而生定朔焉。 至於加時早晚，或速或遲，皆由轉分強弱所致。 舊曆課轉分，以九分之五為強率，一百一分之五十六為弱率，乃於強弱之際而求秒焉。 新曆轉分二百九十八億八千二百二十四萬二千二百五十一，以一百萬平之，得二十七日五十五萬四千六百二十六，最得中平之數。 舊曆置日餘而求朏朒之數，衰次不倫。 今從其度而遲疾有漸，月之課驗，稍符天度。
Monthly rotation fraction: the Commentary on the Hong Fan states: "When the moon is seen in the west on the last day of the month, it is called fei (incipient crescent). The moon has not yet reached conjunction at new moon and still lags behind the sun; now it stands before the sun—too fast. An incipient crescent signaled a lax sovereign and arrogant, overbearing ministers. If the moon appears in the east at new moon, it is called ce ni (hiding aside). At true new moon sun and moon conjoin; if the moon still lags behind the sun, it is too slow. Ce ni betokened a stern sovereign and ministers in peril and dread." When full it advances, when shrunken it retreats; the moon traverses and leaves the nine paths, completing a cycle in thirty days—investigate its changing motion and fixed numbers appear." The Commentary warns that attributing slackness or urgency to the sovereign misses the moon's regular alternation of slow and fast motion. Liu Hong of Later Han grasped the principle in outline. Later calendar-makers mostly followed older methods, reckoning slow-fast divisions and adjusting mean conjunctions until the moon's fixed catch-up with the sun yielded true new moon. Whether the computed hour came early or late, fast or slow, all stemmed from the strength or weakness of the rotation fraction. The old calendar computed rotation fractions with a strong rate of 5/9 and a weak rate of 56/101, deriving seconds at the transition between them. The new calendar's rotation fraction is 29,882,422,251; divided by 1,000,000 this yields 27 days 554,626—the most balanced mean value. The old calendar derived crescent-gibbous figures from day remainders, producing disordered declining ranks. Now following its degrees with gradual slow-fast change, lunar verification aligns more closely with the heavens.
34
轉度母：轉法、會周附。
Rotation-degree mother: see attached rotation method and conjunction cycle.
35
本以朔分並周天，是為會周。 一朔之月常度也，名周本母。
Originally the new-moon fraction was added to the circuit of heaven to form the conjunction cycle. This is one new moon's normal monthly arc, called the circuit root mother.
36
去其朔差為轉終，朔差乃終外之數也。
Subtract the new-moon difference to obtain the rotation endpoint; the new-moon difference is the excess beyond the endpoint.
37
各以等數約之，即得實用之數。 乃以等數約本母為轉度母，齊數也。
Reduce each by common divisors to obtain the practical figures. The root mother is then reduced by common divisors to become the rotation-degree mother—aligned figures.
38
又以等數約月分為轉法，亦名轉日法也。
The monthly fraction is similarly reduced to form the rotation divisor, also called the rotation day divisor.
39
以轉法約轉終，得轉日及餘。 本曆創立此數，皆古曆所未有。 約得八千一百一十二萬為轉度母，二百九十八億八千二百二十四萬二千二百五十一為轉終分，三百二十億二千五百一十二萬九千二百五十一為會周，一十億八千四百四十七萬三千為轉法，二十一億四千二百八十八萬七千為朔差。
Dividing the rotation endpoint by the rotation divisor yields rotation days and remainder. The present calendar first established these quantities, unknown in older calendars. The reduced values are: rotation-degree mother 81,120,000; rotation-end fraction 29,882,422,251; conjunction cycle 32,025,129,251; rotation divisor 1,084,473,000; new-moon difference 2,142,887,000.
40
月離遲疾定差：《皇極》有加減限、朏朒積，《麟德》曰增減率、遲疾積，《大衍》曰損益率、朏朒積，《崇天》亦曰損益率、朏朒積。 所謂日不及平行則損之，過平行則益之，從陽之義也； 月不及平行則益之，過平行則損之，禦陰之道也。 陰陽相錯而以損益、遲疾為名。 新曆以一萬四千八百一十九為遲疾之極，而得五度八分，其數與躔相錯，可以知合食加時之早晚也。
Lunar motion slow-fast fixed difference: the Huangji used increase-decrease limit and crescent-gibbous accumulation; the Lindé used increase-decrease rate and slow-fast accumulation; the Dayan and Chongtian used increase-decrease rate and crescent-gibbous accumulation. When the sun falls short of mean motion it is decreased; when it exceeds mean motion it is increased—following the principle of yang; when the moon falls short it is increased, and when it exceeds it is decreased—the way of governing yin. Yin and yang interlock, hence the names increase-decrease and slow-fast. The new calendar sets 14,819 as the slow-fast maximum, yielding 5°8′; this figure interlocks with lodge motion to determine whether conjunctions and eclipses occur early or late.
41
進朔：進朔之法，興於《麟德》。 自後諸曆，因而立法，互有不同。 假令仲夏月朔月行極疾之時，合朔當於亥正，若不進朔，則晨而月見東方； 若從《大衍》，當戌初進朔，則朔日之夕，月生於西方。 新曆察朔日之餘，驗月行徐疾，變立法率，參驗加時，常視定朔小餘：秋分後四分法之三已上者，進一日； 春分後定朔晨分差如春分之日者，三約之，以減四分之二； 定朔小餘如此數已上者，亦進，以來日為朔。 俾迴圈合度，月不見於朔晨； 交會無差，明必藏於朔夕。 加時在於午中，則晦日之晨同二日之夕，皆合月見； 加時在於酉中，則晦日之晨尚見，二日之夕未生； 加時在於子中，則晦日之晨不見，二日之夕以生。 定晦朔，乃月見之晨夕可知； 課小余，則加時之早晏無失。 使坦然不惑，觸類而明之。
Advancing new moon: the advancing-new-moon method originated in the Lindé calendar. Later calendars each legislated accordingly, with differing rules. Suppose at midsummer new moon, when lunar motion is fastest, conjunction falls at the hai quarter; without advancing the new moon, the moon would be visible in the east at dawn; Following the Dayan method, advancing at xu initial, the moon would rise in the west on new-moon evening. The new calendar examines the new-moon remainder, tests lunar slow-fast motion, sets a variable rate, and checks the hour of conjunction, always judging the fixed new moon's small remainder: after the autumn equinox, if it reaches three-fourths of the quarter method or more, advance one day; after the spring equinox, if the fixed-new-moon dawn difference equals that on the equinox day, take two-thirds of two-fourths as the threshold; if the fixed new moon's small remainder meets this threshold or more, advance likewise, making the following day the new moon. Thus the cycle aligns with combined degrees so the moon is not visible at new-moon dawn; conjunctions stay accurate, and the moon is hidden at new-moon dusk. When conjunction falls at noon, the moon is visible both at dawn on the last day and at dusk on the second day; when it falls at the you quarter, the moon is still visible at dawn on the last day but has not yet risen at dusk on the second day; when it falls at the zi quarter, the moon is not seen at dawn on the last day but has risen by dusk on the second day. With fixed last day and new moon, the moon's visibility at dawn and dusk can be known; and tabulating the small remainder keeps the hour of conjunction accurate. The method is transparent and applies consistently throughout.
42
消息數：因漏刻立名，義通晷景。 《麟德》曆差曰屈伸率。 天晝夜者，《易》進退之象也。 冬至一陽爻生而晷道漸升，夜漏益減，象君子之道長，故曰息； 夏至一陰爻生，而晷道漸降，夜漏益增，象君子之道消，故曰消。 表景與陽為沖，從晦者也，故與夜漏長短。 今以屈伸象太陰之行，而刻差曰消息數。 黃道去極，日行有南北，故晷漏有長短。 然景差徐疾不同者，句股使之然也。 景直晷中則差遲，與句股數齊則差急，隨北極高下，所遇不同。 其黃道去極度數與日景、漏刻、昏晚中星反覆相求，消息用率，步日景而稽黃道，因黃道而生漏刻，而正中星，四術旋相為中，以合九服之變，約而易知，簡而易從。
Waxing-waning numbers: named from clepsydra graduations, equivalent to gnomon shadow measurement. The Lindé calendar termed the difference the bending-stretching rate. Day and night in heaven embody the Changes' image of advance and retreat. At the winter solstice a yang line is born, shadows gradually shorten, and night clepsydra marks decrease—emblematic of the noble Way growing, hence called waxing; at the summer solstice a yin line is born, shadows lengthen, and night clepsydra marks increase—emblematic of the noble Way waning, hence called waning. Gnomon shadow opposes yang and follows the obscure, hence it tracks the length of night clepsydra marks. Now bending-stretching images lunar motion, while clepsydra differences are termed waxing-waning numbers. As the ecliptic departs from the pole, the sun's daily path shifts north and south, so gnomon shadows and clepsydra marks vary in length. Yet shadow differences vary in speed because right-triangle geometry governs them. When the shadow falls straight at the gnomon center the change is slow; when the right-triangle ratio applies it is fast—varying with the height of the north pole. Ecliptic polar distance, solar shadow, clepsydra marks, dusk-dawn, and culminating stars are solved iteratively; waxing-waning rates derive solar shadow to verify the ecliptic, clepsydra marks from the ecliptic, and culminating stars at center—four methods interlocking to harmonize regional variation, concise and easy to apply.
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六十四卦：十二月卦出於孟氏，七十二候原于《周書》。 後宋景業因劉洪傳卦，李淳風據舊曆元圖，皆未睹陰陽之賾。 至開元中，浮屠一行考揚子雲《太玄經》，錯綜其數，索隱周公三統，糾正時訓，參其變通，著在爻象，非深達《易》象，孰能造於此乎！ 今之所修，循一行舊義，至於周策分率，隨數遷變。 夫六十卦直常度全次之交者，諸侯卦也； 竟六日三千四百八、十六秒而大夫受之； 次九卿受之； 次三公受之； 次天子受之。 五六相錯，復協常月之次。 凡九三應上九，則天微然以靜； 六三應上六，則地鬱然而定。 九三應上六即溫，六三應上九即寒。 上爻陽者風，陰者雨。 各視所直之爻，察不刊之象，而知五等與君辟之得失、過與不及焉。 七十二候，李業興以來迄於《麟德》，凡七家曆，皆以雞始乳為立春初候，東風解凍為次候，其餘以次承之。 與《周書》相校，二十餘日，舛訛益甚。 而一行改從古義，今亦以《周書》為正。
Sixty-four hexagrams: the twelve-month hexagrams derive from the Meng school; the seventy-two hou originate in the Documents of Zhou. Later, Song Jingye followed Liu Hong's hexagram transmission and Li Chunfeng relied on older origin diagrams—neither had grasped yin-yang's deeper pattern. In the Kaiyuan era the monk Yixing studied Yang Xiong's Taixuan jing, interwove its numbers, recovered the Duke of Zhou's Triple Concordance, corrected seasonal doctrine, and embodied its transformations in the line images—only one deeply versed in the Changes could have achieved this. The present revision follows Yixing's original scheme; the Zhou policy fractional rates change with the figures. Among the sixty hexagrams, those aligned with whole-degree crossings are the feudal-lord hexagrams; after six days 3,408 and 86 seconds the grandee receives it; next the nine ministers receive it; next the three dukes receive it; next the Son of Heaven receives it. Five and six interlock, returning to the regular monthly sequence. When nine-three corresponds to upper nine, Heaven grows faintly still; when six-three corresponds to upper six, Earth grows dense and settled. Nine-three with upper six brings warmth; six-three with upper nine brings cold. A yang upper line brings wind; a yin upper line brings rain. Observe the governing line and its fixed images to read the five ranks' and the ruler's gains and losses, excess and deficiency. Seventy-two hou: from Li Yexing through the Lindé, seven calendar traditions all took hens beginning to brood as the first Establishing Spring hou and thawing east wind as the second—the rest followed in sequence. Compared with the Documents of Zhou, they were off by more than twenty days, with ever greater errors. Yixing reverted to ancient usage; the present revision likewise takes the Documents of Zhou as authoritative.
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嶽台日晷：岳台者，今京師嶽台坊，地曰浚儀，近古候景之所。 《尚書·洛誥》稱東土是也。 《禮》玉人職：「土圭長尺有五寸以致日。」 此即日有常數也。 司徒職以圭正日晷，「日至之景，尺有五寸，謂之地中。」 此即是地土中致日景與土圭等。 然表長八尺，見於《周髀》。 夫天有常運，地有常中，曆有正象，表有定數。 言日至者，明其日至此也。 景尺有五寸與圭等者，是其景晷之真效。 然夏至之日尺有五寸之景，不因八尺之表將何以得？ 故經見夏至日景者，明表有定數也。 新曆周歲中晷長短，皆以八尺之表測候，所得名中晷常數。 交會日月，成象於天，以辨尊卑之序。 日，君道也； 月，臣道也。 謫食之變，皆與人事相應。 若人君修德以禳之，則或當食而不食。 故太陰有變行以避日，則不食； 五星潛在日下，為太陰禦侮而扶救，則不食； 涉交數淺，或在陽曆，日光著盛，陰氣衰微，則不食； 德之休明而有小眚焉，天為之隱，是以光微蔽之，雖交而不見食。 此四者，皆德感之所繇致也。 按《大衍曆議》：開元十二年七月戊午朔，當食。 時自交阯至朔方，同日度景測候之際，晶明無雲而不食。 以曆推之，其日入交七百八十四分，當食八分半。 十三年，天正南至，東封禮畢，還次梁、宋，史官言：「十二月庚戌朔，當食。」 帝曰：「予方修先後之職，謫見於天，是朕之不敏，無以對揚上帝之休也。」 於是徹膳素服以俟之，而卒不食。 在位之臣莫不稱慶，以謂德之動天，不俟終日。 以曆推之，是月入交二度弱，當食十五分之十三，而陽光自若，無纖毫之變，雖算術乖舛，不宜若是。 凡治曆之道，定分最微，故損益毫釐，未得其正，則上考《春秋》以來日月交食之載，必有所差。 假令治曆者因開元二食變交限以從之，則所協甚少，而差失過多。 由此明之，《詩》云：「此日而微。」 乃非天之常數也。 舊曆直求月行入交，今則先課交初所在，然後與月行更相表裏，務通精數。
Yue Terrace gnomon: Yue Terrace is present-day Yue Terrace ward in the capital, at Junyi—the ancient shadow-observation site. The Documents' Luo Announcement identifies this as the eastern domain. The Rites' jade-worker statute: "The earth gnomon is one chi and five cun long to reach the sun." This shows the sun has fixed measures. The Minister of Works used the gnomon to correct the solar dial: "At solstice the shadow of one chi five cun marks the earth's center." That is, at the earth's center the solar shadow equals the earth gnomon. Yet an eight-chi gnomon appears in the Zhou Bi. Heaven has constant motion, earth a fixed center, the calendar correct images, and the gnomon fixed measures. To speak of solstice day is to mark where solstice occurs. A shadow of one chi five cun matching the gnomon is the true gnomon-shadow effect. Yet how could the summer solstice shadow of one chi five cun be obtained without an eight-chi gnomon? Hence the classic's mention of the summer solstice shadow shows the gnomon has a fixed measure. The new calendar's mid-year shadow lengths are all measured with an eight-chi gnomon, yielding what is called the mid-gnomon constant. Solar and lunar conjunction forms heavenly images to distinguish the order of honor and rank. The sun is the Way of the lord; The moon is the Way of the minister. Eclipse phenomena all correspond to human affairs. If the ruler cultivates virtue to avert them, a predicted eclipse may fail to occur. Thus if the moon alters its course to avoid the sun, there is no eclipse; If the five planets hide below the sun to shield and aid the moon, there is no eclipse; If the crossing depth is shallow, or if it falls in the solar season when sunlight is strong and yin force is weak, there is no eclipse; When virtue shines brightly with only a minor fault, Heaven conceals the sign—dimming the light so that though the bodies cross, no eclipse appears. These four cases all arise from the influence of virtue. The Discourse on the Great Evolution Calendar records: at the wuwu new moon of the seventh month, Kaiyuan year 12, an eclipse was predicted. Observations from Jiaozhi to Shuofang on that same day found clear skies and bright sun—no eclipse occurred. Calendar calculation gave an entry into crossing of 784 parts and a predicted eclipse of eight and a half tenths. In year 13, after the winter solstice and the eastern Feng ceremony, while the emperor halted at Liang and Song, the historiographers reported: 'An eclipse is due on the gengxu new moon of the twelfth month. The emperor replied: 'I am fulfilling the duties of my forebears, yet heaven shows a reproof—this reflects my own inadequacy, unworthy to answer and glorify Heaven's favor.' He then abolished rich fare and wore plain robes to await the event—but no eclipse occurred. Every minister in court rejoiced, declaring that virtue had moved heaven without waiting even a full day. The calendar predicted entry into crossing of just over two degrees and an eclipse of thirteen-fifteenths, yet the sun remained unchanged—not a trace of dimming; reckoning may err, but not to this degree. Calendar-making hinges on minute fractional constants; the slightest adjustment, if wrong, will misalign all solar and lunar eclipse records back through the Spring and Autumn Annals. If calendar-makers adjusted crossing limits to fit those two Kaiyuan eclipses alone, few other predictions would agree and errors would multiply. This makes clear the Odes line: 'On this day the sun grew dim.' That was not heaven's regular course. The old calendar simply computed when the moon entered crossing; the Tianli Calendar first determines the crossing's initial position, then interlocks it with lunar motion to achieve precise agreement.
45
四正食差：正交如累壁，漸減則有差。 在內食分多，在外食分少； 交淺則間遙，交深則相薄； 所觀之地又偏，所食之時亦別。 苟非地中，皆隨所在而漸異。 縱交分正等同在南方，冬食則多，夏食乃少。 假均冬夏，早晚又殊，處南北則高，居東西則下。 視有斜正，理不可均。 月在陽曆，校驗古今交食，所虧不過其半。 合置四正食差，則斜正於卯酉之間，損益於子午之位，務從親密，以考精微。
Eclipse differences at the four cardinal points: at true crossing the paths align like a wall; as they diverge, discrepancies appear. Eclipses nearer the center show greater obscuration; those farther out show less; A shallow crossing widens the gap; a deep crossing brings the bodies close; The observer's location skews the view, and the hour of eclipse varies as well. Except at the earth's center, results differ gradually with location. Even with identical crossing fractions in the south, winter eclipses are larger and summer eclipses smaller. Even averaging winter and summer, morning and evening differ; observers to the north or south see higher paths, those east or west lower. Sight lines are oblique or direct—the phenomenon cannot be uniform. When the moon is in the solar season, comparison of historical eclipses shows obscuration never exceeds half. Applying eclipse corrections at the four cardinal points—adjusting for obliquity between mao and you and for latitude at zi and wu—yields close agreement with fine observation.
46
五星立率：五星之行，亦因日而立率，以示尊卑之義。 日週四時，無所不照，君道也； 星分行列宿，臣道也。 陰陽進退，於此取儀刑焉。 是以當陽而進，當陰而退，皆得其常，故加減之。 古之推步，悉皆順行，至秦方有金、火逆數。
Establishing planetary rates: the five planets' motions are likewise calibrated to the sun's motion to express the hierarchy of noble and base. The sun circles the four seasons and illuminates all—this is the Way of the lord; The stars move separately through the lodges—this is the Way of the minister. Yin and yang advance and retreat—their pattern is taken from this. Thus they advance in yang seasons and retreat in yin seasons, each according to rule—hence the additions and subtractions. Ancient astronomical calculation assumed only direct motion; retrograde tables for Venus and Mars appeared only in the Qin.
47
《大衍》曰：「木星之行與諸星稍異：商、周之際，率一百二十年而超一次； 至戰國之時，其行浸急； 逮中平之後，八十四年而超一次，自此之後，以為常率。」 其行也，初與日合，一十八日行四度，乃晨見東方。 而順行一百八日，計行二十二度強，而留二十七日。 乃退行四十六日半，退行五度強，與日相望。 旋日而退，又四十六日半，退五度強，復留二十七日。 而順行一百八日，行十八度強，乃夕伏西方。 又十八日行四度，復與日合。
The Great Evolution Calendar states: 'Jupiter's motion differs slightly from the other planets: during the Shang and Zhou, it passed one lodge every 120 years; by the Warring States period its motion had gradually accelerated; after the Zhongping era it crossed one lodge every eighty-four years, and that has been the standard rate ever since.' Its cycle begins at solar conjunction; in eighteen days it travels four degrees and appears as the morning star in the east. It then moves direct for 108 days, covering just over 22 degrees, and stations for 27 days. It retrogrades for 46½ days through just over five degrees, reaching opposition with the sun. It continues retrograde another 46½ days for just over five degrees and stations again for 27 days. It moves direct for 108 more days through just over eighteen degrees and sets in the west at dusk. After eighteen more days and four degrees of travel it conjoins the sun again.
48
火星之行：初與日合，七十日行五十二度，乃晨見東方。 而順行二百八十日，計行二百一十六度半弱，而留十一日。 乃退行二十九日，退九度，與日相望。 旋日而退，又二十九日，退九度，復留十一日。 而順行二百八十日，行一百六十四度半弱，而夕伏西方。 又七十日，行五十二度，復與日合。
Mars's cycle: it begins at solar conjunction; in seventy days it travels fifty-two degrees and appears as the morning star in the east. It moves direct for 280 days, covering just under 216½ degrees, and stations for 11 days. It retrogrades for 29 days through nine degrees, reaching opposition with the sun. It continues retrograde another 29 days for nine degrees and stations again for 11 days. It moves direct for 280 more days through just under 164½ degrees and sets in the west at dusk. After seventy more days and fifty-two degrees of travel it conjoins the sun again.
49
土星之行：初與日合，二十一日行二度半，乃晨見東方。 順行八十四日，計行九度半強，而留三十五日。 乃退行四十九日，退三度半，與日相望。 乃旋日而退，又四十九日，退三度少，復留三十五日。 又順行八十四日，行七度強，而夕伏西方。 又二十一日，行二度半，復與日合。
Saturn's cycle: it begins at solar conjunction; in twenty-one days it travels two and a half degrees and appears as the morning star in the east. It moves direct for 84 days, covering just over nine and a half degrees, and stations for 35 days. It retrogrades for 49 days through three and a half degrees, reaching opposition with the sun. It continues retrograde another 49 days for just under three degrees and stations again for 35 days. It moves direct for 84 more days through just over seven degrees and sets in the west at dusk. After twenty-one more days and two and a half degrees of travel it conjoins the sun again.
50
金星之行：初與日合，五十八日半行四十九度太，而夕見西方。 乃順行二百三十一日，計行二百五十一度半，而留七日。 乃退行九日，退四度半，而夕伏西方。 又六日半，退四度太，與日再合。 又六日半，退四度太，而晨見東方。 又退九日，逆行四度半，而復留七日。 而復順行二百三十一日，行二百五十一度半，乃晨伏東方。 又三十八日半，行四十九度太，復與日會。
Venus's cycle: it begins at solar conjunction; in fifty-eight and a half days it travels just over forty-nine degrees and appears as the evening star in the west. It moves direct for 231 days, covering 251½ degrees, and stations for 7 days. It retrogrades for 9 days through four and a half degrees and sets in the west at dusk. After six and a half more days it retrogrades just over four degrees and conjoins the sun again. After another six and a half days it retrogrades just over four degrees and appears as the morning star in the east. It retrogrades again for 9 days through four and a half degrees and stations for 7 days. It moves direct for 231 more days through 251½ degrees and sets in the east at dawn. After thirty-eight and a half more days and just over forty-nine degrees of travel it conjoins the sun again.
51
水星之行：初與日合，十五日行三十三度，乃夕見西方。 而順行三十日，計行六十六度，而留二日，乃夕伏西方。 而退十日，退八度，與日再合。 又退十日，退八度，乃晨見東方，而復留三日。 又順行三十三日，行三十三度，而晨伏東方。 又十五日，行三十三度，與日復會。
Mercury's cycle: it begins at solar conjunction; in fifteen days it travels thirty-three degrees and appears as the evening star in the west. It moves direct for 30 days, covering 66 degrees, stations for 2 days, and sets in the west at dusk. It retrogrades for 10 days through eight degrees and conjoins the sun again. It retrogrades another 10 days for eight degrees, appears as the morning star in the east, and stations for 3 days. It moves direct for 33 more days through thirty-three degrees and sets in the east at dawn. After fifteen more days and thirty-three degrees of travel it conjoins the sun again.
52
一行云：「五星伏、見、留、逆之效，表、裏、盈、縮之行，皆系之于時，驗之於政。 小失則小變，大失則大變； 事微而象微，事章而象章。 蓋皇天降譴以警悟人主。 又或算者昧於象，占者迷於數，睹五星失行，悉謂之曆舛，以數象相參，兩喪其實。 大凡校驗之道，必稽古今注記，使上下相距，反覆相求，苟獨異常，則失行可知矣。」
Yixing said: 'The setting, appearing, stationing, and retrograde of the five planets, their outer and inner courses and waxing-waning motion, all depend on the seasons and are verified against government. Small faults bring small changes; great faults bring great changes; minor affairs yield minor signs; major affairs yield major signs. August Heaven sends reproofs to awaken the ruler. Some calculators ignore celestial signs and some diviners lose themselves in numbers; seeing a planet out of place, they blame the calendar—yet cross-checking figures and omens, both miss the truth. The proper method of verification is to consult records ancient and modern, compare distant eras, and seek repeatedly; only what remains uniquely anomalous can be called a true deviation from motion.'
53
星行盈縮：五星差行，惟火尤甚。 乃有南侵狼坐，北入匏瓜，變化超越，獨異于常，是以日行之分，自有盈縮。 此乃天度廣狹不等，氣序升降有差，考今升降之分，積為盈縮之數。 凡五星入氣加減，興于張子信，以後方士各自增損，以求親密。 而《開元曆》別為四象六爻，均以進退，今則別立盈縮，與舊異。
Planetary waxing and waning: all five planets vary in motion, but Mars most of all. It may invade south to the Wolf asterism or enter north to the Gourd, changing and leaping beyond the norm—hence its daily motion naturally waxes and wanes. This reflects unequal breadth of celestial degrees and varying rise and fall of seasonal qi; the present rise-and-fall fractions are accumulated into waxing-waning numbers. Additions and subtractions for the five planets entering qi began with Zhang Zixin; later specialists each adjusted them to seek closer agreement. The Kaiyuan Calendar used four images and six lines, averaging advance and retreat; the Tianli Calendar now establishes separate waxing-waning tables, differing from the old method.
54
五星見伏：五星見伏，皆以日度為規。 日度之運，既進退不常； 星行之差，亦隨而增損。 是以五星見伏，先考日度之行，今則審日行盈縮，究星躔進退，五星見伏，率皆密近。 舊說，水星晨應見不見在雨水後、穀雨前，夕應見不見在處暑後、霜降前。 又雲，五星在卯酉南則見遲、0伏早，在卯酉北則見早、伏遲，蓋天勢使之然也。
Planetary visibility: the appearing and setting of all five planets are governed by solar degree. Solar motion itself advances and retreats irregularly; planetary motion must be adjusted accordingly. Formerly planetary visibility was reckoned from solar motion alone; the Tianli Calendar now examines the sun's daily waxing and waning and planetary advance and retreat together, so planetary visibility closely agrees with observation. Older doctrine held that Mercury failed to appear as the morning star between Rain Water and Grain Rain, and as the evening star between Limit of Heat and Frost Descent. It was also said that planets south of mao and you appear late and set early, while those north appear early and set late—heaven's geometry makes it so.
55
步氣朔術
Procedure for Stepping Qi and New Moon
56
演紀上元甲子歲，距治平元年甲辰，歲積七十一萬一千七百六十，算外。 上驗往古，每年減一算； 下算將來，每年加一算。
From the upper origin jiazi year of the era cycle to the jiachen year of Zhiping 1, the accumulated years are 711,760, with remainder outside the count. To verify into the past, subtract one count per year; to project into the future, add one count per year.
57
元法：三萬九千。
Origin divisor: 39,000.
58
歲周：一千四百二十四萬四千五百。
Annual circuit: 14,244,500.
59
朔實：一百一十五萬一千六百九十三。
New-moon true value: 1,151,693.
60
歲周：三百六十五日、餘九千五百。
Annual cycle: 365 days, remainder 9,500.
61
朔策：二十九、余二萬六百九十三。
New-moon cycle: 29 days, remainder 20,693.
62
望策：一十四、余二萬九千八百四十六半。
Full-moon cycle: 14 days, remainder 29,846½.
63
弦策：七、余一萬四千九百二十三、秒四半。
Quarter-moon cycle: 7 days, remainder 14,923, seconds 4½.
64
氣策：一十五、餘八千五百二十、秒一十五。
Qi cycle: 15 days, remainder 8,520, seconds 15.
65
中盈分：一萬七千四十一、秒一十二。
Mid-qi excess fraction: 17,041, seconds 12.
66
朔虛分：一萬八千三百七。
New-moon void fraction: 18,307.
67
閏限：一百一十一萬六千三百四十四、秒六。
Intercalation limit: 1,116,344, seconds 6.
68
歲閏：四十二萬四千一百八十四。
Annual intercalation increment: 424,184.
69
月閏：三萬五千三百四十八、秒一十二。
Monthly intercalation increment: 35,348, seconds 12.
70
沒限：三萬四百七十九、秒三。
Extinction limit: 30,479, seconds 3.
71
紀法：六十。
Cycle divisor: 60.
72
秒母：一十八。
Second denominator: 18.
73
求天正冬至：置所求積年，以歲周乘之，為天正冬至氣積分； 滿元法除之為積日，不滿為小餘。 日盈紀法去之，不盡，命甲子，算外，即得所求年前天正冬至日辰及餘。
To find the civil-year winter solstice: set the accumulated years sought, multiply by the annual circuit to obtain the winter-solstice qi accumulated parts; divide by the origin divisor for accumulated days; the remainder is the small remainder. Remove full cycle divisors from the day-count; count the remainder from jiazi beyond the count to obtain the winter solstice date, hour, and remainders for the year before the one sought.
74
求次氣：置天正冬至大、小餘，以氣策加之，即得次氣大、小餘。 若秒盈秒母從小余，小余滿元法從大余，大余滿紀法即去之。
To find successive qi: set the civil winter solstice large and small remainders, add the qi cycle to obtain each next qi's large and small remainders. When seconds fill the second denominator, carry into the small remainder; when the small remainder fills the origin divisor, carry into the large remainder; when the large remainder fills the cycle divisor, discard the excess.
75
命大餘甲子，算外，即次氣日辰及餘。 餘氣累而求之。
Count from jiazi on the large remainder, beyond the count, for each qi's date, hour, and remainder. Find the remaining qi by repeated addition.
76
求天正經朔：置天正冬至氣積分，滿朔實去之為積月，不盡為閏餘； 盈元法為日，不盈為餘； 以減天正冬至大、小餘，為天正經朔大、小餘。 大余不足減，加紀法； 小餘不足減，退大餘，加元法以減之。
To find the civil-year standard new moon: set the winter-solstice qi accumulated parts; remove full new-moon true values for accumulated months; the remainder is the intercalary remainder; full origin divisors convert to days; the remainder is the fractional part; subtract from the civil winter solstice large and small remainders to obtain the standard new moon's large and small remainders. If the large remainder is too small, add the cycle divisor; if the small remainder is too small, borrow one from the large remainder and add the origin divisor, then subtract.
77
命大餘甲子，算外，即得所求年前天正經朔日辰及餘。
Count from jiazi on the large remainder, beyond the count, to obtain the standard new moon date, hour, and remainders for the year before the one sought.
78
求弦望及次朔經日：置天正經朔大、小餘，以弦策累加之，命如前，即得弦、望及次朔經日日辰及餘。
To find first quarter, full moon, and successive standard new moons: set the civil standard new moon's large and small remainders, add the quarter-moon cycle repeatedly, and count as before to obtain each event's date, hour, and remainder.
79
求沒日：置有沒之氣小餘，二十四氣小餘在沒限已上者，為有沒之氣。
To find surplus days: set the small remainder of the qi subject to extinction; any of the twenty-four qi whose small remainder reaches the extinction limit or above is a qi with surplus.
80
以秒母乘之，其秒從之。
Multiply by the second denominator, carrying the seconds.
81
用減七十一萬二千二百二十五，余以一萬二百二十五除之為沒日，不滿為除。 以沒日加其氣大餘，命甲子，算外，即其氣沒日日辰。
Subtract from 712,225; divide the remainder by 10,125 for surplus days; the remainder is the fractional part. Add the surplus days to that qi's large remainder, count from jiazi beyond the count, and obtain the surplus day's date and hour.
82
求減日：置有減經朔小餘，經朔小餘不滿朔虛分者，為有減之朔。
To find void days: set the standard new moon small remainder subject to reduction; any standard new moon whose small remainder falls short of the new-moon void fraction is a month with a void day.
83
以三十乘之，滿朔虛分為減日，不滿為餘。 以減日加經朔大餘，命甲子，算外，即其月減日日辰。
Multiply by 30; divide by the new-moon void fraction for void days; the remainder is the fractional part. Add the void days to the standard new moon's large remainder, count from jiazi beyond the count, and obtain that month's void-day date and hour.
84
步發斂術
Procedure for Stepping Emission and Contraction
85
候策：五、餘二千八百四十、秒五。
Pentad cycle: 5 days, remainder 2,840, seconds 5.
86
卦策：六、餘三千四百八、秒六。
Hexagram cycle: 6 days, remainder 3,408, seconds 6.
87
土王策：三、餘一千七百四、秒三。
Earth-king cycle: 3 days, remainder 1,704, seconds 3.
88
辰法：三千二百五十。
Double-hour divisor: 3,250.
89
刻法：三百九十。
Clepsydra-mark divisor: 390.
90
半辰法：一千六百二十五。
Half double-hour divisor: 1,625.
91
秒母：一十八。
Second denominator: 18.
92
求七十二候：各置中節大、小餘命之，為初候； 以候策加之，為次候； 又加之，為末候。 各命甲子，算外，即得其候日辰。
To find the seventy-two pentads: for each mid-season node, set its large and small remainders and count them as the first pentad; add the pentad cycle for the second pentad; add again for the third pentad. For each, count from jiazi beyond the count to obtain the pentad's date and hour.
93
求六十四卦：各因中氣大、小餘命之，為公卦用事日； 以卦策加之，即次卦用事日； 以土王策加諸侯之卦，得十有二節之初外卦用事日。
To find the sixty-four hexagrams: for each mid-qi, set its large and small remainders and count them as the duke hexagram's dominion day; add the hexagram cycle to obtain the next hexagram's dominion day; add the earth-king cycle to the feudal-lord hexagram to obtain the outer hexagram's dominion day at the opening of each of the twelve seasonal nodes.
94
求五行用事日：各因四立之節大、小餘命之，即春木、夏火、秋金、冬水首用事日； 以土王策減四季中氣大、小餘，命甲子，算外，即其月土始用事日也。
To find the five phases' dominion days: for each of the four establishment nodes, set its large and small remainders and count them—the opening dominion days of spring Wood, summer Fire, autumn Metal, and winter Water; subtract the earth-king cycle from each seasonal mid-qi's large and small remainders, count from jiazi beyond the count, and obtain the day when Earth begins its dominion that month.
95
求發斂加時：各置小余，滿辰法除之為辰數，不滿者，刻法而一為刻，又不滿為分。 命辰數從子正，算外，即得所求加時辰時。 若以半辰之數加而命之，即得辰初後所入刻數。
To find emission-contraction time-of-day: set the small remainder; divide by the double-hour divisor for the double-hour count; the remainder divided by the clepsydra-mark divisor gives marks; any remainder is fractional parts. Count from midnight zi on the double-hour number, beyond the count, to obtain the true hour sought. If one adds the half double-hour value before counting, one obtains the clepsydra marks elapsed after the hour's opening.
96
求發斂去經朔：置天正經朔閏餘，以月閏累加之，即每月閏餘； 滿元法除之為閏日，不盡為小餘，即得其月中氣去經朔日及餘秒。 其閏余滿閏限，即為置閏，以月內無中氣為定。
To find emission-contraction offset from the standard new moon: set the civil standard new moon's intercalary remainder and add the monthly intercalation repeatedly to obtain each month's intercalary remainder; divide by the origin divisor for intercalary days; the remainder is the small remainder—thus obtaining each month's mid-qi offset from the standard new moon in days, parts, and seconds. When the intercalary remainder reaches the intercalation limit, insert an intercalary month, confirmed by a month that contains no mid-qi.
97
求卦候去經朔：各以卦、候策及餘秒累加減之，中氣前，減； 中氣後，加。 即各得卦、候去經朔日及餘秒。
To find hexagram and pentad offset from the standard new moon: for each, cumulatively add or subtract the hexagram and pentad cycles with remainders and seconds; before mid-qi, subtract; after mid-qi, add. Thus obtain each hexagram and pentad's offset from the standard new moon in days, parts, and seconds.
98
步日躔術
Procedure for Stepping Solar Lodge Motion
99
日度母：六百二十四萬。
Day-degree mother: 6,240,000.
100
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.
101
周天：三百六十五度。 余一百六十四萬四百四十七，約分二千五百六十四、秒八十二。
Circuit of heaven: 365°. Remainder 1,640,447, reduced fraction 2,564, seconds 82.
102
歲差：八萬四百四十七。
Precession: 80,447.
103
二至限：一百八十二度。 余二萬四千二百五十，約分六千二百一十八。
Solstitial limit: 182°. Remainder 24,250, reduced fraction 6,218.
104
一象度：九十一。 余一萬二千一百二十五，約分三千一百九。
One quadrant: 91°. Remainder 12,125, reduced fraction 3,119.
105
求朔弦望入盈縮度：置二至限度及餘，以天正閏日及餘減之，余為天正經朔入縮度及餘； 以弦策累加之，滿二至限度及餘去之，則盈入縮，縮入盈而互得之。 即得弦、望及次經朔日所入盈縮度及餘。 其餘以一萬乘之，元法除之，即得約分。
To find entry into expansion-contraction for new moon, quarters, and full moon: set the solstitial limit and remainder, subtract the civil intercalary day and remainder—the remainder is the standard new moon's entry into contraction degrees and parts; add the quarter-moon cycle repeatedly; remove the solstitial limit when full—expansion then yields to contraction and contraction to expansion in alternation. Thus obtain the expansion-contraction degrees and parts for each quarter, full moon, and successive standard new moon. Multiply the remainder by 10,000 and divide by the origin divisor to obtain the reduced fraction.
106
求朔弦望盈縮差及定差：各置朔、弦、望所入盈縮度及約分，如在象度分以下者為在初； 已上者，覆減二至限，餘為在末。 置初、末度分於上，列二至於下，以上減下，餘以下乘上，為積數； 滿四千一百三十五除之為度，不滿，退除為分，命曰盈縮差度及分。 若以四百乘積數，滿五百六十七除之，為盈縮定差。 若用立成者，以其度損益率乘度除，滿元法而一，所得，以損益其度下盈縮積，為定差度； 其損益初、末分為二日者，各隨其初、末以乘除。 其後皆如此例。
To find expansion-contraction difference and true correction for new moon, quarters, and full moon: set the entered expansion-contraction degrees and reduced fraction; if below the quadrant-degree fraction, the position is in the initial portion; if at or above, subtract from the solstitial limit in reverse; the remainder places it in the final portion. Place the initial or final degree-fraction above and the solstitial value below; subtract the lower from the upper; multiply the remainder by the upper to obtain the accumulated product; divide by 4,135 for degrees; the remainder, reduced one place, gives fractional parts—the expansion-contraction difference in degrees and parts. Multiplying the accumulated product by 400 and dividing by 567 yields the true expansion-contraction correction. With precomputed tables, multiply the degree's increase-decrease rate by the degree divisor, divide by the origin divisor, and apply the result to adjust the expansion-contraction accumulation below that degree for the true correction; Where initial and final correction fractions span two days, apply multiplication and division according to each portion. All subsequent cases follow this rule.
107
求定氣日：冬、夏二至，盈縮之端，以常為定。 餘者以其氣所得盈縮差度及分盈減縮加常氣日及約分，即為其氣定日及分。
To find true qi dates: winter and summer solstices mark the endpoints of solar anomaly; the mean date stands as the true date. For the remaining qi, subtract the expansion-contraction difference in expansion and add it in contraction from the mean qi date and reduced fraction to obtain each qi's true date and parts.
108
赤道宿度
Equatorial Lodge Degrees
109
鬥：二十六
Dipper: 26°
110
牛：八
Ox: 8°
111
女：十二
Maid: 12°
112
虛：十及分
Void: 10° and fractional parts
113
危：十七
Rooftop: 17°
114
室：十六
Encampment: 16°
115
壁：九
Wall: 9°
116
北方七宿九十八度。 余一百六十萬四百四十七，約分二千五百六十四。
The seven northern lodges: 98°. Remainder 1,640,447, reduced fraction 2,564.
117
奎：十六
Strider: 16°
118
婁：十二
Bond: 12°
119
胃：十四
Stomach: 14°
120
昴：十一
Hairy Head: 11°
121
畢：十七
Net: 17°
122
觜：一
Turtle Beak: 1°
123
參：十
Three Stars: 10°
124
西方七宿八十一度。
The seven western lodges: 81°.
125
井：三十三鬼：三
Well: 33°; Ghost: 3°
126
柳：十五
Willow: 15°
127
星：七
Seven Stars: 7°.
128
張：十八
Extended Net: 18°.
129
翼：十八
Wings: 18°.
130
軫：十七
Chariot Shaft: 17°.
131
南方七宿一百一十一度。
The seven southern lodges total 111 degrees.
132
角：十二
Horn: 12°.
133
亢：九
Neck: 9°.
134
氐：十五
Root: 15°.
135
房：五
Room: 5°.
136
心：五
Heart: 5°.
137
尾：十八
Tail: 18°.
138
箕：十一
Winnowing Basket: 11°.
139
東方七宿七十五度。
The seven eastern lodges total 75 degrees.
140
前皆赤道度，自《大衍》以下，以儀測定，用為常數。 赤道者，常道也，紘於天半，以格黃道。
The figures above are all equatorial arc-degrees; from the Dayan calendar on, they were determined by instrument observation and taken as fixed constants. The equator is the unchanging track, running round the celestial middle to bracket the ecliptic.
141
求天正冬至赤道日度：以歲差乘所求積年，滿周天分去之，不盡，用減周天分，餘以度母除之，一度為度，不滿為餘。 余以一萬乘之，度母退除為約分。
To find the sun's equatorial longitude at winter solstice of civil new year: multiply the precession constant by the years sought, remove whole circuits of the universal divisor, subtract the remainder from the universal divisor, then divide by the degree mother — the quotient is degrees and the remainder is fractional parts. Multiply the fractional remainder by ten thousand and divide back through the degree mother to obtain reduced fractional parts.
142
命起赤道虛宿六度去之，至不滿宿，即所求年天正冬至加時赤道日躔所在宿度及分。
Count from Void lodge at 6° on the equator and subtract lodge widths until less than one lodge remains — the result is the equatorial lodge, degrees, and parts of the sun's position at hour-added winter solstice for the year sought.
143
求夏至赤道加時日度：置天正冬至加時赤道日度，以二至限度及分加之，滿赤道宿度去之，即得夏至加時赤道日度。 若求二至昏後夜半赤道日度者，各以二至之日約餘減一萬分，餘以加二至加時赤道日度，即為二至初日昏後夜半赤道日度，每日加一度，滿赤道宿度去之，即得每日昏後夜半赤道日度。
For the summer solstice equatorial longitude at hour-added time: take the winter solstice hour-added equatorial degree, add the solstice-interval limit in degrees and parts, and discard whole equatorial lodge circuits to obtain the summer solstice hour-added equatorial degree. To derive after-dusk and midnight equatorial solar longitudes at the solstices: subtract each solstice day's reduced remainder from ten thousand parts and add what remains to the solstice hour-added equatorial degree — this gives the first solstice day's after-dusk and midnight equatorial longitude; then add one degree per day, discarding full lodge circuits, for each subsequent day.
144
求赤道宿積度：置冬至加時赤道宿全度，以冬至赤道加時日度減之，餘為距後度及分； 以赤道宿度累加之，即各得赤道其宿積度及分。
To find accumulated equatorial longitude by lodge: set the full width of the winter solstice hour-added equatorial lodge, subtract the winter solstice hour-added equatorial solar degree — the remainder is the post-distance in degrees and parts; Add equatorial lodge widths cumulatively to obtain each lodge's accumulated equatorial degree and parts.
145
求赤道宿積度入初末限，各置赤道宿積度及分，滿九十一度三十一分去之，餘在四十五度六十五分半以下分以日為母。
To determine whether an accumulated equatorial longitude falls in the first or last limit band: set the accumulated degree and parts, remove multiples of 91° 31′; if the remainder is 45° 65½′ or less, use the day-count as denominator for the fractional parts.
146
為在初限； 以上者，用減九十一度三十一分，餘為入末限度及分。
It lies in the initial limit; if above that threshold, subtract from 91° 31′; the remainder is the entry into the terminal limit in degrees and parts.
147
求二十八宿黃道度：各置赤道宿入初、末限度及分，用減一百一十一度三十七分，余以乘初、末限度及分，進一位，以一萬約之，所得，命曰黃赤道差度及分； 在至後、分前減，在分後、至前加，皆加減赤道宿積度及分，為其宿黃道積度及分； 以前宿黃道積度減其宿黃道積度，為其宿黃道度及分。 其分就近為太、半、少。
To find ecliptic arc-degrees for the twenty-eight lodges: set each lodge's initial- or terminal-limit entry, subtract from 111° 37′, multiply the remainder by that limit entry, shift one decimal place, and reduce by ten thousand — the result is the ecliptic–equator difference in degrees and parts; after a solstice and before the equinox fraction, subtract; after the equinox fraction and before the solstice, add — apply this to the equatorial accumulated longitude to obtain each lodge's ecliptic accumulated degree and parts; Subtract the preceding lodge's ecliptic accumulated longitude from this lodge's to obtain that lodge's ecliptic width in degrees and parts. Fractional parts are rounded to the nearest greater, half, or lesser step.
148
黃道宿度
Ecliptic Lodge Degrees
149
鬥：二十三半
Southern Dipper: 23½°.
150
牛：七半
Ox: 7½°.
151
女：十一半
Maid: 11½°.
152
虛：十少、秒六十四
Emptiness: 10° lesser, 64 seconds.
153
危：十七太
Rooftop: 17° greater.
154
室：十七少
Encampment: 17° lesser.
155
壁：九太
Eastern Wall: 9° greater.
156
北方七宿九十七度半、秒六十四。
The seven northern lodges total 97½ degrees, 64 seconds.
157
奎：十七太
Stride: 17° greater.
158
婁：十二太
Bond: 12° greater.
159
胃：十四半
Stomach: 14½°.
160
昴：十太
Hairy Head: 10° greater.
161
畢：十六
Net: 16°.
162
觜：一
Turtle Beak: 1°.
163
參：九少
Three Stars: 9° lesser.
164
西方七宿八十二度。
The seven western lodges total 82 degrees.
165
井：三十
Eastern Well: 30°.
166
鬼：二太
Ghost: 2° greater.
167
柳：十四少
Willow: 14° lesser.
168
星：七
Seven Stars: 7°.
169
張：十八太
Extended Net: 18° greater.
170
翼：十九半
Wings: 19½°.
171
軫：十八太
Chariot Shaft: 18° greater.
172
南方七宿一百一十一度。
The seven southern lodges total 111 degrees.
173
角：十三
Horn: 13°.
174
亢：九半
Neck: 9½°.
175
氐：十五半
Root: 15½°.
176
房：五
Room: 5°.
177
心：四
Heart: 4°.
178
求朔弦望盈縮差及定差：各置朔、弦、望所入盈縮度及約分，如在象度分以下者為在初； 已上者，覆減二至限，餘為在末。 置初、末度分於上，列二至於下，以上減下，餘以下乘上，為積數； 滿四千一百三十五除之為度，不滿，退除為分，命曰盈縮差度及分。 若以四百乘積數，滿五百六十七除之，為盈縮定差。 若用立成者，以其度損益率乘度除，滿元法而一，所得，以損益其度下盈縮積，為定差度； 其損益初、末分為二日者，各隨其初、末以乘除。 其後皆如此例。
To find the surplus–deficit difference and fixed correction for syzygy and quarters: set the surplus–deficit degree and reduced parts entered for each event; if within the image-degree fraction, it falls in the initial band; If above that, subtract from the solstice-interval limit in reverse — the remainder lies in the terminal band. Place the initial or terminal degree and parts above and the two solstice values below; subtract the lower from the upper and multiply the remainder by the lower to obtain the accumulated product; Divide by 4135 to obtain degrees; treat the remainder as fractional parts — this is the surplus–deficit difference in degrees and parts. Alternatively multiply the accumulated product by 400 and divide by 567 to obtain the fixed surplus–deficit correction. If using the ready-made table: multiply by the increase–decrease rate for that degree, divide by the degree and by the origin divisor; apply the result to adjust the surplus–deficit accumulation under that degree to obtain the fixed correction in degrees; When the increase–decrease at an initial or terminal fraction spans two days, apply multiplication or division according to each day's initial or terminal share. Subsequent cases follow the same rule.
179
求定氣日：冬、夏二至，盈縮之端，以常為定。 餘者以其氣所得盈縮差度及分盈減縮加常氣日及約分，即為其氣定日及分。
To find the fixed qi day: winter and summer solstices are the endpoints of surplus–deficit variation — the mean day is taken as fixed. For the other qi, apply that term's surplus–deficit difference in degrees and parts — subtract in surplus, add in deficit — to the mean qi day and reduced parts to obtain the fixed day and parts.
180
赤道宿度
Equatorial Lodge Degrees
181
鬥：二十六
Southern Dipper: 26°.
182
牛：八
Ox: 8°.
183
女：十二
Maid: 12°.
184
虛：十及分
Emptiness: 10° and fractional parts.
185
危：十七
Rooftop: 17°.
186
室：十六壁：九
Encampment: 16°; Eastern Wall: 9°.
187
北方七宿九十八度。 余一百六十萬四百四十七，約分二千五百六十四。
The seven northern lodges total 98 degrees. Fractional remainder 1,600,447; reduced parts 2,564.
188
奎：十六
Stride: 16°.
189
婁：十二
Bond: 12°.
190
胃：十四
Stomach: 14°.
191
昴：十一
Hairy Head: 11°.
192
畢：十七
Net: 17°.
193
觜：一
Turtle Beak: 1°.
194
參：十
Three Stars: 10°.
195
西方七宿八十一度。
The seven western lodges total 81 degrees.
196
井：三十三
Eastern Well: 33°.
197
鬼：三
Ghost: 3°.
198
柳：十五
Willow: 15°.
199
星：七
Seven Stars: 7°.
200
張：十八
Extended Net: 18°.
201
翼：十八
Wings: 18°.
202
軫：十七
Chariot Shaft: 17°.
203
南方七宿一百一十一度。
The seven southern lodges total 111 degrees.
204
角：十二
Horn: 12°.
205
亢：九
Neck: 9°.
206
氐：十五
Root: 15°.
207
房：五
Room: 5°.
208
心：五
Heart: 5°.
209
尾：十八
Tail: 18°.
210
箕：十一
Winnowing Basket: 11°.
211
東方七宿七十五度。
The seven eastern lodges total 75 degrees.
212
前皆赤道度，自《大衍》以下，以儀測定，用為常數。 赤道者，常道也，紘於天半，以格黃道。
The figures above are all equatorial arc-degrees; from the Dayan calendar on, they were determined by instrument observation and taken as fixed constants. The equator is the unchanging track, running round the celestial middle to bracket the ecliptic.
213
求天正冬至赤道日度：以歲差乘所求積年，滿周天分去之，不盡，用減周天分，餘以度母除之，一度為度，不滿為餘。 余以一萬乘之，度母退除為約分。
To find the sun's equatorial longitude at winter solstice of civil new year: multiply the precession constant by the years sought, remove whole circuits of the universal divisor, subtract the remainder from the universal divisor, then divide by the degree mother — the quotient is degrees and the remainder is fractional parts. Multiply the fractional remainder by ten thousand and divide back through the degree mother to obtain reduced fractional parts.
214
命起赤道虛宿六度去之，至不滿宿，即所求年天正冬至加時赤道日躔所在宿度及分。
Count from Void lodge at 6° on the equator and subtract lodge widths until less than one lodge remains — the result is the equatorial lodge, degrees, and parts of the sun's position at hour-added winter solstice for the year sought.
215
求夏至赤道加時日度：置天正冬至加時赤道日度，以二至限度及分加之，滿赤道宿度去之，即得夏至加時赤道日度。 若求二至昏後夜半赤道日度者，各以二至之日約餘減一萬分，餘以加二至加時赤道日度，即為二至初日昏後夜半赤道日度，每日加一度，滿赤道宿度去之，即得每日昏後夜半赤道日度。
For the summer solstice equatorial longitude at hour-added time: take the winter solstice hour-added equatorial degree, add the solstice-interval limit in degrees and parts, and discard whole equatorial lodge circuits to obtain the summer solstice hour-added equatorial degree. To derive after-dusk and midnight equatorial solar longitudes at the solstices: subtract each solstice day's reduced remainder from ten thousand parts and add what remains to the solstice hour-added equatorial degree — this gives the first solstice day's after-dusk and midnight equatorial longitude; then add one degree per day, discarding full lodge circuits, for each subsequent day.
216
求赤道宿積度：置冬至加時赤道宿全度，以冬至赤道加時日度減之，餘為距後度及分； 以赤道宿度累加之，即各得赤道其宿積度及分。
To find accumulated equatorial longitude by lodge: set the full width of the winter solstice hour-added equatorial lodge, subtract the winter solstice hour-added equatorial solar degree — the remainder is the post-distance in degrees and parts; Add equatorial lodge widths cumulatively to obtain each lodge's accumulated equatorial degree and parts.
217
求赤道宿積度入初末限，各置赤道宿積度及分，滿九十一度三十一分去之，餘在四十五度六十五分半以下分以日為母。
To determine whether an accumulated equatorial longitude falls in the first or last limit band: set the accumulated degree and parts, remove multiples of 91° 31′; if the remainder is 45° 65½′ or less, use the day-count as denominator for the fractional parts.
218
為在初限； 以上者，用減九十一度三十一分，餘為入末限度及分。
It lies in the initial limit; if above that threshold, subtract from 91° 31′; the remainder is the entry into the terminal limit in degrees and parts.
219
求二十八宿黃道度：各置赤道宿入初、末限度及分，用減一百一十一度三十七分，余以乘初、末限度及分，進一位，以一萬約之，所得，命曰黃赤道差度及分； 在至後、分前減，在分後、至前加，皆加減赤道宿積度及分，為其宿黃道積度及分； 以前宿黃道積度減其宿黃道積度，為其宿黃道度及分。 其分就近為太、半、少。
To find ecliptic arc-degrees for the twenty-eight lodges: set each lodge's initial- or terminal-limit entry, subtract from 111° 37′, multiply the remainder by that limit entry, shift one decimal place, and reduce by ten thousand — the result is the ecliptic–equator difference in degrees and parts; after a solstice and before the equinox fraction, subtract; after the equinox fraction and before the solstice, add — apply this to the equatorial accumulated longitude to obtain each lodge's ecliptic accumulated degree and parts; Subtract the preceding lodge's ecliptic accumulated longitude from this lodge's to obtain that lodge's ecliptic width in degrees and parts. Fractional parts are rounded to the nearest greater, half, or lesser step.
220
黃道宿度
Ecliptic Lodge Degrees
221
鬥：二十三半
Southern Dipper: 23½°.
222
牛：七半
Ox: 7½°.
223
女：十一半
Maid: 11½°.
224
虛：十少、秒六十四
Emptiness: 10° lesser, 64 seconds.
225
危：十七太
Rooftop: 17° greater.
226
室：十七少
Encampment: 17° lesser.
227
壁：九太
Eastern Wall: 9° greater.
228
北方七宿九十七度半、秒六十四。
The seven northern lodges total 97½ degrees, 64 seconds.
229
奎：十七太
Stride: 17° greater.
230
婁：十二太
Bond: 12° greater.
231
胃：十四半
Stomach: 14½°.
232
昴：十太
Hairy Head: 10° greater.
233
畢：十六
Net: 16°.
234
觜：一
Turtle Beak: 1°.
235
參：九少
Three Stars: 9° lesser.
236
西方七宿八十二度。
The seven western lodges total 82 degrees.
237
井：三十
Eastern Well: 30°.
238
鬼：二太
Ghost: 2° greater.
239
柳：十四少
Willow: 14° lesser.
240
星：七
Seven Stars: 7°.
241
張：十八太
Extended Net: 18° greater.
242
翼：十九半
Wings: 19½°.
243
軫：十八太
Chariot Shaft: 18° greater.
244
南方七宿一百一十一度。
The seven southern lodges total 111 degrees.
245
角：十三
Horn: 13°.
246
亢：九半
Neck: 9½°.
247
氐：十五半
Root: 15½°.
248
房：五
Room: 5°.
249
心：四
Heart: 4°.
250
尾：十七
Tail: 17°.
251
箕：十
Winnowing Basket: 10°.
252
東方七宿七十四度太。
The seven eastern lodges total 74° greater.
253
七曜循此黃道宿度，准今曆變定。 若上考往古，下驗將來，當據歲差，每移一度，乃依法變從當時宿度，然後可步日、月、五星，知其守犯。
The seven luminaries follow these ecliptic lodge degrees, as fixed by the present calendar's adjustments. To test the past above and the future below, one must use precession: for each degree of shift, adjust by rule to the lodge degrees of that epoch — only then can one compute the sun, moon, and five planets and know their stations and violations.
254
求天正冬至加時黃道日度：以冬至加時赤道日度及分，減一百一十一度三十七分，餘以冬至加時赤道日度及分乘之，進一位，滿一萬約之為度； 不滿為分，命曰黃赤道差； 用減冬至赤道日度及分，即為所求年天正冬至加時黃道日度及分。
To find the ecliptic solar longitude at hour-added civil new-year winter solstice: subtract 111° 37′ from the winter solstice hour-added equatorial degree and parts, multiply the remainder by that equatorial degree and parts, shift one decimal place, and reduce by ten thousand to obtain degrees; the remainder is fractional parts — called the ecliptic–equator difference; subtract this from the winter solstice equatorial degree and parts to obtain the hour-added ecliptic solar degree and parts for the year sought.
255
求冬至之日晨前夜半日度：置一萬分，以其日升分加之，以乘冬至約餘，以一萬約之，所得，以減冬至加時黃道日度，即為冬至之日晨前夜半黃道日度及分。
For the solar longitude before dawn and at midnight on the winter solstice day: set 10,000 parts, add that day's ascending parts, multiply by the winter solstice reduced remainder, and reduce by 10,000; subtract the result from the winter solstice hour-added ecliptic degree to obtain the ecliptic longitude before dawn and at midnight on that day.
256
求逐月定朔之日晨前夜半黃道日度：置其朔距冬至日數，以其度下盈縮積度盈加縮減之，餘以加天正冬至晨前夜半日度，命之，即其月定朔之日晨前夜半日躔所在宿次。
For the ecliptic solar longitude before dawn and at midnight on each month's fixed new moon: set the days from new moon to winter solstice, apply the surplus–deficit accumulation under that degree (add in surplus, subtract in deficit), add the remainder to the civil new-year winter solstice before-dawn and midnight longitude, and assign lodges — this gives the lodge of the sun's station at that fixed new moon.
257
求每日晨前夜半黃道日度：各置其定朔之日晨前夜半黃道日度，每日加一度，以其日升降分升加降減之，滿黃道宿度去之，即各得每日晨前夜半黃道日躔所在宿度及分。 若次年冬至小余滿法者，以升分極數加之。
For daily ecliptic longitudes before dawn and at midnight: set each fixed new moon's before-dawn and midnight ecliptic degree, add one degree per day with that day's ascending or descending parts, discard whole ecliptic lodge circuits, and obtain each day's lodge, degrees, and parts for the sun's station before dawn and at midnight. If the following year's winter solstice fractional remainder fills the divisor, add the maximum ascending-parts correction.
258
尾：十七
Tail: 17°.
259
箕：十
Winnowing Basket: 10°.
260
東方七宿七十四度太。
The seven eastern lodges total 74° greater.
261
七曜循此黃道宿度，准今曆變定。 若上考往古，下驗將來，當據歲差，每移一度，乃依法變從當時宿度，然後可步日、月、五星，知其守犯。
The seven luminaries follow these ecliptic lodge degrees, as fixed by the present calendar's adjustments. To test the past above and the future below, one must use precession: for each degree of shift, adjust by rule to the lodge degrees of that epoch — only then can one compute the sun, moon, and five planets and know their stations and violations.
262
求天正冬至加時黃道日度：以冬至加時赤道日度及分，減一百一十一度三十七分，餘以冬至加時赤道日度及分乘之，進一位，滿一萬約之為度； 不滿為分，命曰黃赤道差； 用減冬至赤道日度及分，即為所求年天正冬至加時黃道日度及分。
To find the ecliptic solar longitude at hour-added civil new-year winter solstice: subtract 111° 37′ from the winter solstice hour-added equatorial degree and parts, multiply the remainder by that equatorial degree and parts, shift one decimal place, and reduce by ten thousand to obtain degrees; the remainder is fractional parts — called the ecliptic–equator difference; subtract this from the winter solstice equatorial degree and parts to obtain the hour-added ecliptic solar degree and parts for the year sought.
263
求冬至之日晨前夜半日度：置一萬分，以其日升分加之，以乘冬至約餘，以一萬約之，所得，以減冬至加時黃道日度，即為冬至之日晨前夜半黃道日度及分。
For the solar longitude before dawn and at midnight on the winter solstice day: set 10,000 parts, add that day's ascending parts, multiply by the winter solstice reduced remainder, and reduce by 10,000; subtract the result from the winter solstice hour-added ecliptic degree to obtain the ecliptic longitude before dawn and at midnight on that day.
264
求逐月定朔之日晨前夜半黃道日度：置其朔距冬至日數，以其度下盈縮積度盈加縮減之，餘以加天正冬至晨前夜半日度，命之，即其月定朔之日晨前夜半日躔所在宿次。
For the ecliptic solar longitude before dawn and at midnight on each month's fixed new moon: set the days from new moon to winter solstice, apply the surplus–deficit accumulation under that degree (add in surplus, subtract in deficit), add the remainder to the civil new-year winter solstice before-dawn and midnight longitude, and assign lodges — this gives the lodge of the sun's station at that fixed new moon.
265
求每日晨前夜半黃道日度：各置其定朔之日晨前夜半黃道日度，每日加一度，以其日升降分升加降減之，滿黃道宿度去之，即各得每日晨前夜半黃道日躔所在宿度及分。 若次年冬至小余滿法者，以升分極數加之。
For daily ecliptic longitudes before dawn and at midnight: set each fixed new moon's before-dawn and midnight ecliptic degree, add one degree per day with that day's ascending or descending parts, discard whole ecliptic lodge circuits, and obtain each day's lodge, degrees, and parts for the sun's station before dawn and at midnight. If the following year's winter solstice fractional remainder fills the divisor, add the maximum ascending-parts correction.