1
嗚呼，五代禮樂文章，吾無取焉。 其後世有欲知之者，不可以遺也。 作《司天職方考》。 ○司天考第一
Alas, I find nothing in the rites, music, or literary culture of the Five Dynasties that I would wish to take as a model. Still, later generations who wish to understand that era must not be left without a record. I therefore wrote the "Examination of the Director of Heaven and the Director of Regions." ○ Part One: Examination of the Director of Heaven
2
司天掌日月星辰之象。 周天一歲，四時，二十四氣，七十二候，行十日十二辰，以為歷。 而謹察其變者，以為占。 占者，非常之兆也，以驗吉兇，以求天意，以覺人事，其術藏於有司。 歷者，有常之數也，以推寒暑，以先天道，以勉人事，其法信於天下。 術有時而用，法不可一日而差。 差之毫厘，則亂天人之序，乖百事之時，蓋有國之所重也。 然自堯命羲、和見於《書》，中星閏余，略存其大法。 而三代中間千有餘歲，遺文曠廢，《六經》無所述。 而孔子之徒，亦未嘗道也。 至於後世，其學一出於陰陽之家，其事則重，其學則末。 夫天人之際，遠哉微矣，而使一藝之士，布算積分，上求數千萬歲之前，必得甲子朔旦夜半冬至，而日、月、五星皆會於子，謂之上元，以為歷始。 蓋自漢而後，其說始詳見於世，其源流所自止於如此。 是果堯、舜、三代之法歟？ 皆不可得而考矣。 然自是以來，歷家之術，雖世多不同，而未始不本於此。
The Director of Heaven was charged with observing the sun, moon, stars, and constellations. One circuit of heaven marks a year, divided into four seasons, twenty-four qi, seventy-two hou, and measured by the ten-day week and twelve double-hours—this is what the calendar records. Careful observation of their variations yields material for divination. Divination reads extraordinary signs to test good and ill fortune, discern Heaven's will, and guide human affairs; its methods were entrusted to the appropriate offices. The calendar rests on fixed numerical cycles to project cold and heat, anticipate the heavenly pattern, and regulate human affairs; its methods commanded universal trust. Astronomical techniques may be applied as needed, yet calendrical rules must not deviate even for one day. A deviation no wider than a hair's breadth can throw Heaven and humanity out of order and upset the timing of every affair of state—which is why rulers treated it with such gravity. Still, from Yao's appointment of Xi and He in the Documents, the cardinal stars and intercalary remainders preserve at least the broad outline of the original method. Yet across the thousand-odd years of the Three Dynasties, the transmitted records fell into neglect, and the Six Classics say nothing on the subject. Nor did Confucius and his disciples ever address it. In later times the subject passed entirely into the hands of the yin-yang specialists: the office remained weighty, but the learning had become a minor craft. The boundary between Heaven and humanity is immeasurably remote and subtle, yet technicians of a single craft would lay out counting-rods and accumulated fractions, reaching back tens of millions of years until they fixed a jiazi year in which new moon, dawn, midnight, and winter solstice aligned, with sun, moon, and the five planets all gathered at zi—the "supreme origin" taken as the calendar's starting point. Only from the Han onward did this doctrine appear in full; its origins extend no further back than this. Can this really have been the method of Yao, Shun, and the Three Dynasties? None of it can be verified by evidence. Yet from that time on, though calendrical schools differed from one generation to the next, none ever departed from this foundation.
3
五代之初，因唐之故，用《崇玄曆》。 至晉高祖時，司天監馬重績始更造新歷，不復推古上元甲子冬至七曜之會，而起唐天寶十四載乙未為上元，用正月雨水為氣首。 初，唐建中時，術者曹士蒍始變古法，以顯慶五年為上元，雨水為歲首，號《符天曆》。 然世謂之小歷，只行於民間。 而重績乃用以為法，遂施於朝廷，賜號《調元曆》。 然行之五年，輒差不可用，而復用《崇玄曆》。 周廣順中，國子博士王處訥私撰《明玄曆》於家。 民間又有《萬分曆》，而蜀有《永昌曆》、《正象曆》，南唐有《齊政曆》。 五代之際，歷家可考見者止於此。 而《調元曆》法既非古，《明玄》又止藏其家，《萬分》止行於民間，其法皆不足紀。 而《永昌》《正象》《齊政曆》，皆止用於其國，今亦亡，不復見。
At the outset of the Five Dynasties period, the court continued Tang practice and adopted the Chongxuan Calendar. Under Jin Emperor Gaozu, Director of Heaven Ma Chongji devised a new calendar that abandoned retrojection to the ancient supreme origin of jiazi, winter solstice, and the seven luminaries in conjunction, and instead took yiwei, the fourteenth year of Tang's Tianbao era, as the supreme origin, with Rain Water in the first month as the year's qi-head. Earlier, during Tang's Jianzhong era, the technician Cao Shibian had first broken with ancient practice, taking the fifth year of Xianqing as the supreme origin and Rain Water as the year's beginning—the calendar known as the Futian Calendar. The world called it the "minor calendar," and it circulated only among commoners. Chongji adopted it as the official standard, and the court promulgated it under the title Tiaoyuan Calendar. After only five years it had drifted out of alignment and was abandoned, and the Chongxuan Calendar was restored. During Later Zhou's Guangshun era, National University academician Wang Chune privately composed the Mingxuan Calendar at home. Commoners also used the Wanfen Calendar; Shu had the Yongchang and Zhengxiang calendars; Southern Tang had the Qizheng Calendar. For the Five Dynasties period, the calendrical traditions that can be documented end here. The Tiaoyuan method was already unorthodox; the Mingxuan never left Wang's household; the Wanfen never left the folk sphere—none of their methods merit recording. The Yongchang, Zhengxiang, and Qizheng calendars were each confined to its own state; all are now lost and no longer extant.
4
世宗即位，外伐僭叛，內修法度。 端明殿學士王樸，通於歷數，乃詔樸撰定。 歲餘，樸奏曰：
When Emperor Shizong came to the throne, he campaigned abroad against rebel regimes while reforming laws and institutions at home. Hanlin academician Wang Pu, who was expert in calendrical reckoning, received an imperial order to draft a new calendar. After more than a year, Pu submitted a memorial that read:
5
◎臣聞聖人之作也，在乎知天人之變者也。 人情之動，則可以言知之； 天道之動，則當以數知之。 數之為用也，聖人以之觀天道焉。 歲月日時，由斯而成； 陰陽寒暑，由斯而節； 四方之政，由斯而行。 夫為國家者，履端立極，必體其元； 布政考績，必因其歲； 禮動樂舉，必正其朔； 三農百工，必順其時； 五刑九伐，必順其氣； 庶務有為，必從其日月。 是以聖人受命，必治歷數。 故五紀有常度，庶征有常應，正朔行之於天下也。
◎ Your servant has heard that what the sage establishes rests on understanding the transformations of Heaven and humanity. The movements of human affairs can be grasped through language; but the movements of the Heavenly Way must be grasped through numbers. It is through number that the sage observes the Heavenly Way. Years, months, days, and hours take their form thereby; yin and yang, cold and heat, are regulated thereby; and the governance of the four quarters proceeds thereby. Whoever governs a state, in inaugurating the year and establishing the cosmic pole, must embody its origin; in promulgating policy and assessing achievement, must follow the year's cycle; in performing rites and music, must align the new moon; farmers and artisans alike must follow its seasons; punishments and punitive campaigns must follow its qi; every affair of state must follow its sun and moon. Therefore when the sage receives the Mandate, he must set the calendar in order. Thus the five chronologies hold their fixed measures, the various portents their fixed responses, and the correct calendar is promulgated throughout the realm.
6
自唐之季，凡歷數朝，亂日失天，垂將百載，天之歷數，汨陳而已。 陛下順考古道，寅畏上天，咨詢庶官，振舉墜典。 臣雖非能者，敢不奉詔。 乃包萬象以為法，齊七政以立元，測圭箭以候氣，審朓朒以定朔，明九道以步月，校遲疾以推星，考黃道之斜正，辨天勢之升降，而交蝕詳焉。
From the end of Tang through successive dynasties, the calendar fell into chaos and Heaven's order was lost; for nearly a century the heavenly reckoning lay in ruins. Your Majesty has followed the ancient Way, stood in reverent awe of Heaven, consulted his officials, and revived fallen institutions. Though your servant lacks ability, how could I dare refuse the imperial command? I have embraced the myriad phenomena as method, aligned the seven regulators to establish the origin, measured the gnomon and clepsydra to observe the qi, examined lunar elongation to fix the new moon, clarified the nine paths to track the moon, calibrated planetary motion to project the stars, investigated the obliquity of the ecliptic, distinguished the rising and falling of celestial disposition, and rendered eclipses precise.
7
夫立天之道，曰陰與陽。 陰陽各有數，合則化成矣。 陽之策三十六，陰之策二十四。 奇偶相命，兩陽三陰，同得七十二。 同則陰陽之數合。 七十二者，化成之數也。 化成則謂之五行之數。 五之，得期數。 過之者謂之氣盈，不及者謂之朔虛。 至於應變分用，無所不通。 故以七十二為經法。 經者，常用之法也。 百者，數之節也，隨法進退，不失舊位，故謂之通法。 以通法進經法，得七千二百，謂之統法。 自元入經，先用此法，統歷之諸法也。 以通法進統法，得七十二萬。 氣朔之下，收分必盡，謂之全率。 以通法進全率，得七千二百萬，謂之大率，而元紀生焉。 元者，歲、月、日、時皆甲子； 日、月、五星合在子； 當盈縮、先後之中，所謂七政齊矣。
The Way established for Heaven is yin and yang. Yin and yang each have their numbers; combined, they produce transformation and completion. The yang tally is thirty-six; the yin tally, twenty-four. Odd and even are paired by command: two yang and three yin alike yield seventy-two. When they match, the numbers of yin and yang unite. Seventy-two is the number of transformation and completion. Transformation and completion constitute the numbers of the Five Phases. Multiply by five to obtain the period number. What exceeds is called qi surplus; what falls short is called new-moon deficit. In responding to change and apportioning use, nothing lies beyond its reach. Therefore seventy-two serves as the canonical method. The canonical method is that in constant use. One hundred is the node of numbers; advancing and retreating with the method without losing position—hence the universal method. Advancing the canonical by the universal method yields 7,200—the comprehensive method. From the origin through the canonical, this method is applied first—it governs all methods of the comprehensive calendar. Advancing the comprehensive by the universal method yields 720,000. Beneath qi and new moon, when collected fractions are fully exhausted, this is the complete rate. Advancing the complete rate by the universal method yields 72 million—the great rate, from which the origin and era are born. The origin is when year, month, day, and hour are all jiazi; sun, moon, and the five planets all gather at zi; standing at the mean of expansion and contraction, precedence and lag—what is called the seven regulators in alignment.
8
古者植圭於陽城，以其近洛也。 蓋尚慊其中，乃在洛之東偏。 開元十二年，遣使天下候影，南距林邑，北距橫野，中得浚儀之嶽臺，應南北弦，居地之中。 大周建國，定都於汴。 樹圭置箭，測嶽臺晷漏，以為中數。 晷漏正，則日之所至，氣之所應，得之矣。
In antiquity the gnomon was erected at Yangcheng because it lay near the Luo. Still dissatisfied with its centrality, they placed it on the Luo's eastern margin. In Kaiyuan 12, envoys were dispatched empire-wide to observe shadows, from Linyi in the south to Hengye in the north; at the center they found the Yuetai at Junyi, on the north-south meridian at the earth's midpoint. When Great Zhou established its state, it fixed the capital at Bian. The gnomon was erected and clepsydra arrows set; the Yuetai sundial and water-clock were measured to establish the central reckoning. When sundial and water-clock are correct, the sun's position and the qi's response can be determined.
9
日月皆有盈縮。 日盈月縮，則後中而朔。 月盈日縮，則先中而朔。 自古朓朒之法，率皆平行之數； 入歷既有前次，而又衰稍不倫。 《皇極》舊術，則迂回而難用。 降及諸歷，則疏遠而多失。 今以月離朓朒，隨歷校定，日躔朓朒，臨用加減。 所得者，入離定日也。 一日之中，分為九限。 每限損益，衰稍有倫。 朓朒之法，可謂審矣。
Both sun and moon exhibit expansion and contraction. When the sun expands and the moon contracts, the median lags and the new moon follows. When the moon expands and the sun contracts, the median comes early and the new moon precedes. From antiquity, methods for lunar elongation relied on uniform-motion numbers; though entering the calendar already had prior sequences, the gradations of increase and decrease were inconsistent. The old Huangji method was circuitous and impractical. Later calendars were crude and prone to error. Now lunar elongation is calibrated against the calendar, and solar motion receives additions and subtractions at the point of application. What is obtained is the day fixed by entering departure. A single day is divided into nine limits. Each limit has its increase and decrease; the gradations follow a coherent order. The method for lunar elongation may truly be called precise.
10
赤道者，天之纮帶也。 其勢圜而平，紀宿度之常數焉。 黃道者，日軌也。 其半在赤道內，半在赤道外，去極二十四度。 當與赤道近，則其勢斜； 當與赤道遠，則其勢直。 當斜，則日行宜遲； 當直，則日行宜速。 故二分前後加其度，二至前後減其度。 九道者，月軌也。 其半在黃道內，半在黃道外，去極遠六度。 出黃道，謂之正交； 入黃道，謂之中交。 若正交在秋分之宿，中交在春分之宿，則比黃道益斜。 若正交在春分之宿，中交在秋分之宿，則比黃道反直。 若正交、中交在二至之宿，則其勢差斜。 故校去二至二分遠近，以考斜正，乃得加減之數。 自古雖有九道之說，蓋亦知而未詳，徒有祖述之文，而無推步之用。 今以黃道一周，分為八節； 一節之中，分為九道； 盡七十二道，而使日月無所隱其斜正之勢焉。 九道之法，可謂明矣。
The celestial equator is Heaven's girding belt. Circular and level in form, it records the constant lodge degrees. The ecliptic is the sun's track. Half lies within the equator, half outside, twenty-four degrees from the pole. When near the equator, its course is oblique; when far from the equator, its course is straight. When oblique, the sun's daily motion should be slow; when straight, the sun's daily motion should be fast. Accordingly, degrees are added before and after the equinoxes and subtracted before and after the solstices. The nine paths are the moon's orbit. Half lies within the ecliptic, half outside, six degrees farther from the pole. Emerging from the ecliptic is called the primary crossing; entering the ecliptic is called the median crossing. If the primary crossing falls at the autumn-equinox lodge and the median at the spring-equinox lodge, the orbit is more oblique than the ecliptic. If the primary crossing falls at the spring-equinox lodge and the median at the autumn-equinox lodge, the orbit is conversely straighter than the ecliptic. If both crossings fall at the solstice lodges, the orbit is moderately oblique. By calibrating distance from the solstices and equinoxes to test obliquity and rectitude, one obtains the numbers for addition and subtraction. Although the nine paths were discussed from antiquity, the tradition knew them only in outline—texts handed down by rote, without computational use. The ecliptic's full circuit is now divided into eight nodes; each node is divided into nine paths; yielding seventy-two paths in all, so that sun and moon conceal nothing of their oblique and straight courses. The method of the nine paths may truly be called clear.
11
星之行也，近日而疾，遠日而遲。 去日極遠，勢盡而留。 自古諸歷，分段失實，隆降無準； 今日行分尚多，次日便留； 自留而退，惟用平行，仍以入段行度為入歷之數； 皆非本理，遂至乖戾。 今校逐日行分積，以為變段。 然後自疾而漸遲，勢盡而留。 自留而行，亦積微而後多。 別立諸段變曆，以推變差，俾諸段變差，際會相合。 星之遲疾，可得而知之矣。
A planet moves swiftly near the sun and slowly when far from it. At greatest distance from the sun its momentum is spent and it stations. Ancient calendars divided segments inaccurately, with no standard for rise and fall; one day its motion-fraction is still large, the next day it stations; from station it retreats using only uniform motion, still taking segment motion-degrees as the calendar entry number; none of this accords with fundamental principle, and contradiction follows. Daily motion-fractions are now accumulated in sequence to form variable segments. Then from swift it gradually slows until momentum is spent and it stations. From station it proceeds again, accumulating minutely before becoming great. Separate variable segments and variable calendars project the variable differences so that segment differences align in meeting. Planetary slow and swift motion can thereby be known.
12
自古相傳，皆謂去交十五度以下，則日月有蝕。 殊不知日月之相掩，與暗虛之所射，其理有異。 今以日月徑度之大小，校去交之遠近，以黃道之斜正，天勢之升降，度仰視、旁視之分數，則交虧得其實矣。
Tradition held that within fifteen degrees of nodal departure, sun and moon would eclipse. They did not grasp that mutual masking of sun and moon and casting of the dark void follow different principles. By comparing sun and moon diameters, calibrating nodal distance, ecliptic obliquity, celestial disposition, and overhead versus side-view fractions, eclipse magnitude is obtained in reality.
13
臣考前世，無食神首尾之文。 近自司天卜祝小術，不能舉其大體，遂為等接之法。 蓋從假用，以求徑捷，於是乎交有逆行之數。 後學者不能詳知，因言歷有九曜，以為註歷之常式。 今並削而去之。 謹以《步日》、《步月》、《歲星》、《步發斂》為四篇，合為《曆經》一卷，《曆》十一卷，《草》三卷，顯德三年《七政細行曆》一卷，以為《欽天曆》。
Your servant finds no earlier text on the head and tail of the eclipse spirit. Recently, minor techniques of the Director of Heaven and diviners, unable to grasp the whole, devised a method of equal connection. Adopted provisionally for convenience, it gave crossing a retrograde number. Later students, ignorant of the details, said the calendar had nine luminaries as the regular annotating form. All of these are now cut away. Respectfully, "Pacing the Sun," "Pacing the Moon," "Jupiter," and "Pacing the Collector" form four chapters, combined into one fascicle of the 《Calendar Classic》, eleven fascicles of 《Calendar》, three of 《Draft》, and one fascicle of the 《Detailed Motion of the Seven Regulators》 for Xiande 3, as the 《Qintian Calendar》.
14
昔在帝堯，欽若昊天。 陛下考歷象日月星辰，唐堯之道也。 天道玄遠，非微臣之所盡知。 世宗嘉之。 詔司天監用之，以明年正月朔旦為始。 《顯德欽天曆》
Of old, Emperor Yao reverently conformed to august Heaven. Your Majesty examines the calendar and the images of sun, moon, stars, and constellations—the Way of Yao and Tang. The Heavenly Way is profound and remote—beyond your humble servant's full knowledge. Emperor Shizong commended it. He ordered the Director of Heaven to adopt it, beginning from new moon and dawn of the first month of the following year. The 《Xiande Qintian Calendar》
15
演紀上元甲子，距今顯德三年丙辰，積七千二百六十九萬八千四百五十二算外。 《欽天》統法：七千二百。 《欽天》經法：七十二。
Projecting the era from supreme origin jiazi to the present Xiande 3 bingchen: accumulated 72,698,452 counts beyond. 《Qintian》 comprehensive method: 7,200. 《Qintian》 canonical method: 72.
16
《欽天》通法：一百。 《欽天》步日躔術歲率：二百六十二萬九千七百六十，四十。 軌率：二百六十二萬九千八百四十四，八十。
《Qintian》 universal method: 100. 《Qintian》 method for pacing the sun's motion, year rate: 2,629,760, 40. Track rate: 2,629,844, 80.
17
朔率：二十一萬二千六百二十，二十八。 歲策：三百六十五，一千七百六十，四十。 軌策：三百六十五，一千八百四十四，八十。 歲中：一百八十二，四千四百八十，二十。
New-moon rate: 212,620, 28. Year tally: 365, 1,760, 40. Track tally: 365, 1,844, 80. Year median: 182, 4,480, 20.
18
軌中：一百八十二，四千五百二十二，四十。 朔策：二十九，三千八百二十，二十八。 氣策：一十五，一千五百七十三，三十五。 象策：七，二千七百五十五，七。
Track median: 182, 4,522, 40. New-moon tally: 29, 3,820, 28. Qi tally: 15, 1,573, 35. Image tally: 7, 2,755, 7.
19
周紀：六十。 歲差：八十四，四十。
Cycle era: 60. Year difference: 84, 40.
20
辰則：六百； 八刻二十四分。 ◎赤道宿次
Double-hour rule: 600; eight quarters and twenty-four minutes. ◎ Red Path lodge sequence
21
鬥：二十六度。 牛：八度。 女：十二度。 虛：一十度少。 危：十七度。 室：十六度。 壁：九度。 北方七宿九十八度少。
Dipper: 26 degrees. Ox: 8 degrees. Girl: 12 degrees. Void: 10 and a fraction degrees. Rooftop: 17 degrees. Room: 16 degrees. Wall: 9 degrees. The seven northern lodges: 98 and a fraction degrees.
22
奎：十六度。 婁：十二度。 胃：十四度。 昴：十一度。 畢：十七度。 觜：一度。 參：一十度。 西方七宿八十一度。
Striding: 16 degrees. Bond: 12 degrees. Stomach: 14 degrees. Mao: 11 degrees. Net: 17 degrees. Turtle Beak: 1 degree. Shen: 10 degrees. The seven western lodges: 81 degrees.
23
井：三十三度。 鬼：三度。 柳：十五度。 星：七度。 張：十八度。 翼：十八度。 軫：十七度。 南方七宿一百一十一度。
Well: 33 degrees. Ghost: 3 degrees. Willow: 15 degrees. Star: 7 degrees. Extended Net: 18 degrees. Wings: 18 degrees. Chariot Shaft: 17 degrees. The seven southern lodges: 111 degrees.
24
角：十二度。 亢：九度。 氐：十五度。 房：五度。 心：五度。 尾：十八度。 箕：十一度。 東方七宿七十五度。 ◎中節
Horn: 12 degrees. Gullet: 9 degrees. Base: 15 degrees. Room: 5 degrees. Heart: 5 degrees. Tail: 18 degrees. Winnowing Basket: 11 degrees. The seven eastern lodges: 75 degrees. ◎ Median nodes
25
置歲率，以演紀上元距所求積年乘之，為氣積。 統法而一，為日。 盈周紀去之，命甲子算外，即天正中氣日辰及分秒也。 以氣策累加之，秒盈通法從分，分盈統法從日，日盈周紀去之，即各得次氣日辰及分秒也。
Set the year rate and multiply by accumulated years from the projecting era's supreme origin to the year sought to obtain qi accumulation. Divide by the comprehensive method to obtain days. Remove full cycle eras; count from jiazi beyond the calculation—the day, double-hour, and fractional parts of the heavenly median qi. Add cumulatively by the qi tally; seconds filling the universal method follow into minutes, minutes filling the comprehensive method into days, days filling the cycle era are removed—each yields the next qi's day, double-hour, and fractional parts.
26
朔弦望
New moon, first quarter, full moon
27
置氣積，以朔率去之，不盡為閏余。 用減氣積，為朔積。 統法而一，為日。 盈周紀去之，命甲子算外，即天正常朔日辰及分秒也。 以象策累加之，即各得弦望及次朔也。
Set the qi accumulation and remove by the new-moon rate; the remainder is intercalary surplus. Subtract it from qi accumulation to obtain new-moon accumulation. Divide by the comprehensive method to obtain days. Remove full cycle eras; count from jiazi beyond the calculation—the day, double-hour, and fractional parts of the heavenly normal new moon. Add cumulatively by the image tally to obtain each quarter, full moon, and successive new moon.
28
日躔入歷
Sun's motion entering the calendar
29
置歲率，以閏余減之，統法而一，為日。 歲中以下為盈； 以上，減去歲中為縮，即天正常朔加時所入也。 累加象策，滿歲中去之，盈縮互命，即四象所入也。
Set the year rate, subtract intercalary surplus, and divide by the comprehensive method to obtain days. Below the year median counts as expansion; above it, subtract the year median for contraction—the entry for the heavenly normal new moon's added time. Add cumulatively by the image tally; when full remove from the year; expansion and contraction alternate in naming—the four images' entry.
30
日躔朓朒
Sun's motion in lunar elongation
31
置加時入歷分秒，以其日損益率乘之，統法而一，損益其日朓朒數，為日躔朓朒定數。 ◎赤道日度
Set added-time entry calendar fractional parts; multiply by that day's increase-decrease rate; divide by the comprehensive method; adjust that day's elongation number for the sun's fixed elongation number. ◎ Red Path solar degrees
32
置氣積，以軌率去之，余統法而一，為度； 命赤道虛八算外，即天正中氣加時日躔赤道宿度及分秒也。 加歲中，以次命之，即夏至之宿也。 ◎黃道宿次
Set qi accumulation; remove by track rate; divide remainder by comprehensive method for degrees; count from Void 8 beyond the Red Path calculation—the sun's Red Path lodge degree and fractional parts at heavenly median qi's added time. Add the year median and name in sequence—the summer solstice lodge. ◎ Yellow Path lodge sequence
33
置二至日躔赤道宿度。 距前後每五度為限，初率八，每限減一，蓋九限，末率空，乃一度少強，亦限率空。 其半當四立之宿。 自後亦五度為限，初率空，每限增一，盡九限，末率八，殷二分之宿。 自二分至二至，亦如之。 各以限率乘所入限度，為分。 經法而一，為度。 二至前後各九限以減、二分前後各九限以加赤道宿，為黃道宿及分。 就其分為少、太、半之數。
Set the Red Path lodge degrees of solar motion at the two solstices. Every five degrees before and after as a limit; initial rate eight, each limit decreasing by one through nine limits, final rate zero; then one degree and a slight surplus, limit rate zero. Its half corresponds to the lodges of the four establishments. Thereafter five degrees as a limit; initial rate zero, each limit increasing by one through nine limits, final rate eight—the equinox lodges. From equinox to solstice, the same applies. Each multiplies limit rate by limit degrees entered for fractional parts. Divide by the canonical method to obtain degrees. Nine limits before and after solstices subtract from, and nine limits before and after equinoxes add to, Red Path lodges for Yellow Path lodges and fractional parts. From fractional parts derive slight, great, and half numbers.
34
黃道日度
Yellow Path solar degrees
35
置天正中氣加時日躔赤道宿度。 各與所入限率相乘，皆以統法通之； 所入限率乘其分，以從之。 經法而一，為分； 盈統法，為度。 用減赤道所躔，即天正中氣加時日躔黃道宿度及分也。 加歲中，以黃道宿次命之，即夏至加時日度及分也。
Set Red Path lodge degree of solar motion at heavenly median qi's added time. Each is multiplied by the limit rate entered, all extended through the comprehensive method; the limit rate entered multiplies its fractional parts and follows along. Divide by the canonical method for fractional parts; when full of comprehensive method, for degrees. Subtract from the Red Path position—the Yellow Path lodge degree and fractional parts at heavenly median qi's added time. Add year median; name by Yellow Path lodge sequence—solar degree and fractional parts at summer solstice added time.
36
午中日躔
Noon solar motion
37
置二至分，減去半法，為午後分； 不足，反減，為午前分。 以乘初日躔分，經法而一，午前以加、午後以減加時黃道日度，為午中日度及分也。 各以次日躔分加之，滿統法從度。 依宿次命之，即次日午中日躔也。
Set solstice fractional parts; subtract half the method for after-noon fractional parts; if insufficient, reverse subtract for before-noon fractional parts. Multiply first day's motion fractional parts; divide by canonical method; before noon add, after noon subtract from added-time Yellow Path degrees—for noon solar degrees. Add each next day's motion fractional parts; when full of comprehensive method follow into degrees. Name by lodge sequence—the next day's noon solar motion.
38
午中日躔入歷
Noon solar motion entering the calendar
39
置天正中氣午前分，便為午中入盈歷日分。 其在午後者，以午後分減歲中，為午中入縮歷日分。 累加一日，滿歲中即去之，盈縮互命，為每日午中入歷也。
Set heavenly median qi before-noon fractional parts as noon entry expansion calendar day fractional parts. If after noon, subtract after-noon fractional parts from year median for noon entry contraction calendar day fractional parts. Add one day cumulatively; when full of year median remove; expansion and contraction alternate—for each day's noon calendar entry.
40
嶽臺中晷
Yuetai median gnomon shadow
41
置午中入歷分，以其日損益率乘之，如統法而一，為分； 分十為寸。 用損益其下中晷數，為定數也。 ◎晨昏分
Set noon entry calendar fractional parts; multiply by that day's increase-decrease rate; as comprehensive method to one for fractional parts; ten fractional parts make one inch. Increase or decrease the lower median gnomon number for the fixed number. ◎ Dawn and dusk fractional parts
42
各置入歷分，以其日損益率乘之，如統法而一，用損益其下晨分，即所求晨定分也。 用損加、益減其下昏分，即所求昏定分也。 ◎日出入辰刻
Set entry calendar fractional parts; multiply by increase-decrease rate; as comprehensive method to one; adjust lower dawn fractional parts for dawn fixed fractional parts. Decrease to add and increase to subtract lower dusk fractional parts for dusk fixed fractional parts. ◎ Day entry and exit double-hours and quarters
43
置晨昏分，以一百八十加晨、減昏，為日出入分。 各以辰則除，為辰數； 余滿經法，為刻； 命辰數子正算外，則日出入辰刻也。 ◎晝夜刻
Set dawn and dusk fractional parts; add 180 to dawn and subtract from dusk for day entry-exit fractional parts. Each divide by double-hour rule for double-hour numbers; remainder filling canonical method for quarters; count double-hours from zi beyond calculation—day entry-exit double-hours and quarters. ◎ Day and night quarters
44
置日入分，以日出分減之，為晝分。 用減統法，為夜分。 各滿經法，為晝夜刻。 ◎五夜辰刻
Set day-entry fractional parts; subtract day-exit for day fractional parts. Subtract from comprehensive method for night fractional parts. Each fills canonical method for day and night quarters. ◎ Five night double-hours and quarters
45
置昏分，以辰則除，為辰數； 經法除，為刻數。 命辰數子正算外，即甲夜辰刻也。 倍晨分，五約之，為更用分。 又五約之，為籌用分。 用累加甲夜，滿辰則為辰，滿經法為刻，即各得五夜辰刻也。
Set dusk fractional parts; divide by double-hour rule for double-hour numbers; divide by canonical method for quarter numbers. Count from zi beyond calculation—the first night double-hours and quarters. Double dawn fractional parts; divide by five for watch-use fractional parts. Again divide by five for tally-use fractional parts. Add cumulatively to first night; when full of double-hour rule for hours, full of canonical method for quarters—five nights' double-hours and quarters.
46
昏曉中星
Median stars at dusk and dawn
47
置昏分，減去半統，用乘軌率，統法除之，為距中分。 盈統法，為度。 加午中日躔，為昏中星； 減之，為曉中星。 ◎赤道內外數
Set dusk fractional parts; subtract half comprehensive method; multiply by track rate; divide by comprehensive method for distance-from-median fractional parts. When full of comprehensive method, for degrees. Add to noon solar motion for dusk median star; subtract for dawn median star. ◎ Red Path inner and outer numbers
48
置入歷分，以其日損益率乘之，如統法而一，用損益其下內外數； 如不足損，則反損之； 內外互命，即得所求赤道內外定數也。 ◎九服距軌數
Set entry calendar fractional parts; multiply by increase-decrease rate; as comprehensive method to one; adjust lower inner-outer numbers; if insufficient to decrease, reverse the decrease; inner and outer alternate in naming—for sought Red Path inner-outer fixed number. ◎ Nine domains' distance-from-track numbers
49
置距嶽臺南北里數，以三百六十通之，為步。 一千七百五十六除之，用北加、南減二千五百一十三，為其地戴中數以赤道內外定數，內減、外加之，即九服距軌數也。
Set li distance north and south from Yuetai; extend through by 360 for paces. Divide by 1,756; north add, south subtract 2,513 for girdle-median number; with Red Path inner-outer fixed number, inner subtract outer add—for nine domains' distance-from-track number.
50
九服中晷
Nine domains' median gnomon
51
置距軌數，二十五乘之，一百三十七除，為天用分。 置之，以二十二乘，六約之，用減四千，為晷法。 又以天用分自相乘，如晷法而一，為地用分。 相從為晷分，分十為寸，即得其地中晷也。
Set distance-from-track number; multiply by 25; divide by 137 for heaven-use fractional parts. Set it; multiply by 22; divide by six; subtract 4,000 for gnomon method. Self-multiply heaven-use fractional parts; as gnomon method to one for earth-use fractional parts. Follow together for gnomon fractional parts; ten parts make one inch—that place's median gnomon.
52
九服刻漏
Nine domains' clepsydra gradations
53
經法通軌中而半之，用自相乘，如其地戴中數而一； 以乘二百六十三，經法除之，為漏法。 通軌中於上，置赤道內外數於下，以下減上，余用乘之； 盈漏法，為漏分。 赤道內以減、赤道外以加一千六百二十，為其地晨分。 減統法，為昏分。 置晨昏分，各如嶽臺術入之，即得其地日出入辰刻、五夜辰刻、昏曉中星也。
Extend comprehensive method through track median and halve; self-multiply; as that place's girdle-median number to one; multiply by 263; divide by canonical method for clepsydra method. Extend track median above; set Red Path inner-outer below; subtract below from above; multiply remainder; when full of clepsydra method, for clepsydra fractional parts. Within Red Path subtract, outside add 1,620 for that place's dawn fractional parts. Subtract from comprehensive method for dusk fractional parts. Set dawn and dusk fractional parts; each enter by Yuetai technique—for that place's day entry-exit, five nights, and dusk-dawn median stars.
54
《欽天》步月離術離率：一十九萬八千三百九十三，九。 交率：一十九萬五千九百二十七，九十七，五十六。 離策：二十七，三千九百九十三，九。
《Qintian》 method for pacing lunar departure, departure rate: 198,393, 9. Crossing rate: 195,927, 97, 56. Departure tally: 27, 3,993, 9.
55
交策：二十七，一千五百二十七，九十七，五十六。 望策：一十四，五千五百一十，一十四。 交中：一十三，四千三百六十三，九十八，七十八。 離朔：一，七千二十七，一十九。
Crossing tally: 27, 1,527, 97, 56. Full-moon tally: 14, 5,510, 14. Crossing median: 13, 4,363, 98, 78. Departure new moon: 1, 7,027, 19.
56
交朔：二，二千二百九十二，三十，四十四。 中準：一千七百三十六。 中限：四千七百八十。 平離：九百六十三。
Crossing new moon: 2, 2,292, 30, 44. Median standard: 1,736. Median limit: 4,780. Level departure: 963.
57
程節：八百。 ◎月離入歷
Procedure interval: 800. ◎ Lunar departure: entering the calendar
58
置朔積，以離率去之，余滿統法為日，即天正常朔加時入歷也。 累加象策，盈離策去之，即弦望及次朔入歷也。 ◎月離朓朒
Set new-moon accumulation, remove the departure rate, and take days as the remainder over the comprehensive method—the calendar entry at added time for the heavenly normal new moon. Add the image tally cumulatively, removing full departure tallies, to obtain calendar entries for quarters, full moons, and successive new moons. ◎ Lunar departure: elongation correction
59
置入歷分，以日躔朓朒定數，朓減、朒加之，程節除之，為限數。 余乘所入限損益率，程節而一，用損益其限朓朒為定數。 ◎朔弦望定日
Set calendar-entry fractional parts; apply the sun's fixed elongation number—subtract for waning elongation, add for waxing—and divide by the procedure interval to obtain the limit number. Multiply the remainder by the limit's increase-decrease rate, divide by the procedure interval, and adjust the limit elongation to obtain the fixed number. ◎ Fixing the days of new moon, quarters, and full moon
60
各以日躔月離朓朒定數，朓減、朒加朔弦望常分，為定日。 定朔加時日入後，則進一日； 有交見初則不進。 弦望加時日未出，則退一日，日雖出有交見初亦如之。 元日有交，則消息定之。 定朔與後朔幹同者，大； 不同者，小； 無中氣者，為閏。
For each phase, apply the fixed solar and lunar elongation numbers—subtract for waning, add for waxing—to the constant fractional parts to obtain the fixed day. If the sun has already set at the computed time of the fixed new moon, advance the date by one day; unless an eclipse is visible at its beginning—in which case do not advance. For a quarter or full moon whose added time falls before sunrise, retreat one day; the same applies if an eclipse is visible at its beginning even when the sun has already risen. When the year's first day bears a crossing, fix the intercalation by the waxing-and-waning rule. If the fixed new moon and the next new moon share the same celestial stem, the month is long; if they differ, it is short; A month without a median qi is intercalary.
61
朔望加時日度
Solar longitude at added time for new and full moon
62
各置日躔入歷，以日躔月離朓朒定數，朓減、朒加之，為定朔加時入歷。 以歷分乘其日損益率，統法而一，損益其下盈縮數，為定數。 置定朔歷分，通法約之，以定數盈加、縮減之。 各命以冬夏至之宿算外，即所求也。
Set the sun's calendar entry and adjust it with the fixed solar and lunar elongation numbers—subtract for waning, add for waxing—to obtain the calendar entry at fixed-new-moon added time. Multiply calendar fractional parts by that day's increase-decrease rate, divide by the comprehensive method, and adjust the lower expansion-contraction figure to obtain the fixed number. Set the fixed-new-moon calendar fractional parts, reduce by the universal method, and add for expansion or subtract for contraction according to the fixed number. Count lodge positions from the winter and summer solstice starting points to obtain the result sought.
63
月離入交
Lunar departure: entering the crossing
64
置朔積，以交率去之，余滿統法為日，即天正常朔入交泛日也。 以望策累加之，盈交策去之，即望及次朔所入也。 各以日躔朓朒定數，朓減、朒加之，為入交常日。 置月離朓朒定數，經法乘之，平離而一，朓減、朒加常分，即入交定日也。
Set new-moon accumulation, remove the crossing rate, and take days as the remainder over the comprehensive method—the general crossing day for the heavenly normal new moon. Add the full-moon tally cumulatively, removing full crossing tallies, to find entries for the full moon and successive new moons. Apply the sun's fixed elongation number—subtract for waning, add for waxing—to obtain the constant crossing day. Set the lunar fixed elongation number, multiply by the canonical method, divide by level departure, and subtract or add to the constant fractional parts to obtain the fixed crossing day.
65
黃道正交月度
Ecliptic longitude at rectified crossing
66
統法通朔交定日，以二百五十四乘之，十九而一。 復以統法除，為入交度。 用減其朔加時日度，即朔前月離正交黃道宿度也。 ◎九道宿次
Extend fixed crossing days through the comprehensive method, multiply by 254, and divide by 19. Divide again by the comprehensive method to obtain crossing-entry degrees. Subtract from the new-moon added-time solar longitude to obtain the moon's ecliptic lodge position at rectified crossing before new moon. ◎ Lodge sequence of the nine paths
67
月離出入黃道六度。 變從八節，斜正不同。 故月有九道。 黃道八節，各有九限。 若正交起，八節後第一限之宿，為月行其節第一道。 起第二限之宿，為月行其節第二道，即以所起限為正交後第一限。 初率八，每限減一，盡九限，末率空。 又九限，初率空，每限增一，末率八，殷半交之宿。 自後亦九限，初率八，每限減一，末率空。 又九限，初率空，每限增一，末率八，復與黃道相會，謂之中交。 自中交至正交，亦如之。 各置所入限度，以限率乘之，為泛差。 其正交、中交前後各九限，以距二至之宿限數乘之。 半交前後各九限，以距二分之宿限數乘之：皆如經法而一，為黃道差。 在冬至之宿後，正交前後各九限為減，中交前後各九限為加。 在夏至之宿後，正交前後各九限為加，中交前後各九限為減。 凡月正交後出黃道外，中交後入黃道內。 其半交前後各九限，在春分之宿後，出黃道外，秋分之宿後，入黃道內：皆以差為加； 在春分之宿後，入黃道內，秋分之宿後，出黃道外：皆以差為減。 四約泛差，以黃道差減之，為赤道差。 正交、中交前後各九限，皆以差為加。 半交前後各九限，皆以差為減。 以黃赤二差加減黃道，為九道宿次； 就其分為少、太、半之數。 八節各九道，七十二道周焉。
Lunar departure crosses the ecliptic by six degrees. The variation follows the eight nodal points, with differing obliquity and alignment. Hence the moon's motion comprises nine paths. Each of the eight ecliptic nodal points has nine limits. From rectified crossing, the lodge of the first limit after each node marks the moon's first path in that node. The lodge where the second limit begins marks the second path, and that starting limit becomes the first limit after rectified crossing. The initial rate is eight, decreasing by one per limit through nine limits, with a final rate of zero. Nine more limits follow, rising from zero to eight and approaching the lodge of half crossing. Another nine limits then descend from eight to zero in the same manner. Nine limits rise again from zero to eight until the path rejoins the ecliptic—the median crossing. The segment from median to rectified crossing follows the same pattern. Set the limit degrees entered and multiply by the limit rate to obtain the general difference. For the nine limits before and after rectified and median crossing, multiply by the solstice limit count. For the nine limits before and after half crossing, multiply by the equinox limit count; divide each by the canonical method to obtain the ecliptic difference. After the winter-solstice lodge, subtract for nine limits around rectified crossing and add for nine limits around median crossing. After the summer-solstice lodge, add for nine limits around rectified crossing and subtract for nine limits around median crossing. In general, after rectified crossing the moon lies outside the ecliptic; after median crossing it lies within. For nine limits around half crossing: after the spring-equinox lodge, exiting the ecliptic, and after the autumn-equinox lodge, entering within—all add the difference; After the spring-equinox lodge, entering within, and after the autumn-equinox lodge, exiting without—all subtract the difference. Reduce the general difference by one quarter and subtract the ecliptic difference to obtain the equatorial difference. For nine limits around rectified and median crossing, always add the difference. For nine limits around half crossing, always subtract the difference. Apply the two ecliptic and equatorial differences to the ecliptic to obtain the nine-path lodge sequence; Express the fractional parts as slight, great, and half. Eight nodal points times nine paths yield seventy-two paths completing the cycle.
68
九道正交月度
Lunar position on the nine paths at rectified crossing
69
置月離正交黃道宿度； 各以所入限率乘之，亦乘其分，經法約之，為泛差。 用求黃赤二差，以加減之，即月離正交九道宿度也。 ◎九道朔月度
Set the moon's ecliptic lodge position at rectified crossing; Multiply by the entered limit rate and its fractional parts, reduce by the canonical method, to obtain the general difference. Derive the ecliptic and equatorial differences and apply them to obtain the moon's nine-path lodge position at rectified crossing. ◎ New-moon lunar position on the nine paths
70
置月離正交九道宿度，以入交度加之，命以九道宿次，即其朔加時月離九道宿度也。 ◎九道望月度
Set the moon's nine-path position at rectified crossing, add crossing-entry degrees, and count through the nine-path sequence for the lunar nine-path position at new-moon added time. ◎ Full-moon lunar position on the nine paths
71
置朔望加時日相距之度，以軌中加之，為加時象積。 用加其朔九道月度，命以其道宿次，既所求也。 自望推朔，亦如之。 ◎月離午中入歷
Set the solar separation at new- and full-moon added time, add track median, to obtain the added-time image accumulation. Add this to the new-moon nine-path lunar position and count through that path's lodge sequence to obtain the result. Project backward from full moon to new moon by the same method. ◎ Lunar departure: noon calendar entry
72
置朔望月離入歷，加半統，減去定分，各以日躔月離朓朒定數，朓減、朒加之，即所求也。 ◎晨昏月度
Set lunar calendar entries for new and full moon, add half the comprehensive method and subtract fixed fractional parts, then adjust with fixed solar and lunar elongation—subtract for waning, add for waxing—to obtain the result. ◎ Dawn and dusk lunar positions
73
置其日晨昏分，以定分減之，為前； 不足，返減，為後。 用乘其日離程，統法而一，滿經法為度，為晨昏前後度。 前加、後減加時月，為晨昏月度。
Set that day's dawn and dusk fractional parts and subtract the fixed parts to obtain the before fraction; if insufficient, reverse the subtraction to obtain the after fraction. Multiply by that day's departure procedure, divide by the comprehensive method, and convert fractions filling the canonical method into degrees for dawn and dusk offsets. add for before and subtract for after from the added-time lunar position to obtain dawn and dusk lunar positions.
74
晨昏象積
Image accumulation at dawn and dusk
75
置加時象積，以前象前後度，前減、後加，又以後象前後度，前加、後減之，即所求也。 ◎每日晨昏月度
Set the added-time image accumulation, apply the prior image's before-after degrees (subtract before, add after), then the latter image's degrees (add before, subtract after), to obtain the result. ◎ Daily dawn and dusk lunar positions
76
累計距後象離度，以減晨昏象積，為加； 不足，反減之，為減。 以距後象日數除之，用加減每日離度，為定度。 累加晨昏月度，命以九道宿次，即所求。
Accumulate departure degrees from the following image and subtract from the dawn-dusk image accumulation for addition; if insufficient, reverse the subtraction for subtraction. Divide by the days to the following image and apply to adjust each day's departure degrees to obtain fixed degrees. Add dawn and dusk lunar positions cumulatively and count through the nine-path lodge sequence to obtain the result.
77
月去黃道度
Lunar latitude relative to the ecliptic
78
置入交定日。 交中以下，月行陽道； 以上，去之，月行陰道：皆以經法通之。 用減九百八十，余以乘之，五百五十六而一，為分； 滿經法為度。 行陽道，在黃道外； 行陰道，在黃道內，即所求月去黃道內外度也。
Set the fixed crossing day. Below the crossing median, the moon travels the yang path; above it, subtract to obtain the yin path—extend both through the canonical method. subtract from 980, multiply the remainder, and divide by 556 for fractional parts; and convert to degrees when the canonical method is filled. on the yang path, outside the ecliptic; on the yin path, inside the ecliptic—the lunar latitude sought.
79
日月食限
Solar and lunar eclipse limits
80
置定交行陰陽道日。 半交中以下，為交後； 以上，用減交中，為交前：皆以統法通之，為距交分。 朔視距交分，陽道四千二百一十九、陰道一萬三百八十三以下，日入食限。 望視距交分陰陽道皆六千九百九十五以下，月入蝕限。
Set the fixed day on the yin or yang crossing path. Below half the crossing median counts as after crossing; above, subtract from the crossing median for before crossing—extend both through the comprehensive method as distance-from-crossing parts. At new moon, if distance-from-crossing parts are 4,219 or less on the yang path or 10,383 or less on the yin path, a solar eclipse is possible. At full moon, if distance-from-crossing parts are 6,995 or less on either path, a lunar eclipse is possible.
81
日月食甚加時定分
Fixed fractional parts for eclipse maximum time
82
置朔定分。 半統以上，以半統減之； 半統以下，用減半統：為距午分。 十一乘之，經法而一。 半統以下，以減半統； 以上，以加朔定分：為日食加時定分。 望以其日晨分與一千六百二十相減，余以二百四十五乘之，三百一十三而一； 用減二百四十五，余以損益望定分，為月食加時定分。
Set the fixed fractional parts for new moon. If above half the comprehensive method, subtract half the comprehensive method; if below, subtract from half the comprehensive method to obtain distance-from-noon parts. Multiply by eleven and divide by the canonical method. If below half the comprehensive method, subtract it; if above, add to the new-moon fixed parts—to obtain fixed fractional parts for solar-eclipse added time. When the value lies above [half comprehensive], add the new-moon fixed fraction to obtain the fixed fraction for the hour of greatest solar eclipse. For full moon, subtract the day's morning fraction from 1,620, multiply the remainder by 245, and divide by 313; subtract that from 245, then add or subtract the remainder to the full-moon fixed fraction to obtain the fixed fraction for the hour of greatest lunar eclipse.
83
日食常準
Constant standard for solar eclipses
84
置中準； 與其日赤道內外數相乘，二千五百一十三除，為黃道出入食差。 以距午分減半晝分以乘之，半晝分而一； 赤道內以減、赤道外以加中準，為日食常準。
Place the median standard; multiply it by the day's equatorial inner-or-outer index, divide by 2,513, and obtain the ecliptic ingress–egress eclipse correction. multiply by (half daytime fraction minus distance-from-noon fraction), then divide by half the daytime fraction; subtract from the median standard when inside the equator, add when outside—this yields the solar eclipse constant standard.
85
日食定準
Fixed standard for solar eclipses
86
置日躔入歷，以經法通之，三千二百八十七以下，用減三千二百八十七，為二至後； 以上，減去三千二百八十七，為二分前。 六千五百七十四以上，用減九千八百六十一，為二分後； 以上，減去九千八百六十一，為二至前。 各三約之，二至前後用減、二分前後用加二千七百七十二，為黃道斜正食差。 以距午分乘之，半晝分而一，以加常準，為定準。
Set solar motion's circuit entry, convert by the canonical method; if below 3,287, subtract from 3,287—this is post-solstitial; if above, subtract 3,287—this is pre-equinoxial. If above 6,574, subtract from 9,861—this is post-equinoxial; if above, subtract 9,861—this is pre-solstitial. Divide each by three; subtract for solstitial intervals, add 2,772 for equinoctial intervals—this is the ecliptic obliquity eclipse correction. Multiply by distance-from-noon, divide by half daytime, add to the constant standard to obtain the fixed standard.
87
日食分
Magnitude of solar eclipse
88
以定準加中限，為陰道定準； 減中限，為陽道定限。 不足減者，反減之，為限外分。 視陰道距交分，定準以上，定限以下，為陰道食； 即置定限，以距交分減之，為距食分。 定準以下，雖曰陰道，亦為陽道食； 即加陽道定限，為距食分。 其有限外分者，即減去限外分，為距食分。 不足減者，不食。 其陽道距交分，定限以下，為入定食限； 即用減陽道定限，為距食分。 各置距食分，皆以四百七十八除，為日食之大分； 余為小分。 命大分以十為限； 命小分以半及強弱。
Add the fixed standard to the median limit to obtain the yin-path fixed standard; subtract the median limit to obtain the yang-path fixed limit. If subtraction is impossible, subtract in reverse—this is the beyond-limit fraction. For yin-path distance-from-crossing: when fixed standard exceeds and fixed limit is not exceeded, a yin-path eclipse applies; set the fixed limit and subtract distance-from-crossing to obtain distance-from-eclipse. Below the fixed standard, though nominally yin-path, treat as yang-path eclipse; add the yang-path fixed limit to obtain distance-from-eclipse. If a beyond-limit fraction exists, subtract it to obtain distance-from-eclipse. If subtraction fails, no eclipse occurs. For yang-path distance-from-crossing below the fixed limit, the eclipse enters the fixed limit; subtract it from the yang-path fixed limit to obtain distance-from-eclipse. Set each distance-from-eclipse fraction and divide by 478 to obtain the solar eclipse major fraction; the remainder is the minor fraction. Express the major fraction in tenths; express the minor fraction in halves and strong/weak gradations.
89
月食分
Magnitude of lunar eclipse
90
視距交分，中準以下，皆既； 以上，用減食限，為距食分。 置之，以五百二十六除，為月食之大分； 余為小分。 命大分以十為限； 命小分以半及強弱。
Inspect distance-from-crossing: at or below the median standard, the eclipse is total; above that, subtract from the eclipse limit to obtain distance-from-eclipse. Set the value and divide by 526 for the lunar eclipse major fraction; the remainder is the minor fraction. Express the major fraction in tenths; express the minor fraction in halves and strong/weak gradations.
91
月食泛用分
General-use fraction for solar eclipse timing
92
置距食分，一千九百一十二以上，用減四千七百八十； 余自相乘，六萬三千二百七十二除之； 以減六百四十七，為泛用分。 九百五十六以下，用減一千九百一十二，余以通法乘之，七百三十五而一； 以減五百一十七，為泛用分。 九百五十六以上，以距食分自相乘，二千三百六十二除之； 用減三百八十七，為泛用分。
Set distance-from-eclipse; if 1,912 or greater, subtract from 4,780; multiply the remainder by itself and divide by 63,272; subtract the result from 647 to obtain the general-use fraction. If 956 or less, subtract from 1,912, multiply the remainder by the universal method, and divide by 735; subtract from 517 to obtain the general-use fraction. If 956 or greater, square distance-from-eclipse and divide by 2,362; subtract from 387 to obtain the general-use fraction.
93
月食泛用分
General-use fraction for lunar eclipse timing
94
置距食分，二千一百四以上，用減五千二百六十； 余自相乘，六萬九千一百六十九除之； 以減七百一十一，為泛用分。 一千五十二以上，用減二千一百四十； 余，七除之； 以減五百六十七，為泛用分。 一千五十二以下，以距食分減之； 余自相乘，二千六百五十四而一； 用減四百一十七，為泛用分。
Set distance-from-eclipse; if 2,140 or greater, subtract from 5,260; multiply the remainder by itself and divide by 69,169; subtract from 711 to obtain the general-use fraction. If 1,052 or greater, subtract from 2,140; divide the remainder by 7; subtract from 567 to obtain the general-use fraction. If 1,052 or less, subtract distance-from-eclipse; multiply the remainder by itself and divide by 2,654; subtract from 417 to obtain the general-use fraction.
95
日月初末加時定分
Fixed fractions for the added times of eclipse first contact and last contact
96
各置泛用分，以平離乘之，其日離程而一，為定用分。 以減朔望定分，為虧初。 加之，為復末。 加時常分，如食甚術推之，得虧初、復末定分。 置初、甚、末定分，各以辰則除之，為辰； 經法除之，為刻：即初、甚、末之辰刻也。
Set each general-use fraction, multiply by mean departure, divide by the day's departure interval to obtain the fixed-use fraction. Subtract it from the syzygy fixed fraction to obtain first contact. Add it to obtain last contact. Apply the added-time constant fraction by the greatest-eclipse procedure to obtain fixed times for first and last contact. Set the first-contact, greatest, and last-contact fixed fractions; divide each by the double-hour divisor to obtain the double-hour; divide the remainder by the canonical method for quarters—the double-hours and quarters of first contact, maximum, and last contact.
97
虧食所起
Direction of eclipse onset
98
日食起虧自西，月食起虧自東。 其食分少者，月行陽道，則日食偏南，月食偏北； 陰道，則日食偏北，月食偏南：此常數也。 立春後，立夏前，食分多，則日食偏南，月食偏北； 立秋後，立冬前，食分多，則日食偏北，月食偏南：此黃道斜正也。 陽道交前，陰道交後，食分多，則日食偏南，月食偏北； 陽道交後，陰道交前，食分多，則日食偏北，月食偏南：此九道斜正也。 黃道比常數所偏差少，九道比黃道所偏又四分之一：皆據午而言之。 若午前午後，一理偏南，一理偏北，及消息所食分數多少，以定初、甚、末之方，即各得所求也。
Solar eclipses first contact in the west; lunar eclipses first contact in the east. For small magnitude, when the Moon is on the yang path, solar eclipses skew south and lunar eclipses north; on the yin path, solar eclipses skew north and lunar south—these are the standard constants. From after Lichun to before Lixia, large eclipses bias solar south and lunar north; from after Liqiu to before Lidong, large eclipses bias solar north and lunar south—this follows ecliptic obliquity. Yang path before the node and yin path after it, large eclipses bias solar south and lunar north; yang path after the node and yin path before it, large eclipses bias solar north and lunar south—this follows lunar-path obliquity. Ecliptic deviation from the standard constants is smaller; lunar-path deviation from the ecliptic is a further quarter—all reckoned from local noon. For morning or afternoon, one rule skews south and one north; combine eclipse magnitude with seasonal phase to fix the azimuths of first contact, maximum, and last contact—each desired result then follows.
99
帶食出入分
Fraction for eclipses at sunrise or sunset
100
視其日出入分，在虧初定分已上，復末定分已下，即帶食出入。 食甚在出入分已下者，以出入分減復末定分，為帶食差。 食甚在出入分已上者，以虧初定分減出入分，為帶食差。 各置帶食差，以距食分乘之，定用分而一，日以四百七十八、月以五百二十六除，為帶食之大分； 余為小分。
If the day's sunrise/sunset fraction falls between the first-contact and last-contact fixed fractions, the eclipse occurs at rising or setting. If greatest eclipse precedes sunrise/sunset, subtract the exit-and-entry fraction from the last-contact fixed fraction for the horizon correction. If greatest eclipse follows sunrise/sunset, subtract the exit-and-entry fraction from the first-contact fixed fraction for the horizon correction. Set each horizon correction, multiply by distance-from-eclipse, divide by fixed-use fraction, then divide by 478 for solar or 526 for lunar eclipses to obtain the horizon eclipse major fraction; the remainder is the minor fraction.
101
食入更籌
Night watches and clepsydra tallies of the eclipse
102
各置初、甚、末定分。 晨分已下，以昏分加之； 昏分已上，昏分減之：皆更用分而一，為更數。 余，籌用分而一，為籌數。
Set the first-contact, greatest, and last-contact fixed fractions. When below the morning fraction, add the evening fraction; if above the evening fraction, subtract it; divide each by the watch divisor to obtain the watch count. Divide the remainder by the tally divisor to obtain the clepsydra tally count.
103
《欽天》步五星術◎歲星周率：二百八十七萬一千九百七十六，六。 變率：二十四萬二千二百一十五，六十六。
《Qintian》 procedure for the five planets ◎ Jupiter circuit rate: 2,871,976, 6. Variation rate: 242,215; remainder 66.
104
曆率：二百六十二萬九千七百六十一，七十八。 周策：三百九十八，六千三百七十六，六。 歷中：一百八十二，四千四百八十，八十九。 變段變日變度變曆
Calendar rate: 2,629,761; remainder 78. Circuit tally: 398 days, 6,376 parts, remainder 6. Circuit median: 182, 4,480, 89. Variable segment, variable days, variable degrees, variable calendar
105
晨見一十七三 〈（三十七）〉 二 〈（二十四）〉
Morning appearance: 17 days, 3 parts (37)〉 2 [parts] (24)〉
106
順疾九十一十六 〈（六十三）〉 一十一 〈（一十三）〉
Swift direct motion: 91 days, 16 parts (63)〉 11 [parts] (13)〉
107
順遲二十五二 （九） 一 〈（二十九）〉 前留二十六 〈（三十二）〉
Direct slow motion: 25, 2. 9 1 (29)〉 Station before retrograde: 26. (32)〉
108
退遲一十四一 〈（一十二）〉 空 〈（二十八）〉
Retrograde slow motion: 14, 1. (12)〉 zero (blank entry) (28)〉
109
退疾二十七四 〈（三十八）〉 一 〈（三十七）〉
Retrograde fast motion: 27, 4. (38)〉 1 (37)〉
110
退疾二十七四 〈（三十八）〉 一 〈（三十七）〉
Retrograde fast motion: 27, 4. (38)〉 1 (37)〉
111
退遲一十四一 〈（一十二）〉 空 〈（二十八）〉 後留二十六 〈（三十二）〉
Retrograde slow motion: 14, 1. (12)〉 zero (blank entry) (28)〉 Station after retrograde: 26. (32)〉
112
順遲二十五二 （九） 一 〈（二十九）〉
Direct slow motion: 25, 2. 9 1 (29)〉
113
順疾九十一十六 〈（六十三）〉 一十一 〈（一十三）〉
Direct fast motion: 91, 16. (63)〉 11 (13)〉
114
夕伏一十七三 〈（三十七）〉 二 〈（二十四）〉 ◎熒惑周率：五百六十一萬五千四百二十二，一十一。 變率：二百九十八萬五千六百六十一，七十一。
Evening disappearance: 17, 3. (37)〉 2 (24)〉 ◎ Mars orbital cycle rate: 5,615,422; remainder 11. Mutation rate: 2,985,661; remainder 71.
115
曆率：二百六十二萬九千七百六十，空。 周策：七百七十九，六千六百二十二，一十一。 歷中：一百八十二，四千四百八十，空。 變段變日變度變曆
Calendar rate: 2,629,760; remainder zero. Circuit tally: 779; 6,622; remainder 11. Mid-cycle: 182; 4,480; remainder zero. Mutation segment | mutation days | mutation degrees | mutation calendar.
116
晨見七十三五十三 〈（六十八）〉 五十 〈（五十八）〉 順疾七十三五十一 （一） 四十八 （三）
Morning appearance: 73, 53. (68)〉 50 (58)〉 Direct fast motion: 73, 51. 1 48 3
117
次疾七十一四十六 〈（六十九）〉 四十四 〈（一十七）〉
Second direct-fast phase: 71, 46. (69)〉 44 (17)〉
118
次遲七十一四十五 〈（三十三）〉 四十二 〈（五十八）〉 順遲六十二一十九 〈（二十九）〉 一十八 〈（二十）〉 前留八 〈（六十九）〉
Second direct-slow phase: 71, 45. (33)〉 42 (58)〉 Direct slow motion: 62, 19. (29)〉 18 (20)〉 Station before retrograde: 8. (69)〉
119
退遲一十一 〈（五十八）〉 空 〈（四十四）〉 退疾二十一七 〈（四十六）〉 二 〈（四十）〉 退疾二十一七 〈（四十六）〉 二 〈（四十）〉
Retrograde slow motion: 11. (58)〉 zero (blank entry) (44)〉 Retrograde fast motion: 21, 7. (46)〉 2 (40)〉 Retrograde, swift: 21 days, 7 degrees (46)〉 2 (40)〉
120
退遲一十一 〈（五十八）〉 空 〈（四十四）〉 後留八 〈（六十九）〉 順遲六十二一十九 〈（二十九）〉 一十八 〈（二十）〉
Retrograde, slow: 11 days, 1 degree (58)〉 — (44)〉 Latter station: 8 days (69)〉 Prograde, slow: 62 days, 19 degrees (29)〉 18 (20)〉
121
次遲七十一四十五 〈（三十三）〉 四十二 〈（五十八）〉
Secondary slow: 71 days, 45 degrees (33)〉 42 (58)〉
122
次疾七十一四十六 〈（六十九）〉 四十四 〈（一十七）〉 順疾七十三五十一 （一） 四十八 （三）
Secondary swift: 71 days, 46 degrees (69)〉 44 (17)〉 Prograde, swift: 73 days, 51 degrees fraction 1 48 fraction 3
123
夕伏七十三五十三 〈（六十八）〉 五十 〈（五十八）〉 ◎鎮星周率：二百七十二萬二千一百七十六，九十。 變率：九萬二千四百一十六，五十。
Evening hiding: 73 days, 53 degrees (68)〉 50 (58)〉 ◎ Saturn cycle rate: 2,722,176; remainder 90. Variation rate: 92,416; remainder 50.
124
曆率：二百六十二萬九千七百五十九，八十。 周策：三百七十八，五右七十六，九十。 歷中：一百八十二，四千四百七十九，九十。 變段變日變度變曆
Calendar rate: 2,629,759; remainder 80. Sidereal period: 378 days; remainder 576; fraction 90. Mid-cycle: 182 days; remainder 4,479; fraction 90. Segment | Days | Degrees | Calendar
125
晨見一十九二 （七） 一 〈（一十四）〉 順疾六十五六 〈（三十八）〉 三 〈（五十一）〉 順遲一十九空 〈（六十三）〉 空 〈（三十五）〉
Morning appearance: 19 days, 2 degrees fraction 7 1 (14)〉 Prograde, swift: 65 days, 6 degrees (38)〉 3 (51)〉 Prograde, slow: 19 days; degrees — (63)〉 — (35)〉
126
前留三十七 （三） 退遲一十六空 〈（四十三）〉 空 〈（一十四）〉 退疾三十三二 〈（三十五）〉 空 〈（六十）〉 退疾三十三二 〈（三十五）〉 空 〈（六十）〉
Prior station: 37 days fraction 3 Retrograde, slow: 16 days; degrees — (43)〉 — (14)〉 Retrograde, swift: 33 days, 2 degrees (35)〉 — (60)〉 Retrograde, swift: 33 days, 2 degrees (35)〉 — (60)〉
127
退遲一十六空 〈（四十三）〉 空 〈（一十四）〉 後留三十七 （三） 順遲一十九空 〈（六十三）〉 空 〈（三十五）〉 順疾六十五六 〈（三十八）〉 三 〈（五十一）〉
Retrograde, slow: 16 days; degrees — (43)〉 — (14)〉 Latter station: 37 days fraction 3 Prograde, slow: 19 days; degrees — (63)〉 — (35)〉 Prograde, swift: 65 days, 6 degrees (38)〉 3 (51)〉
128
夕伏一十九二 （七） 一 〈（一十四）〉 ◎太白周率：四百二十萬四千一百四十三，九十六。 變率：四百二十萬四千一百四十三，九十六。
Evening hiding: 19 days, 2 degrees fraction 7 1 (14)〉 ◎ Venus cycle rate: 4,204,143; remainder 96. Variation rate: 4,204,143; remainder 96.
129
曆率：二百六十二萬九千七百五十，五十六。 周策：五百八十三，六千五百四十三，九十六。 歷中：一百八十二，四千四百七十五，二十八。 變段變日變度變曆
Calendar rate: 2,629,750; remainder 56. Sidereal period: 583 days; remainder 6,543; fraction 96. Mid-cycle: 182 days; remainder 4,475; fraction 28. Segment | Days | Degrees | Calendar
130
夕見四十二五十三 〈（四十）〉 五十一 〈（一十七）〉
Evening appearance: 42 days, 53 degrees (40)〉 51 (17)〉
131
順疾九十六一百二十一 〈（五十七）〉 一百一十六 〈（三十九）〉 次疾七十三八十 〈（三十七）〉 七十七 （二） 次遲三十三三十四 （一） 三十二 〈（四十）〉
Prograde, swift: 96 days, 121 degrees (57)〉 116 (39)〉 Secondary swift: 73 days, 80 degrees (37)〉 77 fraction 2 Secondary slow: 33 days, 34 degrees fraction 1 32 (40)〉
132
順遲二十四一十一 〈（六十一）〉 一十一 〈（二十四）〉 前留六 〈（六十九）〉 退遲四一 〈（二十二）〉 空 〈（三十一）〉
Prograde, slow: 24 days, 11 degrees (61)〉 11 (24)〉 Prior station: 6 days (69)〉 Retrograde, slow: 4 days, 1 degree (22)〉 — (31)〉
133
退疾六三 〈（六十五）〉 一 〈（二十二）〉 夕伏七四 〈（四十）〉 一 〈（三十七）〉 晨見七四 〈（四十）〉 一 〈（三十七）〉
Retrograde, swift: 6 days, 3 degrees (65)〉 1 (22)〉 Evening hiding: 7 days, 4 degrees (40)〉 1 (37)〉 Morning appearance: 7 days, 4 degrees (40)〉 1 (37)〉
134
退疾六三 〈（六十五）〉 一 〈（二十二）〉 退遲四一 〈（二十二）〉 空 〈（三十一）〉 後留六 〈（六十九）〉
Retrograde, swift: 6 days, 3 degrees (65)〉 1 (22)〉 Retrograde, slow: 4 days, 1 degree (22)〉 — (31)〉 Latter station: 6 days (69)〉
135
順遲二十四一十一 〈（六十一）〉 一十一 〈（二十四）〉 次遲三十三三十四 （一） 三十二 〈（四十）〉 次疾七十三八十 〈（三十七）〉 七十七 （二）
Prograde, slow: 24 days, 11 degrees (61)〉 11 (24)〉 Secondary slow: 33 days, 34 degrees fraction 1 32 (40)〉 Secondary swift: 73 days, 80 degrees (37)〉 77 fraction 2
136
順疾九十六一百二十一 〈（五十七）〉 一百一十六 〈（三十九）〉
Prograde, swift: 96 days, 121 degrees (57)〉 116 (39)〉
137
晨伏四十二五十三 〈（四十）〉 五十一 〈（一十七）〉 ◎辰星周率：八十三萬四千三百三十五，五十二。 變率：八十三萬四千三百三十五，五十二。
Morning concealment: 42 days, 53 degrees (40)〉 51 (17)〉 ◎ Mercury cycle rate: 834,335; remainder 52. Variation rate: 834,335; remainder 52.
138
曆率：二百六十二萬九千七百六十，四十四。 周策：一百一十五，六千三百三十五，五十二。 歷中：一百八十二，四千四百八十，二十二。 變段變日變度變曆
Calendar rate: 2,629,760; remainder 44. Sidereal period: 115 days; remainder 6,335; fraction 52. Mid-cycle: 182 days; remainder 4,480; fraction 22. Segment | Days | Degrees | Calendar
139
夕見一十七三十四 （一） 二十九 〈（五十四）〉 順疾一十一一十八 〈（二十四）〉 一十六 （四） 順遲一十六一十一 〈（四十三）〉 一十 〈（一十）〉
Evening appearance: 17 days, 34 degrees fraction 1 29 (54)〉 Prograde, swift: 11 days, 18 degrees (24)〉 16 fraction 4 Prograde, slow: 16 days, 11 degrees (43)〉 10 (10)〉
140
前留二 〈（六十八）〉 夕伏一十一六二晨見一十一六二後留二 〈（六十八）〉
Prior station: 2 days (68)〉 Evening concealment: 11 d, 6°, frac. 2; morning appearance: 11 d, 6°, frac. 2; latter station: 2 d (68)〉
141
順遲一十六一十一 〈（四十三）〉 一十 〈（一十）〉 順疾一十一一十八 〈（二十四）〉 一十六 （四）
Prograde, slow: 16 days, 11 degrees (43)〉 10 (10)〉 Prograde, swift: 11 days, 18 degrees (24)〉 16 fraction 4
142
晨伏一十七三十四 （一） 二十九 〈（五十四）〉 ◎中日中星
Morning concealment: 17 days, 34 degrees fraction 1 29 (54)〉 ◎ Mean conjunction: day and central star
143
置氣積，以其星周率除之，為周數； 不盡為天正中氣積前合。 用減歲率，為前年天正中氣後合。 如不足減，則加歲率以減之，為次前年天正中氣後合。 各以統法約之，為日、為度，即所求平合中日、中星也。 置中日，以逐段變日累加之，即逐段中日也。 置中星，以逐段變度順加、退減之，即得逐段中星。 金水夕伏晨見，皆退變也。
Set the qi accumulation and divide by the planet's cycle rate to obtain the number of complete cycles; the remainder is the accumulation before conjunction at celestial mid-qi. Subtract it from the annual rate to obtain the post-conjunction at celestial mid-qi in the preceding year. If the subtraction cannot be completed, add one annual rate and subtract again to obtain the post-conjunction two years earlier. Reduce each by the unification divisor to obtain days and degrees—the mean-conjunction mid-day and central star sought. Set the mid-day and add the segment mutation-days cumulatively to obtain each segment's mid-day. Set the central star and apply each segment's mutation-degrees, adding in direct motion and subtracting in retrograde, to obtain each segment's central star. Venus and Mercury evening concealment and morning appearance all use retrograde mutation.
144
置變率。 以周數乘之，以曆率去之，余滿統法為度。 歷中以下，為先； 以上，減去歷中，為後：即所求平合入歷。 以逐段變曆累加之，得逐段入歷也。
Set the variation rate. Multiply by the cycle count, remove multiples of the calendar rate, and convert the remainder to degrees by the unification divisor. If below mid-cycle, it is prior; if above, subtract the mid-cycle value to obtain the posterior portion—the mean-conjunction's entry into the epicycle sought. Add the segment mutation-calendar values cumulatively to obtain each segment's epicycle entry.
145
後後定數
Prior-posterior fixed corrections
146
置入歷分，以其度損益率乘之，經法而一，用損益其下先後數，即所求也。 ◎常日定星
Set the epicycle parts, multiply by the degree's variation rate, divide by the canonical divisor, and apply the result to adjust the prior-posterior constants below—the value sought. ◎ Mean day and corrected star
147
置中日中星，各以先後定數，先加、後減之，留用前段先後數，太白順伏見及前順疾次疾後次遲次疾、辰星順伏見及前疾後遲，並先減、後加之，即各為其段常日定星。 置定星，以其年天正中氣日躔黃道宿次加而命之，得逐段末日加時宿度也。
Set the mean day and central star and apply the prior-posterior corrections, adding when prior and subtracting when posterior; at stations use the prior segment's values. For Venus in direct concealment/appearance and the segments prior swift, secondary swift, posterior secondary slow, and secondary swift, and for Mercury in direct concealment/appearance with prior swift and posterior slow, reverse the rule (subtract when prior, add when posterior) to obtain each segment's mean day and corrected star. Set the corrected star, add the year's celestial mid-qi solar longitude on the ecliptic and name the lodge, to obtain each segment's terminal-day added-time lodge position.
148
盈縮定數
Expansion and contraction corrections
149
置常日，如歲中以下，為在盈； 以上，減去歲中，余為在縮：即常日入盈縮歷也。 置歷分。 以其日損益率乘之，經法而一，用損益其下盈縮數，即得所求也。
Set the mean day; if it does not exceed the annual mid-cycle, it lies in the expansion phase; if above, subtract the annual mid-cycle; the remainder lies in contraction—the mean day's entry into the expansion-contraction epicycle. Set the epicycle parts. Multiply by the day's variation rate, divide by the canonical divisor, and adjust the expansion-contraction constant below to obtain the value sought.
150
置常日，以盈縮定數盈減、縮加之，為定日。 以其年天正中氣加而命之，即逐段末日加時日辰也。 ◎入中節
Set the mean day and subtract in expansion or add in contraction according to the correction to obtain the corrected day. Add the year's celestial mid-qi and name the result to obtain each segment's terminal-day added-time date and hour. ◎ Entry into qi and solar terms
151
置定日，以氣策除之，命起冬至，即所入氣日數也。 ◎平行分
Set the corrected day, divide by the qi divisor, and count from the winter solstice to obtain the qi and day entered. ◎ Mean daily motion
152
置定日，以前段定日減之，為日率； 定星與前段定星相減，為度率。 通度率，以經法乘之，通日率而一，為平行分。 ◎初末行分
Set the corrected day and subtract the prior segment's corrected day to obtain the day rate; subtract the prior segment's corrected star from the current corrected star to obtain the degree rate. Convert the degree rate, multiply by the canonical divisor, divide by the converted day rate, to obtain the mean daily motion. ◎ Initial and final daily motion
153
近伏段與伏段平行分，合而半之，為其段近伏行分。 以平行分減之，余減平行分，為其段遠伏行分。 近留段近留行分空。 倍平行分為其段遠留行分。 其不近伏留段，皆以順行二段平行分，合而半之，為前段末日、後段初日行分。 各與其段平行分相減，平行分多，則加平行分； 平行分少，則減平行分，即前段初日、後段末日行分。 其不近伏留段，退行則以遲段近疾行分，為疾段近遲行分，所得與平行分相減，平行分多，則加之，少則減之：皆為遠遲行分也。
For a segment near concealment, average its mean motion with that of the concealment segment to obtain the near-concealment daily motion. Subtract from the mean motion; subtract the remainder again from the mean motion to obtain the far-concealment daily motion. Near-station segment: near-station daily motion is blank. Double the mean daily motion to obtain the far-station daily motion. For segments not near concealment or station, average the mean motions of the two adjacent direct-motion segments to obtain the last day of the prior segment and the first day of the next. Compare each with its segment's mean motion; if the mean motion is greater, add the mean motion; if less, subtract the mean motion—to obtain the first day of the prior segment and the last day of the posterior segment. For segments not near concealment or station in retrograde motion, take the slow segment's near-swift motion as the swift segment's near-slow motion; compare with the mean motion and add if the mean is greater or subtract if less—all yielding the far-slow daily motion.
154
初行夜半宿次
Initial motion: midnight lodge position
155
置經法，以前段末日加時分減之：余乘前段末日行分，經法而一； 用順加、退減前段末日加時宿度，為其段初行昏後夜半宿度也。 ◎每日行分
Set the canonical divisor and subtract the prior segment's terminal-day added-time parts; multiply the remainder by that day's motion and divide by the canonical divisor; add in direct motion or subtract in retrograde from the prior segment's terminal-day added-time lodge position to obtain this segment's initial dusk-after-midnight lodge position. ◎ Daily motion
156
初末行分相減，為差率。 累計其段初行昏後夜半距後段初行昏後夜半日數除之，為日差。 半日差，以減多、加少為其段初末定行分。 置初定行分，用日差末多則累加、末少則累減，為每日行分。 以每日行分順加、退減初行昏後夜半宿度，為每日昏後夜半星所至宿度也。
Subtract the initial from the final daily motion to obtain the difference rate. Count the days from this segment's initial dusk-after-midnight to the next segment's initial dusk-after-midnight and divide to obtain the daily increment. Halve the daily increment and apply it to reduce the larger and increase the smaller to obtain the segment's corrected initial and final motion. Set the corrected initial motion and accumulate the daily increment, adding when the final motion is greater and subtracting when less, to obtain each day's motion. Apply each day's motion, adding in direct motion and subtracting in retrograde from the initial dusk-after-midnight position, to obtain the star's lodge at dusk after midnight for each day.
157
先定日昏後夜半宿次
Corrected day: dusk-after-midnight lodge position
158
自初日累計距所求日數，以乘其段日差； 末多用加、末少用減初日行分，為其日行分。 合初日而半之，以所累計日乘之，用順加、退減其段初行昏後夜半宿次，即所求也。
Count the days from the segment's first day to the day sought and multiply by the segment's daily increment; add to or subtract from the first day's motion according to whether the final motion is greater or less to obtain that day's motion. Average with the first day's motion, multiply by the elapsed days, and add or subtract from the segment's initial dusk-after-midnight lodge position in direct or retrograde motion to obtain the result sought.
159
《欽天》步發斂術候策：五，五百二十四，四十五。 卦策：六，六百二十九，三十四。 外策：三，三百一十四，六十七。
Qintian Calendar, procedure for pacing emergence and retraction—hou divisor: 5; remainder 524; fraction 45. Hexagram divisor: 6; remainder 629; fraction 34. Outer divisor: 3; remainder 314; fraction 67.
160
維策：一十二，一千二百五十八，六十八。 氣盈：一千五百七十三，三十五。
Intercalary divisor: 12; remainder 1,258; fraction 68. Solar-term surplus: 1,573; fraction 35.
161
朔虛：三千三百九十九，七十二。 ◎氣候圖冬至十一月中蚯蚓結麋角解水泉動小寒十二月節雁北鄉鵲始巢雉始雊
New-moon void: 3,399; fraction 72. ◎ Climate chart — Winter Solstice (11th month, mid-month): earthworms coil; elk shed antlers; springs stir. Minor Cold (12th month, first node): wild geese turn north; magpies begin to nest; pheasants first call.
162
大寒十二月中雞始乳鷙鳥厲疾水澤腹堅立春正月節東風解凍蟄蟲始振魚上冰雨水正月中獺祭魚鴻雁來草木萌動
Major Cold (12th month, mid-month): hens begin brooding; birds of prey grow swift; ice hardens deep in the marshes. Start of Spring (1st month, first node): the east wind thaws the ice; dormant insects stir; fish rise beneath the ice. Rain Water (1st month, mid-month): otters offer fish; wild geese arrive; grass and trees put forth shoots.
163
驚蟄二月節桃始華倉庚鳴鷹化為鳩春分二月中玄鳥至雷乃發聲始電清明三月節桐始華田鼠化為鴽虹始見
Awakening of Insects (2nd month, first node): peaches blossom; orioles sing; hawks turn into cuckoos. Spring Equinox (2nd month, mid-month): swallows arrive; thunder sounds; lightning begins. Pure Brightness (3rd month, first node): paulownia blooms; field mice become quails; rainbows first appear.
164
谷雨三月中萍始生鳴鳩拂其羽戴勝降於桑立夏四月節螻蟈鳴蚯蚓出王瓜生小滿四月中苦菜秀靡草死小暑至
Grain Rain (3rd month, mid-month): duckweed sprouts; turtle-doves preen their feathers; hoopoes alight on mulberry trees. Start of Summer (4th month, first node): tree crickets chirp; earthworms emerge; gourds grow. Minor Fullness (4th month, mid-month): bitter herbs flourish; tender grasses wither; lesser heat draws near.
165
芒種五月節螗螂生鵙始鳴反舌無聲夏至五月中鹿角解蜩始鳴半夏生小暑六月節溫風至蟋蟀居壁鷹乃學習
Grain in Ear (5th month, first node): praying mantises hatch; shrikes begin to call; mockingbirds fall silent. Summer Solstice (5th month, mid-month): deer shed antlers; cicadas begin to sing; pinellia sprouts. Minor Heat (6th month, first node): warm winds blow; crickets move to the walls; eagles learn to strike.
166
大暑六月中腐草為螢土潤溽暑大雨時行立秋七月節涼風至白露降寒蟬鳴處暑七月中鷹祭鳥天地始肅禾乃登
Major Heat (6th month, mid-month): rotten grass turns to fireflies; the earth grows damp in oppressive heat; heavy rains fall in season. Start of Autumn (7th month, first node): cool winds arrive; white dew descends; cold cicadas cry. End of Heat (7th month, mid-month): eagles offer birds; heaven and earth begin to harden; grain ripens.
167
白露八月節鴻雁來玄鳥歸群鳥養羞秋分八月中雷乃收聲蟄蟲坯戶水始涸寒露九月節鴻雁來賓雀入水為蛤菊有黃華
White Dew (8th month, first node): wild geese arrive; swallows return; birds lay up provisions. Autumn Equinox (8th month, mid-month): thunder fades; hibernating creatures block their burrows; waters begin to shrink. Cold Dew (9th month, first node): geese come as guests; sparrows enter the water as clams; chrysanthemums bloom yellow.
168
霜降九月中豺祭獸草木黃落蟄蟲咸俯立冬十月節水始冰地始凍雉入水為蜃
Frost Descent (9th month, mid-month): jackals sacrifice game; plants yellow and shed their leaves; dormant insects all lie low. Start of Winter (10th month, first node): water begins to freeze; the earth begins to chill; pheasants enter the water as oysters.
169
小雪十月中虹藏不見天氣上騰地氣下降閉塞成冬
Minor Snow (10th month, mid-month): rainbows vanish from sight; the qi of heaven rises and that of earth sinks; all closes into winter.
170
大雪十一月節鹖鳥不鳴虎始交荔挺出◎爻象圖
Major Snow (11th month, first node): the crossbill ceases to sing; tigers begin to mate; spring onions sprout. ◎ Hexagram-image chart.
171
冬至《坎》初六公《中孚》辟《復》侯《屯》 （內）
Winter Solstice: 《Kan》, first six; duke 《Zhong Fu》; emperor 《Fu》; marquis 《Zhun》. (inner hexagrams)
172
小寒《坎》九二侯《屯》 （外） 大夫《謙》卿《睽》
Minor Cold: 《Kan》, nine in the second; marquis 《Zhun》. (outer hexagrams) Great officer 《Qian》; minister 《Kui》.
173
大寒《坎》六三公《升》辟《臨》侯《小過》 （內）
Major Cold: 《Kan》, six in the third; duke 《Sheng》; emperor 《Lin》; marquis 《Xiao Guo》. (inner hexagrams)
174
立春《坎》六四侯《小過》 （外） 大夫《蒙》卿《益》
Start of Spring: 《Kan》, six in the fourth; marquis 《Xiao Guo》. (outer hexagrams) Great officer 《Meng》; minister 《Yi》.
175
雨水《坎》九五公《漸》辟《泰》侯《需》 （內）
Rain Water: 《Kan》, nine in the fifth; duke 《Jian》; emperor 《Tai》; marquis 《Xu》. (inner hexagrams)
176
驚蟄《坎》上六侯《需》 （外） 大夫《隨》卿《晉》
Awakening of Insects: 《Kan》, top six; marquis 《Xu》. (outer hexagrams) Great officer 《Sui》; minister 《Jin》.
177
春分《震》初九公《解》辟《大壯》侯《豫》 （內）
Spring Equinox: 《Zhen》, first nine; duke 《Xie》; emperor 《Da Zhuang》; marquis 《Yu》. (inner hexagrams)
178
清明《震》六二侯《豫》 （外） 大夫《訟》卿《蠱》
Pure Brightness: 《Zhen》, six in the second; marquis 《Yu》. (outer hexagrams) Great officer 《Song》; minister 《Gu》.
179
谷雨《震》六三公《革》辟《》侯《旅》內
Grain Rain: 《Zhen》, six in the third; duke 《Ge》; emperor 《》; marquis 《Lü》; inner.
180
立夏《震》九四侯《旅》 （外） 大夫《師》卿《比》
Start of Summer: 《Zhen》, nine in the fourth; marquis 《Lü》. (outer hexagrams) Great officer 《Shi》; minister 《Bi》.
181
小滿《震》六五公《小畜》辟《乾》侯《大有》 （內）
Minor Fullness: 《Zhen》, six in the fifth; duke 《Xiao Xu》; emperor 《Qian》; marquis 《Da You》. (inner hexagrams)
182
芒種《震》上六侯《大有》 （外） 大夫《家人》卿《井》
Grain in Ear: 《Zhen》, top six; marquis 《Da You》. (outer hexagrams) Great officer 《Jia Ren》; minister 《Jing》.
183
夏至《離》初九公《咸》辟《後》侯《鼎》內
Summer Solstice: 《Li》, first nine; duke 《Xian》; emperor 《Hou》; marquis 《Ding》; inner.
184
小暑《離》六二侯《鼎》 （外） 大夫《豐》卿《渙》
Minor Heat: 《Li》, six in the second; marquis 《Ding》. (outer hexagrams) Great officer 《Feng》; minister 《Huan》.
185
大暑《離》九三公《履》辟《遁》侯《恒》 （內）
Major Heat: 《Li》, nine in the third; duke 《Lü》; emperor 《Dun》; marquis 《Heng》. (inner hexagrams)
186
立秋《離》九四侯《恒》 （外） 大夫《節》卿《同人》
Start of Autumn: 《Li》, nine in the fourth; marquis 《Heng》. (outer hexagrams) Great officer 《Jie》; minister 《Tong Ren》.
187
處暑《離》六五公《損》辟《否》侯《巽》 （內）
End of Heat: 《Li》, six in the fifth; duke 《Sun》; emperor 《Pi》; marquis 《Xun》. (inner hexagrams)
188
白露《離》上九侯《巽》 （外） 大夫《萃》卿《大畜》
White Dew: 《Li》, top nine; marquis 《Xun》. (outer hexagrams) Great officer 《Cui》; minister 《Da Xu》.
189
秋分《兌》初九公《賁》辟《觀》侯《歸妹》 （內）
Autumn Equinox: 《Dui》, first nine; duke 《Bi》; emperor 《Guan》; marquis 《Gui Mei》. (inner hexagrams)
190
寒露《兌》九二侯《歸妹》 （外） 大夫《無妄》卿《明夷》
Cold Dew: 《Dui》, nine in the second; marquis 《Gui Mei》. (outer hexagrams) Great officer 《Wu Wang》; minister 《Ming Yi》.
191
霜降《兌》六三公《困》辟《剝》侯《艮》 （內）
Frost Descent: 《Dui》, six in the third; duke 《Kun》; emperor 《Bo》; marquis 《Gen》. (inner hexagrams)
192
立冬《兌》九四侯《艮》 （外） 大夫《既濟》卿《噬嗑》
Start of Winter: 《Dui》, nine in the fourth; marquis 《Gen》. (outer hexagrams) Great officer 《Ji Ji》; minister 《Shi He》.
193
小雪《兌》九五公《大過》辟《坤》侯《未濟》 （內）
Minor Snow: 《Dui》, nine in the fifth; duke 《Da Guo》; emperor 《Kun》; marquis 《Wei Ji》. (inner hexagrams)
194
大雪《兌》上六侯《未濟》 （外） 大夫《蹇》卿《頤》◎七十二候
Major Snow: 《Dui》, top six; marquis 《Wei Ji》. (outer hexagrams) Great officer 《Jian》; minister 《Yi》. ◎ Seventy-two hou.
195
各置中節，即初候也。 以候策累加之，即次候也。 ◎六十四卦
Set each at the mid-node of its solar term; that yields the first phenological hou. Add the hou divisor cumulatively to obtain each successive hou. ◎ The sixty-four hexagrams.
196
置中氣，即公卦也。 以卦策累加之，即次卦也。 置侯卦，以外策加之，即外卦也。 ◎五行用事
Set at the central qi; that is the duke hexagram. Add the hexagram divisor cumulatively to obtain each successive hexagram. From the marquis hexagram, add the outer divisor to obtain the outer hexagram. ◎ Governance of the Five Phases.
197
置四立之節而命之，即春木、夏火、秋金、冬水用事之初也。 置四季之節，各以維策加之，即土用事也。 ◎沒日
Fix the four establishment nodes and name them: that marks when Wood governs in spring, Fire in summer, Metal in autumn, and Water in winter. Fix the four seasonal nodes and add the intercalary divisor to each; that is when Earth governs. ◎ Submergence days (mo days).
198
中節分五千六百二十六秒六十五已上者，用減統法，為有沒分。 通氣策以乘之，氣盈而一，滿統法為日； 用加其氣而命之，即所求沒日也。 ◎滅日
When the mid-node fraction is 5,626 seconds 65 or more, subtract the epoch divisor to obtain the submergence remainder. Multiply by the communicated qi divisor, divide by the qi surplus plus one, and convert remainders filling the epoch divisor into days; add the result to the qi and name the day to obtain the submergence day sought. ◎ Extinction days (extinguished days).
199
常朔分朔虛已下者，為滅分。 以朔率乘之，朔虛而一，盈統法為日； 用加其朔而命之，即所求滅日也。
When the regular new-moon fraction falls at or below the new-moon void, that is the extinction remainder. multiply by the new-moon rate, divide by the new-moon void plus one, and convert remainders filling the epoch divisor into days; add the result to the new moon and name the day to obtain the extinction day sought.
200
右樸所撰《欽天曆經》四篇。 《舊史》亡其《步發斂》一篇，而在者三篇，簡略不完，不足為法。 樸歷世既罕傳，予嘗問於著作佐郎劉羲候叟，羲叟為予求得其本經，然後樸之歷大備。 羲叟好學知書史，尤通於星歷，嘗謂予曰：「前世造歷者，其法不同而多差。 至唐一行始以天地之中數作《大衍曆》，最為精密。 後世善治歷者，皆用其法，惟寫分擬數而已。 至樸亦能自為一家。 樸之曆法，總日躔差為盈縮二歷，分月離為遲疾二百四十八限，以考衰殺之漸，以審朓朒，而朔望正矣。 校赤道九限，更其率數，以步黃道，使日躔有常度； 分黃道八節，辨其內外，以揆九道，使月行如循環，而二曜協矣。 觀天勢之升降，察軌道之斜正，以制食差，而交會密矣。 測嶽臺之中晷，以辨二至之日夜，而軌漏實矣。 推星行之逆順、伏留，使舒亟有漸，而五緯齊矣。 然不能宏深簡易，而徑急是取。 至其所長，雖聖人出不能廢也。」 羲叟之言蓋如此，覽者得以考焉。
At right, four chapters composed by Pu: the 《Qintian Calendar Classic》. The 《Old History》 lost its one chapter 《Pacing Emergence and Retraction》; of those remaining, three chapters are brief and incomplete, insufficient to serve as method. Pu's calendar had scarcely been handed down. I once asked Liu Xisou, Assistant in the Directorate of Authorship; he obtained the original classic for me, and only then did Pu's calendar stand complete. Xisou was a devoted scholar of history and letters, especially expert in astronomy and calendrics. He once told me: "Calendar-makers of earlier ages used different methods, and most of them were in error. Reaching Tang, Yixing first used heaven-and-earth central numbers to make the 《Dayan Calendar》, most precise and close. Later masters of calendrics all followed his method, merely copying fractions and adjusting figures. Pu, in his turn, was able to establish a school of his own. In Pu's method, solar motion discrepancies are summed into paired surplus-and-wane tables; lunar motion is divided into 248 slow-and-swift limits to track diminishing and waxing, to judge the moon's phases—so that new and full moons align correctly. He calibrated nine equatorial limits and revised their rates to pace the ecliptic, giving the sun's motion a fixed measure; he divided the ecliptic into eight nodes, distinguished inner from outer paths, and gauged the nine courses of the moon so that her motion runs in cycles and the two luminaries keep accord. He observed how celestial momentum rises and falls, examined whether orbits run oblique or true, and regulated eclipse discrepancy so that conjunctions grew exact. He measured the central gnomon at the Imperial Observatory to fix the day-lengths of the solstices, and the clepsydra matched the solar path. He computed planets' retrograde and direct motion, their hiding and station, giving slow and swift motion proper gradation so that the five planets kept in step. Yet he could not make the system broad, deep, and simple; he took what was quick and direct. In what he did best, not even a sage could set his work aside." Such, in sum, was Xisou's judgment, which readers may consult for themselves.