1
隋书卷一十八志第十三律历下
Book of Sui, Volume 18, Treatises 13: Measures and the Calendar, Part Three.
2
开皇二十年，袁充奏日长影短，高祖因以历事付皇太子，遣更研详著日长之候。 太子征天下历算之士，咸集于东宫。 刘焯以太子新立，复增修其书，名曰《皇极历》，驳正胄玄之短。 太子颇嘉之，未获考验。 焯为太学博士，负其精博，志解胄玄之印，官不满意，又称疾罢归。 至仁寿四年，焯言胄玄之误于皇太子：
In the twentieth year of Kaihuang, Yuan Chong reported that daylight was lengthening and shadows were shortening. Emperor Gaozu therefore entrusted calendrical affairs to the Crown Prince and ordered a further investigation into the signs of lengthening days. The Crown Prince summoned calendrical and computational experts from across the empire, and all assembled at the Eastern Palace. Because the Crown Prince had just been installed, Liu Chuo revised and expanded his work once more, naming it the Imperial Ultimate Calendar and refuting the flaws in Zhang Zhouxuan's system. The Crown Prince was greatly impressed, but the calendar had not yet undergone empirical verification. Chuo held the post of Doctor of the Imperial Academy. Confident in his mastery and erudition, he sought to win Zhouxuan's official endorsement. Unhappy with his rank, he also pleaded illness and resigned to return home. By the fourth year of Renshou, Chuo presented the Crown Prince with an account of Zhouxuan's errors:
3
其一曰，张胄玄所上见行历，日月交食，星度见留，虽未尽善，得其大较，官至五品，诚无所愧。 但因人成事，非其实录，就而讨论，违舛甚众。
First: Zhang Zhouxuan's calendar now in use, though not flawless in predicting solar and lunar eclipses or the stations, appearances, and retrogradations of the planets, does capture the broad outline. That he attained fifth-rank office is nothing to be ashamed of. Yet he achieved this by building on others' work rather than through his own authentic scholarship. On closer examination, the errors are extremely numerous.
4
其二曰，胄玄弦望晦朔，违古且疏，气节闰候，乖天爽命。 时不从子半，晨前别为后日。 日躔莫悟缓急，月逡妄为两种，月度之转，辄遗盈缩，交会之际，意造气差。 七曜之行，不循其道，月星之度，行无出入，应黄反赤，当近更远，亏食乖准，阴阳无法。 星端不协，珠璧不同，盈缩失伦，行度愆序。 去极晷漏，应有而无，食分先后，弥为烦碎。 测今不审，考古莫通，立术之疏，不可纪极。 今随事纠驳，凡五百三十六条。
Second: Zhouxuan's reckoning of new moons, full moons, and the first and last days of the month departs from ancient practice and is imprecise; his treatment of qi nodes, intercalary periods, and seasonal terms violates celestial order and corrupts the calendar. He does not reckon the day from midnight at the zi hour; time before dawn is arbitrarily assigned to the following day. He fails to account for the sun's varying speed through the lunar lodges. For the moon he recklessly devises two separate methods. In computing the moon's monthly motion he routinely omits apogee and perigee corrections. At conjunction he arbitrarily fabricates qi discrepancies. The seven luminaries do not follow their proper courses. The degrees of the moon and stars lack proper ingress and egress. Positions that should be yellow appear red; bodies that should be nearer appear farther away. Eclipses miss their predicted times. There is no coherent method for yin and yang. Stellar positions fail to align. Celestial markers do not correspond. Apogee and perigee lose their proper ordering. Degrees of motion fall out of sequence. Calculations for distance from the celestial pole and gnomon shadow lengths are missing where they should appear. The ordering of eclipse phases is needlessly convoluted. Present observations are not scrutinized carefully; comparisons with ancient records cannot be reconciled. The defects in the established methods are beyond numbering. I now submit corrections and refutations totaling five hundred thirty-six items.
5
其三曰，胄玄以开皇五年，与李文琮于张宾历行之后，本州贡举，即赍所造历拟以上应。 其历在乡阳流布，散写甚多，今所见行，与焯前历不异。 玄前拟献，年将六十，非是忽迫仓卒始为，何故至京未几，即变同焯历，与旧悬殊？ 焯作于前，玄献于后，舍己从人，异同暗会。 且孝孙因焯，胄玄后附孝孙，历术之文，又皆是孝孙所作，则元本偷窃，事甚分明。 恐胄玄推讳，故依前历为驳，凡七十五条，并前历本俱上。
Third: In the fifth year of Kaihuang, after Zhang Bin's calendar had been adopted through Li Wencong, Zhouxuan was nominated by his prefecture for the civil examination and immediately brought the calendar he had devised, intending to submit it to the throne. His calendar had circulated widely in his home prefecture, copied and distributed in great numbers. The version now in use is identical to Chuo's earlier calendar. Zhouxuan had been preparing to submit his calendar for nearly sixty years—it was not composed in sudden haste. Why then, shortly after arriving at the capital, did it suddenly change to match Chuo's calendar, differing vastly from the earlier version? Chuo composed his calendar first; Zhouxuan submitted later. Having abandoned his own work to follow another's, their differences and similarities align with suspicious precision. Moreover, Xiaosun drew upon Chuo's work, and Zhouxuan later attached himself to Xiaosun. The calendrical texts were all written by Xiaosun. The original plagiarism is therefore quite evident. Fearing Zhouxuan would evade the issue, he refuted the calendar according to the earlier version—seventy-five items in all—and submitted both together with the original text.
6
其四曰，玄为史官，自奏亏食，前后所上，多与历违，今算其乖舛有一十三事。 又前与太史令刘晖等校其疏密五十四事，云五十三条新。 计后为历应密于旧，见用算推，更疏于本。 今纠发并前，凡四十四条。
Fourth: As a historiographical official, Zhouxuan himself reported eclipse discrepancies. His submissions before and after largely contradicted the calendar. I now calculate thirteen instances of contradiction. Moreover, he had previously joined Director of Astronomy Liu Hui and others in comparing points of laxity and precision: fifty-four matters, of which he claimed fifty-three were improvements. By calculation, his revised calendar should have been more precise than the old one, but when actually computed, it proves less accurate than the original. Together with earlier corrections, there are forty-four items in all.
7
其五曰，胄玄于历，未为精通。 然孝孙初造，皆有意，徵天推步，事必出生，不是空文，徒为臆断。
Fifth: Zhouxuan's mastery of calendrical science is not yet complete. Yet Xiaosun's original work was deliberate throughout—observing Heaven and calculating celestial motion, each element grounded in reality rather than empty formulas or idle speculation.
8
其六曰，焯以开皇三年，奉敕修造，顾循记注，自许精微，秦汉以来，无所与让。 寻圣人之迹，悟曩哲之心，测七曜之行，得三光之度，正诸气朔，成一历象，会通今古，符允经传，稽于庶类，信而有徵。 胄玄所违，焯法皆合，胄玄所阙，今则尽有，隐括始终，谓为总备。
Sixth: In the third year of Kaihuang, Chuo received an imperial order to compile a calendar. Drawing on historical records and annotations, he claimed unmatched precision—since the Qin and Han dynasties, none could rival him. Tracing the footsteps of the sages, he grasped the minds of past masters; measured the motion of the seven luminaries and determined the degrees of the sun, moon, and stars; corrected all qi nodes and new moons into a unified calendrical system; harmonized past and present; accorded with canonical texts; and verified his results against the myriad phenomena—trustworthy and well substantiated. Where Zhouxuan's methods diverged, Chuo's all accord with truth. Where Zhouxuan's system was deficient, Chuo's now supplies everything. Encompassing beginning and end, he declared it fully complete.
9
仍上启曰：“自木铎寝声，绪言成烬，群生荡析，诸夏沸腾，曲技云浮，畴官雨绝，历纪废坏，千百年矣。 焯以庸鄙，谬荷甄擢，专精艺业，耽玩数象，自力群儒之下，冀睹圣人之意。 开皇之初，奉敕修撰，性不谐物，功不克终，犹被胄玄窃为己法，未能尽妙，协时多爽，尸官乱日，实玷皇猷。 请征胄玄答，验其长短。”
He further submitted a memorial stating: "Since the wooden clapper fell silent and transmitted learning turned to ashes, the people were scattered and the realm seethed with turmoil. Minor arts floated like clouds; calendrical offices were extinguished like rain; calendrical records lay in ruins for more than a thousand years. Chuo, mediocre and humble, was undeservedly favored with selection and promotion. He devoted himself to his craft and immersed himself in numerical and celestial signs. Working from below the ranks of scholars, he hoped to glimpse the intent of the sages. At the beginning of Kaihuang, he received an order to compile a calendar. His temperament did not accord with others, and the work was never completed. Yet Zhouxuan stole it as his own method. Unable to master its subtleties, it frequently failed to harmonize with the seasons. Holding office in name only while corrupting the calendar truly disgraced the imperial design. I request that Zhouxuan be summoned to respond and that the strengths and weaknesses of both calendars be verified."
10
焯又造历家同异，名曰《稽极》。 大业元年，著作郎王邵、诸葛颍二人，因入侍宴，言刘焯善历，推步精审，证引阳明。 帝曰：“知之久矣。 ”仍下其书与胄玄参校。 胄玄驳难云：“焯历有岁率、月率，而立定朔，月有三大、三小。 案岁率、月率者，平朔之章岁、章月也。 以平朔之率而求定朔，值三小者，犹以减三五为十四； 值三大者，增三五为十六也。 校其理实，并非十五之正。 故张衡及何承天创有此意，为难者执数以校其率，率皆自败，故不克成。 今焯为定朔，则须除其平率，然后为可。 ”互相驳难，是非不决，焯又罢归。
Chuo also compiled a comparison of agreements and differences among calendar masters, titling it Verification of the Ultimate. In the first year of Daye, Supervising Secretaries Wang Shao and Zhuge Ying, while attending an imperial banquet, spoke of Liu Chuo's mastery of calendrical science—his astronomical calculations being precise and rigorously verified, supported by clear demonstrations. The Emperor said, "I have known this for a long time. " He then ordered Chuo's work sent to Zhouxuan for joint comparison and verification. Zhouxuan raised objections, saying: "Chuo's calendar employs year rate and month rate, yet establishes fixed new moons, producing months with three long and three short intervals. Examining the matter: year rate and month rate are the cyclic year and cyclic month used for mean new moons. Using mean-new-moon rates to calculate fixed new moons—when three short months occur, one still subtracts three-fifths to obtain fourteen; when three long months occur, one adds three-fifths to obtain sixteen. Checking against principle and fact, these do not yield the correct fifteen. Zhang Heng and He Tiankong first proposed this approach; critics who tested the rates by computation found the rates self-contradictory—thus the method could not succeed. Now that Chuo employs fixed new moons, the mean rates must be removed—only then is the method valid." They exchanged refutations back and forth, and no resolution could be reached. Chuo again resigned and returned home.
11
四年，驾幸汾阳宫，太史奏曰：“日食无效。 ”帝召焯，欲行其历。 袁允方幸于帝，左右胄玄，共排焯历，又会焯死，历竟不行。 术士咸称其妙，故录其术云。 甲子元，距大隋仁寿四年甲子积一百万八千八百四十算。
In the fourth year, when the Emperor visited the Fenyang Palace, the Director of Astronomy reported: "The predicted solar eclipse did not occur. " The Emperor summoned Chuo, intending to adopt his calendar. Yuan Yun was then in the Emperor's favor and, together with Zhouxuan at court, jointly blocked Chuo's calendar. Chuo then died, and the calendar was never adopted. Calendrical experts all praised its excellence; therefore its methods are recorded here. Jiazi Origin: from the Great Sui, Renshou year 4 (jiazi), accumulated count 1,008,840.
12
岁率，六百七十六。
Year rate: 676.
13
月率，八千三百六十一。
Month rate: 8,361.
14
朔日法，千二百四十二。
New moon day divisor: 1,242.
15
朔实，三万六千六百七十七。
New moon constant: 36,677.
16
旬周，六十。
Ten-day cycle: 60.
17
朔辰，百三半。
New moon chronology: 103 and a half.
18
日干元，五十二。
Day stem origin: 52.
19
日限，十一。
Day limit: 11.
20
盈泛，十六。
Surplus general: 16.
21
亏总，十七。
Deficit total: 17.
22
推经朔术：
Method for calculating canonical new moons:
23
置入元距所求年，月率乘之，如岁率而一，为积月，不满为闰衰。 朔实乘积月，满朔日法得一，为积日，不满为朔余。 旬周去积日，不尽为日，即所求年天正经朔日及余。
Enter the accumulated count from the origin to the year sought. Multiply by the month rate and divide by the year rate to obtain accumulated months; the remainder is the intercalary deficit. Multiply accumulated months by the new moon constant; when the product fills the new moon day divisor, obtain one accumulated day; the remainder is the new moon remainder. Remove the ten-day cycle from accumulated days; the remainder is the day—this gives the canonical new moon day and remainder for the year sought.
24
求上下弦、望：加经朔日七、余四百七十五小，即上弦经日及余。 又加得望、下弦及后月朔。 就径求望者，加日十四、余九百五十半； 下弦加日二十二、余百八十三大； 后月朔加日二十九，余六百五十九。 每月加闰衰二十大，即各其月闰衰也。
To find the first quarter and full moon: add 7 days and 475 small to the canonical new moon day and remainder—this yields the first quarter canonical day and remainder. Adding again yields the full moon, last quarter, and the following month's new moon. To find the full moon directly, add 14 days and 950 and a half to the remainder; for the last quarter add 22 days and 183 large; for the following month's new moon add 29 days and 659. Each month add 20 large to the intercalary deficit—this gives each month's intercalary deficit.
25
凡月建子为天正，建丑为地正，建寅为人正。 即以人正为正月，统求所起，本于天正。 若建岁历从正月始，气、候、月、星，所值节度，虽有前却，并亦随之。 其前地正为十二月，天正为十一月，并诸气度皆属往年。 其日之初，亦从星起，晨前多少，俱归昨日。 若气在夜半之后，量影以后日为正。 诸因加者，各以其余减法，残者为全余。 若所因之余满全余以上，皆增全一而加之，减其全余； 即因余少于全余者，不增全加，皆得所求。 分度亦尔。 凡日不全为余，积以成余者曰秒； 度不全为分，积以成分者曰篾； 其有不成秒曰麽，不成篾曰幺。 其分、余、秒、篾，皆一为小，二为半，三为大，四为全，加满全者从一。 其三分者，一为少，二为太。 若加者，秒篾成法，从分余。 分余满法从日度一，日度有所满，则从去之。 而日命以日辰者，满旬周则亦除； 命有连分、余、秒、篾者，亦随全而从去。 其日度虽满，而分秒不满者，未可从去，仍依本数。 若减者，秒篾不足，减分余一，加法而减之； 分余不足减者，加所从去或前日度乃减之。 即其名有总，而日度全及分余共者，须相加除，当皆连全及分余共加除之。 若须相乘，有分余者，母必通全内子，乘讫报除。 或分余相并，母不同者，子乘而并之。 母相乘为法，其并，满法从一为全，此即齐同之也。 既除为分余而有不成，若例有秒篾，法乘而又法除，得秒篾数。 已为秒篾及正有分余，而所不成不复须者，须过半从一，无半弃之。 若分余其母不等，须变相通，以彼所法之母乘此分余，而此母除之，得彼所须之子。 所有秒篾者，亦法乘，不满此母，又除而得其数。 麽幺亦然。 其所除去而有不尽全，则谓之不尽，亦曰不如。 其不成全，全乃为不满分、余、秒、篾，更曰不成。 凡以数相减，而有小及半、太须相加减，同于分余法者，皆以其母三四除其气度日法，以半及太、大本率二三乘之，少、小即须因所除之数随其分余而加减焉。 秋分后春分前为盈泛，春分后秋分前为亏总，须取其数。 泛总为名，指用其时，春分为主，亏日分后，盈日分前。 凡所不见，皆放于此。
In general, a month established at zi is Heaven's first month; at chou, Earth's first month; at yin, Man's first month. Taking Man's first month as the first month of the year, all calculations trace back to Heaven's first month as the foundation. If the year calendar begins from the first month, the qi nodes, seasonal terms, months, and stars—the degrees and positions they occupy—may shift forward or backward, but all follow accordingly. The preceding Earth's first month becomes the twelfth month; Heaven's first month becomes the eleventh month—all stellar degrees belong to the previous year. At the day's beginning, reckoning starts from the stars; however much time remains before dawn—all is assigned to the previous day. If a qi node falls after midnight, the gnomon shadow measurement assigns it to the following day. For all additive operations, subtract each remainder from the divisor; what remains is the full remainder. If the inherited remainder equals or exceeds the full remainder, increase the whole number by one and subtract from the full remainder; when the inherited remainder is less than the full remainder, do not advance the whole number—all yield the desired result. Degree divisions follow the same rule. In general, a fractional part of a day is called a remainder; fractional parts that accumulate into a remainder are called seconds; a fractional part of a degree is called a fraction; fractional parts that accumulate into a fraction are called hairs; fractions too small to form a second are called mo; fractions too small to form a hair are called yao. For fractions, remainders, seconds, and hairs—one is small, two is half, three is large, four is full; when a full unit is reached, advance by one. For thirds—one is lesser, two is greater. When adding, if seconds and hairs reach the divisor, carry over to fractions and remainders. When fractions and remainders reach the divisor, advance one day or degree; when the day or degree is full, remove the excess. When designating days by chronology, when the ten-day cycle is full, also remove the excess; when consecutive fractions, remainders, seconds, and hairs are present, they also carry over and are removed accordingly. Even if the day or degree count is full, if fractions and seconds are not complete, do not yet remove—retain the original value. When subtracting, if seconds and hairs are insufficient, borrow one from fractions and remainders, add the divisor, and subtract; when fractions and remainders are insufficient, borrow from the next higher unit—the day or degree—and then subtract. When a value includes a total, with whole days or degrees plus fractions and remainders combined, add and divide them together—all whole units and fractional parts must be combined in the operation. When multiplication is required and fractions or remainders are present, convert whole units into the numerator, multiply, and then divide back. When combining fractions or remainders with different denominators, multiply the numerators and combine them. Multiply the denominators to obtain the divisor; when the combined value fills the divisor, advance one to a whole unit—this is the method of equalization. When division yields fractions and remainders with incomplete parts, and seconds or hairs are required by rule, multiply by the divisor and divide again to obtain the seconds and hairs. When the value is already expressed in seconds and hairs with proper fractions and remainders, and further incomplete parts are not needed—round up if more than half, discard if less. When denominators differ, convert by cross-multiplication: multiply each fraction by the other denominator and divide by its own denominator to obtain the required numerators. For seconds and hairs, multiply by the divisor; when the product does not fill the denominator, divide again to obtain the value. The same applies to mo and yao. The remainder after division that does not form a whole unit is called incomplete, or insufficient. When a whole unit is not yet complete in fractions, remainders, seconds, and hairs, it is likewise called incomplete. When subtracting numbers involving small, half, and greater fractional parts, apply the fraction-remainder method: divide the qi-day degree divisor by three or four, multiply half and greater parts by the basic rates of two or three, and adjust lesser and small parts according to their fractional remainders. From after the autumn equinox to before the spring equinox use surplus general; from after the spring equinox to before the autumn equinox use deficit total—take the appropriate value for each period. General and total are designations referring to the season of application. The spring equinox is the reference point: use deficit total after the equinox day-fraction, surplus general before it. All invisible quantities are accounted for here.
26
气日法，四万六千六百四十四。
Qi day divisor: 46,644.
27
岁数，千七百三万六千四百六十六半。
Year constant: 17,036,466 and a half.
28
度准，三百三十八。
Degree standard: 338.
29
约率，九。
Reduction rate: 9.
30
气辰，三千八百八十七。
Qi chronology: 3,887.
31
余通，八百九十七。
Remainder common: 897.
32
秒法，四十八。
Second divisor: 48.
33
麽法，五。
Mo divisor: 5.
34
推气术：
Method for calculating qi nodes:
35
半闰衰乘朔实，又度准乘朔余，加之，如约率而一，所得满气日法为去经朔日，不满为气余。 以去经朔日，即天正月冬至恆日定余，乃加夜数之半者，减日一，满者因前，皆为定日。 命日甲子算外，即定冬至日。 其余如半气辰千九百四十三半以下者，为气加子半后也； 过以上，先加此数，乃气辰而一，命以辰算外，即气所在辰。 十二辰外，为子初以后余也。 又十二乘辰余：
Multiply half the intercalary deficit by the new moon constant; multiply the degree standard by the new moon remainder and add. Divide by the reduction rate. When the result fills the qi day divisor, this gives days from canonical new moon; the remainder is the qi remainder. The days from canonical new moon give the mean day and fixed remainder of winter solstice in Heaven's first month. Add half the night count and subtract one day—carrying forward when full—to obtain the fixed day. Count beyond the jiazi cycle to name the day—this is the fixed winter solstice date. If the remainder is at or below 1,943 and a half (half the qi chronology), the qi node falls after midnight at zi; if above, first add this number, then divide by the qi chronology and count beyond the chronology marker—this gives the hour in which the qi node falls. Beyond the twelve chronologies lies the remainder after the start of zi. Also multiply the chronology remainder by twelve:
36
四为小太，亦曰少； 五为半步； 六为半；
Four is small-greater, also called lesser; Five is half-step; Six is half;
37
七为半太； 八为大少，亦曰太； 九为太；
Seven is half-greater; Eight is large-lesser, also called greater; Nine is greater;
38
十为大太； 十一为穷辰少。
Ten is large-greater; Eleven is exhausted-chronology lesser.
39
其又不成法者，半以上为进，以下为退。 退以配前为强，进以配后为弱。 即初不成一而有退者，谓之沾辰； 初成十一而有进者，谓之穷辰。 未旦其名有重者，则于间可以加之，命辰通用其余，辨日分辰而判诸日。 因别亦皆准此。 因冬至有减日者，还加之。 每加日十五、余万一百九十二、秒三十七，即各次气恆日及余。 诸月齐其闰衰，如求冬至法，亦即其月中气恆日去经朔数。 其求后月节气恆日，如次之求前节者减之。
When the value does not reach the divisor, round up if above half, down if below half. Rounding down paired with the preceding hour is called strong; rounding up paired with the following hour is called weak. When initially not reaching one but rounding down, it is called touching chronology; when initially reaching eleven but rounding up, it is called exhausted chronology. Before dawn, when duplicate names occur, they may be combined in the interval; chronology naming uses the common remainder to distinguish day-fraction hours and assign each day. Separate cases all follow this same standard. When a day is subtracted at winter solstice, restore it by addition. Each time add 15 days, remainder 192, seconds 37—this yields each subsequent qi mean day and remainder. For each month, align the intercalary deficit; using the winter solstice method, this also gives the mean day of the month's central qi measured from canonical new moon. To find the next month's solar term mean day, subtract as when seeking the preceding term.
40
推每日迟速数术：
Method for calculating daily acceleration and retardation numbers:
41
见求所在气陟降率，并后气率半之，以日限乘而泛总除，得气末率。 又日限乘二率相减之残，泛总除，为总差。 其总差亦日限乘而泛总除，为别差。 率前少者，以总差减末率，为初率，乃别差加之； 前多者，即以总差加末率，皆为气初日陟降数。 以别差前多者日减，前少者日加初数，得每日数。 所历推定气日随算其数，陟加、降减其迟速，为各迟速数。 其后气无同率及有数同者，皆因前末，以末数为初率，加总差为末率，及差渐加初率，为每日数，通计其秒，调而御之。
For the qi in question, take the ascent-descent rate at its position and half the following qi's rate, multiply by the day limit and divide by the general/total factor to obtain the terminal qi rate. Multiply the day limit by the difference between the two rates and divide by the general/total factor—this is the total difference. For the total difference: multiply by the day limit and divide by the general/total factor to obtain the separate difference. When the prior rate is smaller, subtract the total difference from the terminal rate to obtain the initial rate, then add the separate difference; when the prior rate is greater, add the total difference to the terminal rate—both yielding the qi-initial-day ascent-descent values. By separate difference: subtract daily when the prior rate is greater, add to the initial value when smaller—yielding each day's value. Traversing the fixed qi days, compute accordingly; add for ascending motion, subtract for descending motion to the slow-fast values—obtaining each acceleration-retardation number. When the following qi lacks an identical rate or shares the same value, follow the prior terminal: use the terminal value as the initial rate, add the total difference for the terminal rate, gradually add the separate difference to the initial rate for each day's value, tally the seconds, and adjust.
42
求月朔弦望应平会日所入迟速：各置其经余为辰，以入气辰减之，乃日限乘日，日内辰为入限，以乘其气前多之末率，前少之初率，日限而一，为总率。 其前多者，入限减泛总之残，乘总差，泛总而一，为入差，并于总差，入限乘，倍日限除，加以总率； 前少者，入限自乘再乘别差，日限自乘，倍而除，亦加总率，皆为总数。 乃以陟加、降减其气迟速数为定，即速加、迟减其经余，各其月平会日所入迟速定日及余。
To find the slow-fast correction at mean conjunction days for new moon, first quarter, full moon, and last quarter: set each canonical remainder as chronology and subtract the entering-qi chronology; multiply the day limit by the day count, with the chronology within the day as the entering limit; multiply by the terminal rate if prior is greater or the initial rate if prior is less; divide by the day limit to obtain the total rate. When prior is greater: subtract the entering limit from the general/total remainder, multiply by the total difference and divide by the general/total factor for the entering difference; combine with the total difference, multiply by the entering limit, divide by double the day limit, and add to the total rate; when prior is less: square the entering limit and multiply by the separate difference; square the day limit, double and divide; add to the total rate—all yielding total values. Then apply ascent to add and descent to subtract, fixing the qi's slow-fast number; add for fast motion, subtract for slow motion from the canonical remainder—each yielding the slow-fast fixed day and remainder at the month's mean conjunction.
43
求每日所入先后：各置其气躔衰与衰总，皆以余通乘之，所乃躔衰如陟降率； 衰总如迟速数，亦如求迟速法，即得每所入先后及定数。
To find daily advance and retardation: set each qi's lodge deficit and total deficit, multiply both by the remainder common; the lodge deficit corresponds to the ascent-descent rate; the total deficit corresponds to the slow-fast number; apply the slow-fast method likewise—obtaining each entering advance-retardation and fixed value.
44
求定气：其每日所入先后数即为气余，其所历日皆以先加之，以后减之，随算其日，通准其余，满一恆气，即为二至后一气之数。 以加二气，如法用别其日而命之。 又算其次，每相加命，各得其定气日及余也。 亦以其先后已通者，先减后加其恆气，即次气定日及余。 亦因别其日，命以甲子，各得所求。
To find fixed qi: the daily entering advance-retardation number is the qi remainder; for each day traversed, add the prior and subtract the following; compute accordingly and convert the remainder—when one mean qi is full, this gives the count of one qi after a solstice. Add to the two qi nodes and apply the rule separately to name the days. Calculate the next in sequence; each time add and name the day—obtaining each fixed qi date and remainder. Using the converted advance-retardation: first subtract then add to the mean qi—yielding the next qi's fixed day and remainder. Name each day separately by the jiazi cycle to obtain the desired results.
45
求土王：距四立各四气外所入先后加减，满二十二日、余八千一百五十四、秒十、麽二。 除所满日外，即土始王日。
To find Earth dominance: from each of the four establishment nodes, four qi outward, apply the entering advance-retardation corrections—when full of 22 days, remainder 8,154, seconds 10, mo 2. Remove the full days—the remainder is the day Earth begins its dominance.
46
求侯日：定气即初候日也。 三除恆气，各为平候日。 余亦以所入先后数为气余，所历之日皆以先加、后减，随计其日，通准其余，每满其平，以加气日而命之，即得次候日。 亦算其次，每相加命，又得末候及次气日。
To find pentad days: the fixed qi date is the first pentad day. Divide the mean qi by three—each portion is a mean pentad day. The remainder also uses the entering advance-retardation as qi remainder; for each day traversed, add the prior and subtract the following; compute accordingly and convert the remainder—each time the mean is full, add to the qi day and name it—obtaining the next pentad day. Calculate the next in sequence; each time add and name—also obtaining the last pentad and the next qi day.
47
倍夜半之漏，得夜刻也。 以减百刻，不尽为昼刻。 每减昼刻五，以加夜刻，即其昼为日见、夜为不见刻数。 刻分以百为母。
Double the midnight clepsydra measurement—this gives the night clepsydra count. Subtract from 100 clepsydra counts; the remainder is the day clepsydra count. Subtract five from the day clepsydra and add to the night clepsydra—this gives the visible-sun day clepsydra and the invisible-sun night clepsydra. Clepsydra fractions use 100 as the denominator.
48
求日出入辰刻：十二除百刻，得辰刻数，为法。 半不见刻以半辰加之，为日出实，又加日出见刻，为日入实。 如法而一，命子算外，即所在辰，不满法，为刻及分。
To find sunrise and sunset chronology clepsydra: divide 100 clepsydra counts by 12 to obtain the chronology clepsydra count—this is the divisor. Half the invisible clepsydra plus half a chronology gives the sunrise constant; adding the visible-sun clepsydra gives the sunset constant. Divide by the divisor; count beyond zi—this gives the chronology; the remainder is clepsydra count and fraction.
49
求辰前余数：气、朔日法乘夜半刻，百而一，即其余也。
To find the remainder before chronology: multiply the qi/new moon day divisor by the midnight clepsydra and divide by 100—this is the remainder.
50
求每日刻差：每气准为十五日，全刻二百二十五为法。 其二至各前后于二分，而数因相加减，间皆六气； 各尽于四立，为三气。 至与前日为一，乃每日增太； 又各二气，每日增少； 其末之气，每日增少之小，而末六日，不加而裁焉。 二望至前后一气之末日，终于十少； 二气初日，稍增为十二半，终于二十太，三气初日，二十一，终于三十少； 四立初日，三十一，终于三十五太； 五气亦少增，初日三十六太，终四十一少； 末气初日，四十一少，终于四十二。 每气前后累算其数，又百八十乘为实，各泛总乘法而除，得其刻差。 随而加减夜刻而半之，各得入气夜定刻。 其分后十五日外，累算尽日，乃副置之，百八十乘，亏总除，为其所因数。 以减上位，不尽为所加也。 不全日者，随辰率之。
To find daily clepsydra difference: each qi standard is 15 days; 225 full clepsydra counts serve as the divisor. The two solstices each fall before and after the two equinoxes, with numbers adding and subtracting accordingly; six qi lie between each pair; each group ending at the four establishment nodes—three qi per group. The solstice and the preceding day count as one; then increase daily by greater increments; for each pair of qi, increase daily by lesser increments; for the last qi, increase daily by small-lesser amounts; for the last six days, do not add but trim. The last day of one qi before and after the two solstices ends at ten-lesser; the first day of the second qi increases slightly to twelve and a half, ending at twenty-greater; the third qi starts at twenty-one, ending at thirty-lesser; the four establishment nodes start at thirty-one, ending at thirty-five-greater; the fifth qi also increases slightly; first day thirty-six-greater, ending at forty-one-lesser; the last qi starts at forty-one-lesser, ending at forty-two. For each qi, accumulate the numbers before and after; multiply by 180 for the dividend; divide by the general/total multiplier to obtain the clepsydra difference. Add and subtract to the night clepsydra and halve—each yielding the entering-qi fixed night clepsydra. Beyond fifteen days after the equinox, accumulate to exhaust the days; set aside, multiply by 180, divide by deficit total—this is the inherited number. Subtract from the upper position; the remainder is the amount added. When less than a full day, follow the chronology rate.
51
求晨去中星：加周度一，各昏去中星减之，不尽为晨去度。
To find dawn distance from culmination: add one to the circuit degree; subtract each dusk distance from culmination—the remainder is the dawn distance in degrees.
52
求每日度差：准日因增加裁，累算所得，百四十三之，四百而一，亦百八十乘，泛总除，为度差数。 满转法为度，随日加减，各得所求。 分后气间，亦求准外与前求刻，至前加减，皆因日数逆算求之。 亦可因至向背其刻，冬减夏加，而度冬加夏减。 若至前，以入气减气间，不尽者，因后气而反之，以不尽日累算乘除所定，从后气而逆以加减，皆得其数。 此但略校其总，若精存于《稽极》云。
To find daily degree difference: according to the daily inherited increase and trim, accumulate the result; divide by 143, take one of 400; multiply by 180 and divide by general/total—obtaining the degree difference. When full of the rotation divisor, this gives degrees; add and subtract daily to obtain the desired values. Between qi nodes after the equinox, also seek beyond the standard as with the prior clepsydra method; before the solstice add and subtract—all computed in reverse by day count. Alternatively, follow the solstice direction for clepsydra: subtract in winter, add in summer; for degrees, add in winter, subtract in summer. Before the solstice, subtract entering qi from the qi interval; for the remainder, reverse from the following qi, multiply and divide by the accumulated incomplete days, and add or subtract in reverse from the following qi—all yielding the correct values. This only provides a rough verification of the totals; finer details are preserved in the Verification of the Ultimate.
53
转终日，二十七； 余，千二百五十五。
Rotation full days: 27; Remainder: 1,255.
54
终法，二千二百六十三。
Terminal divisor: 2,263.
55
终实，六万二千三百五十六。
Terminal constant: 62,356.
56
终全余，千八。
Terminal full remainder: 1,008.
57
转法，五十二。
Rotation divisor: 52.
58
篾法，八百九十七。
Hair divisor: 897.
59
闰限，六百七十六。
Intercalary limit: 676.
60
推入转术：终实去积日，不尽，以终法乘而又去，不如终实者，满终法得一日，不满为余，即其年天正经朔夜半入转日及余。
Method for calculating entering rotation: Remove accumulated days from the terminal constant; multiply the remainder by the terminal divisor and remove again; when less than the terminal constant, divide by the terminal divisor for whole days—the remainder gives the entering rotation day and remainder at midnight of the year's canonical new moon.
61
求次日：加一日，每日满转终则去之，其二十八日者加全余为夜半入初日余。
To find the next day: add one day; remove when the rotation terminal is full; on the twenty-eighth day, add the full remainder for the midnight entering-initial-day remainder.
62
求弦望：皆因朔加其经日，各得夜半所入日余。
To find first quarter and full moon: add the canonical days from new moon—each yielding the midnight entering day and remainder.
63
求次月：加大月二日，小月一日，皆及全余，亦其夜半所入。
To find the next month: add two days for long months, one for short months—all including the full remainder, likewise at midnight entering.
64
求经辰所入朔弦望：经余变从转，不成为秒，加其夜半所入，皆其辰入日及余。 因朔辰所入，每加日七、余八百六十五、秒千一百六十大，秒满日法成余，亦得上弦。 望、下弦、次朔经辰所入径求者，加望日十四、余千七百三十一、秒千七十九半，下弦日二十二、余三百三十四、秒九百九十八小，次朔日一、余二千二百八、秒九百一十七。 亦朔望各增日一，减其全余，望五百三十一、秒百六十二半，朔五十四、秒三百二十五。
To find canonical chronology entering for new moon, first quarter, full moon, and last quarter: convert the canonical remainder from rotation, with incomplete parts as seconds; add to midnight entering—obtaining chronology entering day and remainder. From new moon chronology entering, add 7 days, remainder 865, seconds 1,160 large each time; when seconds fill the day divisor they form the remainder—also yielding first quarter. For direct calculation of full moon, last quarter, and next new moon chronology entering: full moon add 14 days, remainder 1,731, seconds 1,079 and a half; last quarter 22 days, remainder 334, seconds 998 small; next new moon 1 day, remainder 2,208, seconds 917. For new moon and full moon, each add one day and subtract the full remainder: full moon 531, seconds 162 and a half; new moon 54, seconds 325.
65
求月平应会日所入：以月朔弦望会日所入迟速定数，亦变从转余，乃速加、迟减其经辰所入余，即各平会所入日余。
To find the moon's mean conjunction entering: take the slow-fast fixed number at new moon, first quarter, full moon, and last quarter mean conjunction days, convert from rotation remainder; add for fast motion, subtract for slow motion from the canonical chronology entering remainder—obtaining each mean conjunction entering day and remainder.
66
推朔弦望定日术：
Method for calculating fixed new moon, first quarter, and full moon days:
67
各以月平会所入之日加减限，限并后限而半之，为通率； 又二限相减，为限衰。 前多者，以入余减终法，残乘限衰，终法而一，并于限衰而半之； 前少者，半入余乘限衰，亦终法而一，减限衰。 皆加通率，入余乘之，日法而一，所得为平会加减限数。 其限数又别从转余为变余，朓减、朒加本入余。 限前多者，朓以减与未减，朒以加与未加，皆减终法，并而半之，以乘限衰； 前少者，亦朓朒各并二入余，半之，以乘限衰； 皆终法而一，加于通率，变余乘之，日法而一。 所得以朓减、朒加限数，加减朓朒积而定朓朒。 乃朓减、朒加其平会日所入余，满若不足进退之，即朔弦望定日及余。 不满晨前数者，借减日算，命甲子算外，各其日也。 不减与减，朔日立算与后月同。 若俱无立算者，月大，其定朔算后加所借减算。 闰衰限满闰限，定朔无中气者为闰，满之前后，在分前若近春分后、秋分前，而或月有二中者，皆量置其朔，不必依定。 其后无同限者，亦因前多以通率数为半衰而减之，二前少，即为通率。 其加减变余进退日者，分为一日，随余初末如法求之，所得并以加减限数。 凡分余秒篾，事非因旧，文不著母者，皆十为法。 若法当求数，用相加减，而更不过通远，率少数微者，则不须算。 其入七日余二千一十一，十四日余千七百五十九，二十一日余千五百七，二十八日始终余以下为初数，各减终法以上为末数。 其初末数皆加减相返，其要各为九分，初则七日八分，十四日七分，二十一日六分，二十八日五分； 末则七日一分，十四日二分，二十一日三分，二十八日四分。 虽初稍弱而末微强，余差止一，理势兼举，皆今有转差，各随其数。 若恆算所求，七日与二十一日得初衰数，而末初加隐而不显，且数与平行正等。 亦初末有数而恆算所无，其十四日、二十八日既初末数存，而虚衰亦显，其数当去，恆法不见。
Take the limit added and subtracted from each mean conjunction entering day; combine the limits and halve—this is the common rate; Subtract the two limits—this is the limit deficit. When prior is greater: subtract entering remainder from terminal divisor, multiply the remainder by limit deficit, divide by terminal divisor; combine with limit deficit and halve; When prior is less: halve the entering remainder, multiply by limit deficit, divide by terminal divisor; subtract from limit deficit. Add the common rate in all cases; multiply by entering remainder and divide by day divisor—the result is the mean conjunction add-subtract limit number. The limit number also converts separately from rotation remainder to changed remainder; subtract for waxing, add for waning to the original entering remainder. When the prior limit is greater: for waxing subtract both reduced and unreduced portions, for waning add both added and unadded portions—all subtract terminal divisor, combine and halve, multiply by limit deficit; When prior is less: for waxing and waning combine both entering remainders, halve, and multiply by limit deficit; All divide by terminal divisor; add to common rate; multiply by changed remainder and divide by day divisor. The result subtracts for waxing and adds for waning to the limit number; add and subtract the waxing-waning accumulation to fix waxing and waning. Then subtract for waxing and add for waning to the mean conjunction entering remainder; carry forward or borrow when full or insufficient—obtaining the fixed new moon, first quarter, and full moon day and remainder. When less than the before-dawn count, borrow and subtract from the day count; name beyond the jiazi cycle—each giving its date. Whether subtracted or not, the new moon establishment count matches the following month. If neither has an establishment count, in a long month add the borrowed subtraction count after the fixed new moon count. When intercalary deficit fills the intercalary limit, a fixed new moon without central qi is intercalary; when filling before or after, if before the equinox fraction or near after spring equinox and before autumn equinox, and the month has two central qi—all adjust new moon placement, not necessarily following the fixed date. When no identical limit follows, follow the prior greater case: use the common rate as half deficit and subtract; when both prior are less, this becomes the common rate. When add-subtract changed remainder advances or retreats the day, divide into one day; compute by remainder initial and terminal according to rule—all results add and subtract to the limit number. For fractions, remainders, seconds, and hairs not inherited from prior usage, where no denominator is shown—all use ten as the divisor. If numbers must be computed for mutual addition and subtraction, but conversion is excessive and rates are minute—calculation may be omitted. Entering seven days remainder 2,011; fourteen days remainder 1,759; twenty-one days remainder 1,507; below the twenty-eighth day terminal remainder is the initial number; above the terminal divisor is the terminal number. Initial and terminal numbers add and subtract in reverse; the key is nine parts each: initially—seven days eight parts, fourteen days seven parts, twenty-one days six parts, twenty-eight days five parts; Terminally—seven days one part, fourteen days two parts, twenty-one days three parts, twenty-eight days four parts. Though the initial is slightly weaker and the terminal slightly stronger, the remainder difference is only one—the principle accommodates both; all now have rotation difference, each following its value. By mean calculation, seven and twenty-one days obtain the initial deficit number, while terminal-initial addition is hidden—and the value equals mean motion. Initial and terminal numbers exist but mean calculation lacks them; on fourteen and twenty-eight days initial and terminal numbers are stored while empty deficit appears—the numbers should be removed, invisible in the mean method.
68
求朔弦望之辰所加：
To find the chronology added at new moon, first quarter, and full moon:
69
定余半朔辰五十一大以下，为加子过； 以上，加此数，乃朔辰而一，亦命以子，十二算外，又加子初。 以后其求入辰强弱，如气。
Fixed remainder at or below half new moon chronology 51 large—this is passing zi in addition; Above, add this number, divide by new moon chronology; name from zi; beyond twelve count; also add zi initial. Thereafter, seeking entering chronology strength and weakness follows the qi method.
70
求入辰法度：
Method for calculating entering chronology degrees:
71
度法，四万六千六百四十四。
Degree divisor: 46,644.
72
周数，千七百三万七千七十六。
Circuit number: 17,037,076.
73
推黄道术：
Method for calculating the ecliptic:
74
准冬至所在为赤道度，后于赤道四度为限。 初数九十七，每限增一，以终百七。 其三度少弱，平。 乃初限百九，亦每限增一，终百一十九，春分所在。 因百一十九每限损一，又终百九。 亦三度少弱，平。 乃初限百七，每限损一，终九十七，夏至所在。 又加冬至后法，得秋分、冬至所在数。 各以数乘其限度，百八而一，累而总之，即皆黄道度也。 度有分者，前后辈之，宿有前却，度亦依体，数逐差迁，道不常定，准令为度，见步天行，岁久差多，随术而变。 斗二十四牛七女十一半虚十危十七室十七壁十北方九十六度半。 奎十七娄十三胃十五昴十一毕十五半觜二参九西方八十二度半。 井三十鬼四柳十四半星七张十七翼十九轸十八南方一百九度半。 角十三亢十氐十六房五心五尾十七箕十半东方七十六度半。 前皆黄道度，步日所行。 月与五星出入，循此。
Take the winter solstice position as equatorial degrees; four degrees past the equator is the limit. Initial number 97; each limit increases by one, ending at 107. At three degrees small-weak, level. Then initial limit 109; each limit increases by one, ending at 119—spring equinox position. From 119 each limit decreases by one, again ending at 109. Also three degrees small-weak, level. Then initial limit 107; each limit decreases by one, ending at 97—summer solstice position. Also add the post-winter-solstice method to obtain autumn equinox and winter solstice position numbers. Multiply each limit degree, divide by 108; accumulate and sum—all yielding ecliptic degrees. When degrees have fractions, round accordingly; lodges advance and retreat, degrees follow the celestial body; values shift with accumulated difference; the path is not constant—take the standard as degree, visible in stepping celestial motion; over long periods the discrepancy grows, changing with the method. Dipper 24, Ox 7, Woman 11½, Emptiness 10, Rooftop 17, Room 17, Wall 10—Northern Region 96½ degrees. Striding 17, Bond 13, Stomach 15, Hairy Head 11, Net 15½, Turtle Beak 2, Three Stars 9—Western Region 82½ degrees. Well 30, Ghost 4, Willow 14½, Star 7, Extended Net 17, Wings 19, Chariot Base 18—Southern Region 109½ degrees. Horn 13, Gullet 10, Base 16, Room 5, Heart 5, Tail 17, Winnowing Basket 10½—Eastern Region 76½ degrees. Above are all ecliptic degrees, for stepping the sun's daily motion. The moon and five planets in their ingress and egress follow this.
75
推月道所行度术：
Method for calculating degrees traversed on the moon's path:
76
准交定前后所在度半之，亦于赤道四度为限，初十一，每限损一，以终于一。 其三度强，平。 乃初限数一，每限增一，亦终十一，为交所在。 即因十一，每限损一，以终于一。 亦三度强，平。 又初限数一，每限增一，终于十一，复至交半，返前表里。 仍因十一增损，如道得后交及交半数。 各积其数，百八十而一，即道所行每与黄道差数。 其月在表，半后交前，损减增加； 交后半前，损加增减于黄道。 其月在里，各返之，即得月道所行度。 其限未尽四度，以所直行数乖入度，四而一。 若月在黄道度，增损于黄道之表里，不正当于其极，可每日准去黄道度，增损于黄道，而计去赤道之远近，准上黄道之率以求之，遁伏相消，朓朒互补，则可知也。 积交差多，随交为正。 其五星先候，在月表里出入之渐，又格以黄仪，准求其限。 若不可推明者，依黄道命度。
Take half the degree before and after the fixed node position; also four degrees from the equator as limit; initial 11, each limit decreases by one, ending at 1. At three degrees strong, level. Then initial limit 1; each limit increases by one, ending at 11—node position. From 11 each limit decreases by one, ending at 1. Also three degrees strong, level. Again initial limit 1; each limit increases by one, ending at 11; return to node half, reversing front and back. Still following 11 in increase and decrease, as the path obtains the following node and half-node numbers. Accumulate each number, divide by 180—this is the difference from the ecliptic for each degree the path traverses. When the moon is on the outer side, half after node before node, subtract deficit and add increase; After node half before node, add deficit and subtract increase from the ecliptic. When the moon is on the inner side, reverse each—obtaining the degree the moon's path traverses. When the limit has not exhausted four degrees, by the straight-travel number deviate into degrees, one of four. If the moon is at ecliptic degree, increase and decrease from the ecliptic's inner and outer sides—not directly at its pole—one may daily take the departed ecliptic degree, adjust from the ecliptic, and calculate distance from the equator; follow the ecliptic rate above; hidden and visible cancel, waxing and waning complement—then the result can be known. Accumulated node difference is substantial; follow the node as standard. The five planets are first observed in the moon's inner and outer gradual ingress and egress; also calibrated by the yellow instrument to determine the limit. If it cannot be deduced clearly, assign degrees according to the ecliptic.
77
推日度术：
Method for calculating solar degrees:
78
置入元距所求年岁数乘之，为积实，周数去之，不尽者，满度法得积度，不满为分。 以冬至余减分； 命积度以黄道起于虚一宿次除之，不满宿算外，即所求年天正冬至夜半日所在度及分。
Enter the accumulated count from origin to the year sought; multiply by the year constant for the accumulated dividend; remove the circuit number; when the remainder fills the degree divisor, obtain accumulated degrees; the remainder is the fraction. Subtract the fraction by the winter solstice remainder; Name accumulated degrees starting the ecliptic from Emptiness, first lodge, removing in sequence; the remainder beyond the lodge count gives the year's Heaven-regular winter solstice midnight solar position in degrees and fractions.
79
求年天正定朔度：
To find the year's Heaven-regular fixed new moon degree:
80
以定朔日至冬至每日所入先后余为分，日为度，加分以减冬至度，即天正定朔夜半日在所度分。 亦去朔日乘衰总已通者，以至前定气除之，又如上求差加以并去朔日乃减度，亦即天正定朔日所在度。 皆日为度，余为分。 其所入先后及衰总用增损者，皆分前增、分后损其平日之度。
Take the daily entering advance-retardation remainder from fixed new moon day to winter solstice as fraction, day as degree; add fraction to subtract from winter solstice degree—giving Heaven-regular fixed new moon midnight solar position in degrees and fractions. Also remove new moon day multiplied by total deficit already converted, divide by fixed qi before solstice; likewise seek difference, add, combine, remove new moon day, then subtract degree—also giving Heaven-regular fixed new moon solar position. Days as degrees, remainder as fraction. The entering advance-retardation and total deficit used for increase and decrease—all increase before the fraction, decrease after the fraction from the mean day's degree.
81
求次日：
To find the next day:
82
每日所入先后分增损度，以加定朔度，得夜半。
Apply daily entering advance-retardation fraction to increase and decrease degrees; add to fixed new moon degree—obtaining midnight position.
83
求弦望：
To find first quarter and full moon:
84
去定朔每日所入分，累而增损去定朔日，乃加定朔度，亦得其夜半。
Remove fixed new moon daily entering fraction; accumulate to increase and decrease days from fixed new moon; then add fixed new moon degree—also obtaining midnight position.
85
求次月：
To find the next month:
86
历算大月三十日，小月二十九日，每日所入先后分增损其月，以加前朔度，即各夜半所在至虚去周分。
By calendar calculation, long months have 30 days and short months 29; apply daily entering advance-retardation fraction to adjust the month; add to the prior new moon degree—each midnight position, removing circuit fraction to Emptiness.
87
求朔弦望辰所加：
To find the chronology added at new moon, first quarter, and full moon:
88
各以度准乘定余，约率而一，为平分。 又定余乘其日所入先后分，日法而一，乃增损其平分，以加其夜半，即各辰所加。 其分皆篾法约之，为转分，不成为篾。 凡朔辰所加者，皆为合朔日月同度。
Multiply each fixed remainder by the degree standard and divide by the reduction rate—this is the even fraction. Multiply fixed remainder by the daily entering advance-retardation fraction and divide by the day divisor; adjust the even fraction accordingly; add to midnight—obtaining each chronology added. Reduce all fractions by the hair divisor to obtain rotation fraction; incomplete parts become hairs. All new moon chronology added positions mark conjunction—sun and moon at the same degree.
89
推月而与日同度术：
Method for calculating when the moon reaches the same degree as the sun:
90
各以朔平会加减限数加减朓朒，为平会朓朒。 以加减定朔，度准乘，约率除，以加减定朔辰所加日度，即平会辰日所在。 又平会余乘度准，约率除，减其辰所在，为平会夜半日所在。 乃以四百六十四半乘平会余，亦以周差乘，朔实除，从之，以减夜半日所在，即月平会夜半所在。 三十七半乘平会余，增其所减，以加减半，得月平会辰平行度。 五百二乘朓棵，亦以周差乘，朔实除而从之，朓减、朒加其平行，即月定朔辰所在度，而与日同。 若即以平会朓朒所得分加减平会辰所在，亦得同度。
Take each mean conjunction add-subtract limit and apply waxing-waning corrections—this is mean conjunction waxing-waning. Apply to fixed new moon; multiply by degree standard and divide by reduction rate; add and subtract from fixed new moon chronology-added solar degree—giving mean conjunction chronology day position. Multiply mean conjunction remainder by degree standard and divide by reduction rate; subtract its chronology location—giving mean conjunction midnight solar position. Then multiply mean conjunction remainder by 464½; also multiply circuit difference; divide by new moon constant and apply; subtract from midnight solar position—giving moon mean conjunction midnight position. Multiply mean conjunction remainder by 37½; increase the subtracted amount; add and subtract half—obtaining moon mean conjunction chronology mean-travel degree. Multiply waxing-waning by 502; also multiply circuit difference; divide by new moon constant and apply; subtract for waxing, add for waning to mean travel—giving moon fixed new moon chronology position, same as the sun. If directly add and subtract the mean conjunction waxing-waning fraction to mean conjunction chronology position—same degree is also obtained.
91
求月弦望定辰度：
To find moon first quarter and full moon fixed chronology degree:
92
各置其弦望辰所加日度及分，加上弦度九十一，转分十六，篾三百一十三； 望度百八十二，转分三十二，篾六百二十六； 下弦度二百七十三，转分四十九，篾四十二，皆至虚，去转周求之。
Set each first quarter and full moon chronology-added solar degree and fraction; add first quarter degree 91, rotation fraction 16, hair 313; Full moon degree 182, rotation fraction 32, hair 626; Last quarter degree 273, rotation fraction 49, hair 42—all to Emptiness, remove rotation circuit to compute.
93
定朔夜半入转：
Fixed new moon midnight entering rotation:
94
经朔夜半所入准于定朔日有增损者，亦以一日加减之，否者因经朔为定。
Compare canonical new moon midnight entering to fixed new moon day—if increase or decrease applies, add or subtract one day; otherwise follow canonical new moon as fixed.
95
其因定求朔次日、弦望、次月夜半者，如于经月法为之。
For fixed new moon next day, first quarter, full moon, and next month midnight—follow the canonical month method.
96
推月转日定分术：
Method for calculating moon rotation daily fixed fraction:
97
以夜半入转余乘逡差，终法而一，为见差。 以息加、消减其日逡分，为月每日所行逡定分。
Multiply midnight entering rotation remainder by deferral difference; divide by terminal divisor—this is appearance difference. Add to rest, subtract from wane to daily deferral fraction—this is the moon's daily deferral fixed fraction.
98
求次日：
To find the next day:
99
各以逡定分加转分，满转法从度，皆其夜半。 因日转若各加定日，皆得朔、弦望夜半月所在定度。 其就辰加以求夜半，各以半逡差减逡分，消者，定余乘差，终法除，并差而半之； 息者，半定余以乘差，终法而一。 皆加所减，乃以定余乘之，日法而一，各减辰所加度，亦得其夜半度。 因夜半亦如此求逡分，以加之，亦得辰所加度。 诸转可初以逡分及差为篾，而求其次，皆讫，乃除为转分。 因经朔夜半求定辰度者，以定辰去经朔夜半减，而求其增损数，乃以数求逡定分，加减其夜半，亦各定辰度。
Add deferral fixed fraction to rotation fraction each day; when full of rotation divisor, advance degree—all yielding midnight positions. Following daily rotation or adding each fixed day—all yielding new moon, first quarter, full moon midnight moon fixed degree. To seek midnight by adding to chronology: halve deferral difference and subtract deferral fraction; for wane, multiply fixed remainder by difference, divide by terminal divisor, combine difference and halve; For rest, half fixed remainder multiply difference, one of terminal divisor. Add all subtracted amounts; multiply fixed remainder, one of day divisor; subtract from chronology-added degree—also obtaining midnight degree. From midnight likewise seek deferral fraction and add—also obtaining chronology-added degree. All rotations may initially use deferral fraction and difference as hairs to seek the next; when complete, divide to rotation fraction. From canonical new moon midnight seek fixed chronology degree: subtract fixed chronology from canonical new moon midnight and seek increase-decrease number; by number seek deferral fixed fraction to add-subtract midnight—also each fixed chronology degree.
100
求月晨昏度：
To find moon dawn and dusk degree:
101
如前气与所求每日夜漏之半，以逡定分乘之，百而一，为晨分； 减逡定分，为昏分。 除为转度，望前以昏，后以晨，加夜半定度，得所在。
As with prior qi and the sought daily night clepsydra half; multiply deferral fixed fraction, one of 100—this is dawn fraction; Subtract deferral fixed fraction—this is dusk fraction. Divide as rotation degree; before full moon use dusk, after use dawn; add to midnight fixed degree—obtaining position.
102
求晨昏中星：
To find dawn and dusk culmination stars:
103
各以度数加夜半定度，即中星度。 其朔、弦、望，以百刻乘定余，满日法得一刻，即各定辰近入刻数。 皆减其夜半漏，不尽为晨，初刻不满者属昨日。 复月，五千四百五十八。 交月，二千七百二十九。 交率，四百六十五。 交数，五千九百二十三。 交法，七百三十五万六千三百六十六。 会法，五十七万七千五百三十。 交复日，二十七。 余，二百六十三。 秒，三千四百三十五。 交日，十三。 余，七百五十二。 秒，四千六百七十九。 交限，日，十二。 余，五百五十五。 秒，四百七十三半。 望差，日，一。 余，百九十七。 秒，四千二百五半。 朔差，日，二。 余，三百九十五。 秒，二千四百八十八。 会限，百五十八。 余，六百七十六。 秒，五十半。 会日，百七十三。 余，三百八十四。 秒，二百八十三。
Add degree count to midnight fixed degree—this is culmination star degree. For new moon, first quarter, full moon: multiply fixed remainder by 100 clepsydra counts; when full of day divisor obtain one clepsydra—each fixed chronology near entering clepsydra count. Subtract from midnight clepsydra; remainder is dawn; initial clepsydra not full belongs to yesterday. Circuit moon: 5,458. Node moon: 2,729. Node rate: 465. Node number: 5,923. Node divisor: 7,356,366. Conjunction divisor: 577,530. Node return day: 27. Remainder: 263. Seconds: 3,435. Node day: 13. Remainder: 752. Seconds: 4,679. Node limit, day: 12. Remainder: 555. Seconds: 473 and a half. Full moon difference, day: 1. Remainder: 197. Seconds: 4,205 and a half. New moon difference, day: 2. Remainder: 395. Seconds: 2,488. Conjunction limit: 158. Remainder: 676. Seconds: 50 and a half. Conjunction day: 173. Remainder: 384. Seconds: 283.
104
推月行入交表里术：
Method for calculating moon entering node inner and outer:
105
置入元积月，复月去之，不尽。 交率乘而复去，不如复月者，满交月去之，为在里数； 不满为在表数，即所求年天正经入交表里数。
Enter accumulated months from origin; remove circuit moon; remainder. Multiply by node rate and remove again; when less than circuit moon, divide by node moon—this is inner count; Not full is outer count—this is the year's Heaven-canonical entering node inner-outer count.
106
求次月：
To find next month:
107
以交率加之，满交月去之，前表者在里，前里者在表。
Add node rate; remove when full of node moon; prior outer becomes inner, prior inner becomes outer.
108
推月入交日术：
Method for calculating moon entering node day:
109
以朔实乘表里数，为交实； 满交法为日，不满者交数而一，为余，不成为秒，命日算外，即其经朔月平入交日余。
Multiply inner-outer number by new moon constant—this is node dividend; When full of node divisor, obtain days; remainder one of node number; incomplete becomes seconds; name day beyond count—giving canonical new moon mean entering node day and remainder.
110
求望：以望差加之，满交日去之，则月在表里与朔同； 不满者与朔返。 其月食者，先交与当月朔，后交与月朔表里同。
To find full moon: add full moon difference; remove when full of node day—moon inner-outer same as new moon; Not full reverses with new moon. For lunar eclipse: prior node with current month new moon, following node with month new moon inner-outer same.
111
求次月：朔差加月朔所入，满交日去之，表里与前月返； 不满者，与前月同。
To find next month: add new moon difference to month new moon entering; remove when full of node day; inner-outer reverses from prior month; Not full same as prior month.
112
求经朔望入交常日：
To find canonical new moon and full moon entering node mean day:
113
以月入气朔望平会日迟速定数，速加、迟减其平入交日余，为经交常日及余。
Take moon entering qi new moon and full moon mean conjunction slow-fast fixed number; add for fast, subtract for slow from mean entering node day remainder—giving canonical node mean day and remainder.
114
求定朔望入交定日：
To find fixed new moon and full moon entering node fixed day:
115
以交率乘定朓朒，交数而一，所得以朓减、朒加常日余，即定朔望所入定日及余。 其去交如望差以下、交限以上者月食，月在里者日食。
Multiply fixed waxing-waning by node rate, one of node number; subtract for waxing, add for waning to mean day remainder—giving fixed new moon and full moon entering fixed day and remainder. When distance from node is at or below full moon difference and above node limit—lunar eclipse; moon in inner—solar eclipse.
116
推日入会日术：
Method for calculating sun entering conjunction day:
117
会法除交实为日，不满者，如交率为余，不成为秒，命日算外，即经朔日入平会日及余。
Divide node dividend by conjunction divisor for days; remainder as node rate; incomplete as seconds; name day beyond count—giving canonical new moon entering mean conjunction day and remainder.
118
求望：加望日及余，次月加经朔，其表里皆准入交。
To find full moon: add full moon day and remainder; next month add canonical new moon; inner-outer all follow entering node.
119
求入会常日：以交数乘月入气朔望所平会日迟速定数，交率而一，以速加、迟减其入平会日余，即所入常日余。 亦以定朓朒，而朓减、朒加其常日余，即日定朔望所入会日及余。 皆满会日去之，其朔望去会，如望以下、会限以上者，亦月食； 月日道表在日道里则日食。
To find entering conjunction mean day: multiply moon entering qi new moon and full moon mean conjunction slow-fast fixed number by node number, one of node rate; add for fast, subtract for slow from entering mean conjunction day remainder—giving entering mean day remainder. Also apply fixed waxing-waning; subtract for waxing, add for waning to mean day remainder—giving sun fixed new moon and full moon entering conjunction day and remainder. Remove when full of conjunction day; new moon and full moon distance from conjunction below full moon, above conjunction limit—also lunar eclipse; Moon path outer, sun path inner—solar eclipse.
120
求月定朔望入交定日夜半：交率乘定余，交数而一，以减定朔望所入定日余，即其夜半所定入。
To find moon fixed new moon and full moon entering node fixed day midnight: multiply fixed remainder by node rate, one of node number; subtract from fixed new moon and full moon entering fixed day remainder—giving midnight fixed entering.
121
求次日：
To find the next day:
122
以每日迟速数，分前增、分后损定朔所入定日余，以加其日，各得所入定日及余。 求次月：加定朔，大月二日，小月一日，皆余九百七十八，秒二千四百八十八。 各以一月迟速数，分前增、分后损其所加，为定。 其入七日，余九百九十七，秒二千三百三十九半以下者，进； 其入此以上，尽全余二百四十四，秒三千五百八十三半者，退。 其入十四日，如交余及秒以下者，退； 其入此以上，尽全余四百八十九，秒千二百四十四者，进而复也。 其要为五分，初则七日四分，十四日三分； 末则七日后一分，十四日后二分，虽初强末弱，衰率有检。 求月入交去日道：皆同其数，以交余为秒积，以后衰并去交衰，半之，为通数。 进则秒积减衰法，以乘衰，交法除，而并衰以半之； 退者，半秒积以乘衰，交法而一； 皆加通数，秒积乘，交法除，所得以进退衰积，十而一为度，不满者求其强弱，则月去日道数。 月朔望入交，如限以上，减交日，残为去后交数； 如望差以下即为去先交数。 有全日同为余，各朔辰而一，得去交辰。 其月在日道里，日应食而有不食者； 月日道表在日不应食而亦有食者。 推应食不食术：朔先后在夏至十日内，去交十二辰少； 二十日内，十二辰半； 一月内，十二辰大； 闰四月、六月，十三辰以上，加南方三辰。 若朔在夏至二十日内，去交十三辰，以加辰申半以南四辰； 闰四月、六月，亦加四辰； 谷雨后、处暑前，加三辰； 清明后、白露前，加巳半以西、未半以东二辰； 春分后秋分前，加午一辰。 皆去交十三辰半以上者，并或不食。
Apply daily slow-fast number; increase before fraction, decrease after fraction to fixed new moon entering fixed day remainder; add to its day—each yielding entering fixed day and remainder. To find next month: add fixed new moon, long month 2 days, short month 1 day, all remainder 978, seconds 2,488. Apply one month's slow-fast number; increase before fraction, decrease after fraction to what is added—fixed. Entering seven days, remainder 997, seconds 2,339 and a half or below—advance; Entering above this level, when full remainder reaches 244 and seconds 3,583½—retreat. Entering fourteen days, at or below node remainder and seconds—retreat; Entering above this level, when full remainder reaches 489 and seconds 1,244—advance and return. The key is five parts: initially seven days four parts, fourteen days three parts; Terminally one part after seven days, two parts after fourteen days—though initial is strong and terminal weak, deficit rates are verified. To find the moon's distance from the solar path at node entry: use the same value; take node remainder as second accumulation; combine following deficit with departing node deficit and halve—this is the common number. When advancing: subtract deficit divisor from second accumulation, multiply by deficit; divide by node divisor; combine deficit and halve; When retreating: halve second accumulation and multiply by deficit, one of node divisor; Add the common number; multiply second accumulation; divide by node divisor; advance or retreat the deficit accumulation; one of ten gives degrees; for incomplete values determine strength or weakness—obtaining the moon's distance from the solar path. When the moon at new moon or full moon enters the node above the limit—subtract node day; the remainder is the distance from the following node; At or below full moon difference, this is the distance from the prior node. When full days equal the remainder, divide each by new moon chronology—obtaining the distance from node chronology. When the moon is within the solar path, an eclipse should occur yet sometimes does not; When the moon is outside the solar path, no eclipse should occur yet one sometimes does. Method for calculating when an eclipse is expected but does not occur: new moon within ten days before or after summer solstice, distance from node twelve chronologies lesser; Within twenty days, twelve chronologies half; Within one month, twelve chronologies greater; In intercalary fourth and sixth months, above thirteen chronologies—add three southern chronologies. If new moon within twenty days of summer solstice, distance from node thirteen chronologies—add four chronologies from shen half southward; In intercalary fourth and sixth months, also add four chronologies; After Grain Rain and before End of Heat, add three chronologies; After Clear Bright and before White Dew, add two chronologies from si half west to wei half east; After spring equinox and before autumn equinox, add one wu chronology. All cases with distance from node of thirteen chronologies half or above—eclipse may not occur.
123
推不应食而食术：
Method for calculating when an eclipse occurs unexpectedly:
124
朔在夏至前后一月内，去交二辰； 四十六日内，一辰半，以加二辰； 又一月内，亦一辰半，加三辰及加四辰，与四十六日内加三辰； 谷雨后、处暑前，加巳少后、未太前； 清明后、白露前，加二辰； 春分后、秋分前，加一辰。 皆去交半辰以下者，并得食。
New moon within one month before and after summer solstice, distance from node two chronologies; Within forty-six days, one and a half chronologies, add to two chronologies; Within another month, also one and a half chronologies, add three and four chronologies, same as adding three within forty-six days; After Grain Rain and before End of Heat, add after si lesser, before wei greater; After Clear Bright and before White Dew, add two chronologies; After spring equinox and before autumn equinox, add one chronology. All cases with distance from node of half a chronology or below—eclipse may occur.
125
推月食多少术：
Method for calculating lunar eclipse magnitude:
126
望在分后，以去夏至气数三之； 其分前，又以去分气数倍而加分后者； 皆又以十加去交辰倍而并之，减其去交余，为不食定余。 乃以减望差，残者九十六而一，不满者求其强弱，亦如气辰法，以十五为限，命之，即各月食多少。
Full moon after equinox: take distance from summer solstice qi number and divide by three; Before equinox, also multiply distance from equinox qi number and add to after equinox; Also add ten plus double the distance from node chronology and combine; subtract distance from node remainder—this is the fixed non-eclipse remainder. Subtract full moon difference; take one of ninety-six from the remainder; for incomplete values determine strength or weakness by the qi chronology method; use fifteen as limit and name—obtaining each lunar eclipse magnitude.
127
推日食多少术：
Method for calculating solar eclipse magnitude:
128
月在内者，朔在夏至前后二气，加南二辰，增去交余一辰太； 加三辰，增一辰少，加四辰，增太。 三气内，加二辰，增一辰； 加三辰，增太； 加四辰，增少。 四气内，加二辰，增太； 加三辰及五气内，加二辰，增少。 自外所加辰，立夏后、立秋前，依本其气内加四辰，五气内加三辰，六气内加二辰。 六气内加二辰者，亦依平。 自外所加之北诸辰，各依其去立夏、立秋、清明、白露数，随其依平辰，辰北每辰以其数三分减去交余； 雨水后、霜降前，又半其去分日数，以加二分去二立之日，乃减去交余； 其在冬至前后，更以去霜降、雨水日数三除之，以加霜降雨水当气所得之数； 而减去交余，皆为定不食余。 以减望差，乃如月食法。 月在外者，其去交辰数，若日气所系之限，止一而无等次者，加所去辰一，即为食数。 若限有等次，加别系同者，随所去交辰数而返其衰，以少为多，以多为少，亦加其一，以为食数。 皆以十五为限，乃以命之，即各日之所食多少。
Moon inner: new moon within two qi before and after summer solstice, add two southern chronologies, increase distance from node remainder by one chronology greater; Add three chronologies, increase by one chronology lesser; add four chronologies, increase by greater. Within three qi, add two chronologies, increase one chronology; Add three chronologies, increase greater; Add four chronologies, increase lesser. Within four qi, add two chronologies, increase greater; Add three chronologies and within five qi add two chronologies, increase lesser. Southern chronologies added from outside: after Start of Summer and before Start of Autumn, follow original—within four qi add four chronologies, within five qi add three, within six qi add two. Those adding two chronologies within six qi also follow the mean. Northern chronologies added from outside: each follow distance from Start of Summer, Start of Autumn, Clear Bright, and White Dew; follow mean chronologies; north of each chronology subtract one-third its number from node remainder; After Rain Water and before Frost Descent, also halve distance from equinox day number, add to days from equinox to two establishments, then subtract node remainder; Near winter solstice, further divide distance from Frost Descent and Rain Water day numbers by three, add to numbers obtained at those qi nodes; And subtract node remainder—all yielding fixed non-eclipse remainder. Subtract full moon difference—then apply the lunar eclipse method. Moon outer: its distance from node chronology—if the solar qi limit has only one value without gradations, add one to distance chronology—this is the eclipse number. If the limit has gradations, add the same category separately; reverse deficit according to distance from node chronology—less becomes more, more becomes less; also add one—this is the eclipse number. All use fifteen as limit and name—obtaining each day's eclipse magnitude.
129
凡日食，月行黄道，体所映蔽，大较正交如累璧，渐减则有差，在内食分多，在外无损。 虽外全而月下，内损而更高，交浅则闲遥； 交深则相搏而不淹。 因遥而蔽多，所观之地又偏，所食之时亦别。 月居外道，此不见亏，月外之人反以为食。 交分正等，同在南方，冬损则多，夏亏乃少。 假均冬夏，早晚又殊。 处南辰体则高，居东西傍而下视有邪正。 理不可一，由准率若实而违。 古史所详，事有纷互，今故推其梗概，求者知其指归。 苟地非于阳城，皆随所而渐异。 然月食以月行虚道，暗气所冲，日有暗气，天有虚道，正黄道常与日对，如镜居下，魄耀见阴，名曰暗虚，奄月则食，故称“当月月食，当星星亡。 ”虽夜半之辰，子午相对，正隔于地，虚道即亏。 既月兆日光，当午更耀，时亦隔地，无废禀明。 谅以天光神妙，应感玄通，正当夜半，何害亏禀。 月由虚道，表里俱食。 日之与月，体同势等，校其食分，月尽为多，容或形差，微增亏数，疏而不漏，纲要克举。
For solar eclipses generally, the moon travels the ecliptic; the body's shadow—in true conjunction like stacked jade disks; gradual reduction introduces difference; inner eclipses have greater magnitude, outer ones no loss. Though outer appears total yet the moon is below, inner damaged yet higher; shallow conjunction yields distant gap; Deep conjunction yields mutual collision without full obscuration. Because of distance, much obscuration; the observed location is also biased; the eclipse time also differs. Moon on the outer path—here no visible deficiency; observers outside the moon's path believe an eclipse occurred. Conjunction fraction equally correct, both in the south—winter eclipses are greater, summer eclipses lesser. Even assuming equal winter and summer, morning and evening also differ. At southern chronology the body is high; at east and west sides, viewing angle is oblique or direct. The principle cannot be uniform—because standard rates seem accurate yet deviate. Ancient histories detail this; matters are complex and interwoven; we now deduce the outline so readers know the direction. If the location is not at Yangcheng, all gradually differ according to place. Lunar eclipses occur because the moon travels the void path, struck by dark qi; the sun has dark qi; Heaven has a void path; the true ecliptic constantly faces the sun, like a mirror below; the soul's light sees yin, called dark void; covering the moon causes eclipse—hence the saying: "when the moon is covered, the moon eclipses; when the star is covered, the star vanishes. " Though at midnight zi and wu are opposite, directly blocked by earth, the void path is deficient. Since the moon reflects sunlight, at noon it shines brighter; time is also blocked by earth, without losing received brightness. Surely Heaven's light is wondrous, responding to mystery; even at midnight, what harm to received brightness. Moon through the void path, inner and outer both eclipsed. Sun and moon, bodies of equal potential; comparing eclipse fractions, the moon's is entirely greater; perhaps shape differs, slightly increase deficiency number—broad yet not leaking, outline fully captured.
130
推日食所在辰术：
Method for calculating solar eclipse chronology:
131
置定余，倍日限，克减之，月在里，三乘朔辰为法，除之，所得以艮巽坤乾为次。 命艮算外，不满法者半法减之，无可减者为前，所减之残为后，前则因余，后者减法，各为其率。 乃以十加去交辰，三除之，以乘率，十四而一，为差。 其朔所在气二分前后一气内，即为定差。 近冬至，以去寒露、惊蛰，近夏至，以去清明、白露气数，倍而三除去交辰，增之。 近冬至，艮巽以加，坤乾以减； 近夏至，艮巽以减，坤乾以加其差为定差。 乃艮以坤加，巽以乾减定余。 月在外，直三除去交辰，以乘率，十四而一，亦为定差。 艮坤以减，巽乾以加定余，皆为食余。 如气求入辰法，即日食所在辰及大小。 其求辰刻，以辰克乘辰余，朔辰而一，得刻及分。 若食近朝夕者，以朔所入气日之出入刻，校食所在，知食见否之少多所在辰，为正见。
Set fixed remainder; double day limit and subtract; moon inner, use three times new moon chronology as divisor and divide; result uses gen, xun, kun, qian as sequence. Name beyond gen count; if not full of divisor, halve divisor and subtract; nothing to subtract is prior; subtracted remainder is following; prior follows remainder, following subtracts divisor—each is its rate. Add ten to distance from node chronology, divide by three, multiply rate, one of fourteen—this is the difference. New moon located within one qi before and after equinox—this is fixed difference. Near winter solstice, use distance from Cold Dew and Awakening of Insects; near summer solstice, use distance from Clear Bright and White Dew qi numbers; double, divide by three the distance from node chronology, and increase. Near winter solstice, add for gen and xun, subtract for kun and qian; Near summer solstice, subtract for gen and xun, add for kun and qian—the difference is fixed difference. Then gen with kun add, xun with qian subtract fixed remainder. Moon outer: directly divide by three the distance from node chronology, multiply rate, one of fourteen—also fixed difference. Gen and kun subtract, xun and qian add fixed remainder—all are eclipse remainder. Like qi method seek entering chronology—this is solar eclipse chronology and magnitude. Seeking chronology clepsydra: chronology overcomes multiply chronology remainder, one of new moon chronology—obtain clepsydra and fraction. If eclipse near dawn or dusk: use new moon entering qi day's sunrise-sunset clepsydra, verify eclipse location, determine visibility and magnitude—location chronology is true appearance.
132
推月食所在辰术：
Method for calculating lunar eclipse chronology:
133
三日阻减望定余半。 置望之所入气日，不见刻，朔日法乘之，百而一，所得若食余与之等、以下，又以此所得减朔日法，其残食余与之等、以上，为食正见数。 其食余亦朔辰而一，如求加辰所在。 又如前求刻校之，月在冲辰食，日月食既有起讫晚早，亦或变常进退，皆于正见前后十二刻半候之。
Three-day obstruction subtract half full moon fixed remainder. Set full moon entering qi day invisible clepsydra; multiply by new moon day divisor; one of 100; if eclipse remainder equals or is below, subtract this from new moon day divisor; if remainder eclipse remainder equals or is above—this is true appearance number. Eclipse remainder also one of new moon chronology—like seeking added chronology location. Also like prior seek clepsydra verify; moon at opposite chronology eclipses; solar and lunar eclipses have start-end early-late, may change normal advance-retreat—all observe twelve and a half clepsydra before and after true appearance.
134
推日月食起讫辰术：
Method for calculating solar and lunar eclipse start-end chronology:
135
准其食分十五分为率，全以下各为衰。 十四分以上，以一为衰，以尽于五分。 每因前衰每降一分，积衰增二，以加于前，以至三分。 每积增四。 二分每增四，二分增六，一分增十九，皆累算为各衰。 三百为率，各衰减之，各以其残乘朔日法，皆率而一，所得为食衰数。 其率全，即以朔日法为衰数，以衰数加减食余，其减者为起，加者为讫，数亦如气。
Take eclipse fraction fifteen parts as rate; below full each is deficit. Fourteen parts and above, one as deficit, exhaust at five parts. Each following prior deficit decrease one part; accumulated deficit increase two, add to prior—until three parts. Each accumulation increase four. Two parts each increase four; two parts increase six; one part increase nineteen—all accumulated as each deficit. Three hundred as rate; decrease each deficit; each remainder multiply new moon day divisor; all one of rate—result is eclipse deficit number. Rate full then directly use new moon day divisor as deficit number; add-subtract eclipse remainder by deficit number; subtract is start, add is end—number also like qi.
136
求入辰法及求刻：以加减食所刻等，得起讫晚早之辰，与校正见多少之数。 史书亏复起讫不同，今以其全一辰为率。
Seek entering chronology method and seek clepsydra: add-subtract eclipse clepsydra etc.—obtain start-end early-late chronology, and verify true appearance magnitude. Historical records of eclipse deficiency and restoration start-end differ; we now use one full chronology as the rate.
137
推日月食所起术：
Method for calculating solar and lunar eclipse onset:
138
月在内者，其正南，则起右上，亏左上。 若正东，月自日上邪北而下。 其在东南维前，东向望之，初不正，横月高日下； 乃月稍西北，日渐东南，过于维后，南向望之，月更北，日差西南； 以至于午之后，亦南望之，月欹西北，日复东南。 西南维后，西向而望，月为东北，日则西南。 正西，自日北下邪亏，而亦后不正，横月高日下。 若食十二分以上，起右亏左。 其正东，起上近亏下而北，午前则渐自上邪下。 维西，起西北，亏东南。 维北，起西南，亏东北； 午后则稍从下傍下。 维东，起西南，亏东北。 维南，起西北，亏东南。 在东则以上为东，在西则以下为西。
Moon inner: facing due south, onset is upper right, deficiency upper left. If due east, the moon passes obliquely northward downward from above the sun. Before the southeast corner, looking east, initially not direct, the moon horizontal and high, the sun low; Then the moon shifts slightly northwest, the sun gradually southeast; past the corner, looking south, the moon is more north, the sun slightly southwest; Until after noon, looking south, the moon tilts northwest, the sun again southeast. After the southwest corner, looking west, the moon is northeast, the sun southwest. Due west, deficiency obliquely downward from north of the sun; also afterward not direct, moon horizontal and high, sun low. If the eclipse is twelve parts or above, onset right and deficiency left. Due east, onset upper near deficiency lower and north; before noon gradually from upper oblique downward. West corner, onset northwest, deficiency southeast. North corner, onset southwest, deficiency northeast; After noon then slightly from the lower side downward. East corner, onset southwest, deficiency northeast. South corner, onset northwest, deficiency southeast. In the east, above is east; in the west, below is west.
139
月在外者，其正南，起右下，亏左上。 在正东，月自日南邪下而映。 维北，则月微东南，日返西。 维西南，日稍移东北，以至于午，月南日北，过午之后，月稍东南，日更西北。 维北，月有西南，日复东北。 正西，月自日下邪南而上。 皆准此体以定起亏，随其所处，每用不同。 其月之所食，皆依日亏起，每随类反之，皆与日食限同表里，而与日返其逆顺，上下过其分。
Moon outer: due south, onset lower right, deficiency upper left. Due east, the moon passes obliquely downward from south of the sun and reflects. North corner, the moon slightly southeast, the sun returns west. Southwest corner, the sun gradually shifts northeast; until noon moon south and sun north; after noon the moon slightly southeast, the sun more northwest. North corner, the moon southwest, the sun again northeast. Due west, the moon passes obliquely southward upward from below the sun. All follow this pattern to determine onset and deficiency; according to location, each case differs. Lunar eclipses all follow solar deficiency onset; each reverses by category—all share the same inner-outer limit as solar eclipses, yet reverse direction from the sun, above and below exceeding its fraction.
140
五星：
The Five Planets:
141
岁为木
Jupiter is Wood
142
荧惑为火
Mars is Fire
143
镇为土
Saturn is Earth
144
太白为金
Venus is Metal
145
辰为水
Mercury is Water
146
木数，千八百六十万五千四百六十八。
Wood number: 18,605,468.
147
伏半平，八十三万六千八百四十八。
Hidden half-mean: 836,848.
148
复日，三百九十八； 余，四万一千一百五十六。
Return day: 398; Remainder: 41,156.
149
岁一，残日，三十三； 余，二万九千七百四十九半。
One per year, residual day: 33; Remainder: 29,749 and a half.
150
见去日，十四度。
Appearance distance from sun: 14 degrees.
151
平见，在春分前，以四乘去立春日； 小满前，又三乘去春分日，增春分所乘者； 白露后，亦四乘去寒露日； 小暑，加七日； 小雪前，以八乘去寒露日； 冬至后，以八乘去立春日，为减，小雪至冬至减七日。
Mean appearance: before spring equinox, multiply by four the distance from Start of Spring day; Before Lesser Fullness, also multiply by three the distance from spring equinox day, increasing the spring equinox multiplier; After White Dew, also multiply by four the distance from Cold Dew day; Lesser Heat, add seven days; Before Lesser Snow, multiply by eight the distance from Cold Dew day; After winter solstice, multiply by eight the distance from Start of Spring day as decrease; from Lesser Snow to winter solstice decrease seven days.
152
见，初日行万一千八百一十八分，日益迟七十分，百一十日行十八度、分四万七百三十八而留。 二十八日乃逆，日退六千四百三十六分，八十七日退十二度、分二百四。 又留二十八日。 初日行四千一百八十八分，日益疾七十分，百一十日亦行十八度、分四万七百三十八而伏。
At appearance: initial daily motion 10,118 parts, daily retardation increasing by 70 parts; in 110 days travels 18 degrees, fraction 40,738 then stations. After twenty-eight days retrogrades, daily retreat 6,436 parts; in 87 days retreats 12 degrees, fraction 204. Again stations for twenty-eight days. Initial daily motion 4,188 parts, daily acceleration increasing by 70 parts; in 110 days also travels 18 degrees, fraction 40,738 then becomes hidden.
153
火数，三千六百三十七万七千五百九十五。
Fire number: 36,377,595.
154
伏半平，三百三十七万九千三百二十七半。
Hidden half-mean: 3,379,327 and a half.
155
复日，七百七十九； 余，四万一千九百一十九。
Return day: 779; Remainder: 41,919.
156
岁再，残日，四十九； 余，万九千一百六。
Twice per year, residual day: 49; Remainder: 19,106.
157
见去日，十六度。
Appearance distance from sun: 16 degrees.
158
平见，在雨水前，以十九乘去大寒日：清明前，又十八乘去雨水日，增雨水所乘者； 夏至后，以十六乘去处暑日； 小满后，又十五日； 寒露前，以十八乘去白露日； 小雪前，又十七乘去寒露日，增寒露所乘者； 大雪后，二十九乘去大寒日，为减，小雪至大雪减二十五日。
Mean appearance: before Rain Water, multiply by nineteen the distance from Great Cold day; before Clear Bright, also multiply by eighteen the distance from Rain Water day, increasing the Rain Water multiplier; After summer solstice, multiply by sixteen the distance from End of Heat day; After Lesser Fullness, also fifteen days; Before Cold Dew, multiply by eighteen the distance from White Dew day; Before Lesser Snow, also multiply by seventeen the distance from Cold Dew day, increasing the Cold Dew multiplier; After Great Snow, multiply by twenty-nine the distance from Great Cold day as decrease; from Lesser Snow to Great Snow decrease twenty-five days.
159
见，初在冬至，则二百三十六日行百五十八度，以后日度随其日数增损各一； 尽三十日，一日半损一； 又八十六日，二日损一； 复三十八日，同； 又十五日，三日损一； 复十二日，同； 又三十九日，三日增一； 又二十四日，二日增一； 又五十八日，一日增一； 复三十三日，同； 又三十日，二日损一，还终至冬至，二百三十六日行百五十八度。 其立春尽春分，夏至尽立夏，八日减一日； 春分至立夏，减六日； 立秋至秋分，减五度，各其初行日及度数。 白露至寒露，初日行半度，四十日行二十度。 以其残日及度，计充前数，皆差行，日益迟二十分，各尽其日度乃迟，初日行分二万二千六百六十九，日益迟一百一十分，六十一日行二十五度、分万五千四百九。 初减度五者，于此初日加分三千八百二十三、篾十七； 以迟日为母，尽其迟日行三十度，分同，而留十三日。
At appearance: initially at winter solstice, then 236 days travel 158 degrees; afterward daily degrees follow day count, increasing or decreasing by one each; After thirty days, decrease one every one and a half days; Also eighty-six days, decrease one every two days; Again thirty-eight days, unchanged; Also fifteen days, decrease one every three days; Again twelve days, unchanged; Also thirty-nine days, increase one every three days; Also twenty-four days, increase one every two days; Also fifty-eight days, increase one each day; Again thirty-three days, unchanged; Also thirty days, decrease one every two days, returning to winter solstice—236 days travel 158 degrees. Start of Spring to spring equinox, Start of Summer to Start of Summer, decrease one day every eight days; Spring equinox to Start of Summer, decrease six days; Start of Autumn to autumn equinox, decrease five degrees—each initial travel day and degree number. White Dew to Cold Dew, initial daily motion half a degree, forty days travel twenty degrees. With residual days and degrees, calculate to fill prior number—all differential motion, daily retardation increasing by 220 parts; initial daily fraction 22,669, daily retardation increasing by 110 parts; in 61 days travels 25 degrees, fraction 15,409. Initial five-degree decrease, at this initial day add fraction 3,823, hair 17; Using retardation days as denominator, exhaust retardation daily travel of thirty degrees, same fraction, then stations thirteen days.
160
前减日分于二留，乃逆，日退分万二千五百二十六，六十三日退十六度、分四万二千八百三十四。 又留十三日而行，初日万六千六十九，日益疾百一十分，六十一日行二十五度、分万五千四百九。 立秋尽秋分，增行度五，加初日分同前，更疾。 在冬至则二百一十三日行百三十五度； 尽三十六日，一日损一； 又二十日，二日损一； 复二十四日，同； 又五十四日，三日日增一； 又十二日，二日增一； 又四十二日，一日增一； 又十四日，一日增一半； 又十二日，增一； 复四十五日，同； 又一百六日，二日损一，亦终冬至二百一十三日，行百三十五度。
Prior subtracted day fraction at second station, then retrogrades, daily retreat fraction 12,526; in 63 days retreats 16 degrees, fraction 42,834. Again stations thirteen days and travels, initial daily 16,069, daily acceleration increasing by 110 parts; in 61 days travels 25 degrees, fraction 15,409. Start of Autumn to autumn equinox, increase travel five degrees, add initial day fraction same as prior, more rapid. At winter solstice then 213 days travel 135 degrees; After thirty-six days, decrease one each day; Also twenty days, decrease one every two days; Again twenty-four days, unchanged; Also fifty-four days, increase one every three days; Also twelve days, increase one every two days; Also forty-two days, increase one each day; Also fourteen days, increase one and a half each day; Also twelve days, increase one; Again forty-five days, unchanged; Also 106 days, decrease one every two days, also ending at winter solstice—213 days travel 135 degrees.
161
前增行度五者，于此亦减五度，为疾日及数。 其立夏尽夏至初，日行半度，六十日行三十度。 夏至尽立秋，亦初日行半度，四十日行二十度。 其残亦计充如前，皆差行，日益疾二十分，各尽其日度而伏。
Prior increased travel five degrees, here also decrease five degrees, as rapid day and number. Start of Summer to summer solstice initially, daily travel half a degree, sixty days travel thirty degrees. Summer solstice to Start of Autumn, also initial daily travel half a degree, forty days travel twenty degrees. Residual also calculate to fill as prior—all differential motion, daily acceleration increasing by 220 parts; each exhausts daily degrees then becomes hidden.
162
土数，千七百六十三万五千五百九十四。
Earth number: 17,635,594.
163
伏半平，八十六万四千九百九十五。
Hidden half-mean: 864,995.
164
复日，三百七十八； 余，四千一百六十二。
Return day: 378; Remainder: 4,162.
165
岁一，残日，十二； 余，三万九千三百九十九半。
One per year, residual day: 12; Remainder: 39,399 and a half.
166
见去日，十六度半。
Appearance distance from sun: 16½ degrees.
167
平见，在大暑前，以七乘去小满日； 寒露后，九乘去小雪日，为加，大暑至寒露加八日。 小寒前，以九乘去小雪日； 雨水后，以四乘去小满日； 立春后，又三乘去雨水日，增雨水所乘者，为减，小寒至立春减八日。
Mean appearance: before Great Heat, multiply by seven the distance from Lesser Fullness day; After Cold Dew, multiply by nine the distance from Lesser Snow day as increase; Great Heat to Cold Dew add eight days. Before Lesser Cold, multiply by nine the distance from Lesser Snow day; After Rain Water, multiply by four the distance from Lesser Fullness day; After Start of Spring, also multiply by three the distance from Rain Water day, increasing the Rain Water multiplier, as decrease; Lesser Cold to Start of Spring decrease eight days.
168
见，日行分四千三百六十四，八十日行七度、分二万二千六百一十二而留三十九日乃逆，日退分二千八百二十，百三日退六度、分万五百九十六。 又留三十九日，亦行分日四千三百六十四，八十日行七度、分二万二千六百一十二而伏。
At appearance: daily travel fraction 4,364; in 80 days travels 7 degrees, fraction 22,612 then stations thirty-nine days then retrogrades, daily retreat fraction 2,820; in 103 days retreats 6 degrees, fraction 10,596. Again stations thirty-nine days, also daily travel fraction 4,364; in 80 days travels 7 degrees, fraction 22,612 then becomes hidden.
169
金数，二千七百二十三万六千二百八。
Metal number: 27,236,208.
170
晨伏半平，百九十五万七千一百四。
Dawn hidden half-mean: 1,957,104.
171
复日，五百八十三； 余，四万二千七百五十六。
Return day: 583; Remainder: 42,756.
172
岁一，残日，二百一十八； 余，三万一千三百四十九半。
One per year, residual day: 218; Remainder: 31,349 and a half.
173
夕见伏，二百五十六日。
Evening appearance and hiding: 256 days.
174
晨见伏，三百二十七日； 余与复同。
Dawn appearance and hiding: 327 days; Remainder same as return.
175
见去日，十一度。
Appearance distance from sun: 11 degrees.
176
夕平见，在立秋前，以六乘去芒种日； 秋分后，以五乘去小雪日； 小雪后，又四乘去大雪日，增小雪所乘者，为加，立秋至秋分加七日。 立春前，以五乘去大雪日； 雨水前，又四乘去立春日，增立春所乘者； 清明后，以六乘去芒种日，为减，雨水至清明减七日。
Evening mean appearance: before Start of Autumn, multiply by six the distance from Grain in Ear day; After autumn equinox, multiply by five the distance from Lesser Snow day; After Lesser Snow, also multiply by four the distance from Great Snow day, increasing the Lesser Snow multiplier, as increase; Start of Autumn to autumn equinox add seven days. Before Start of Spring, multiply by five the distance from Great Snow day; Before Rain Water, also multiply by four the distance from Start of Spring day, increasing the Start of Spring multiplier; After Clear Bright, multiply by six the distance from Grain in Ear day as decrease; Rain Water to Clear Bright decrease seven days.
177
晨平见，在小寒前，以六乘去冬至日； 立春前，又五乘去小寒日，增小寒所乘者； 芒种前，以六乘去夏至日； 立夏前，又五乘去芒种日，增芒种所乘者，为加，立春至立夏加五日。 小暑前，以六乘去夏至日； 立秋前，又五乘去小暑日； 增小暑所乘者； 大雪后，以六乘去冬至日； 立冬后，又五乘去大雪日，增大雪所乘者，为减，立秋至立冬减五日。
Dawn mean appearance: before Lesser Cold, multiply by six the distance from winter solstice day; Before Start of Spring, also multiply by five the distance from Lesser Cold day, increasing the Lesser Cold multiplier; Before Grain in Ear, multiply by six the distance from summer solstice day; Before Start of Summer, also multiply by five the distance from Grain in Ear day, increasing the Grain in Ear multiplier, as increase; Start of Spring to Start of Summer add five days. Before Lesser Heat, multiply by six the distance from summer solstice day; Before Start of Autumn, also multiply by five the distance from Lesser Heat day; Increase the Lesser Heat multiplier; After Great Snow, multiply by six the distance from winter solstice day; After Start of Winter, also multiply by five the distance from Great Snow day, increasing the Great Snow multiplier, as decrease; Start of Autumn to Start of Winter decrease five days.
178
夕见，百七十一日行二百六度。 其谷雨至小满、白露至寒露，皆十日加一度； 小满至白露，加三度。 乃十二日行十二度。 冬至后，十二日减日度各一，雨水尽夏至，日度七； 夏至后六日增一。 大暑至立秋，还日度十二； 至寒露，日度二十二，后六日减一。 自大雪尽冬至，又日度十二而迟。 日益疾五百二十分，初日行分二万三千七百九十一、篾三十五，行日为母，四十三日行三十二度。
Evening appearance: in 171 days travels 206 degrees. From Grain Rain to Lesser Fullness, White Dew to Cold Dew, add one degree every ten days; Lesser Fullness to White Dew, add three degrees. Then twelve days travel twelve degrees. After winter solstice, decrease daily degree by one every twelve days; Rain Water to summer solstice, daily degree seven; After summer solstice increase one every six days. Great Heat to Start of Autumn, restore daily degree twelve; To Cold Dew, daily degree twenty-two, afterward decrease one every six days. From Great Snow to winter solstice, again daily degree twelve then slows. Daily acceleration increasing by 520 parts; initial daily fraction 23,791, hair 35; travel days as denominator; in 43 days travels 32 degrees.
179
前加度者，此依减之。 留九日乃逆，日退太半度，九日退六度，而夕伏晨见。 日退太半度，九日退六度。 复留，九日而行，日益迟五百二十分，初日行分四万五千六百三十一、篾三十五，四十三日行三十二度。 芒种至小暑，大雪至立冬，十五日减一度； 小暑至立冬，减二度。 又十二日行十二度。 冬至后，十五日增日度各一。 惊蛰至春分，日度十七，后十五日减一，尽夏至，还日度十二。 后六日减一，至白露，日度皆尽。 霜降后，五日增一，尽冬至，又日度十二。 乃疾，百七十一日行二百六度。 前减者，此亦加之，而晨伏。
Prior added degrees, here decrease accordingly. Stations nine days then retrogrades, daily retreat greater half degree; in nine days retreats six degrees, then evening hidden and dawn appearance. Daily retreat greater half degree, nine days retreat six degrees. Again stations, nine days and travels, daily retardation increasing by 520 parts; initial daily fraction 45,631, hair 35; in 43 days travels 32 degrees. Grain in Ear to Lesser Heat, Great Snow to Start of Winter, decrease one degree every fifteen days; Lesser Heat to Start of Winter, decrease two degrees. Again twelve days travel twelve degrees. After winter solstice, increase daily degree by one every fifteen days. Awakening of Insects to spring equinox, daily degree seventeen, afterward decrease one every fifteen days, exhaust summer solstice, restore daily degree twelve. Afterward decrease one every six days, to White Dew, daily degrees all exhausted. After Frost Descent, increase one every five days, exhaust winter solstice, again daily degree twelve. Then rapid, in 171 days travels 206 degrees. Prior decreased, here also increase, then dawn hidden.
180
水数，五百四十万五千六。
Water number: 5,405,006.
181
晨伏半平，七十九万九十九。
Dawn hidden half-mean: 790,099.
182
复日，百一十五； 余，四万九百四十六。
Return day: 115; Remainder: 40,946.
183
夕见伏，五十一日。
Evening appearance and hiding: 51 days.
184
晨见伏，六十四日； 余与复同。
Dawn appearance and hiding: 64 days; Remainder same as return.
185
见去日，十七度。
Appearance distance from sun: 17 degrees.
186
夕应见，在立秋后小雪前者不见； 其白露前立夏后，时有见者。
Evening should appear: after Start of Autumn and before Lesser Snow not visible; Before White Dew and after Start of Summer, sometimes visible.
187
晨应见，在立春后小满前者不见； 其惊蛰前立冬后，时有见者。
Dawn should appear: after Start of Spring and before Lesser Fullness not visible; Before Awakening of Insects and after Start of Winter, sometimes visible.
188
夕见，日行一度太，十二日行二十度。 小暑至白露，行度半，十二日行十八度，乃八日行八度。 大暑后，二日去度一，讫十六日，而日度俱尽。 而迟，日行半度，四日行二度。 益迟，日行少半度，三日行一度。 前行度半者，去此益迟。 乃留四日而夕伏晨见，留四日，为日行少半度，三日行一度。 大寒至惊蛰，无此行，更疾，日行半度； 四日行二度； 又八日行八度。 亦大寒后，二日去度一； 讫十六日，亦日度俱尽。 益疾，日行一度太，十二日行二十度。 初无迟者，此行度半，十二日行十八度而晨伏。
Evening appearance: daily travel one and a half degrees, twelve days travel twenty degrees. Lesser Heat to White Dew, travel half a degree, twelve days travel eighteen degrees, then eight days travel eight degrees. After Great Heat, remove one degree every two days, exhaust sixteen days, then daily degrees all exhausted. Then slow, daily travel half a degree, four days travel two degrees. Increasingly slow, daily travel lesser half degree, three days travel one degree. Prior travel half degree, remove this increasingly slow. Then stations four days then evening hidden and dawn appearance, stations four days, daily travel lesser half degree, three days travel one degree. Great Cold to Awakening of Insects, no this travel, more rapid, daily travel half a degree; Four days travel two degrees; Again eight days travel eight degrees. Also after Great Cold, remove one degree every two days; Exhaust sixteen days, also daily degrees all exhausted. Increasingly rapid, daily travel one and a half degrees, twelve days travel twenty degrees. Initially without slow phase, this travel half degree, twelve days travel eighteen degrees then dawn hidden.
189
推星平见术：
Method for calculating planetary mean appearance:
190
各以伏半减积半实，乃以其数去之； 残返减数，满气日法为日，不满为余，即所求年天正冬至后平见日余。 金、水满晨见伏日者，去之，晨平见。 求平见月日：以冬至去定朔日、余，加其后日及余，满复日又去，起天正月，依定大小朔除之，不尽算外日，即星见所在。 求后平见，因前见去其岁一、再，皆以残日加之，亦可。 其复日，金水准以晨夕见伏日，加晨得夕，加夕得晨。
Each subtract hidden half from accumulated half dividend, then remove by its number; Remainder reverse subtract number, when full of qi day divisor obtain days, remainder otherwise—year's Heaven-regular winter solstice after mean appearance day and remainder. Metal and Water: when full of dawn appearance hidden days remove—dawn mean appearance. Seek mean appearance month day: remove fixed new moon day and remainder from winter solstice, add following days and remainder, remove when full of return day, start Heaven first month, follow fixed long-short new moon and remove, remainder beyond count gives day—star appearance location. Seek following mean appearance, follow prior appearance and remove its year once or twice, all add residual days—also acceptable. Return day: Metal and Water follow dawn and evening appearance hidden days, add dawn obtain evening, add evening obtain dawn.
191
求常见日：以转法除所得加减者，为日； 其不满，以余通乘之，为余； 并日，皆加减平见日、余，即为常见日及余。
Seek common appearance day: rotation divisor divide obtained add-subtract as day; Not full, remainder common multiply as remainder; Combine days, all add-subtract mean appearance day and remainder—this is common appearance day and remainder.
192
求定见日：以其先后已通者，先减后加常见日，即得定见日余。
Seek fixed appearance day: using converted advance-retardation, first subtract then add common appearance day—obtain fixed appearance day and remainder.
193
求星见所在度：
Seek star appearance location degree:
194
置星定见、其日夜半所在宿度及分，以其日先后余，分前加、分后减气日法，而乘定见余，气日法而一所得加夜半度分，乃以星初见去日度数，晨减夕加之，即星初见所在宿度及分。
Set star fixed appearance that day's midnight location lodge degree and fraction; that day's advance-retardation remainder, increase before fraction and decrease after fraction by qi day divisor and multiply fixed appearance remainder, one of qi day divisor and add to midnight degree fraction; then star initial appearance distance from sun degrees, subtract for dawn and add for evening—star initial appearance location lodge degree and fraction.
195
求次日：
Seek next day:
196
各加一日所行度及分。 其有益疾、迟者副置一日行分，各以其分疾增、迟损，乃加之。 有篾者，满法从分，其母有不等，齐而进退之。 留即因前，逆则依减入虚去分，逆出先加。 皆以篾法除，为转分； 其不尽者，仍谓之篾，各得每日所在知去日度。 增以日所入先后分，定之。 诸行星度求水其外内，准月行增损黄道而步之； 不明者，依黄道而求所去日度。 先后分亦分明前加后减。 其金、火诸日度，计数增损定之者。 其日少度多，以日减度之残者，与日多度少之度，皆度法乘之，日数而一，所得为分。 不满篾，以日数为母。 日少者以分并减之一度，日多者直为度分，即皆一日平行分。 其差行者，皆减所行日数一，乃半其益疾、益迟分而乘之，益疾以减、益迟以加一日平行分，皆初日所行分。 有计日加减，而日数不满未得成度者，以气日法若度法乘，见已所行日即日数除之，所得以增损其气日疾法，为日及度。 其不成者，亦即为篾。 其木、火、土，晨有见而夕有伏； 金、水即夕见还夕伏，晨见即晨伏。 然火之初行及后疾，距冬至日计日增损日度者，皆当先置从冬至日余数，累加于位上，以知其去冬至远近，乃以初见与后疾初日去冬至日数而增损定之，而后依其所直日度数行之也。
Each add one day's travel degree and fraction. Those with increasing rapid and slow motion, set aside one day's travel fraction, increase for rapid and decrease for slow by its fraction, then add. Having hairs, when full of divisor advance fraction; denominators unequal, equalize and carry forward or borrow. At station follow prior, retrograde then decrease by entering void and remove fraction, retrograde exit first add. All divide by hair divisor, as rotation fraction; Not exhausted still called hair, each obtain daily location knowing distance from sun in degrees. Adjust by daily entering advance-retardation fraction to fix it. All planetary degrees seek inner and outer water limits, follow moon travel increase-decrease on the ecliptic and step; When unclear, follow the ecliptic and seek distance from sun in degrees. Advance-retardation fraction also: increase before fraction, decrease after fraction. Metal and Fire daily degrees, count increase-decrease and fix. When days are few and degrees many, day subtracts degree remainder; when days many and degrees few, multiply all by degree divisor, one of day number, result is fraction. Not full hair, use day number as denominator. Few days combine fraction and subtract one degree, many days directly as degree fraction—all one day mean travel fraction. Differential travelers, all subtract travel day number by one, then halve increasing rapid and increasing slow fraction and multiply; increasing rapid subtract, increasing slow add to one day mean travel fraction—all initial day travel fraction. Having counted day add-subtract, when day number not full and degree not yet formed, multiply by qi day divisor or degree divisor, use already traveled days as day number and divide, result increase-decrease its qi day rapid divisor, as day and degree. Not forming, also directly as hair. Wood, Fire, and Earth—dawn has appearance and evening has hiding; Metal and Water—evening appearance returns to evening hiding, dawn appearance then dawn hiding. Yet Fire's initial travel and afterward rapid phase, from winter solstice day counting daily increase-decrease of degrees—all should first set the remainder number from winter solstice day, accumulate on the position, know distance from winter solstice, then by initial appearance and afterward rapid initial day distance from winter solstice day number increase-decrease and fix, afterward follow its corresponding daily degree number and travel.