Volume 27A Treatises 17A: Calendar 3A

開元九年，麟德曆署日蝕比不效，詔僧一行作新曆，推大衍數立術以應之，較經史所書氣朔、日名、宿度可考者皆合。十五年，草成而一行卒，詔特進張說與曆官陳玄景等次為曆術七篇、略例一篇、曆議十篇，玄宗顧訪者則稱制旨。明年，說表上之，起十七年頒于有司。時善算瞿曇譔者，怨不得預改曆事，二十一年，與玄景奏：「大衍寫九執曆，其術未盡。」太子右司禦率南宮說亦非之。詔侍御史李麟、太史令桓執圭較靈臺候簿，大衍十得七、八，麟德纔三、四，九執一、二焉。乃罪說等，而是否決。

In Kaiyuan 9 the Lindé Calendar office's predicted solar eclipses failed to match observation. The emperor ordered the monk Yi Xing to draft a new calendar, deriving the Dayan numbers and establishing computational methods to fit the heavens; checked against verifiable qi-periods, new moons, day-names, and lodge-degrees in the classics and histories, everything agreed. In year 15 the draft was finished but Yi Xing died. An edict ordered Zhang Yue, as specially advanced minister, and the calendar officials Chen Xuanjing and others to arrange in sequence seven treatises on calendrical method, one summary of principles, and ten calendar discourses; passages on which Emperor Xuanzong had personally consulted were labeled as imperial directives. The following year Zhang Yue memorialized and presented it; from year 17 it was issued to the responsible offices. At the time the skilled calculator Qutan Zhan, resentful that he had not been allowed to join the calendar reform, memorialized jointly with Chen Xuanjing in year 21: 'The Dayan calendar merely copies the Nine Foundations calendar; its methods are incomplete.' Nangong Shuo, commandant of the crown prince's right guard bureau, also opposed it. An edict ordered Attending Censor Li Lin and Grand Astrologer Huan Zhigui to check the Spirit Terrace observation records against the calendars: the Dayan scored seven or eight cases in ten, the Lindé only three or four, and the Nine Foundations one or two. Nangong Shuo and his allies were then punished, and their objections were rejected.

自太初至麟德，曆有二十三家，與天雖近而未密也。至一行，密矣，其倚數立法固無以易也。後世雖有改作者，皆依倣而已，故詳錄之。略例，所以明述作本旨也；曆議，所以考古今得失也。其說皆足以為將來折衷。略其大要，著于篇者十有二。

From the Taichu calendar down to the Lindé there had been twenty-three systems; they approached the heavens but were not yet precise. With Yi Xing it became precise; founded on numerical principles, his system could not be improved upon. Later reformers only followed his model, which is why it is set down here at length. The summary of principles states the author's original purpose; the calendar discourses weigh what past and present calendars got right and wrong. Their arguments are sufficient to guide future judgments between competing methods. Selecting the main points, twelve topics are set out in the treatises.

其一曆本議曰：

The first discourse, on the calendar's foundation, reads:

易：「天數五，地數五，五位相得而各有合，所以成變化而行鬼神也。」天數始於一，地數始於二，合二始以位剛柔。天數終於九，地數終於十，合二終以紀閏餘。天數中於五，地數中於六，合二中以通律曆。天有五音，所以司日也。地有六律，所以司辰也。參伍相周，究於六十，聖人以此見天地之心也。自五以降，為五行生數；自六以往，為五材成數。錯而乘之，以生數衍成位。一、六而退極，五、十而增極；一、六為爻位之統，五、十為大衍之母。成數乘生數，其算六百，為天中之積。生數乘成數，其算亦六百，為地中之積。合千有二百，以五十約之，則四象周六爻也；二十四約之，則太極包四十九用也。綜成數，約中積，皆十五。綜生數，約中積，皆四十。兼而為天地之數，以五位取之，復得二中之合矣。蓍數之變，九、六各一，乾坤之象也。七、八各三，六子之象也。故爻數通乎六十，策數行乎二百四十。是以大衍為天地之樞，如環之無端，蓋律曆之大紀也。

The Book of Changes says: 'Heaven's numbers are five, earth's numbers are five; when the five positions pair with one another each attains its conjunction, and thereby change is wrought and spirits and ghosts set in motion.' Heaven's numbers begin at one and earth's numbers at two; joining the two beginnings sets yin and yang in place. Heaven's numbers end at nine and earth's numbers at ten; joining the two endings records the intercalary remainder. Heaven's numbers center at five and earth's numbers at six; joining the two centers links pitch-pipes and calendar. Heaven has the five tones, and thereby governs the day. Earth has the six pitch-standards, and thereby governs the chronogram. Three and five interweave until they reach sixty; by this the sage perceives the mind of heaven and earth. From five downward come the generating numbers of the five phases; from six upward come the completion numbers of the five materials. Interlaced and multiplied, the generating numbers extend into positional values. One and six retreat to the limit, five and ten advance to the limit; one and six govern the line-positions, five and ten are the mother of the Dayan numbers. Multiplying completion numbers by generating numbers gives six hundred in calculation—the accumulation at heaven's center. Multiplying generating numbers by completion numbers likewise gives six hundred—the accumulation at earth's center. Together they make twelve hundred; divided by fifty, this yields the four images completing the six lines; divided by twenty-four, the Supreme Ultimate embraces the forty-nine uses. Aggregating the completion numbers and reducing the central accumulation, each is fifteen. Aggregating the generating numbers and reducing the central accumulation, each is forty. Taken together as the numbers of heaven and earth and selected by the five positions, one again obtains the conjunction of the two centers. In the changes of yarrow numbers, nine and six each occur once—the image of Qian and Kun. Seven and eight each occur three times—the image of the six children. Therefore the line-numbers extend through sixty, and the stalk-numbers run through two hundred forty. Thus the Dayan is the pivot of heaven and earth, like a ring without end—the great norm of pitch-pipes and calendar.

夫數象微於三、四，而章於七、八。卦有三微，策有四象，故二微之合，在始中之際焉。蓍以七備，卦以八周，故二章之合，而在中終之際焉。中極居五六間，由闢闔之交，而在章微之際者，人神之極也。天地中積，千有二百，揲之以四，為爻率三百；以十位乘之，而二章之積三千；以五材乘八象，為二微之積四十。兼章微之積，則氣朔之分母也。以三極參之，倍六位除之，凡七百六十，是謂辰法，而齊於代軌。以十位乘之，倍大衍除之，凡三百四，是謂刻法，而齊于德運。半氣朔之母，千五百二十，得天地出符之數，因而三之，凡四千五百六十，當七精返初之會也。易始于三微而生一象，四象成而後八卦章。三變皆剛，太陽之象。三變皆柔，太陰之象。一剛二柔，少陽之象。一柔二剛，少陰之象。少陽之剛，有始、有壯、有究。少陰之柔，有始、有壯、有究。兼三才而兩之，神明動乎其中。故四十九象，而大業之用周矣。數之德圓，故紀之以三而變於七。象之德方，故紀之以四而變于八。

Numbers and images are subtle at three and four, and fully manifest at seven and eight. The hexagram has three subtleties and the stalks four images; thus the conjunction of the two subtleties lies at the juncture of beginning and center. The stalks are complete at seven and the hexagrams cycle at eight; thus the conjunction of the two manifestations lies at the juncture of center and end. The central pole stands between five and six; passing through the juncture of opening and closing, at the juncture of manifestation and subtlety—this is the pole of human and spirit. The central accumulation of heaven and earth is twelve hundred; sorted by four this gives the line-rate of three hundred; multiplied by the ten positions, the accumulation of the two manifestations is three thousand; multiplying the five materials by the eight images gives the accumulation of the two subtleties, forty. Combining the accumulations of manifestation and subtlety yields the denominator for qi-period and new moon. With the three poles as reference, doubled and divided by the six positions, the total is seven hundred sixty—called the chronogram method, aligning with the generation-track. Multiplied by the ten positions, doubled and divided by the Dayan, the total is three hundred four—called the clepsis method, aligning with the virtue-cycle. Half the mother of qi-period and new moon is 1,520, obtaining the number when heaven and earth issue their tokens; tripling it gives 4,560, matching the conjunction when the seven luminaries return to their starts. The Changes begins with three subtleties to generate one image; once the four images are formed, the eight trigrams become manifest. Three changes all yang—the image of great yang. Three changes all yin—the image of great yin. One yang and two yin—the image of lesser yang. One yin and two yang—the image of lesser yin. In the yang of lesser yang there are beginning, flourishing, and completion. In the yin of lesser yin there are beginning, flourishing, and completion. Embracing the three powers and doubling them, spirit and clarity move within. Therefore with forty-nine images, the function of the great enterprise is complete. The virtue of number is round, so it is recorded in threes and changes at seven. The virtue of image is square, so it is recorded in fours and changes at eight.

人在天地中，以閱盈虛之變，則閏餘之初，而氣朔所虛也。以終合通大衍之母，虧其地十，凡九百四十為通數。終合除之，得中率四十九，餘十九分之九，終歲之弦，而斗分復初之朔也。地於終極之際，虧十而從天，所以遠疑陽之戰也。夫十九分之九，盈九而虛十也。乾盈九，隱乎龍戰之中，故不見其首。坤虛十，以導潛龍之氣，故不見其成。周日之朔分，周歲之閏分，與一章之弦，一蔀之月，皆合於九百四十，蓋取諸中率也。

Humankind stands between heaven and earth to observe fullness and emptiness; at the beginning of intercalary remainder lies what qi-period and new moon leave vacant. Using the end-conjunction to penetrate the mother of the Dayan, diminishing earth's ten, nine hundred forty in all is the universal number. Divided by the end-conjunction, one obtains the central rate of forty-nine, remainder nine-nineteenths—the year's quarter-moon, and the new moon when the Dipper's fraction returns to the start. Earth at the juncture of ultimate end diminishes ten to follow heaven, thereby keeping far from the battle of doubtful yang. Nine-nineteenths means full at nine and empty at ten. Qian, full at nine, is hidden within the dragon's battle; therefore its head is not seen. Kun, empty at ten, conducts the qi of the hidden dragon; therefore its completion is not seen. The new-moon fraction of the circuit-day, the intercalary fraction of the circuit-year, one chapter's quarter-moon, and one obscuration's month all combine at nine hundred forty—taken from the central rate.

一策之分十九，而章法生；一揲之分七十六，而蔀法生。一蔀之日二萬七千七百五十七，以通數約之，凡二十九日餘四百九十九，而日月相及於朔，此六爻之紀也。以卦當歲，以爻當月，以策當日，凡三十二歲而小終，二百八十五小終而與卦運大終，二百八十五，則參伍二終之合也。數象既合，而遯行之變在乎其間矣。

One stalk's fraction is nineteen, and from it the chapter method is born; one sorting's fraction is seventy-six, and from it the obscuration method is born. One obscuration's days are 27,757; reduced by the universal number, this is twenty-nine days remainder 499, when sun and moon meet at new moon—this is the record of the six lines. Hexagrams stand for years, lines for months, stalks for days—in all thirty-two years make a small completion; two hundred eighty-five small completions match the hexagram-cycle's great completion; two hundred eighty-five is the conjunction of the two completions of three and five. Once numbers and images have combined, the changes of receding motion lie between them.

所謂遯行者，以爻率乘朔餘，為十四萬九千七百，以四十九用、二十四象虛之，復以爻率約之，為四百九十八、微分七十五太半，則章微之中率也。二十四象，象有四十九蓍，凡千一百七十六。故虛遯之數七十三，半氣朔之母，以三極乘參伍，以兩儀乘二十四變，因而幷之，得千六百一十三，為朔餘。四揲氣朔之母，以八氣九精遯其十七，得七百四十三，為氣餘。歲八萬九千七百七十三而氣朔會，是謂章率。歲二億七千二百九十萬九百二十而無小餘，合于夜半，是謂蔀率。歲百六十三億七千四百五十九萬五千二百而大餘與歲建俱終，是謂元率。此不易之道也。

What is called receding motion: multiply new-moon remainder by the line-rate to get 149,700; void it with the forty-nine uses and twenty-four images, then reduce again by the line-rate to 498, minute-parts 75½—this is the central rate within chapter-subtlety. Twenty-four images; each image has forty-nine yarrow-stalks, 1,176 in all. Therefore the voided receding number is seventy-three; half the mother of qi-period and new moon, multiplied by the three poles and three-and-five, multiplied by the two instruments and twenty-four changes, then combined, gives 1,613 as new-moon remainder. Four sortings of the mother of qi-period and new moon, with eight qi and nine luminaries receding seventeen, gives 743 as qi-remainder. At 89,773 years qi and new moon meet—this is called the chapter rate. At 272,900,920 years without small remainder, matching at midnight—this is called the obscuration rate. At 16,374,595,200 years the great remainder and year-establishment both end—this is called the origin rate. This is the unchanging Way.

策以紀日，象以紀月。故乾坤之策三百六十，為日度之準。乾坤之用四十九象，為月弦之檢。日之一度，不盈全策；月之一弦，不盈全用。故策餘萬五千九百四十三，則十有二中所盈也。用差萬七千一百二十四，則十有二朔所虛也。綜盈虛之數，五歲而再閏。中節相距，皆當三五；弦望相距，皆當二七。升降之應，發斂之候，皆紀之以策而從日者也。表裏之行，朓朒之變，皆紀之以用而從月者也。

Stalks record days, images record months. Therefore the stalks of Qian and Kun are three hundred sixty—the standard of daily motion. The uses of Qian and Kun are forty-nine images—the gauge of lunar quarters. One day-degree does not fill the full stalk-count; one lunar quarter does not fill the full use. Therefore stalk-remainder 15,943 is what the twelve centers exceed. Use-difference 17,124 is what the twelve new moons leave vacant. Aggregating the numbers of excess and vacancy, intercalation occurs again every five years. Mid-season nodes are separated, all matching three-five; quarters and full moons are separated, all matching two-seven. Responses of ascent and descent, signs of issuing and gathering—all are recorded by stalks and follow the day. Inner and outer motion, changes of waxing and waning—all are recorded by uses and follow the moon.

積算曰演紀，日法曰通法，月氣曰中朔，朔實曰揲法，歲分曰策實，周天曰乾實，餘分曰虛分。氣策曰三元，一元之策，則天一遯行也。月策曰四象，一象之策，則朔、弦、望相距也。五行用事，曰發斂。候策曰天中，卦策曰地中，半卦曰貞悔。旬周曰爻數，小分母曰象統。日行曰躔，其差曰盈縮，積盈縮曰先後。古者平朔，月朝見曰朒，夕見曰朓。今以日之所盈縮、月之所遲疾損益之，或進退其日，以為定朔。舒亟之度，乃數使然，躔離相錯，偕以損益，故同謂之朓朒。月行曰離，遲疾曰轉度，母曰轉法。遲疾有衰，其變者勢也。月逶迤馴屈，行不中道，進退遲速，不率其常。過中則為速，不及中則為遲。積遲謂之屈，積速謂之伸。陽，執中以出令，故曰先後；陰，含章以聽命，故曰屈伸。日不及中則損之，過則益之。月不及中則益之，過則損之。尊卑之用睽，而及中之志同。觀晷景之進退，知軌道之升降。軌與晷名舛而義合，其差則水漏之所從也。總名曰軌漏。中晷長短謂之陟降。景長則夜短，景短則夜長。積其陟降，謂之消息。遊交曰交會，交而周曰交終。交終不及朔，謂之朔差。交中不及望，謂之望差。日道表曰陽曆，其裏曰陰曆。五星見伏周，謂之終率。以分從日謂之終日，其差為進退。

Accumulated counts are called evolving eras; day-divisor is called universal method; month-qi is called central-new-moon; new-moon substance is called sorting method; year-fraction is called stalk substance; circuit of heaven is called Qian substance; remainder-fraction is called void fraction. Qi-stalk is called three origins; one origin's stalks are heaven-one receding motion. Month-stalk is called four images; one image's stalks are the spacing of new, quarter, and full moon. The five phases' dominion is called issuing and gathering. Qi-stalk is called heaven-center, hexagram-stalk is called earth-center, half-hexagram is called correct and repentant. Ten-day cycle is called line-number, small denominator is called image-unifier. Daily motion is called progression; its difference is called excess and deficiency; accumulated excess and deficiency is called before and after. Ancient calendars used mean new moons: a moon visible at dawn was called waning visibility, one visible at dusk waxing visibility. Today one adjusts for the sun's excess and deficiency and the moon's slow/fast anomaly, advancing or delaying the day as needed to obtain the fixed new moon. Slow and fast degrees arise from numerical relations; solar progression and lunar departure cross and are corrected together, which is why both are grouped under waxing/waning visibility. Lunar travel is termed departure; the slow/fast variation is rotation-degree, and its denominator the rotation-factor. The slow/fast cycle declines over time; what actually shifts is the underlying tendency. The moon's path is winding and irregular; it does not keep to the mean track, and its speed varies rather than holding steady. When motion passes the mean rate it is fast; when it falls short, slow. Built-up lag is termed bending; built-up speed, extending. Yang holds the center to give orders, hence the term before and after; Yin holds its pattern and receives command, hence bending and extending. If the sun falls short of the mean, one subtracts; if it overshoots, one adds. If the moon falls short of the mean, one adds; if it overshoots, one subtracts. Superior and subordinate work in opposite ways, but both aim at the same mean. By watching how the gnomon shadow lengthens and shortens, one learns whether the solar track is ascending or descending. Orbit and gnomon are differently named but refer to the same thing; the clepsydra measures the interval between them. Taken together, these are called orbit-clepsydra. Variations in the midday gnomon shadow are termed ascent and descent. Long shadows mean short nights; short shadows mean long nights. The running total of ascent and descent is known as ebb and flow. A crossing encounter is termed conjunction; when the crossing completes a full cycle, conjunction-end. If conjunction-end does not reach new moon, the offset is called new-moon discrepancy. If the crossing midpoint falls short of full moon, the offset is called full-moon discrepancy. The sun's outward track is the yang calendar; its inward track the yin calendar. The period in which the five planets appear and hide is termed the terminal rate. Tracking the sun by fractional degrees yields the terminal day; the resulting offset is advance and retreat.

其二中氣議曰：

The second treatise on qi-periods states:

曆氣始于冬至，稽其實，蓋取諸晷景。春秋傳僖公五年正月辛亥朔，日南至。以周曆推之，入壬子蔀第四章，以辛亥一分合朔冬至，殷曆則壬子蔀首也。昭公二十年二月己丑朔，日南至。魯史失閏，至不在正。左氏記之，以懲司曆之罪。周曆得己丑二分，殷曆得庚寅一分。殷曆南至常在十月晦，則中氣後天也。周曆蝕朔差經或二日，則合朔先天也。傳所據者周曆也，緯所據者殷曆也。氣合于傳，朔合于緯，斯得之矣。戊寅曆月氣專合于緯，麟德曆專合于傳，偏取之，故兩失之。又命曆序以為孔子脩春秋用殷曆，使其數可傳於後。考其蝕朔不與殷曆合，及開元十二年，朔差五日矣，氣差八日矣。上不合於經，下不足以傳於後代，蓋哀、平間治甲寅元曆者託之，非古也。又漢太史令張壽王說黃帝調曆以非太初。有司劾：「官有黃帝調曆不與壽王同，壽王所治乃殷曆也。」漢自中興以來，圖讖漏泄，而考靈曜、命曆序皆有甲寅元，其所起在四分曆庚申元後百一十四歲。延光初中謁者亶誦，靈帝時五官郎中馮光等，皆請用之，卒不施行。緯所載壬子冬至，則其遺術也。魯曆南至，又先周曆四分日之三，而朔後九百四十分日之五十一。故僖公五年辛亥為十二月晦，壬子為正月朔。又推日蝕密於殷曆，其以閏餘一為章首，亦取合於當時也。

Calendrical qi-periods start from the winter solstice; in practice they derive from gnomon measurements. The Zuoshi Commentary records that in Duke Xi's fifth year, on the xinhai new moon of the first month, the sun reached its southern limit. Reckoned by the Zhou calendar, this falls in the fourth chapter of the renzi obscuration, with winter solstice and conjunction new moon at xinhai plus one part; the Yin calendar places it at the head of the renzi obscuration. The Commentary also records Duke Zhao's twentieth year, second month, jichou new moon, with the sun at its southern limit. Lu's chronicle had failed to intercalate, so the solstice did not fall in the correct month. Zuoshi preserved the entry to rebuke the calendar officers for their error. By the Zhou calendar the solstice falls at jichou plus two parts; by the Yin calendar, at gengyin plus one part. In the Yin calendar the solstice usually lands on the last day of the tenth month, meaning qi-periods run late. The Zhou calendar's eclipse and new-moon dates can be off by as much as two days, so conjunction new moons come early. The Commentary follows the Zhou calendar; the apocryphal texts follow the Yin calendar. Match qi-periods to the Commentary and new moons to the apocrypha, and the reckoning holds. The Wuyin calendar aligned only with the apocrypha, the Lindé only with the Commentary; by taking one source each, both went wrong. The Calendar Order also asserts that Confucius compiled the Spring and Autumn Annals with the Yin calendar so its figures could be handed down. Checked against eclipses and new moons, the figures fail to agree with the Yin calendar; by Kaiyuan 12 the new-moon error had reached five days and the qi error eight. It fits neither the canonical texts nor any workable tradition for posterity; it was probably ascribed by compilers of the Jiayin origin calendar in the Ai–Ping period and is not genuinely old. The Han Grand Astrologer Zhang Shouwang also cited the Yellow Emperor's adjusted calendar to attack the Taichu system. Officials impeached him: "The directorate holds a Yellow Emperor adjusted calendar that differs from Shouwang's; Shouwang's work is in fact the Yin calendar." After the Han restoration, prognostic charts circulated widely; both Examining the Luminous Glaze and the Calendar Order posit a Jiayin origin starting 114 years after the Quarter-Remainder calendar's gengshen epoch. In early Yanguang the palace attendant Shan Song, and under Emperor Ling the five-offices gentleman Feng Guang and others, all petitioned to adopt it, but it was never put into use. The renzi winter solstice preserved in the apocrypha is a remnant of that system. In the Lu calendar the solstice comes three-quarters of a day earlier than in the Zhou calendar, while new moon falls fifty-one nine-hundred-fortieths of a day later. Hence in Duke Xi's fifth year xinhai was the last day of the twelfth month and renzi the new moon of the first month. Eclipse calculations also fit the Yin calendar more closely, and setting the intercalary remainder at one for the chapter head suited conditions of the day.

開元十二年十一月，陽城測景，以癸未極長，較其前後所差，則夜半前尚有餘分。新曆大餘十九，加時九十九刻，而皇極、戊寅、麟德曆皆得甲申，以玄始曆氣分二千四百四十三為率，推而上之，則失春秋辛亥，是減分太多也。以皇極曆氣分二千四百四十五為率，推而上之，雖合春秋，而失元嘉十九年乙巳冬至及開皇五年甲戌冬至、七年癸未夏至；若用麟德曆率二千四百四十七，又失春秋己丑。是減分太少也。故新曆以二千四百四十四為率，而舊所失者皆中矣。

In the eleventh month of Kaiyuan 12 Yang Cheng measured the gnomon shadow; on guiwei the shadow was longest, and comparing readings before and after showed fractional parts still remaining before midnight. The new calendar gives a great remainder of 19 and an added time of 99 quarters, while the Huangji, Wuyin, and Lindé calendars all yield jiashen. Using the Xuanshi calendar's qi fraction of 2,443 as the rate and extrapolating backward into the past misses the Spring and Autumn xinhai date—meaning the reduced fraction is too large. Using the Huangji calendar's qi fraction of 2,445 as the rate and extrapolating backward matches the Spring and Autumn but misses the winter solstice of Yuanjia year 19 (yisi), the winter solstice of Kaihuang year 5 (jiaxu), and the summer solstice of Kaihuang year 7 (guimwei); With the Lindé calendar's rate of 2,447, the Spring and Autumn jichou date is missed again. That means the reduced fraction is too small. The new calendar therefore adopts 2,444 as the rate, and every date the older calendars missed now agrees.

漢會稽東部尉劉洪以四分疎闊，由斗分多。更以五百八十九為紀法，百四十五為斗分，減餘太甚，是以不及四十年而加時漸覺先天。韓翊、楊偉、劉智等皆稍損益，更造新術，而皆依讖緯「三百歲改憲」之文，考經之合朔多中，較傳之南至則否。玄始曆以為十九年七閏，皆有餘分，是以中氣漸差。據渾天，二分為東西之中，而晷景不等；二至為南北之極，而進退不齊。此古人所未達也。更因劉洪紀法，增十一年以為章歲，而減閏餘十九分之一。春秋後五十四年，歲在甲寅，直應鍾章首，與景初曆閏餘皆盡。雖減章閏，然中氣加時尚差，故未合于春秋。其斗分幾得中矣。

During the Han, Liu Hong of Kuaiji's eastern commandery found the Quarter-Remainder calendar too coarse, owing to an excessively large dipper fraction. He revised the era method to 589 and the dipper fraction to 145, but cut the remainder too drastically; within forty years the added times were noticeably running ahead of heaven. Han Yi, Yang Wei, Liu Zhi, and others each tweaked the parameters and devised new methods, all citing the apocryphal maxim that "the calendar should be revised every three hundred years". Against the classics their new moons mostly agreed, but against the Annals' winter solstices they did not. The Xuanshi calendar treated the seven intercalations in nineteen years as each carrying a surplus fraction, causing mid-qi to drift over time. On the armillary-heaven model the equinoxes mark the midpoint between east and west, yet the gnomon shadows at those points are unequal; The solstices mark the northern and southern extremes, yet the sun's advance and retreat are uneven. The ancients had not grasped this. Building on Liu Hong's era method, he added eleven years to the chapter year and cut the intercalary remainder by one nineteenth. Fifty-four years after the Spring and Autumn era, in the jiayin year, the cycle aligned directly with the zhong chapter head, at which point the Jingchu calendar's intercalary remainder ran out entirely. Despite trimming the intercalary chapter, the times of mid-qi still diverged, so it failed to agree with the Spring and Autumn. Its dipper fraction came very close to the correct value.

後代曆家，皆因循玄始，而損益或過差。大抵古曆未減斗分，其率自二千五百以上。乾象至于元嘉曆，未減閏餘，其率自二千四百六十以上。玄始、大明至麟德曆皆減分破章，其率自二千四百二十九以上。較前代史官注記，惟元嘉十三年十一月甲戌景長，皇極、麟德、開元曆皆得癸酉，蓋日度變常爾。祖沖之既失甲戌冬至，以為加時太早，增小餘以附會之。而十二年戊辰景長，得己巳；十七年甲午景長，得乙未；十八年己亥景長，得庚子。合一失三，其失愈多。劉孝孫、張冑玄因之，小餘益強，又以十六年己丑景長為庚寅矣。治曆者糾合眾同，以稽其所異，苟獨異焉，則失行可知。今曲就其一，而少者失三，多者失五，是捨常數而從失行也。周建德六年，以壬辰景長，而麟德、開元曆皆得癸巳。開皇七年，以癸未景短，而麟德、開元曆皆得壬午。先後相戾，不可叶也，皆日行盈縮使然。

Later calendrists all followed the Xuanshi model, but their adjustments sometimes overshot or undershot. Broadly speaking, older calendars that had not reduced the dipper fraction used rates of 2,500 and above. From the Qianxiang through the Yuanjia calendar, which did not reduce the intercalary remainder, rates ranged from 2,460 upward. The Xuanshi, Daming, and Lindé calendars all reduced fractions and split chapters, with rates from 2,429 upward. Compared with annotated records from earlier dynasties' historiographers, only for the long-shadow solstice of Yuanjia the thirteenth year, month 11, jiaxu do the Huangji, Lindé, and Kaiyuan calendars all yield guiyou—presumably because the sun's apparent position varied from its mean rate. After missing the jiaxu winter solstice, Zu Chongzhi decided the added time was too early and inflated the small remainder to make the numbers fit. Yet in year 12 the long-shadow solstice fell on wuchen while his calendar gave jisi; In year 17 the long-shadow solstice was jiawu but he obtained yiwei; In year 18 the long-shadow solstice was jihai but he obtained gengzi. He matched one and missed three—the errors piled up. Liu Xiaosun and Zhang Zhouxuan followed suit, pushing the small remainder even harder, and likewise placed the year 16 jichou long-shadow solstice on gengyin. A calendrist should gather what most calendars agree on and scrutinize the outliers; if a calendar stands alone in disagreement, its faulty reckoning becomes evident. Bending the constants to fit a single case yields three misses in one instance and five in another—abandoning the stable numbers to chase an anomaly. At the long-shadow solstice of Northern Zhou Jiande year 6, observed on renchen, the Lindé and Kaiyuan calendars both give guisi. At the short-shadow solstice of Kaihuang year 7, observed on guimwei, the Lindé and Kaiyuan calendars both give renwu. The discrepancies between earlier and later cases cannot be reconciled—they all arise from the sun's variable apparent speed.

凡曆術在於常數，而不在於變行。既叶中行之率，則可以兩齊先後之變矣。麟德已前，實錄所記，乃依時曆書之，非候景所得。又比年候景，長短不均，由加時有早晏，行度有盈縮也。

Calendar making rests on fixed constants, not on chasing variable motion. Once the mean-motion rate is established, both early and late deviations can be balanced together. Before the Lindé era, the Veritable Records simply copied the reigning calendar, not actual gnomon observations. Recent shadow observations have also been inconsistent, since added times vary and solar motion is uneven.

自春秋以來，至開元十二年，冬、夏至凡三十一事，戊寅曆得十六，麟德曆得二十三，開元曆得二十四。

From the Spring and Autumn through Kaiyuan year 12, there are thirty-one verifiable winter and summer solstices. The Wuyin calendar matches sixteen, the Lindé twenty-three, and the Kaiyuan twenty-four.

其三合朔議曰：

The third discourse, on new-moon conjunction, states:

日月合度謂之朔。無所取之，取之蝕也。春秋日蝕有甲乙者三十四。殷曆、魯曆先一日者十三，後一日者三；周曆先一日者二十二，先二日者九。其偽可知矣。

When the sun and moon share the same celestial degree, that is called the new moon. Having no other anchor to rely on, one turns to eclipses. The Spring and Autumn records thirty-four solar eclipses falling on jia or yi days. Against the Yin and Lu calendars, thirteen eclipses fall one day early and three one day late; Against the Zhou calendar, twenty-two fall one day early and nine two days early. Its spuriousness is obvious.

莊公三十年九月庚午朔，襄公二十一年九月庚戌朔，定公五年三月辛亥朔，當以盈縮、遲速為定朔。殷曆雖合，適然耳，非正也。僖公五年正月辛亥朔，十二月丙子朔，十四年三月己丑朔；文公元年五月辛酉朔，十一年三月甲申晦；襄公十九年五月壬辰晦；昭公元年十二月甲辰朔，二十年二月己丑朔，二十三年正月壬寅朔、七月戊辰晦：皆與周曆合。其所記多周、齊、晉事，蓋周王所頒，齊、晉用之。僖公十五年九月己卯晦，十六年正月戊申朔；成公十六年六月甲午晦；襄公十八年十月丙寅晦、十一月丁卯朔，二十六年三月甲寅朔，二十七年六月丁未朔：與殷曆、魯曆合。此非合蝕，故仲尼因循時史，而所記多宋、魯事，與齊、晉不同可知矣。

The new moons of Duke Zhuang year 30 (9th month, gengwu), Duke Xiang year 21 (9th month, gengxu), and Duke Ding year 5 (3rd month, xinhai) should be determined as fixed new moons, accounting for solar excess and deficiency and lunar slow and fast motion. The Yin calendar may appear to agree, but that is coincidence, not proof of correctness. Duke Xi's year 5 (1st month xinhai, 12th month bingzi) and year 14 (3rd month jichou) new moons; Duke Wen's year 1 (5th month xinyou new moon) and year 11 (3rd month jiashen, last day of month); Duke Xiang year 19, 5th month renchen, last day of month; Duke Zhao's year 1 (12th month jiachen), year 20 (2nd month jichou), and year 23 (1st month renyin new moon and 7th month wuchen last day)—all agree with the Zhou calendar. Its entries chiefly concern Zhou, Qi, and Jin—likely calendars issued by the Zhou court and adopted by Qi and Jin. Duke Xi's year 15 (9th month, jimao, last day) and year 16 (1st month, wushen new moon); Duke Cheng's year 16 (6th month, jiawu, last day); Duke Xiang's year 18 (10th month bingyin last day and 11th month dingmao new moon), year 26 (3rd month jiayin new moon), and year 27 (6th month dingwei new moon)—all agree with the Yin and Lu calendars. These are not records tied to eclipses, so Confucius followed contemporary annals; since those annals chiefly record Song and Lu, they naturally differ from Qi and Jin.

昭公十二年十月壬申朔，原輿人逐原伯絞，與魯曆、周曆皆差一日，此丘明即其所聞書之也。僖公二十二年十一月己巳朔，宋、楚戰于泓。周、殷、魯曆皆先一日，楚人所赴也。昭公二十年六月丁巳晦，衞侯與北宮喜盟；七月戊午朔，遂盟國人。三曆皆先二日，衞人所赴也。此則列國之曆不可以一術齊矣。而長曆日子不在其月，則改易閏餘，欲以求合。故閏月相距，近則十餘月，遠或七十餘月，此杜預所甚繆也。夫合朔先天，則經書日蝕以糾之。中氣後天，則傳書南至以明之。其在晦、二日，則原乎定朔以得之。列國之曆或殊，則稽於六家之術以知之。此四者，皆治曆之大端，而預所未曉故也。

Duke Zhao's year 12 (10th month, renshen new moon) records the expulsion of Lord Yuan Jiao by Yuanyu—one day off from both the Lu and Zhou calendars; Qiu Ming recorded it as he had heard it. Duke Xi's year 22 (11th month, jisi new moon): Song and Chu fought at Hong. The Zhou, Yin, and Lu calendars all run one day ahead—reflecting the date the Chu envoys reported. Duke Zhao's year 20 (6th month, dingsi, last day): the Marquis of Wei made a covenant with Beigong Xi; On the wuwu new moon of the 7th month he covenanted with the state's people. All three calendars are two days ahead—what the Wei envoys reported. This shows that the states' calendars cannot be forced into a single system. When the Long Calendar places a day outside its proper month, calendrists tamper with the intercalary remainder trying to force a match. Intercalary months therefore end up spaced anywhere from a little over ten to more than seventy months apart—a grave error in Du Yu's approach. If conjunction dates run ahead of heaven, solar eclipses in the classics serve as a corrective. If mid-qi dates lag heaven, the Annals' winter solstice entries clarify the error. When dates fall on the last day of the month or the second day, fixed-new-moon calculation supplies the answer. Where state calendars diverge, comparison with the six schools' methods reveals the truth. These four points are the main pillars of calendrical science—points Du Yu failed to grasp.

新曆本春秋日蝕、古史交會加時及史官候簿所詳，稽其進退之中，以立常率。然後以日躔、月離、先後、屈伸之變，偕損益之。故經朔雖得其中，而躔離或失其正；若躔離各得其度，而經朔或失其中，則參求累代，必有差矣。三者迭相為經，若權衡相持，使千有五百年間朔必在晝，望必在夜，其加時又合，則三術之交，自然各當其正，此最微者也。若乾度盈虛，與時消息，告譴於經數之表，變常於潛遯之中，則聖人且猶不質，非籌曆之所能及矣。

The new calendar is founded on Spring and Autumn eclipses, ancient records of conjunction times, and detailed observation logs; from their advance and retreat it derives mean rates. It then applies compensating adjustments for the sun's progression, the moon's departure, sequencing, and waxing and waning. Mean new moon may therefore be accurate while solar progression or lunar departure still miss their marks; If progression and departure are correct but mean new moon is off, cross-checking across dynasties will inevitably reveal gaps. The three elements must work as mutual checks, like paired scales: over 1,500 years new moons should fall by day, full moons by night, and added times should agree—then the three methods naturally align. This is the finest calibration. When celestial degrees wax and wane seasonally, sending warnings beyond what standard rates predict and altering constants in ways hidden from ordinary reckoning, even sages would not pretend to nail it down—no arithmetic calendar can capture that.

昔人考天事，多不知定朔。假蝕在二日，而常朔之晨，月見東方；食在晦日，則常朔之夕，月見西方。理數然也。而或以為朓朒變行，或以為曆術疎闊，遇常朔朝見則增朔餘，夕見則減朔餘，此紀曆所以屢遷也。漢編訢、李梵等又以晦猶月見，欲令蔀首先大。賈逵曰：「春秋書朔晦者，朔必有朔，晦必有晦，晦朔必在其月前也。先大，則一月再朔，後月無朔，是朔不可必也。訢、梵等欲諧偶十六日，月朓昏，晦當滅而已。又晦與合朔同時，不得異日。」考逵等所言，蓋知之矣。晦朔之交，始終相際，則光盡明生之限，度數宜均。故合於子正，則晦日之朝，猶朔日之夕也，是以月皆不見。若合於午正，則晦日之晨，猶二日之昏也，是以月或皆見。若陰陽遲速，軌漏加時不同，舉其中數率，去日十三度以上而月見，乃其常也。且晦日之光未盡也，如二日之明已生也。一以為是，一以為非。又常朔進退，則定朔之晦、二也。或以為變，或以為常。是未通於四三交質之論也。

Earlier scholars studying heaven mostly did not understand fixed new moon. Suppose an eclipse falls on the second day, yet on the morning of the regular new moon the moon is still visible in the east; When an eclipse falls on the last day, on the evening of the regular new moon the moon can still be seen in the west. That follows from the underlying logic. Some blamed irregular crescent motion, others blamed crude calendar methods—adding to the new-moon remainder when the moon appeared at dawn on regular new-moon day, subtracting when it appeared at dusk. That is why era calendars were revised again and again. In the Han, Bian Xin, Li Fan, and others, seeing the moon still visible on the last day, tried to make the obscuration-cycle head run ahead. Jia Kui said: "When the Spring and Autumn records a new moon or last day, each must genuinely occur—the last day and new moon must fall within the month they belong to. If the cycle runs too far ahead, one month gets two new moons and the next none—making new moon unreliable. Xin and Fan wanted to align with the sixteenth day—the crescent visible at dusk, the last day when it should disappear. Moreover, last day and conjunction occur at the same time—they cannot fall on different days." Considering what Kui and his colleagues said, they essentially understood the point. At the junction of last day and new moon, where darkness ends and light begins, the degree-count should be symmetrical. When conjunction falls at midnight (zi), the last day's dawn matches the new moon day's dusk, so the moon is invisible throughout. When conjunction falls at noon (wu), the last day's dawn resembles the second day's dusk, so the moon may be visible at both times. Because yin and yang advance unevenly and clepsydra times differ, as a rule the moon becomes visible when it stands more than thirteen degrees from the sun. On the last day its light is not yet fully gone, just as on the second day light has already begun. One school treats it as correct, another as erroneous. Shifts in the regular new moon produce the fixed new moon's last-day and second-day cases. Some call it an anomaly, others call it normal. That shows a failure to grasp the four-three exchange principle.

綜近代諸曆，以百萬為率齊之，其所差，少或一分，多至十數失一分。考春秋纔差一刻，而百數年間不足成朓朒之異。施行未幾，旋復疎闊，由未知躔離經朔相求耳。李業興、甄鸞等欲求天驗，輒加減月分，遷革不已，朓朒相戾，又未知昏明之限與定朔故也。楊偉採乾象為遲疾陰陽曆，雖知加時後天，蝕不在朔，而未能有以更之也。

Comparing recent calendars on a denominator of one million, their errors range from one part to over ten parts short of one full unit. Against the Spring and Autumn the discrepancy is only about one quarter—but over a few centuries that is too small to produce visible crescent anomalies. Implemented briefly, each calendar soon grew coarse again because its makers did not understand how progression, departure, and mean new moon must be solved together. Li Yexing, Zhen Luan, and others, seeking celestial confirmation, kept adjusting lunar fractions and revising endlessly, producing contradictory crescent readings—because they did not understand twilight limits and fixed new moon. Yang Wei adapted the Qianxiang into a slow-fast yin-yang calendar; although he knew added times lagged heaven and eclipses did not always fall on new-moon day, he could not fix it.

何承天欲以盈縮定朔望小餘。錢樂之以為：「推交會時刻雖審，而月頻三大二小。日蝕不唯在朔，亦有在晦、二者。」皮延宗又以為：「紀首合朔，大小餘當盡，若每月定之，則紀首位盈，當退一日，便應以故歲之晦為新紀之首。立法之制，如為不便。」承天乃止。虞𠠎曰：「所謂朔在會合，苟躔次既同，何患於頻大也？日月相離，何患於頻小也？」春秋日蝕不書朔者八，公羊曰：「二日也。」穀梁曰：「晦也。」左氏曰：「官失之也。」劉孝孫推俱得朔日，以丘明為是，乃與劉焯皆議定朔，為有司所抑不得行。傅仁均始為定朔，而曰「晦不東見，朔不西朓」，以為昏晦當滅，亦訢、梵之論。淳風因循皇極，皇極密於麟德，以朔餘乘三千四十，乃一萬除之，就全數得千六百一十三。又以九百四十乘之，以三千四十而一，得四百九十八秒七十五太彊，是為四分餘率。

He Chengtian wanted to use solar expansion and contraction to fix the small remainders for new and full moons. Qian Yuezhi argued: "Even if conjunction times are precise, months will too often run three long and two short. Solar eclipses occur not only on new-moon day but also on the last day or the second day." Pi Yanzong added: "At era-head conjunction the great and small remainders should read zero; if the new moon is fixed month by month, the era head overflows and must retreat a day—meaning the previous year's last day becomes the new era head. Such a legislative arrangement would be impractical." Chengtian abandoned the idea. Yu Xi said: "They say new moon should coincide with conjunction—if celestial positions already agree, why worry about months being too often long? When sun and moon diverge, why worry about months being too often short?" Eight Spring and Autumn eclipses omit the new-moon marker; the Gongyang Commentary says: "The second day." The Guliang Commentary says: "The last day." The Zuozhuan says: "The record-keepers erred." Liu Xiaosun calculated that all eight should fall on new-moon day and sided with Qiu Ming; he and Liu Chuo both advocated fixed new moon but were blocked by the authorities. Fu Renjun pioneered fixed new moon but held that "the last day should not appear in the east, nor new moon wax in the west"—the moon should vanish at last-day dusk—the same logic as Xin and Fan. Chunfeng followed the Huangji calendar, which is finer than the Lindé; multiplying the new-moon remainder by 3,040 and dividing by 10,000 gives the whole number 1,613. Multiplying again by 940 and dividing by 3,040 yields 498 seconds plus 75/96—forming the Quarter-Remainder surplus rate.

劉洪以古曆斗分太彊，久當後天，乃先正斗分，而後求朔法，故朔餘之母煩矣。韓翊以乾象朔分太弱，久當先天，乃先考朔分，而後覆求度法，故度餘之母煩矣。何承天反覆相求，使氣朔之母合簡易之率，而星數不得同元矣。李業興、宋景業、甄鸞、張賓欲使六甲之首眾術同元，而氣朔餘分，其細甚矣。麟德曆有總法，開元曆有通法，故積歲如月分之數，而後閏餘偕盡。

Liu Hong found the ancient calendar's dipper fraction too large, which would eventually make dates lag heaven; he corrected the dipper fraction first and then derived the new-moon method, making the new-moon denominator unwieldy. Han Yi found the Qianxiang new-moon fraction too small, which would eventually make dates run ahead of heaven; he fixed the new-moon fraction first and then reworked the degree method, making the degree denominator unwieldy. He Chengtian iterated until qi and new-moon denominators matched a simpler rate, but star-cycle numbers could no longer share a common origin. Li Yexing, Song Jingye, Zhen Luan, and Zhang Bin wanted all methods to share the same origin at the six jia heads, but the qi, new-moon, and remainder fractions were extraordinarily fine. The Lindé calendar has a general method and the Kaiyuan a universal method, so accumulated years match the month-fraction count and the intercalary remainder is fully exhausted together.

考漢元光已來史官注記，日蝕有加時者凡三十七事，麟德曆得五，開元曆得二十二。

Reviewing historiographers' notes from Han Yuanguang onward, thirty-seven solar eclipses record an added time; the Lindé calendar matches five, the Kaiyuan twenty-two.

其四沒滅略例曰：

The fourth section, summary rules for obscuration and extinction, states:

古者以中氣所盈之日為沒，沒分偕盡者為滅。開元曆以中分所盈為沒，朔分所虛為滅。綜終歲沒分，謂之策餘。終歲滅分，謂之用差。皆歸于揲易再扐而後掛也。

Anciently, the day when mid-qi filled the month was called obscuration; when obscuration fractions were fully exhausted it was called extinction. The Kaiyuan calendar treats what the mid-fraction fills as obscuration and what the new-moon fraction empties as extinction. The year's total obscuration fraction is called the divination remainder. The year's total extinction fraction is called the use difference. Both derive from the Yijing practice of dividing stalks twice before hanging the remainder.

其五卦候議曰：

The fifth section, the hexagram-season discourse, states:

七十二候，原于周公時訓。月令雖頗有增益，然先後之次則同。自後魏始載于曆，乃依易軌所傳，不合經義。今改從古。

The seventy-two seasons originate in the Duke of Zhou's seasonal instructions. The Monthly Ordinances adds some entries, but their sequence is unchanged. Northern Wei first embedded them in calendars, following a transmitted Yijing track that does not match canonical doctrine. The new calendar reverts to the ancient arrangement.

其六卦議曰：

The sixth section, the hexagram discourse, states:

十二月卦出於孟氏章句，其說易本於氣，而後以人事明之。京氏又以卦爻配期之日，坎、離、震、兌，其用事自分、至之首，皆得八十分日之七十三。頤、晉、井、大畜，皆五日十四分，餘皆六日七分，止於占災眚與吉凶善敗之事。至於觀陰陽之變，則錯亂而不明。自乾象曆以降，皆因京氏。惟天保曆依易通統軌圖。自入十有二節、五卦、初爻，相次用事，及上爻而與中氣偕終，非京氏本旨及七略所傳。按郎顗所傳，卦皆六日七分，不以初爻相次用事，齊曆謬矣。又京氏減七十三分，為四正之候，其說不經，欲附會緯文「七日來復」而已。

The twelve-month hexagrams come from Meng's commentary, grounding the Changes in seasonal qi and then explaining them through human affairs. Jing Fang further assigned hexagram lines to period days—Kan, Li, Zhen, and Dui, governing from equinox and solstice heads, each receive seventy-three parts in eighty day-parts. Yi, Jin, Jing, and Da Chu each get five days and fourteen parts; the rest six days and seven parts—limited to divining calamities and matters of fortune, success, and failure. For tracking yin-yang transformation, however, the system becomes confused and unclear. From the Qianxiang calendar onward, all follow Jing Fang. Only the Tianbao calendar follows the Comprehensive Yijing Track Diagram. Starting from the twelve minor nodes, five hexagrams, and the first line governing in turn until the top line ends with mid-qi—this is neither Jing Fang's original intent nor what the Seven Summaries records. Per Lang Yi's transmission, each hexagram governs six days and seven parts without first-line succession—the Qi calendar erred. Jing Fang also cut seventy-three parts for the four cardinal seasons—an unorthodox theory meant only to fit the apocryphal line "it returns in seven days."

夫陽精道消，靜而無迹，不過極其正數，至七而通矣。七者，陽之正也，安在益其小餘，令七日而後雷動地中乎？當據孟氏，自冬至初，中孚用事，一月之策，九六、七八，是為三十。而卦以地六，候以天五，五六相乘，消息一變，十有二變而歲復初。坎、震、離、兌，二十四氣，次主一爻，其初則二至、二分也。坎以陰包陽，故自北正，微陽動於下，升而未達，極於二月，凝涸之氣消，坎運終焉。春分出於震，始據萬物之元，為主於內，則羣陰化而從之，極于南正，而豐大之變窮，震功究焉。離以陽包陰，故自南正，微陰生於地下，積而未章，至于八月，文明之質衰，離運終焉。仲秋陰形于兌，始循萬物之末，為主於內，羣陽降而承之，極於北正，而天澤之施窮，兌功究焉。故陽七之靜始於坎，陽九之動始于震，陰八之靜始于離，陰六之動始于兌。故四象之變，皆兼六爻，而中節之應備矣。易爻當日，十有二中，直全卦之初；十有二節，直全卦之中。齊曆又以節在貞，氣在悔，非是。

When yang essence wanes along its path, still and traceless, it does not exceed its proper count—at seven it completes its cycle. Seven is yang's proper number—why inflate the small remainder to make thunder stir underground only after seven days? One should follow Meng: from winter solstice Zhong Fu governs; one month's divination tally of nine-six and seven-eight totals thirty. Hexagrams use earth's six, seasons heaven's five—five times six yields one wax-wane shift; twelve shifts and the year begins anew. Kan, Zhen, Li, and Dui govern the twenty-four qi, each presiding over one line; at the outset come the two solstices and two equinoxes. Kan wraps yang within yin—starting from the northern cardinal, faint yang stirs below, rising but not yet fulfilled; by the second month frozen qi melts and Kan's turn ends. Spring equinox arises from Zhen, taking hold of the origin of all things and governing inward; yin transforms and follows until the southern cardinal, where abundant growth reaches its limit and Zhen's work is done. Li wraps yin within yang—starting from the southern cardinal, faint yin forms underground, gathering but not yet visible; by the eighth month cultured brilliance fades and Li's turn ends. At mid-autumn yin manifests in Dui, tracing the waning of all things and governing inward; yang falls away and yields, reaching the northern cardinal where heaven's moist bounty ends and Dui's turn is complete. Thus yang-seven's quiescence starts with Kan, yang-nine's activity with Zhen, yin-eight's quiescence with Li, and yin-six's activity with Dui. The four images' transformations each encompass six lines, so the responses at mid-nodes are fully accounted for. Each Yijing line corresponds to a day—the twelve mid-qi align with the start of the full hexagram; the twelve minor nodes align with its midpoint. The Qi calendar wrongly places minor nodes at Zhen and qi at Hui.

其七日度議曰：

The seventh section, the solar-degree discourse, states:

古曆，日有常度，天周為歲終，故係星度于節氣。其說似是而非，故久而益差。虞喜覺之，使天為天，歲為歲，乃立差以追其變，使五十年退一度。何承天以為太過，乃倍其年，而反不及。皇極取二家中數為七十五年，蓋近之矣。考古史及日官候簿，以通法之三十九分太為一歲之差。自帝堯演紀之端，在虛一度。及今開元甲子，却三十六度，而乾策復初矣。日在虛一，則鳥、火、昴、虛皆以仲月昏中，合于堯典。

Ancient calendars assumed the sun moved at a fixed rate and that a full circuit of heaven marked the year's end, so star degrees were tied to seasonal nodes. The theory seems plausible but is wrong, so error accumulates over time. Yu Xi saw the problem: treat heaven and the year separately, then introduce a correction tracking the drift—one degree every fifty years. He Chengtian thought that too fast and doubled the interval, but that proved too slow. The Huangji calendar split the difference at seventy-five years per degree—close to the mark. Reviewing ancient histories and solar observation logs, the Kaiyuan universal method's 39 parts plus 39/96 yields one degree of precession per year. At the start of Emperor Yao's era-count, the sun stood at one degree of Xu. By the Kaiyuan jiazi year it has retreated thirty-six degrees, and the qian tally has cycled back to its origin. With the sun at one degree of Xu, Bird, Fire, Mao, and Xu all culminate at dusk in their proper months, matching the Canon of Yao.

劉炫依大明曆四十五年差一度，則冬至在虛、危，而夏至火已過中矣。梁武帝據虞𠠎曆，百八十六年差一度，則唐、虞之際，日在斗、牛間，而冬至昴尚未中。以為皆承閏後節前，月却使然。而此經終始一歲之事，不容頓有四閏，故淳風因為之說曰：「若冬至昴中，則夏至秋分星火、星虛，皆在未正之西。若以夏至火中，秋分虛中，則冬至昴在巳正之東。互有盈縮，不足以為歲差證。」是又不然。今以四象分天，北正玄枵中，虛九度；東正大火中，房二度；南正鶉火中，七星七度；西正大梁中，昴七度。總晝夜刻以約周天，命距中星，則春分南正中天，秋分北正中天。冬至之昏，西正在午東十八度；夏至之昏，東正在午西十八度：軌漏使然也。冬至，日在虛一度，則春分昏張一度中；秋分虛九度中；冬至胃二度中，昴距星直午正之東十二度；夏至尾十一度中，心後星直午正之西十二度。四序進退，不逾午正間。而淳風以為不叶，非也。又王孝通云：「如歲差自昴至壁，則堯前七千餘載，冬至，日應在東井。井極北，故暑；斗極南，故寒。寒暑易位，必不然矣。」所謂歲差者，日與黃道俱差也。假冬至日躔大火之中，則春分黃道交於虛九，而南至之軌更出房、心外，距赤道亦二十四度。設在東井，差亦如之。若日在東井，猶去極最近，表景最短，則是分、至常居其所。黃道不遷，日行不退，又安得謂之歲差乎？孝通及淳風以為冬至日在斗十三度，昏東壁中，昴在巽維之左，向明之位，非無星也。水星昏正可以為仲冬之候，何必援昴於始覿之際，以惑民之視聽哉！

Liu Xuan, using the Daming calendar's rate of one degree every forty-five years, places winter solstice at Xu and Wei while summer solstice Fire has already passed culmination. Emperor Wu of Liang, using Yu Xi's rate of one degree every 186 years, puts the sun between Dou and Niu in the Tang-Yu era while winter solstice Mao has not yet culminated. He argued this all stemmed from intercalation lagging behind the nodes—the month falling back caused the mismatch. But a full year's events in this classic cannot involve four intercalations at once, so Chunfeng argued: "If winter solstice Mao culminates, then summer solstice and autumn equinox Fire and Void stars all lie west of the wei meridian. If summer solstice Fire and autumn equinox Void culminate instead, winter solstice Mao lies east of the si meridian. The two cases contradict each other and cannot prove precession." That argument is also wrong. Dividing heaven by the four images: the northern cardinal Xuanxiao centers on nine degrees of Xu; the eastern cardinal Great Fire on two degrees of Fang; the southern cardinal Quail Fire on seven degrees of Seven Stars; the western cardinal Great Bridge on seven degrees of Mao. Summing day-night quarters to approximate the full circuit and assigning distance from culmination stars, spring equinox finds the southern cardinal at zenith and autumn equinox the northern cardinal at zenith. At winter solstice dusk the western cardinal lies eighteen degrees east of the wu meridian; at summer solstice dusk the eastern cardinal lies eighteen degrees west of wu—the clepsydra track requires it. At winter solstice with the sun at one degree of Xu, Zhang at one degree culminates at spring equinox dusk; Xu at nine degrees culminates at autumn equinox; Wei at two degrees culminates at winter solstice, with Mao's distance star twelve degrees east of the wu meridian; Wei at eleven degrees culminates at summer solstice, with the rear star of Xin twelve degrees west of the wu meridian. The four seasons' advance and retreat stay within the wu meridian zone. Chunfeng thought this did not harmonize—but he was wrong. Wang Xiaotong also argued: "If precession moved from Mao to Bi, then 7,000-plus years before Yao, winter solstice should find the sun at Dongjing. Jing is farthest north, so it would be hot; Dou is farthest south, so it would be cold. Heat and cold would swap places—which cannot be right." Precession means both the sun and the ecliptic shift together. Suppose winter solstice finds the sun in Great Fire: spring equinox still places the ecliptic at Xu nine, and the south-arrival track still passes beyond Fang and Xin, twenty-four degrees from the equator. The same holds if the sun stood at Dongjing. If the sun at Dongjing were still nearest the pole with the shortest gnomon shadow, equinoxes and solstices would stay fixed in place. With the ecliptic unmoved and the sun's daily motion unchanged, how could that be precession? Xiaotong and Chunfeng held that at winter solstice the sun stands at thirteen degrees of Dou, Dongbi culminates at dusk, and Mao lies in the bright southeast quadrant—stars are plainly visible there. Mercury culminating at dusk suffices as a mid-winter marker—why drag in Mao at first visibility to mislead the people?

夏后氏四百三十二年，日却差五度。太康十二年戊子歲冬至，應在女十一度。

Over the Xia dynasty's 432 years the sun's position retreated five degrees. Winter solstice of Taikang year 12, a wuzi year, should fall at eleven degrees of Nü.

書曰：「乃季秋月朔，辰弗集于房。」劉炫曰：「房，所舍之次也。集，會也。會，合也。不合則日蝕可知。或以房為房星，知不然者，且日之所在正可推而知之。君子慎疑，寧當以日在之宿為文？近代善曆者，推仲康時九月合朔，已在房星北矣。」按古文「集」與「輯」義同。日月嘉會，而陰陽輯睦，則陽不疚乎位，以常其明，陰亦含章示沖，以隱其形。若變而相傷，則不輯矣。房者，辰之所次，星者，所次之名，其揆一也。又春秋傳「辰在斗柄」、「天策焞焞」、「降婁之初」、「辰尾之末」，君子言之，不以為繆，何獨慎疑於房星哉？新曆仲康五年癸巳歲九月庚戌朔，日蝕在房二度。炫以五子之歌，仲康當是其一，肇位四海，復脩大禹之典，其五年，羲、和失職，則王命徂征。虞𠠎以為仲康元年，非也。

The Documents record: "In the third month of autumn, at new moon, the chen did not gather at Fang." Liu Xuan said: "Fang is the station where the sun lodges. Ji means to meet. Hui means conjunction. If they fail to conjoin, an eclipse is implied. Some take Fang to mean the Fang star, but the sun's position can be calculated directly. A careful scholar should read the text as naming the sun's lodge, not the star Fang. Recent skilled calendrists calculate that Zhongkang's ninth-month conjunction already lay north of the Fang star." In the ancient text, "ji" (gather) and "ji" (harmonize) share the same sense. When sun and moon meet properly and yin-yang harmonize, yang keeps its place and brightness while yin modestly hides its form. When they clash and harm each other, harmony is lost. Fang names the chen station; "star" names the station itself—the principle is the same. The Zuozhuan likewise says "chen at the Dipper's handle," "Heaven's tally blazing," "at Jianglou's start," "at chen-tail's end"—nobody treats those as absurd; why doubt only the Fang star? The new calendar places Zhongkang year 5, a guisi year, ninth month gengxu new moon, with a solar eclipse at two degrees of Fang. Xuan, citing the Song of the Five Sons, identifies Zhongkang as one of them; he took the throne, restored Yu's institutions, and in his fifth year, when Xi and He failed in duty, the king sent an expedition against them. Yu Xi's claim that this was Zhongkang's first year is wrong.

國語單子曰：「辰角見而雨畢，天根見而水涸，本見而草木節解，駟見而隕霜，火見而清風戒寒。」韋昭以為夏后氏之令，周人所因。推夏后氏之初，秋分後五日，日在氐十三度，龍角盡見，時雨可以畢矣。又先寒露三日，天根朝覿，時訓「爰始收潦」，而月令亦云「水涸」。後寒露十日，日在尾八度而本見，又五日而駟見。故隕霜則蟄蟲墐戶。鄭康成據當時所見，謂天根朝見，在季秋之末，以月令為謬。韋昭以仲秋水始涸，天根見乃竭。皆非是。霜降六日，日在尾末，火星初見，營室昏中，於是始脩城郭、宮室。故時儆曰：「營室之中，土功其始。火之初見，期于司理。」麟德曆霜降後五日，火伏。小雪後十日，晨見。至大雪而後定星中，日且南至，冰壯地坼。又非土功之始也。

In the Discourses of the States, Master Shan said: "When the Horn rises, the rains finish; when the Sky Root appears, waters recede; when the Root appears, plants shed their leaves; when Si appears, frost falls; when Fire appears, the clear wind warns of cold." Wei Zhao held this to be a Xia seasonal ordinance that the Zhou adopted. Extrapolating to early Xia, five days after autumn equinox the sun stood at thirteen degrees of Di and the Dragon Horn was fully visible—the seasonal rains could then end. Three days before Cold Dew the Sky Root appeared at dawn—the seasonal instruction says "then begin gathering the floodwaters," and the Monthly Ordinances likewise say "waters dry up." Ten days after Cold Dew the sun reached eight degrees of Wei and the Root appeared; five days later Si appeared. When frost falls, hibernating insects seal their burrows. Zheng Xuan, judging by what he could see in his own day, placed Sky Root's dawn visibility at late autumn and called the Monthly Ordinances wrong. Wei Zhao held that waters begin to recede in mid-autumn and are fully dry when the Sky Root appears. Both are incorrect. Six days after Frost Descent the sun stood at the end of Wei, Mars first appeared, and Encampment culminated at dusk—time to begin repairing walls and palaces. Hence the seasonal warning: "When Encampment is centered, earthworks begin. When Fire first appears, the Minister of Works takes charge." The Lindé calendar has Fire hidden five days after Frost Descent. It becomes visible at dawn ten days after Lesser Snow. Only by Greater Snow are the fixed stars centered, the sun at south arrival, and ice thick enough to crack the ground. That is far too late to mark the start of earthworks.

夏曆十二次，立春，日在東壁三度，於太初星距壁一度太也。

In the Xia calendar's twelve stations, at Start of Spring the sun stood at three degrees of Dongbi—one degree great from the Taichu star-distance at Bi.

顓頊曆上元甲寅歲正月甲寅晨初合朔立春，七曜皆直艮維之首。蓋重黎受職於顓頊，九黎亂德，二官咸廢，帝堯復其子孫，命掌天地四時，以及虞、夏。故本其所由生，命曰顓頊，其實夏曆也。湯作殷曆，更以十一月甲子合朔冬至為上元。周人因之，距羲、和千祀，昏明中星率差半次。夏時直月節者，皆當十有二中，故因循夏令。其後呂不韋得之，以為秦法，更考中星，斷取近距，以乙卯歲正月己巳合朔立春為上元。洪範傳曰：「曆記始於顓頊上元太始閼蒙攝提格之歲，畢陬之月，朔日己巳立春，七曜俱在營室五度。」是也。秦顓頊曆元起乙卯，漢太初曆元起丁丑，推而上之，皆不值甲寅，猶以日月五緯復得上元本星度，故命曰閼蒙攝提格之歲，而實非甲寅。

The Zhuanxu calendar's upper origin is jiayin year, first month: jiayin dawn, initial conjunction at Start of Spring, all seven luminaries aligned at gen-wei. Chongli received his post from Zhuanxu; the Nine Li corrupted order and both offices lapsed until Emperor Yao restored their descendants to govern heaven, earth, and the four seasons through Yu and Xia. Named after its originator Zhuanxu, it is in fact the Xia calendar. Tang created the Yin calendar, making the eleventh month's jiazi conjunction at winter solstice the upper origin. The Zhou followed it; a thousand reign-years from Xi and He, dusk-and-dawn culmination stars differed by half a station on average. In Xia times the monthly nodes aligned with the twelve mid-qi, so the Zhou followed Xia seasonal practice. Later Lü Buwei adopted it as Qin law, re-examined culmination stars, and took the nearest fit—yimao year, first month jisi conjunction at Start of Spring—as upper origin. The Hongfan Commentary says: "Calendars begin at Zhuanxu's upper origin, the year Taishi Yuemeng Shetige, month Biyou, new moon on jisi at Start of Spring—all seven luminaries at five degrees of Encampment." So it records. The Qin Zhuanxu origin begins at yimao and the Han Taichu at dingchou; extrapolating backward neither lands on jiayin, yet by sun, moon, and five planets they still recover the upper origin's star positions—so the year is called Yuemeng Shetige though it is not truly jiayin.

夏曆章蔀紀首，皆在立春，故其課中星，揆斗建與閏餘之所盈縮，皆以十有二節為損益之中。而殷、周、漢曆，章蔀紀首皆直冬至，故其名察發斂，亦以中氣為主。此其異也。

The Xia calendar's rules, obscurations, and era heads all begin at Start of Spring, so its culmination checks and Dipper Establishment and intercalary adjustments all use the twelve minor nodes as their pivot. The Yin, Zhou, and Han calendars instead anchor rules, obscurations, and era heads at winter solstice, so their "examining emission and gathering" also centers on mid-qi. That is the difference between them.

夏小正雖頗疎簡失傳，乃羲、和遺迹。何承天循大戴之說，復用夏時，更以正月甲子夜半合朔雨水為上元，進乖夏曆，退非周正，故近代推月令、小正者，皆不與古合。開元曆推夏時立春，日在營室之末，昏東井二度中。古曆以參右肩為距，方當南正。故小正曰：「正月初昏，斗杓懸在下。」魁枕參首，所以著參中也。季春，在昴十一度半，去參距星十八度，故曰：「三月，參則伏。」立夏，日在井四度，昏角中。南門右星入角距西五度，其左星入角距東六度，故曰：「四月初昏，南門正。昴則見。」五月節，日在輿鬼一度半。參去日道最遠，以渾儀度之，參體始見，其肩股猶在濁中。房星正中。故曰：「五月，參則見。初昏，大火中。」「八月，參中則曙」，失傳也。辰伏則參見，非中也。「十月初昏，南門見」，亦失傳也。定星方中，則南門伏，非昏見也。

The Xia Small Calendar, though coarse and imperfectly transmitted, preserves Xi and He's legacy. He Chengtian, following the Dadai Liji, revived Xia reckoning with first month jiazi midnight conjunction at Rain Water as upper origin—ahead of the Xia calendar yet not Zhou's proper first month—so recent reconstructions of the Monthly Ordinances and Small Calendar all fail to match antiquity. The Kaiyuan calendar extrapolates Xia Start of Spring with the sun at the end of Encampment and Dongjing at two degrees culminating at dusk. The ancient calendar takes Shen's right shoulder as the reference point, right at the southern cardinal. Hence the Small Calendar says: "At first dusk of the first month, the Dipper's handle hangs low." The Dipper's head rests on Shen's head, marking Shen at culmination. In late spring the sun stands at eleven and a half degrees of Mao, eighteen degrees from Shen's reference star—hence: "In the third month Shen sets." At Start of Summer the sun is at four degrees of Jing and Horn culminates at dusk. The Southern Gate's right star lies five degrees west of Horn's reference point and its left six degrees east—hence: "At first dusk of the fourth month the Southern Gate is centered. Then Mao appears." At the fifth-month node the sun stands at one and a half degrees of Yugu. Shen is farthest from the sun's path; on the armillary Shen's body first appears while its shoulder and thigh remain below the horizon. Fang is at culmination. Hence: "In the fifth month Shen appears. At first dusk Great Fire is centered." "In the eighth month Shen culminates at dawn"—a corrupted transmission. When the sun enters chen, Shen appears—it is not at culmination. "At first dusk of the tenth month the Southern Gate appears"—also corrupted. When the fixed stars are centered the Southern Gate sets—it is not a dusk sighting.

商六百二十八年，日却差八度。太甲二年壬午歲冬至，應在女六度。

Over the Shang dynasty's 628 years the sun's position retreated eight degrees. Winter solstice of Taikang year 12, a renwu year, should fall at six degrees of Nü.

國語曰：「武王伐商，歲在鶉火，月在天駟，日在析木之津，辰在斗柄，星在天黿。」舊說歲在己卯，推其朏魄，迺文王崩，武王成君之歲也。其明年，武王即位，新曆孟春定朔丙辰，於商為二月，故周書曰：「維王元祀二月丙辰朔，武王訪于周公。」竹書「十一年庚寅，周始伐商」。而管子及家語以為十二年，蓋通成君之歲也。先儒以文王受命九年而崩；至十年，武王觀兵盟津；十三年，復伐商。推元祀二月丙辰朔，距伐商日月，不為相距四年。所說非是。武王十年，夏正十月戊子，周師始起。於歲差日在箕十度，則析木津也。晨初，月在房四度。於易，雷乘乾曰大壯，房、心象焉。心為乾精，而房，升陽之駟也。房與歲星實相經緯，以屬靈威仰之神，后稷感之以生。故國語曰：「月之所在，辰馬農祥，我祖后稷之所經緯也。」又三日得周正月庚寅朔，日月會南斗一度。故曰「辰在斗柄」。壬辰，辰星夕見，在南斗二十度。其明日，武王自宗周次于師所。凡月朔而未見曰「死魄」，夕而成光則謂之「朏」。朏或以二日，或以三日，故武成曰：「維一月壬辰，旁死魄。翌日癸巳，王朝步自周，于征伐商。」是時辰星與周師俱進，由建星之末，歷牽牛、須女，涉顓頊之虛。戊午，師度盟津，而辰星伏于天黿。辰星，汁光紀之精，所以告顓頊而終水行之運，且木帝之所繇生也。故國語曰：「星與日辰之位皆在北維，顓頊之所建也，帝嚳受之。我周氏出自天黿；及析木，有建星、牽牛焉，則我皇妣太姜之姪、伯陵之後逢公之所憑神也。」是歲，歲星始及鶉火。其明年，周始革命。歲又退行，旅於鶉首，而後進及鳥帑，所以返復其道，經綸周室。鶉火直軒轅之虛，以爰稼穡，稷星繫焉，而成周之大萃也。鶉首當山河之右，太王以興，后稷封焉，而宗周之所宅也。歲星與房實相經緯，而相距七舍；木與水代終，而相及七月。故國語曰：「歲之所在，則我有周之分也。自鶉及駟七列，南北之揆七月。」其二月戊子朔，哉生明，王自克商還，至于酆，於周為四月。新曆推定望甲辰，而乙巳旁之。故武成曰：「維四月，既旁生魄，粵六日庚戌，武王燎于周廟。」麟德曆，周師始起，歲在降婁，月宿天根，日躔心而合辰在尾，水星伏於星紀，不及天黿。又周書，革命六年而武王崩。管子、家語以為七年，蓋通克商之歲也。

The Discourses record: "When King Wu attacked Shang, Jupiter was in Quail Fire, the moon at Heavenly Si, the sun at Ximu's ford, chen at the Dipper's handle, and the star at Heavenly Turtle." Older accounts place the year at jimao; extrapolating its lunar phases, that was the year King Wen died and Wu became ruler. The following year Wu took the throne. The new calendar gives first-month fixed new moon on bingchen—in Shang terms the second month—hence the Book of Zhou: "In the king's first year, second month bingchen new moon, King Wu consulted the Duke of Zhou." The Bamboo Annals record: "Year 11 gengyin, Zhou first attacked Shang." Guanzi and the Family Discourses say year 12—probably counting through the year Wu became lord. Earlier scholars held that King Wen received the mandate and died in nine years; in year 10 King Wu reviewed troops at Mengjin; and in year 13 attacked Shang again. Extrapolating from first-year second-month bingchen new moon to the attack on Shang, the interval is not four years. That chronology is wrong. In King Wu's tenth year, Xia tenth month wuzi, the Zhou army first mobilized. By precession the sun stood at ten degrees of Ji—at Ximu's ford. At dawn the moon stood at four degrees of Fang. In the Yijing, thunder over qian is Da Zhuang—Fang and Xin embody it. Xin is qian's essence; Fang is the si of rising yang. Fang and Jupiter truly cross in longitude, belonging to the spirit Lingweiyang; Hou Ji was moved by this to be born. Hence the Discourses: "Where the moon stands—chen-horse, agricultural auspice—the longitude of our ancestor Hou Ji." Three days later came Zhou first month gengyin new moon, sun and moon conjoining at one degree of Southern Dipper. Hence "chen at the Dipper's handle." On renchen Mercury appeared at dusk at twenty degrees of Southern Dippler. The next day King Wu left Zongzhou and halted at the army camp. When the new moon is not yet visible it is called "dead soul"; when light forms at dusk it is called "crescent." The crescent may fall on the second or third day—hence Completion of War: "First month renchen, beside dead soul. Next day guisi, the king at dawn marched from Zhou to attack Shang." Mercury then advanced with the Zhou army from Jianxing's end through Ox, Maiden, and Zhuanxu's void. On wuwu the army crossed Mengjin ford and Mercury set at Heavenly Turtle. Mercury is Zhiguangji's essence, announcing Zhuanxu and ending water's reign—the birthplace of the Wood Emperor. Hence the Discourses: "Star and sun-chen positions all lie at north wei—what Zhuanxu established and Di Ku received. Our Zhou clan descends from Heavenly Turtle; reaching Ximu with Jianxing and Ox—the spirit on whom our royal mother's niece, Lord Feng of Earl Ling's line, relies." That year Jupiter first reached Quail Fire. The next year Zhou completed the revolution. Jupiter retreated again to Quail Head, then advanced to Bird Tail—reversing course to weave Zhou's destiny. Quail Fire lies at Xuanyuan's void where grain is cherished; Ji's star is attached, forming Zhou's great gathering. Quail Head lies west of the Yellow River where the Great Ancestor rose, Hou Ji was enfeoffed, and Zongzhou was settled. Jupiter and Fang truly cross in longitude, seven lodges apart; wood and water succeed each other across seven months. Hence: "Where Jupiter stands is Zhou's allotted domain. From Quail to Si, seven columns; south to north, seven months." On second-month wuzi new moon at crescent brightness the king returned from conquering Shang to Feng—in Zhou reckoning the fourth month. The new calendar places fixed full moon on jiachen with yisi adjacent. Hence Completion of War: "Fourth month, beside full soul, sixth day gengxu, King Wu burned offerings at the Zhou temple." The Lindé calendar has the Zhou army mobilize with Jupiter at Jianglou, the moon at Sky Root, the sun at Xin with chen at Tail, Mercury hidden at Star Record—not reaching Heavenly Turtle. The Book of Zhou also says King Wu died six years after the revolution. Guanzi and Family Discourses say seven years—probably counting through the conquest year.

周公攝政七年二月甲戌朔，己丑望，後六日乙未。三月定朔甲辰，三日丙午。故召誥曰：「惟二月既望，越六日乙未，王朝步自周，至于酆」，「三月，惟丙午朏，越三日戊申，太保朝至于洛。」其明年，成王正位。三十年四月己酉朔甲子，哉生魄。故書曰：「惟四月，才生魄。」甲子，作顧命。康王十二年，歲在乙酉，六月戊辰朔，三日庚午。故畢命曰：「惟十有二年，六月庚午朏。越三日壬申，王以成周之眾命畢公。」自伐紂及此，五十六年，朏魄日名，上下無不合。而三統曆以己卯為克商之歲，非也。夫有効於古者，宜合於今。三統曆自太初至開元，朔後天三日。推而上之，以至周初，先天，失之蓋益甚焉。是以知合於歆者，必非克商之歲。

During the Duke of Zhou's regency, seventh year second month jiaxu new moon, jichou full moon, six days later yimwei. Third month fixed new moon jiachen, third day bingwu. Hence the Announcement: "Second month after full moon, six days yimwei, the king at dawn marched from Zhou to Feng," and "Third month, bingwu crescent, three days wushen, the Grand Guardian at dawn reached Luo." The next year King Cheng took the throne. The thirtieth year, fourth month jiyou new moon jiazi, crescent brightness. Hence the Documents: "In the fourth month, at first crescent." On jiazi he composed the Testamentary Charge. King Kang's year 12, a yiyou year, sixth month wuchen new moon, third day gengwu. Hence Charge to Bi: "In the twelfth year, sixth month gengwu crescent. Three days later on renshen the king commanded Duke of Bi with Chengzou's forces." From the attack on Zhou to this point, fifty-six years—crescent and full-moon day-names all agree. Yet the Three Systems calendar takes jimao as the conquest year—which is wrong. What works for antiquity should match the present. From Taichu to Kaiyuan the Three Systems calendar's new moons lag heaven by three days. Extrapolating back to early Zhou they run ahead of heaven—the error grows worse. Thus what agrees with Liu Xin cannot be the conquest year.

自宗周訖春秋之季，日却差八度。康王十一年甲申歲冬至，應在牽牛六度。

From Zongzhou to the Spring and Autumn's end the sun retreated eight degrees. Winter solstice of King Kang year 11, a jiashen year, should fall at six degrees of Ox.

周曆十二次，星紀初，南斗十四度，於太初星距斗十七度少也。

The Zhou calendar's twelve stations begin at fourteen degrees of Southern Dipper—seventeen degrees small from Taichu star-distance at Dipper.

古曆分率簡易，歲久輒差。達曆數者隨時遷革，以合其變。故三代之興，皆揆測天行，考正星次，為一代之制。正朔既革，而服色從之。及繼體守文，疇人代嗣，則謹循先王舊制焉。

Ancient fractional rates were simple; over long periods they drifted. Skilled calendrists revise with the times to match celestial change. Each of the Three Dynasties measured heaven's motion and verified star stations to establish its calendar. When the calendar's first month changed, dress and color followed. Under succession they preserved former kings' systems through generations of calendar officers.

國語曰：「農祥晨正，日月厎于天廟，土乃脈發。先時九日，太史告稷曰，自今至于初吉，陽氣俱蒸，土膏其動。弗震不渝，脈其滿眚，穀乃不殖。」周初，先立春九日，日至營室。古曆距中九十一度，是日晨初，大火正中，故曰「農祥晨正，日月厎于天廟」也。於易象，升氣究而臨受之，自冬至後七日，乾精始復。及大寒，地統之中，陽洽於萬物根柢，而與萌芽俱升，木在地中之象，升氣已達，則當推而大之，故受之以臨。於消息，龍德在田，得地道之和澤，而動於地中，升陽憤盈，土氣震發，故曰：「自今至於初吉，陽氣俱蒸，土膏其動。」又先立春三日，而小過用事，陽好節止於內，動作于外，矯而過正，然後返求中焉。是以及于艮維，則山澤通氣，陽精闢戶，甲坼之萌見，而莩穀之際離，故曰：「不震不渝，脈其滿眚，穀乃不殖。」君子之道，必擬之而後言，豈億度而已哉！韋昭以為日及天廟，在立春之初，非也。於麟德曆則又後立春十五日矣。

The Discourses say: "When agricultural auspice is correct at dawn and sun and moon reach heaven's temple, the earth's pulse stirs. Nine days earlier the Grand Astrologer told Hou Ji: from now until the first auspicious day yang qi will steam and the earth's ointment stir. If it does not shake and shift, the pulse fills with calamity and grain will not sprout." At Zhou's beginning, nine days before Start of Spring the sun reached Encampment. The ancient calendar placed culmination ninety-one degrees from the sun; that day at dawn Great Fire was centered—hence "agricultural auspice correct at dawn, sun and moon at heaven's temple." In Yijing imagery, ascending qi completes and Lin receives it—from seven days after winter solstice qian essence first returns. By Greater Cold, earth's midpoint, yang harmonizes with roots and sprouts rising together—wood underground; when ascending qi has risen it expands—hence Lin follows. In the wax-wane cycle, dragon virtue is in the field; earth's harmony stirs underground, yang surges, earth qi shakes—hence: "From now to the first auspicious day yang steams and earth stirs." Three days before Start of Spring Xiao Guo governs—yang restrains within while acting without, correcting excess to find the mean. Reaching gen-wei, mountain and marsh connect, yang opens the gate, buds appear and grain husks part—hence: "Without shaking, the pulse fills with calamity and grain fails." The noble person compares before speaking—how could mere estimation of degrees suffice! Wei Zhao placed sun-at-heaven's-temple at Start of Spring's opening—that is wrong. The Lindé calendar places it fifteen days after Start of Spring.

春秋「桓公五年，秋，大雩」。傳曰：「書不時也。凡祀，啟蟄而郊，龍見而雩。」周曆立夏日在觜觽二度。於軌漏，昏角一度中，蒼龍畢見。然則當在建巳之初，周禮也。至春秋時，日已潛退五度，節前月却，猶在建辰。月令以為五月者，呂氏以顓頊曆芒種亢中，則龍以立夏昏見，不知有歲差，故雩祭失時。然則唐禮當以建巳之初，農祥始見而雩。若據麟德曆，以小滿後十三日，則龍角過中，為不時矣。傳曰：「凡土功，龍見而畢務，戒事。火見而致用，水昏正而栽，日至而畢。」十六年冬，城向。十有一月，衞侯朔出奔齊。「冬，城向，書時也。」以歲差推之，周初霜降，日在心五度，角、亢晨見。立冬，火見營室中。後七日，水星昏正，可以興板幹。故祖沖之以為定之方中，直營室八度。是歲九月六日霜降，二十一日立冬。十月之前，水星昏正，故傳以為得時。杜氏據晉曆，小雪後定星乃中，季秋城向，似為大早。因曰功役之事，皆總指天象，不與言曆數同。引詩云「定之方中」，乃未正中之辭，非是。麟德曆，立冬後二十五日火見，至大雪後營室乃中。而春秋九月書時，不已早乎。大雪，周之孟春，陽氣靜復，以繕城隍，治宮室，是謂發天地之房，方於立春斷獄，所失多矣。然則唐制宜以玄枵中天興土功。

The Spring and Autumn records: "Duke Huan's year 5, autumn, great rain-prayer." The commentary says: "Recorded because the season was wrong. Sacrifices generally: suburban rite at Awakening of Insects, rain-prayer when the dragon appears." The Zhou calendar places Start of Summer with the sun at two degrees of Zizi. On the clepsydra track Horn culminates at one degree at dusk and the azure dragon is fully visible. That falls at the start of jian-si—per Zhou ritual. By Spring and Autumn the sun had retreated five degrees; the month lagged before the node and still fell in jian-chen. The Monthly Ordinances place it in the fifth month—Lü's Zhuanxu calendar has Grain in Ear with Kang centered, making the dragon appear at Start of Summer dusk without accounting for precession, so the rain-prayer season was wrong. Tang ritual should therefore rain-pray at jian-si's start when agricultural auspice first appears. By the Lindé calendar, thirteen days after Lesser Fullness puts Horn past culmination—out of season. The commentary says: "For earthworks generally: when the dragon appears finish the work and warn of tasks. When Fire appears put tools to use; when water is correct at dusk, plant; when the sun arrives, finish." In year 16 winter they walled Xiang. In the eleventh month Marquis Wei Shuo fled to Qi. "Winter: walled Xiang—recorded as timely." By precession, at early Zhou Frost Descent the sun stood at five degrees of Xin with Horn and Kang visible at dawn. At Start of Winter Fire appeared with Encampment centered. Seven days later Mercury was correct at dusk—time to raise beams and timbers. Zu Chongzhi therefore placed "Ding's square centered" at eight degrees of Encampment. That year Frost Descent fell on the 6th of the ninth month and Start of Winter on the 21st. Before the tenth month Mercury was correct at dusk—hence the commentary judged it timely. Du, using the Jin calendar, has fixed stars centered only after Lesser Snow—walling Xiang in late autumn seems far too early. He argues labor affairs point generally to celestial signs, not to calendar arithmetic. Citing the Odes' "Ding's square centered" as not-yet-centered language—that is wrong. The Lindé calendar has Fire appear twenty-five days after Start of Winter and Encampment centered only after Greater Snow. Yet the Spring and Autumn records ninth-month work as timely—is that not already too early? Greater Snow is Zhou's first spring month when yang is still recovering—repairing walls and palaces opens heaven-earth's chamber; compared with Start of Spring for ending punishments, much is lost. Tang practice should therefore start earthworks when Xuanxiao is centered.

僖公五年，晉侯伐虢。卜偃曰：「克之。童謠云：『丙之辰，龍尾伏辰，袀服振振，取虢之旂，鶉之賁賁，天策焞焞，火中成軍。』其九月十月之交乎！丙子旦，日在尾，月在策，鶉火中，必是時。」策，入尾十二度。新曆是歲十月丙子定朔，日月合尾十四度於黃道。古曆日在尾，而月在策，故曰「龍尾伏辰」，於古距張中而曙，直鶉火之末，始將西降，故曰「賁賁」。

Duke Xi's year 5: the Marquis of Jin attacked Guo. Diviner Yan said: "Victory is assured. A children's song runs: 'On bing-chen, Dragon Tail hides chen, sacrificial robes flutter, seize Guo's banner, Quail blazes, Heaven's tally blazes, Fire centered forms the army.' It falls at the ninth-tenth month crossing! At bingzi dawn the sun is at Tail, the moon at Ce, Quail Fire centered—that must be the time." Ce enters twelve degrees of Tail. The new calendar gives that year's tenth month bingzi fixed new moon with sun and moon conjoined at fourteen degrees of Tail on the ecliptic. The ancient calendar has the sun at Tail and moon at Ce—hence "Dragon Tail hidden chen"; at ancient distance Zhang culminates at dawn near Quail Fire's end, just descending west—hence "blazing."

昭公七年四月甲辰朔，日蝕。士文伯曰：「去衞地，如魯地。於是有災，魯實受之。」新曆是歲二月甲辰朔入常，雨水後七日，在奎十度。周度為降婁之始，則魯、衞之交也。自周初至是已退七度，故入雨水。七日方及降婁，雖日度潛移，而周禮未改，其配神主祭之宿，宜書於建國之初。淳風駁戊寅曆曰：「漢志降婁初在奎五度，今曆日蝕在降婁之中，依無歲差法，食於兩次之交。」是又不然。議者曉十有二次之所由生，然後可以明其得失。且劉歆等所定辰次，非能有以覩陰陽之賾，而得於鬼神，各據當時中節星度耳。歆以太初曆冬至日在牽牛前五度，故降婁直東壁八度。李業興正光曆，冬至在牽牛前十二度，故降婁退至東壁三度。及祖沖之後，以為日度漸差，則當據列宿四正之中，以定辰次，不復係於中節。淳風以冬至常在斗十三度，則當以東壁二度為降婁之初，安得守漢曆以駁仁均耶？又三統曆昭公二十年，己丑，日南至，與麟德及開元曆同。然則入雨水後七日，亦入降婁七度，非魯、衞之交也。三十一年十二月辛亥朔，日蝕。史墨曰：「日月在辰尾，庚午之日，日始有讁。」開元曆是歲十月辛亥朔，入常立冬。五日，日在尾十三度，於古距辰尾之初。麟德曆日在心三度於黃道，退直于房矣。

Duke Zhao's the seventh year, fourth month jiachen new moon: solar eclipse. Shi Wenbo said: "Leaving Wei's territory it is like Lu's. Disaster falls there, and Lu truly receives it." The new calendar places that year's second month jiachen new moon in regular sequence, seven days after Rain Water, at ten degrees of Kui. In Zhou reckoning that is Jianglou's start—the border of Lu and Wei. From early Zhou to then the sun had retreated seven degrees—hence it falls after Rain Water. Seven days later it reaches Jianglou; though the sun's degree shifts, Zhou ritual is unchanged—the spirit-host lodge should be recorded at the state's founding. Chunfeng refuted the Wuyin calendar: "The Han Treatise places Jianglou's start at five degrees of Kui—the current calendar's eclipse falls mid-Jianglou; by a no-precession method it is at a station crossing." That too is wrong. Only after understanding how the twelve stations arose can one judge their merits. Liu Xin and others fixed the chen stations not by penetrating yin-yang but from then-current mid-node star positions. Xin, using the Taichu calendar's winter solstice at five degrees before Ox, placed Jianglou at eight degrees of Dongbi. Li Yexing's Zhengguang calendar puts winter solstice twelve degrees before Ox, retreating Jianglou to three degrees of Dongbi. After Zu Chongzhi, as solar degrees drift, chen stations should be fixed by the four cardinal lodges' centers, not mid-nodes. Chunfeng fixed winter solstice at thirteen degrees of Dipper, making Dongbi two degrees Jianglou's start—how can he use the Han calendar to refute Renjun? The Three Systems calendar also gives Duke Zhao year 20 jichou south arrival—matching Lindé and Kaiyuan. Seven days after Rain Water also enters seven degrees of Jianglou—not the Lu-Wei border. The thirty-first year, twelfth month xinhai new moon: solar eclipse. Shi Mo said: "Sun and moon at chen-tail; on gengwu day the sun first showed reproof." The Kaiyuan calendar gives that year's tenth month xinhai new moon, entering regular Start of Winter. Five days later the sun is at thirteen degrees of Tail—at ancient distance from chen-tail's start. The Lindé calendar places the sun at three degrees of Xin on the ecliptic, retreated to Fang.

哀公十二年冬十有二月，螽。開元曆推置閏當在十一年春，至十二年冬，失閏已久。是歲九月己亥朔，先寒露三日，於定氣，日在亢五度，去心近一次。火星明大，尚未當伏。至霜降五日，始潛日下。乃月令「蟄蟲咸俯」，則火辰未伏，當在霜降前。雖節氣極晚，不得十月昏見。故仲尼曰：「丘聞之，火伏而後蟄者畢。今火猶西流，司曆過也。」方夏后氏之初，八月辰伏，九月內火，及霜降之後，火已朝覿東方，距春秋之季千五百餘年，乃云「火伏而後蟄者畢」。向使冬至常居其所，則仲尼不得以西流未伏，明是九月之初也。自春秋至今又千五百歲，麟德曆以霜降後五日，日在氐八度，房、心初伏，定增二日，以月蝕衝校之，猶差三度。閏餘稍多，則建亥之始，火猶見西方。向使宿度不移，則仲尼不得以西流未伏，明非十月之候也。自羲、和已來，火辰見伏，三覩厥變。然則丘明之記，欲令後之作者參求微象，以探仲尼之旨。是歲失閏寖久，季秋中氣後天三日，比及明年仲冬，又得一閏。寤仲尼之言，補正時曆，而十二月猶可以螽。至哀公十四年五月庚申朔，日蝕。以開元曆考之，則日蝕前又增一閏，魯曆正矣。長曆自哀公十年六月，迄十四年二月，纔置一閏，非是。

Duke Ai's year 12, winter twelfth month: locusts. The Kaiyuan calendar places intercalation in year 11 spring; by year 12 winter intercalation had been missing too long. That year ninth month jihai new moon, three days before Cold Dew: by fixed qi the sun stood at five degrees of Kang, one station from Xin. Mars was bright and large and had not yet should set. Only five days after Frost Descent did it first set below the sun. The Monthly Ordinances say "hibernating insects all bow"—but Mars had not yet set and should have been before Frost Descent. Even with very late nodes it could not appear at dusk in the tenth month. Confucius therefore said: "I have heard that after Fire hides the hibernators finish. Now Fire still moves west—the calendar officers erred." At early Xia, chen hid in the eighth month and Fire within in the ninth; by Frost Descent Fire was already visible at dawn in the east—1,500 years before Spring and Autumn—yet the text says "after Fire hides hibernators finish." If winter solstice never moved, Confucius could not use west-flowing Fire to show this was early ninth month. From Spring and Autumn to now another 1,500 years: the Lindé calendar has the sun at eight degrees of Di five days after Frost Descent with Fang and Xin first setting; adding two fixed days and checking by lunar eclipse opposition it still falls three degrees short. With slightly too much intercalary remainder, at jian-hai's start Fire still appears in the west. If lodge degrees never moved, Confucius could not use west-flowing Fire to show this was not a tenth-month marker. From Xi and He onward Fire's appearances and settings changed three times. Qiu Ming's record invites later calendrists to compare subtle signs and grasp Confucius's intent. That year intercalation had been missing too long; mid-autumn mid-qi lagged heaven three days—by the next mid-winter another intercalation was needed. Heeding Confucius's words and correcting the calendar, locusts could still appear in the twelfth month. By Duke Ai's the fourteenth year, fifth month gengshen new moon: solar eclipse. The Kaiyuan calendar adds another intercalation before the eclipse—the Lu calendar is then correct. The Long Calendar places only one intercalation from Duke Ai the tenth year, sixth month to the fourteenth year, second month—that is wrong.

戰國及秦，日却退三度。始皇十七年辛未歲冬至，應在斗二十二度。秦曆上元正月己巳朔，晨初立春，日、月、五星俱起營室五度，蔀首日名皆直四孟。假朔退十五日，則閏在正月前。朔進十五日，則閏在正月後。是以十有二節，皆在盈縮之中，而晨昏宿度隨之。以顓頊曆依月令自十有二節推之，與不韋所記合。而潁子嚴之倫謂月令晨昏距宿，當在中氣，致雩祭太晚，自乖左氏之文，而杜預又據春秋以月令為否。皆非是。梁大同曆夏后氏之初，冬至日在牽牛初，以為明堂、月令乃夏時之記，據中氣推之不合，更以中節之間為正，迺稍相符。不知進在節初，自然契合。自秦初及今，又且千歲，節初之宿，皆當中氣。淳風因為說曰：「今孟春中氣，日在營室，昏明中星，與月令不殊。」按秦曆立春，日在營室五度。麟德曆以啟蟄之日迺至營室，其昏明中宿十有二建，以為不差，妄矣。

From Warring States through Qin the sun retreated three degrees. Winter solstice of the First Emperor's year 17, a xinwei year, should fall at twenty-two degrees of Dipper. The Qin calendar's upper origin is first month jisi new moon at dawn initial Start of Spring with sun, moon, and five planets all at five degrees of Encampment; obscuration-head day-names all align with the four meng. If new moon retreats fifteen days, intercalation falls before the first month. If new moon advances fifteen days, intercalation falls after the first month. Thus all twelve minor nodes fall within expansion-contraction and dawn-dusk lodge degrees follow. Extrapolating from twelve minor nodes per the Zhuanxu calendar matches Lü's record. Yet Ying Ziyan and others held Monthly Ordinances' dawn-dusk distances should fall at mid-qi—making rain-prayer too late, contradicting the Zuozhuan, while Du Yu rejected the Monthly Ordinances using the Spring and Autumn. Both are wrong. The Liang Datong calendar placed early Xia winter solstice at Ox's start, treating Bright Hall and Monthly Ordinances as Xia records; mid-qi extrapolation failed, so they used mid-node intervals instead and got rough agreement. They did not see that advancing to the node's start naturally matches. From Qin to now, another thousand years: lodges at node beginnings should align with mid-qi. Chunfeng argued: "Now at first-spring mid-qi the sun is at Encampment and dusk-dawn stars match the Monthly Ordinances." The Qin calendar places Start of Spring with the sun at five degrees of Encampment. The Lindé calendar reaches Encampment only on Awakening of Insects and claims twelve jian dusk-dawn stars match—absurd.

古曆，冬至昏明中星去日九十二度，春分、秋分百度，夏至百一十八度，率一氣差三度，九日差一刻。

The ancient calendar placed winter-solstice dusk-dawn stars ninety-two degrees from the sun, equinoxes at one hundred, summer solstice at one hundred eighteen—three degrees per qi, one quarter per nine days.

秦曆十二次，立春在營室五度，於太初星距危十六度少也。昏，畢八度中，月令參中，謂肩股也。晨，心八度中，月令尾中，於太初星距尾也。仲春昏，東井十四度中，月令弧中，弧星入東井十八度。晨，南斗二度中，月令建星中，於太初星距西建也。甄耀度及魯曆，南方有狼、弧，無東井、鬼，北方有建星，無南斗，井、斗度長，弧、建度短，故以正昏明云。

The Qin calendar's twelve stations place Start of Spring at five degrees of Encampment—sixteen degrees small from Taichu distance at Wei. At dusk Bi at eight degrees is centered—the Monthly Ordinances' Shen centered means shoulder and thigh. At dawn Xin at eight degrees is centered—the Monthly Ordinances' Wei centered uses Taichu distance at Tail. Mid-spring dusk: Dongjing at fourteen degrees centered—the Monthly Ordinances' Hu centered has Hu entering eighteen degrees of Dongjing. At dawn Southern Dipper at two degrees is centered—the Monthly Ordinances' Jianxing centered uses Taichu distance at West Jian. Zhenyaodu and Lu calendars have Wolf and Hu in the south but not Dongjing and Ghost, Jianxing in the north but not Southern Dipper—Jing and Dipper degrees long, Hu and Jian short—hence correct dusk-dawn wording.

古曆星度及漢落下閎等所測，其星距遠近不同，然二十八之宿體不異。古以牽牛上星為距，太初改用中星，入古曆牽牛太半度，於氣法當三十二分日之二十一。故洪範傳冬至日在牽牛一度，減太初星距二十一分，直南斗二十六度十九分也。顓頊曆立春起營室五度，冬至在牽牛一度少。洪範傳冬至所起無餘分，故立春在營室四度太。祖沖之自營室五度，以太初星距命之，因云秦曆冬至，日在牽牛六度。虞𠠎等襲沖之之誤，為之說云：「夏時冬至，日在斗末，以歲差考之，牽牛六度乃顓頊之代。漢時雖覺其差，頓移五度，故冬至還在牛初。」按洪範古今星距，僅差四分之三，皆起牽牛一度。𠠎等所說，亦非是。魯宣公十五年，丁卯歲，顓頊曆第十三蔀首與麟德曆俱以丁巳平旦立春。至始皇三十三年丁亥，凡三百八十歲，得顓頊曆壬申蔀首。是歲秦曆以壬申寅初立春，而開元曆與麟德曆俱以庚午平旦，差二日，日當在南斗二十二度。古曆後天二日，又增二度。然則秦曆冬至，定在牛前二度。氣後天二日，日不及天二度，微而難覺，故呂氏循用之。

Ancient star degrees and Han Luoxia Hong's measurements differ in reference distances, yet the twenty-eight lodges' bodies are the same. Anciently Ox's upper star was the reference; Taichu switched to the center star, entering the ancient calendar at Ox by great half a degree—in qi reckoning that is twenty-one thirty-seconds of a day. The Hongfan Commentary therefore places winter solstice at one degree of Ox; subtracting Taichu's star-distance of twenty-one parts yields twenty-six degrees nineteen parts of Southern Dipper. The Zhuanxu calendar begins Start of Spring at five degrees of Encampment and winter solstice at one degree small of Ox. The Hongfan Commentary's winter solstice start has no surplus fraction, so Start of Spring falls at four degrees great of Encampment. Zu Chongzhi, starting from five degrees of Encampment and using Taichu star-distance, concluded the Qin calendar's winter solstice stood at six degrees of Ox. Yu Xi and others repeated Zu Chongzhi's error, arguing: "At Xia winter solstice the sun stood at Dipper's end; by precession that corresponds to six degrees of Ox in Zhuanxu's era. Han scholars noticed the drift and shifted five degrees, placing winter solstice back at Ox's beginning." Hongfan's ancient and modern star distances differ by only three-fourths—all starting at one degree of Ox. Yu Xi and colleagues are also wrong. Lu Duke Xuan's year 15, a dingmao year: the Zhuanxu calendar's thirteenth obscuration head and the Lindé calendar both give dingsi dawn at Start of Spring. By the First Emperor's year 33 dinghai, 380 years later, the Zhuanxu calendar reaches the renshen obscuration head. That year the Qin calendar has renshen yin initial Start of Spring while Kaiyuan and Lindé give gengwu dawn—two days apart—with the sun at twenty-two degrees of Southern Dipper. The ancient calendar lagged heaven two days and added two degrees. The Qin calendar therefore fixed winter solstice two degrees before Ox. Qi lagged heaven two days while the sun fell two degrees short—too subtle to notice—so Lü Buwei kept using it.

及漢興，張蒼等亦以為顓頊曆比五家疎闊中最近密。今考月蝕衝，則開元冬至，上及牛初正差一次。淳風以為古術疎舛，雖弦望、昏明，差天十五度而猶不知。又引呂氏春秋，黃帝以仲春乙卯日在奎，始奏十二鍾，命之曰咸池。至今三千餘年，而春分亦在奎，反謂秦曆與今不異。按不韋所記，以其月令孟春在奎，謂黃帝之時亦在奎，猶淳風曆冬至斗十三度，因謂黃帝時亦在建星耳。經籍所載，合於歲差者，淳風皆不取，而專取於呂氏春秋。若謂十二紀可以為正，則立春在營室五度，固當不易，安得頓移，使當啟蟄之節？此又其所不思也。

At the Han founding Zhang Cang and others also found the Zhuanxu calendar coarsest among five schools yet nearest to accuracy. Checking lunar eclipse opposition, Kaiyuan winter solstice extrapolated upward reaches just one station short of Ox's beginning. Chunfeng called ancient methods coarse and erroneous—they missed a fifteen-degree discrepancy in quarter-moons and dusk-dawn. He also cited Lüshi Chunqiu: at mid-spring on yimao with the sun at Kui the Yellow Emperor first played the twelve bells and named the pool Xianchi. After 3,000-plus years spring equinox still falls at Kui—yet he claimed the Qin calendar matches the present. Lü placed first spring at Kui in his Monthly Ordinances and assumed the Yellow Emperor's era did too—just as Chunfeng fixed winter solstice at thirteen degrees of Dipper and assumed the Yellow Emperor's era matched Jianxing. Classical records that fit precession Chunfeng ignored, relying instead on Lüshi Chunqiu. If the twelve chronicles were authoritative, Start of Spring at five degrees of Encampment should never move—how could it jump to Awakening of Insects? He never considered that contradiction.

漢四百二十六年，日却差五度。景帝中元三年甲午歲冬至，應在斗二十一度。

Over the Han's 426 years the sun retreated five degrees. Winter solstice of Emperor Jing's Zhongyuan year 3, a jiawu year, should fall at twenty-one degrees of Dipper.

太初元年，三統曆及周曆皆以十一月夜半合朔冬至，日月俱起牽牛一度。古曆與近代密率相較，二百年氣差一日，三百年朔差一日。推而上之，久益先天；引而下之，久益後天。僖公五年，周曆正月辛亥朔，餘四分之一，南至。以歲差推之，日在牽牛初。至宣公十一年癸亥，周曆與麟德曆俱以庚戌日中冬至，而月朔尚先麟德曆十五辰。至昭公二十年己卯，周曆以正月己丑朔日中南至，麟德曆以己丑平旦冬至。哀公十一年丁巳，周曆入己酉蔀首，麟德曆以戊申禺中冬至。惠王四十三年己丑，周曆入丁卯蔀首，麟德曆以乙丑日昳冬至。呂后八年辛酉，周曆入乙酉蔀首，麟德曆以壬午黃昏冬至；其十二月甲申，人定合朔。太初元年，周曆以甲子夜半合朔冬至，麟德曆以辛酉禺中冬至，十二月癸亥晡時合朔。氣差三十二辰，朔差四辰。此疎密之大較也。

the first year of Taichu: the Three Systems and Zhou calendars both place eleventh-month midnight conjunction at winter solstice with sun and moon at one degree of Ox. Compared with recent precise rates, the ancient calendar drifts one day in qi every 200 years and one day in new moon every 300 years. Extrapolating backward it runs increasingly ahead of heaven; extrapolating forward it falls increasingly behind. Duke Xi's year 5: Zhou calendar first month xinhai new moon with one-quarter remainder at south arrival. By precession the sun stood at Ox's beginning. By Duke Xuan year 11 guihai, Zhou and Lindé calendars both give gengxu midday winter solstice, yet the month's new moon still precedes Lindé by fifteen chen. By Duke Zhao year 20 jimao, the Zhou calendar has first month jichou new moon at midday south arrival while Lindé has jichou dawn winter solstice. Duke Ai year 11 dingsi: Zhou calendar enters jiyou obscuration head; Lindé has wushen mid-afternoon winter solstice. King Hui year 43 jichou: Zhou enters dingmao obscuration head; Lindé has yichou mid-afternoon winter solstice. Empress Lü year 8 xinyou: Zhou enters yiyou obscuration head; Lindé has renwu dusk winter solstice; twelfth month jiashen conjunction at the human-ding hour. the first year of Taichu: Zhou has jiazi midnight conjunction at winter solstice; Lindé has xinyou mid-afternoon winter solstice and twelfth month guihai late-afternoon conjunction. Qi differs by thirty-two chen, new moon by four chen. That is the broad comparison of coarse versus precise.

僖公五年，周曆、漢曆、唐曆皆以辛亥南至。後五百五十餘歲，至太初元年，周曆、漢曆皆得甲子夜半冬至，唐曆皆以辛酉，則漢曆後天三日矣。祖沖之、張冑玄促上章歲至太初元年，沖之以癸亥雞鳴冬至，而冑玄以癸亥日出。欲令合於甲子，而適與魯曆相會。自此推僖公五年，魯曆以庚戌冬至，而二家皆以甲寅。且僖公登觀臺以望而書雲物，出於表晷天驗，非時史億度。乖丘明正時之意，以就劉歆之失。今考麟德元年甲子，唐曆皆以甲子冬至，而周曆、漢曆皆以庚午。然則自太初下至麟德差四日，自太初上及僖公差三日，不足疑也。

Duke Xi's year 5: Zhou, Han, and Tang calendars all give xinhai south arrival. 550-plus years later at the first year of Taichu, Zhou and Han both give jiazi midnight winter solstice while Tang gives xinyou—the Han calendar lags heaven three days. Zu Chongzhi and Zhang Zouxuan shortened the upper chapter year to the first year of Taichu—Chongzhi gives guihai cock-crow winter solstice and Zouxuan guihai sunrise. They wanted to match jiazi while aligning with the Lu calendar. Extrapolating back to Duke Xi year 5, the Lu calendar gives gengxu winter solstice while both give jiayin. Duke Xi observed from the tower and recorded cloud signs—gnomon verification, not chronicle estimates. That betrays Qiu Ming's correct timing to fit Liu Xin's error. In the first year of Lindé jiazi the Tang calendars give jiazi winter solstice while Zhou and Han give gengwu. From Taichu down to Lindé differs four days; from Taichu up to Duke Xi three days—not enough to doubt.

以歲差考太初元年辛酉冬至加時，日在斗二十三度。漢曆，氣後天三日，而日先天三度，所差尚少。故落下閎等雖候昏明中星，步日所在，猶未覺其差。然洪範、太初所揆，冬至昏奎八度中，夏至昏氐十三度中，依漢曆，冬至，日在牽牛初太半度，以昏距中命之，奎十一度中；夏至，房一度中。此皆閎等所測，自差三度，則劉向等殆已知太初冬至不及天三度矣。

By precession the first year of Taichu xinyou winter solstice added time places the sun at twenty-three degrees of Dipper. The Han calendar lagged three days in qi while the sun ran three degrees ahead—the net error was still small. Luoxia Hong and colleagues therefore did not notice the discrepancy even when observing dusk-dawn stars and tracking the sun. Hongfan and Taichu measured winter-solstice dusk Kui at eight degrees and summer-solstice dusk Di at thirteen; by the Han calendar winter solstice is at great half of Ox beginning, naming dusk distance gives Kui at eleven degrees; summer solstice Fang at one degree centered. These were Hong's measurements, off by three degrees—Liu Xiang and others probably already knew Taichu winter solstice fell three degrees short of heaven.

及永平中，治曆者考行事，史官注日，常不及太初曆五度。然諸儒守讖緯，以為當在牛初，故賈逵等議：「石氏星距，黃道規牽牛初直斗二十度，於赤道二十一度也。尚書考靈耀斗二十二度，無餘分。冬至，日在牽牛初，無牽牛所起文。編訢等據今日所去牽牛中星五度，於斗二十一度四分一，與考靈耀相近。」遂更曆從斗二十一度起。然古曆以斗魁首為距，至牽牛為二十二度，未聞移牽牛六度以就太初星距也。逵等以末學僻於所傳，而昧天象，故以權誣之，而後聽從他術，以為日在牛初者，由此遂黜。

By Yongping, calendar makers checking records found historiographers' dates often five degrees short of the Taichu calendar. Scholars clinging to apocrypha insisted on Ox's beginning; Jia Kui argued: "Shi's star distance places ecliptic Ox beginning at twenty degrees of Dipper, twenty-one on the equator. Examining Spirit's Radiance gives Dipper twenty-two degrees with no remainder. Winter solstice at Ox beginning lacks Ox-starting documentation. Bian Xin, using today's five degrees from Ox's center star, gets twenty-one and a quarter degrees of Dipper—near Examining Spirit's Radiance." They therefore revised the calendar to start at twenty-one degrees of Dipper. The ancient calendar used the Dipper's head as reference to twenty-two degrees of Ox—never shifting Ox six degrees to match Taichu distance. Kui and colleagues, ignorant of the heavens, used authority to mislead; later followers held the sun at Ox beginning and the correct view was suppressed.

今歲差，引而退之，則辛酉冬至，日在斗二十度，合於密率，而有驗於今。推而進之，則甲子冬至，日在斗二十四度，昏奎八度中，而有證於古。其虛退之度，又適及牽牛之初。而沖之雖促減氣分，冀符漢曆，猶差六度，未及於天。而麟德曆冬至不移，則昏中向差半次。淳風以為太初元年得本星度，日月合璧，俱起建星。賈逵考曆，亦云古曆冬至皆起建星。兩漢冬至，日皆後天，故其宿度多在斗末。今以儀測，建星在斗十三四度間，自古冬至無差，審矣。

With precession, xinyou winter solstice places the sun at twenty degrees of Dipper—matching precise rates and verified today. Pushing forward, jiazi winter solstice is at twenty-four degrees of Dipper with Kui at eight degrees centered at dusk—confirmed in antiquity. The retreating degrees again reach Ox's beginning. Zu Chongzhi shortened the qi fraction hoping to match the Han calendar yet still fell six degrees short of heaven. The Lindé calendar kept winter solstice fixed, so dusk culmination stars differ by half a station. Chunfeng held that the first year of Taichu recovered original star degrees with sun and moon conjoined, both rising at Jianxing. Jia Kui also said ancient calendars all start winter solstice at Jianxing. Both Han winter solstices lagged heaven, so lodge degrees mostly fell at Dipper's end. Instrument measurement places Jianxing between thirteen and fourteen degrees of Dipper—winter solstice has not shifted since antiquity; that is clear.

按古之六術，並同四分。四分之法，久則後天。推古曆之作，皆在漢初，却較春秋，朔並先天，則非三代之前明矣。

The ancient six methods all match the Quarter-Remainder system. The Quarter-Remainder method eventually lags heaven. Ancient calendars were all devised in early Han; compared with the Spring and Autumn their new moons run ahead—clearly not pre-dating the Three Dynasties.

古曆，南斗至牽牛上星二十一度，入太初星距四度，上直西建之初。故六家或以南斗命度，或以建星命度。方周、漢之交，日已潛退，其襲春秋舊曆者，則以為在牽牛之首；其考當時之驗者，則以為入建星度中。然氣朔前後不逾一日，故漢曆冬至，當在斗末。以為建星上得太初本星度，此其明據也。四分法雖疎，而先賢謹於天事，其遷革之意，俱有効於當時，故太史公等觀二十八宿疎密，立晷儀，下漏刻，以稽晦朔、分至、躔離、弦望，其赤道遺法，後世無以非之。故雜候清臺，太初最密。若當時日在建星，已直斗十三度，則壽王調曆宜允得其中，豈容頓差一氣而未知其謬，不能觀乎時變，而欲厚誣古人也。

The ancient calendar counts twenty-one degrees from Southern Dipper to Ox's upper star—four degrees into Taichu distance, directly at West Jian's start. Hence some schools name degrees from Southern Dipper, others from Jianxing. At the Zhou-Han transition the sun had already retreated; those following the Spring and Autumn calendar placed it at Ox's beginning; those checking contemporary observation placed it within Jianxing. Yet qi and new moon differ by no more than a day—so Han winter solstice should fall at Dipper's end. Placing Jianxing upper at Taichu's original star degree is the clear evidence. Though coarse, the Quarter-Remainder method reflects careful celestial work effective in its day; the Grand Historian and others observed lodge density, set gnomons and clepsydrae, and tracked new and full moons, solstices, and lunar motion—equatorial methods later generations cannot refute. Mixed observations at the Clear Terrace rank Taichu densest. If the sun were at Jianxing, already at thirteen degrees of Dipper, Shouwang's calendar adjustment should have hit center—how could it miss an entire qi unknowingly, fail to observe change, yet slander the ancients?

後百餘歲，至永平十一年，以麟德曆較之，氣當後天二日半，朔當後天半日。是歲四分曆得辛酉蔀首，已減太初曆四分日之三，定後天二日太半。開元曆以戊午禺中冬至，日在斗十八度半弱，潛退至牛前八度。進至辛酉夜半，日在斗二十一度半弱。續漢志云：「元和二年冬至，日在斗二十一度四分之一。」是也。

A century later, by Yongping year 11 compared with Lindé, qi should lag two and a half days and new moon half a day. That year the Quarter-Remainder reached xinyou obscuration head, having cut three-quarters day from Taichu, fixing lag at two and a half days. The Kaiyuan calendar gives wuwu mid-afternoon winter solstice with the sun at eighteen and a half weak degrees of Dipper, retreated to eight degrees before Ox. Advancing to xinyou midnight places the sun at twenty-one and a half weak degrees of Dipper. The Continued Han Treatise says: "Yuanhe year 2 winter solstice, sun at twenty-one and a quarter degrees of Dipper." So it records.

祖沖之曰：「四分曆立冬景長一丈，立春九尺六寸，冬至南極日晷最長。二氣去至日數既同，則中景應等。而相差四寸，此冬至後天之驗也。二氣中景，日差九分半弱，進退調均，略無盈縮。各退二日十二刻，則景皆九尺八寸。以此推冬至後天亦二日十二刻矣。」東漢晷漏定於永元十四年，則四分法施行後十五歲也。

Zu Chongzhi said: "The Quarter-Remainder calendar gives Start of Winter shadow one zhang and Start of Spring nine chi six cun; winter solstice has the longest south-pole gnomon shadow. The two qi are equally distant from solstice, so midday shadows should match. Yet they differ four cun—proof that winter solstice lags heaven. Midday shadows at the two qi differ by nine and a half weak parts per day; evenly adjusted there is almost no surplus or deficit. Retreat two days twelve quarters each and shadows become nine chi eight cun. Extrapolating, winter solstice also lags heaven by two days twelve quarters." The Eastern Han gnomon and clepsydra were fixed in Yongyuan year 14—fifteen years after the Quarter-Remainder method was implemented.

二十四氣加時，進退不等，其去午正極遠者四十九刻有餘。日中之晷，頗有盈縮，故治曆者皆就其中率，以午正言之。而開元曆所推氣及日度，皆直子半之始。其未及日中，尚五十刻。因加二日十二刻，正得二日太半。與沖之所算及破章二百年間輒差一日之數，皆合。

Twenty-four qi added times advance and retreat unevenly, farthest from wu midday by forty-nine-plus quarters. Midday gnomon shadows vary; calendrists therefore use the mean rate and speak relative to wu midday. The Kaiyuan calendar's derived qi-periods and solar degrees all align with the start of midnight at Zi. Before reaching midday it still fell short by fifty ke. Adding two days and twelve ke gives exactly two and a half days. Compared with Chong's calculations and the one-day drift every two hundred years when a rule cycle breaks, everything matched.

自漢時辛酉冬至，以後天之數減之，則合於今曆歲差斗十八度。自今曆戊午冬至，以後天之數加之，則合於賈逵所測斗二十一度。反復僉同。而淳風冬至常在斗十三度，豈當時知不及牽牛五度，而不知過建星八度耶？

From the Han-era Xinyou winter solstice, subtracting the later-heaven numbers matches the present calendar's precession of eighteen degrees in the Dipper. From the present calendar's Wuwu winter solstice, adding the later-heaven numbers matches the twenty-one degrees in the Dipper measured by Jia Kui. Repeated checks all agreed. Yet Chunfeng placed the winter solstice constantly at thirteen degrees in the Dipper—could he have known it fell five degrees short of Ox yet not known it overshot Jian by eight degrees?

晉武帝太始三年丁亥歲冬至，日當在斗十六度。晉用魏景初曆，其冬至亦在斗二十一度少。

At the Dinghai winter solstice in Jin Emperor Wu's Taishi year 3, the sun should have stood at sixteen degrees in the Dipper. Jin used Wei's Jingchu calendar, whose winter solstice also fell at just over twenty-one degrees in the Dipper.

太元九年，姜岌更造三紀術，退在斗十七度。曰：「古曆斗分彊，故不可施於今；乾象斗分細，故不可通於古。景初雖得其中，而日之所在；乃差四度，合朔虧盈，皆不及其次。假月在東井一度蝕，以日檢之，乃在參六度。」岌以月蝕衝知日度，由是躔次遂正，為後代治曆者宗。

In Taiyuan 9 Jiang Ji remade the Three Era method, placing the solstice at seventeen degrees in the Dipper. He said: 'Ancient calendars used too large a Dipper fraction, so they cannot be applied today; the Supernal Icon calendar's Dipper fraction was too fine, so it cannot connect with antiquity. Jingchu though it hit the mean, the sun's position was four degrees off; conjunction new moons and eclipse phases all missed their proper nodes. Suppose the moon eclipsed at one degree in Well—checking against the sun, it was actually at six degrees in Three Stars.' Ji determined the sun's degree from lunar eclipse opposition; thereby the lodge positions were corrected, and later calendar makers took him as their authority.

宋文帝時，何承天上元嘉曆，曰：「四分、景初曆，冬至同在斗二十一度，臣以月蝕檢之，則今應在斗十七度。又土圭測二至，晷差三日有餘，則天之南至，日在斗十三四度矣。」事下太史考驗，如承天所上。以開元曆考元嘉十年冬至，日在斗十四度，與承天所測合。

Under Emperor Wen of Song, He Chengtian presented the Shangyuan-Jia calendar, saying: 'The Four Quarters and Jingchu calendars both placed the winter solstice at twenty-one degrees in the Dipper; I checked by lunar eclipse and found it should now be at seventeen degrees. Measuring the two solstices with the gnomon, the shadow difference exceeded three days—meaning at heaven's southern extreme the sun stood at thirteen or fourteen degrees in the Dipper.' The matter was referred to the Grand Astrologer for verification, and matched Chengtian's figures. Checking Yuanjia 10's winter solstice with the Kaiyuan calendar, the sun stood at fourteen degrees in the Dipper, matching Chengtian's measurement.

大明八年，祖沖之上大明曆，冬至在斗十一度，開元曆應在斗十三度。梁天監八年，沖之子員外散騎侍郎暅之上其家術。詔太史令將作大匠道秀等較之，上距大明又五十年，日度益差。其明年，閏月十六日，月蝕，在虛十度，日應在張四度。承天曆在張六度，沖之曆在張二度。

In Daming 8 Zu Chongzhi submitted the Daming calendar, placing the winter solstice at eleven degrees in the Dipper; the Kaiyuan calendar places it at thirteen degrees. In Liang Tiantian 8, Zu Chongzhi's son Xu, Attendant Outside the Scattereds, submitted their family method. An edict ordered Grand Astrologer and Master of Works Dao Xiu and others to compare it; fifty years after Daming, solar degrees had diverged still further. The next year, on the sixteenth of the intercalary month, the moon eclipsed at ten degrees in Void—the sun should have stood at four degrees in Extended Net. Chengtian's calendar placed it at six degrees in Extended Net; Chongzhi's at two degrees.

大同九年，虞𠠎等議：「姜岌、何承天俱以月蝕衝步日所在。承天雖移岌三度，然其冬至亦上岌三日。承天在斗十三四度，而岌在斗十七度。其實非移。祖沖之謂為實差，以推今冬至，日在斗九度，用求中星不合。自岌至今，將二百年，而冬至在斗十二度。然日之所在難知，驗以中星，則漏刻不定。漢世課昏明中星，為法已淺。今候夜半中星，以求日衝，近於得密。而水有清濁，壺有增減，或積塵所擁，故漏有遲疾。臣等頻夜候中星，而前後相差或至三度。大略冬至遠不過斗十四度，近不出十度。」又以九年三月十五日夜半，月在房四度蝕。九月十五日夜半，月在昴三度蝕。以其衝計，冬至皆在斗十二度。自姜岌、何承天所測，下及大同，日已却差二度。而淳風以為晉、宋以來三百餘歲，以月蝕衝考之，固在斗十三四度間，非矣。

In Datong 9 Yu Xiao and others argued: 'Jiang Ji and He Chengtian both determined the sun's position from lunar eclipse opposition. Chengtian though he shifted Ji by three degrees, his winter solstice also advanced Ji's date by three days. Chengtian placed it at thirteen or fourteen degrees in the Dipper, while Ji at seventeen degrees. In fact there was no true shift. Zu Chongzhi treated it as a real discrepancy; projecting the present winter solstice to nine degrees in the Dipper and seeking the culminating stars from that, the result did not agree. From Ji to the present, nearly two hundred years have passed, and the winter solstice stands at twelve degrees in the Dipper. Yet the sun's position is hard to know directly; verifying by culminating stars leaves clepsydra graduations uncertain. Han practice gauged twilight and dawn culminating stars—a method already too coarse. Observing culminating stars at midnight to find solar opposition is closer to precision. Yet water varies in clarity, vessels in capacity, and dust may clog them—so clepsydra flow runs slow or fast. We repeatedly observed culminating stars at night, yet successive readings sometimes differed by as much as three degrees. Broadly speaking, the winter solstice never recedes beyond fourteen degrees in the Dipper nor falls short of ten degrees.' They also took midnight on the fifteenth of the third month in year 9, when the moon eclipsed at four degrees in Room. Midnight on the fifteenth of the ninth month, when the moon eclipsed at three degrees in Hairy Head. Computing from their opposition, both winter solstices fell at twelve degrees in the Dipper. From Jiang Ji and He Chengtian's measurements down to Datong, the sun had already retreated two degrees. Yet Chunfeng held that for three hundred-odd years from Jin and Song, checked by lunar eclipse opposition, it firmly stood between thirteen and fourteen degrees in the Dipper—which was wrong.

劉孝孫甲子元曆，推太初冬至在牽牛初，下及晉太元、宋元嘉皆在斗十七度。開皇十四年，在斗十三度。而劉焯曆仁壽四年冬至，日在黃道斗十度，於赤道斗十一度也。其後孝孫改從焯法，而仁壽四年冬至，日亦在斗十度。焯卒後，冑玄以其前曆上元起虛五度，推漢太初，猶不及牽牛，乃更起虛七度，故太初在斗二十三度，永平在斗二十一度，並與今曆合。而仁壽四年，冬至在斗十三度，以驗近事，又不逮其前曆矣。戊寅曆，太初元年辛酉冬至，進及甲子，日在牽牛三度。永平十一年，得戊午冬至，進及辛酉，在斗二十六度。至元嘉，中氣上景初三日，而冬至猶在斗十七度。欲以求合，反更失之。又曲循孝孫之論，而不知孝孫已變從皇極，故為淳風等所駁。歲差之術，由此不行。

Liu Xiaosun's Jiazi Origin calendar projected Taichu's winter solstice at the start of Ox, and down through Jin Taiyuan and Song Yuanjia at seventeen degrees in the Dipper. In Kaihuang 14 it stood at thirteen degrees in the Dipper. Yet Liu Chuo's calendar for Renshou 4's winter solstice placed the sun at ten degrees in the Dipper on the ecliptic—eleven degrees in the Dipper on the equator. Later Xiaosun adopted Chuo's method, and Renshou 4's winter solstice also placed the sun at ten degrees in the Dipper. After Chuo died, Zouxuan used his earlier calendar's upper origin at five degrees in Void to project Han Taichu—still falling short of Ox—then restarted at seven degrees in Void; thus Taichu fell at twenty-three degrees in the Dipper and Yongping at twenty-one degrees, both matching the present calendar. Yet Renshou 4's winter solstice stood at thirteen degrees in the Dipper—checked against recent events, it fell even further behind his earlier calendar. The Wuyin calendar: the first year of Taichu's Xinyou winter solstice advanced to Jiazi—the sun at three degrees in Ox. Yongping 11 obtained the Wuwu winter solstice, advancing to Xinyou—at twenty-six degrees in the Dipper. By Yuanjia the mid-qi had advanced three days ahead of Jingchu, yet the winter solstice still stood at seventeen degrees in the Dipper. Seeking agreement, they lost it further still. Again bending to Xiaosun's argument, unaware that Xiaosun had already switched to Huangji—therefore Chunfeng and others refuted it. The method of annual precession thereby fell out of use.

以太史注記月蝕衝考日度，麟德元年九月庚申，月蝕在婁十度。至開元四年六月庚申，月蝕在牛六度。較麟德曆率差三度，則今冬至定在赤道斗十度。

Checking solar degrees by lunar eclipse opposition in the Grand Astrologer's records: Lindé 1, ninth month Gengshen, the moon eclipsed at ten degrees in Bond. By Kaiyuan 4, sixth month Gengshen, the moon eclipsed at six degrees in Ox. Compared with the Lindé calendar rate, the difference is three degrees—fixing the present winter solstice at ten degrees in the Dipper on the equator.

又皇極曆歲差皆自黃道命之，其每歲周分，常當南至之軌，與赤道相較，所減尤多。計黃道差三十六度，赤道差四十餘度，雖每歲遯之，不足為過。然立法之體，宜盡其原，是以開元曆皆自赤道推之，乃以今有術從變黃道。

The Huangji calendar's annual precession is reckoned from the ecliptic; its yearly circuit fraction should match the southern extreme's track—compared with the equator, the reduction is especially large. The ecliptic difference totals thirty-six degrees and the equator difference forty-odd—though each year it retreats, that is not excessive. Yet method-making should reach back to the root; therefore the Kaiyuan calendar derives everything from the equator, then converts to the ecliptic by present methods.

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