Volume 31 Treatises 7: Calendar 1

志第七曆一

Treatise Seven: Calendrics, Part One.

後世法勝於古，而屢改益密者，惟曆為最著。《唐志》謂天為動物，久則差忒，不得不屢變其法以求之。此說似矣，而不然也。《易》曰：「天地之道，貞觀者也。」蓋天行至健，確然有常，本無古今之異。其歲差盈縮遲疾諸行，古無今有者，因其數甚微，積久始著。古人不覺，而後人知之，而非天行之忒也。使天果久動而差忒，則必差參淩替而無典耍，安從修改而使之益密哉？觀傳志所書，歲失其次、日度失行之事，不見於近代，亦可見矣。夫天之行度多端，而人之智力有限，持尋尺之儀錶，仰測穹蒼，安能洞悉無遺。惟合古今人心思，踵事增修，庶幾符合。故不能為一成不易之法也。

Among the sciences, calendrics alone shows most clearly how later methods surpass the ancient and how repeated revision makes computation ever tighter. The Tang dynastic treatises hold that heaven is in motion and that, over long intervals, error accumulates—hence calendar law must be changed again and again. That argument sounds plausible; it is not correct. The Book of Changes says: "The Way of Heaven and Earth is constancy in correct observation. Heaven moves with supreme vigor yet with firm constancy; in principle it does not differ between antiquity and the present. Precession, solar elongation, lunar anomaly, and the like—phenomena the ancients did not notice but we do—arise because the underlying quantities are minute and only become visible after centuries of accumulation. The ancients did not perceive them; later astronomers did. That is not a deviation in heaven's motion itself. If heaven truly drifted over long ages, observations would fall into chaotic disorder and no stable canon would remain—how then could repeated revision make calculation ever more precise? Dynastic histories record years that lost their proper season and solar motion that missed its track; such failures do not appear in recent centuries—which itself tells the tale. Heaven's motions are manifold and human intellect finite; armed only with foot-scale instruments and gazing upward at the vault, who can compass every detail? Agreement is possible only when ancient and modern insight are combined and each generation, building on its predecessors, adds refinements. Hence no calendar law can remain fixed for all time.

黃帝迄秦，曆凡六改。漢凡四改。魏迄隋，十五改。唐迄五代，十五改。宋十七改。金迄元，五改。惟明之《大統曆》，實即元之《授時》，承用二百七十餘年，未嘗改憲。成化以後，交食往往不驗，議改曆者紛紛。如俞正己、冷守中不知妄作者無論已，而華湘、周濂、李之藻、刑雲路之倫頗有所見。鄭世子載堉撰《律曆融通》，進《聖壽萬年曆》，其說本之南部御史何瑭，深得《授時》之意，而能補其不逮。臺官泥於舊聞，當事憚於改作，並格而不行。崇禎中，議用西洋新法，命閣臣徐光啟、光祿卿李天經先後董其事，成《曆書》一百三十餘卷，多發古人所未發。時布衣魏文魁上疏排之，詔立兩愕扒驗。累年校測，新法獨密，然亦未及頒行。由是觀之，曆固未有行之久而差者，烏可不隨時修改，以求合天哉。

From the Yellow Emperor through Qin, the calendar was revised six times. Under Han: four revisions. From Wei through Sui: fifteen revisions. From Tang through the Five Dynasties: fifteen revisions. Under Song: seventeen revisions. From Jin through Yuan: five revisions. Ming's Datong Calendar was in substance the Yuan Shoushi Calendar, kept in force for more than two hundred seventy years without statutory revision. After the Chenghua reign, predicted eclipses and conjunctions repeatedly failed verification, and calls for calendar reform multiplied. Yu Zhengji and Leng Shouzhong, who wrote without understanding, may be passed over; but Hua Xiang, Zhou Lian, Li Zhizao, Xing Yunlu, and others of that company had real insight. Prince of Zheng Zhu Zaiyu compiled the Harmonization of Pitch and Calendar and presented the Sacred Longevity Ten-Thousand-Year Calendar. His theory drew on the southern censor He Tang, grasped the intent of the Shoushi system, and remedied its gaps. Observatory officers clung to received tradition; decision-makers shrank from reform; every proposal was blocked. Under Chongzhen, Western calendrical methods were adopted. Grand Secretary Xu Guangqi and Director of the Court of Imperial Entertainments Li Tianjing directed the project in turn and produced the Calendar Treatises in more than 130 juan, opening many questions the ancients had not addressed. The commoner Wei Wenkui then memorialized against the new methods; the throne ordered two bureaus to compare and verify them. Years of comparative testing showed the new method alone to be precise, yet it was still not promulgated. From this it follows that no calendar grows inaccurate merely from long use; revision must track the times so that computation again matches heaven.

今採扣家論說，有裨於曆法者，著於篇端。而《大統曆》則述立法之原，以補《元志》之未備。《回回曆》始終隸於欽天監，與《大統》參用無形之質，形而下為有形之物。南宋朱熹認為，理為形而上，亦附錄焉。

Here we collect discussions from various schools that bear on calendrics and place them at the head of the chapter. The Datong Calendar section sets out the origins of its rules, supplementing omissions in the Yuan dynastic treatises. The Islamic (Huihui) calendar remained under the Imperial Observatory throughout and was employed alongside the Datong calendar. Qi lacks a formless substrate; what lies below form is the realm of embodied things. Zhu Xi of the Southern Song held that li (pattern) belongs to the realm above form; that doctrine is appended as well.

▲曆法沿革

▲ Evolution of Calendar Methods

吳元年十一月乙未冬至，太史院使劉基率其屬高翼上戊申《大統曆》。太祖諭曰：「古者季冬頒曆，太遲。今於冬至，亦未善。宜以十月朔，著為令。」洪武元年改院為司天監，又置回回司天監。詔徵元太史院使張佑、回回司天太監黑的兒等共十四人，尋召回回司天臺官鄭阿裡等十一有至京，議曆法。三年改監為欽天，設四科：曰天文，曰漏刻，曰《大統曆》，曰《回回曆》。以監令、少監統之。歲造《大統民曆》、《御覽月令曆》、《七政躔度曆》、《六壬遁甲曆》、《四季天象佔驗曆》、《御覽天象錄》，各以時上。其日月交食分秒時刻、起復方位，先期以聞。十年三月，帝與羣臣論天與七政之行，皆以蔡氏旋之說對。帝曰：「朕自起以來，仰觀乾象，天左旋，七政右旋，曆家之論，確然不易。爾等猶守蔡氏之說，豈所謂格物致知學乎？」十五年九月，詔翰林李翀、吳伯宗譯《回回曆書》。

In the first year of Wu, on the winter solstice (day yimao, eleventh month), Astrological Academy Director Liu Ji led his subordinate Gao Yi in presenting the Datong Calendar. The Founder instructed: "In antiquity the calendar was issued in the last month of winter—that is too late. Issuing it at the winter solstice is also unsatisfactory. Fix the first day of the tenth month as the date of issue and make it law. In Hongwu 1 the Academy was renamed the Directorate of Astronomy, and an Islamic Directorate of Astronomy was also established. An edict summoned the Yuan Astrological Academy director Zhang You, the Islamic chief astronomical supervisor Hei De'er, and twelve others—fourteen in all. Soon eleven Islamic observatory officers including Zheng Ali reached the capital to deliberate on calendar methods. In year 3 the directorate was renamed the Imperial Observatory and divided into four sections: astronomy, clepsydra, the Datong calendar, and the Islamic calendar. A director and deputy director supervised all four. Each year the observatory produced the Datong popular calendar, the imperial monthly-command calendar, the seven-luminaries ephemeris, the Liuren Dunjia calendar, the four-seasons omen-verification calendar, and the imperial celestial record, each submitted on schedule. For solar and lunar eclipses—magnitude, seconds, time, and the azimuths of first contact and recovery—reports were filed in advance. In year 10, third month, the Emperor debated with his ministers the motion of heaven and the seven luminaries; all answered in terms of Master Cai's rotation theory. The Emperor said: "Since I took power I have watched the sky: heaven rotates to the left and the seven luminaries to the right. The astronomers' doctrine is settled and not to be altered. Yet you still cling to Master Cai—is that what you call investigating things to extend knowledge? In year 15, ninth month, Hanlin academicians Li Chong and Wu Bozong were ordered to translate the Islamic Calendar Treatise.

十七年閏十月，漏刻博士元統言：「曆以《大統》為名，而積分猶踵《授時》之數，非所以重始敬正也。況《授時》以元辛巳為曆元，至洪武甲子積一百四年，年遠數盈，漸差天度，合修改。七政運行不齊，其理深奧。聞有郭伯玉者，精明九數之理，宜徵令推算，以成一代之制。」報可。擢統為監令。統乃取《授時曆》，去其歲實消長之說，析其條例，得四卷，以洪武十七年甲子為曆元，命曰《大統曆法通軌》。二十二年改監令、丞為監正、副。二十六年，監副李德芳言：「監正統孜作洪武甲子曆元，不用消長之法，以考魯獻公十五年戊寅歲天正冬至，比辛巳為元，差四日半強。今當復用辛巳為元及消長之法。」疏入，元統奏辨。太祖曰：「二說皆難憑，但驗七政交會行度無差者為是。」自是《大統曆》元以洪武甲子，而推算仍依《授時》法。三十一年在罷回回欽天監，其《回回曆》科仍舊。

In year 17, intercalary tenth month, Clepsydra Doctor Yuan Tong said: "The calendar is called Datong, yet its accumulated parts still follow Shoushi numbers. That is not how to honor a new beginning and revere the correct alignment. Moreover the Shoushi calendar took the Yuan year xinsi as its epoch; by Hongwu jiazi one hundred four years had elapsed. With time the parameters overflow and drift from celestial degrees; revision is due. The seven luminaries do not move uniformly; the theory is profound. I hear that Guo Boyu is expert in the mathematics of the Nine Chapters; summon him to compute and establish a system for this dynasty. The memorial was approved. Yuan Tong was promoted to director. Yuan Tong took the Shoushi Calendar, discarded its theory of varying the tropical year, analyzed its rules into four juan, set Hongwu 17 jiazi as the epoch, and titled the work Comprehensive Tracks of Datong Calendar Methods. In year 22 the titles director and assistant director were changed to director and deputy director. In year 26 Deputy Director Li Defang said: "Director Yuan Tong set the Hongwu jiazi epoch and abandoned the variation method. Reckoning the celestial winter solstice of Duke Ai of Lu's fifteenth year (wuyin), against an xinsi epoch the discrepancy is four and a fraction days. The xinsi epoch and the variation method should be restored. The memorial was received; Yuan Tong rebutted it in a counter-memorial. The Founder said: "Neither argument can be trusted on its own; accept whichever yields no error in the conjunctions and motions of the seven luminaries. Thereafter the Datong epoch remained Hongwu jiazi, but computation still followed Shoushi procedure. In year 31 the Islamic Imperial Observatory was abolished again; the Islamic calendar section continued unchanged.

永樂遷都順天，仍用應天冬夏晝夜時刻，至正統十四年始改用順天之數。其冬，景帝即位，天文生馬軾奏，晝夜時刻不宜改。下廷臣集議。監正許惇等言：「前監正彭德清測驗得北京北極出地四十度，比南京高七度有奇，冬至晝三十八刻，夏至晝六十二刻。奏準改入《大曆》，永為定式。軾言誕妄，不足聽。」帝曰：「太陽出入度數，當用四方之中。今京師在堯幽都之地，寧可為準。此後造曆，仍用洪、永舊制。」

When Yongle moved the capital to Shuntian, the observatory still used Yingtian's winter and summer day-length divisions; Shuntian values were adopted only in Zhengtong 14. That winter, when the Jing Emperor succeeded, astronomy student Ma Shi argued that the day-night divisions should not be altered. The matter was referred to the ministries for joint deliberation. Director Xu Dun and others said: "Former director Peng Deqing measured Beijing's polar altitude at forty degrees—more than seven degrees higher than Nanjing—with winter-solstice daylength of thirty-eight clepsydra divisions and summer of sixty-two. The throne approved incorporating these values into the great calendar as a permanent standard. Ma Shi's claims are absurd and should not be followed. The Emperor said: "Degrees of sunrise and sunset should be reckoned from the geographic center of the realm. The capital stands in the region of Yao's Youdu—how can it be the standard? Henceforth calendar-making shall follow the Hongwu and Yongle conventions."

景泰元年正月辛卯，卯正三刻月食。監官誤推辰初初刻，致失救護。下法司，論徒。詔宥之。成化十年，以監官多不職，擢雲南提學童軒為太常寺少卿，掌監事。十五年十一月戊戍望，月食，監推又誤，帝以天象微渺，不之罪也。十七年，真定教論俞正己上《改曆議》，詔禮部及軒參考。尚書周洪謨等言：「正己止據《皇極經世書》及歷代天文、曆志推算氣朔，又以己意創為八十七年約法，每月大小相間。輕率狂妄，宜正其罪。」遂下正己詔獄。十九年，天文生張升上言改曆。欽天監謂祖制不可變，升說遂寢。弘治中，月食屢不應，日食亦舛。

Jingtai 1, first month, day xinmao: lunar eclipse at the third quarter of mao upright. Observatory officers had predicted chen initial first quarter and failed the ritual response. The case went to the judiciary; they were sentenced to penal servitude. An edict pardoned them. Chenghua 10: with many observatory officers negligent, Yunnan education intendant Tong Xuan was promoted to vice minister of the Court of Imperial Sacrifices to oversee the observatory. Year 15, eleventh month, full moon (day wuxu): lunar eclipse; the observatory erred again. The Emperor held celestial phenomena too subtle for blame and imposed no punishment. Year 17: Zhending instructor Yu Zhengji submitted his Discussion on Calendar Reform; the Ministry of Rites and Tong Xuan were ordered to review it. Minister Zhou Hongmo and others said: "Zhengji relies only on the Huangji Jingshi and successive astronomical treatises to compute qi and syzygy, then on his own authority devises an eighty-seven-year rule with alternating long and short months. This is rash and presumptuous; he should be punished. Yu Zhengji was thereupon imprisoned. Year 19: astronomy student Zhang Sheng memorialized for calendar reform. The Imperial Observatory held that ancestral institutions must not change; Zhang Sheng's proposal was shelved. Under Hongzhi, lunar eclipses repeatedly failed verification; solar eclipses were wrong as well.

正德十二、三年，連推日食起復，皆弗合。於是漏刻博士硃裕上言：「至元辛巳距今二百三十七年，歲久不能無差，若不量加損益，恐愈久愈舛。乞簡大臣總理其事，令本監官生半推古法，半推新法，兩相交驗，回回科推驗西域《九執曆法》。仍遣官至各省，候土圭以測節氣早晚。往復參較，則交食可正，而七政可齊。」部覆言：「裕及監官曆學未必皆精，今十月望月食，中官正周濂等所推算，與古法及裕所奏不同，請至期考驗。」既而濂等言：「日躔歲退之差一分五十秒。今正德乙亥，距至元辛巳二百三十五年，赤道歲差，當退天三度五十二分五十秒。不經改正，推步豈能有合。臣參較德驗，得正德丙子歲前天正冬至氣應二十七日四百七十五分，命得辛卯日醜初初刻，日躔赤道箕宿六度四十七五十秒，黃道箕宿五度九十六分四十三秒為曆元。其氣閏轉交四應，並周天黃赤道，諸類立成，悉從歲差，隨時改正。望敕禮臣並監正董其事。」部奏：「古法未可輕變，請仍舊法。別選精通曆學者，同濂等以新法參驗，更為奏請。」從之。

In Zhengde 12–13, successive predictions of solar-eclipse first contact and recovery all missed. Clepsydra Doctor Zhu Yu then memorialized: "From the Yuan epoch xinsi to the present is 237 years; error must accumulate. Without measured adjustments, the drift will only worsen. Select a senior minister to direct the project; let half the observatory staff compute by the old method and half by the new, cross-checking the results; let the Islamic section verify the Western Regions' Nine Executions methods. Send officers to each province to set up gnomons and measure whether the solar terms arrive early or late. With repeated comparison back and forth, eclipses can be corrected and the seven luminaries brought into agreement. The ministry replied: "Yu and the observatory staff are not all experts. For the coming tenth-month eclipse, central palace corrector Zhou Lian's figures differ from both the old method and Yu's proposal; verify on the day." Soon Zhou Lian and others said: "The sun's annual precession amounts to one minute fifty seconds. In Zhengde yihai, 235 years from Yuan xinsi, equatorial precession should amount to 3°52′50″ of retreat. Without correction, how can sequential computation ever agree? We compared and verified: for the winter solstice before Zhengde bingzi, qi fell on day 27 with 475 parts; syzygy fell on xinmao, chou initial first quarter; the sun stood at equatorial Lodge 6°47′50″ and ecliptic Lodge 5°96′43″ as the new epoch. Qi, intercalation, rotation, and conjunction tables, together with full-circle ecliptic and equatorial parameters, should all incorporate precession and be revised as times require. We ask that ritual officials and the observatory director jointly oversee the work. The ministry replied: "The old method must not be changed lightly; continue the established system. Separately select experts in calendrics to cross-check with Zhou Lian's new method and report again. Approved.

十五年，禮部員外郎鄭善夫言：「日月交食，日食最為難測。蓋月食分數，但論距交遠近，別無四時加減，且月小暗虛大，八方所見皆同。若日為月所掩，則日大而月小，日上而月下，日遠而月近。日行有四時之異，月行有九道之分。故南北殊觀，時刻亦異。必須據地定表，因時求合。如正德九年八月辛卯日食，曆官報食八分六十七秒，而閩、廣之地，遂至食既。時刻分秒，安得而同？今宜按交食以更曆元，時刻分秒，必使奇零剖析詳盡。不然，積以歲月，躔離朓朒，又不合矣。」不報。十六年以南京戶科給事中樂頀、工部主事華湘通曆法，俱擢光祿少卿，管監事。

Year 15: Ministry of Rites outer-section officer Zheng Shanfu said: "Of solar and lunar eclipses, the solar is hardest to predict. Lunar eclipse magnitude depends only on distance from the node, without seasonal corrections; the moon is small and the earth's shadow large, so observers everywhere see essentially the same event. When the sun is veiled by the moon, the sun is the larger body and the moon the smaller; the sun lies above and the moon below, the sun distant and the moon near. The sun's course varies with the four seasons; the moon's course divides among the nine paths. North and south therefore behold different sights, and the hours differ as well. One must set the shadow-table according to locality and seek agreement according to season. On the xinmao solar eclipse of the eighth month, Zhengde 9, the calendar officers reported an obscuration of eight parts and sixty-seven seconds—yet in Fujian and Guangdong the eclipse ran to totality. How could the hours, minutes, and seconds be one and the same? The calendar origin should now be revised from eclipse data, and every hour, minute, and second parsed down to the last odd fraction. Otherwise, as the years pile up, solar lodges, lunar distance, and the moon's waxing and waning will again fall out of accord." No reply was made. In the sixteenth year, Yue Bin of the Nanjing Household Section and Hua Xiang, principal secretary of the Ministry of Works—both skilled in calendrical computation—were alike promoted to Vice Minister of Imperial Sacrifices and charged with supervising the directorate.

嘉靖二年，湘言：「古今善治曆者三家，漢《太初》以鐘律，唐《大衍》以蓍策，元《授時》以晷景為近。欲正曆而不登臺測景，皆空言臆見也。望許臣暫朝參，督中官正周濂等，及冬至前詣觀象臺，晝夜推測，日記月書，至來年冬至，以驗二十四氣、分至合朔、日躔月離、黃赤二道、昏旦中星、七政四餘之度，視元辛巳所測，離合何如，差次錄聞。更敕禮部延訪精通理數者徵赴京師，令詳定歲差，以成一代之制。」下禮部集議，而護謂曆不可改，與湘頗異。禮部言：「湘欲自行測候，不為無識。請二臣各盡所見，窮極異同，以協天道。」從之。

Jiajing 2: Xiang memorialized, "Of those who have truly mastered the calendar, antiquity and the present offer three exemplars: the Han Taichu rested on pitch-pipes and musical regulation; the Tang Dayan on yarrow-casting; the Yuan Season Granting, grounded in gnomon shadows, comes nearest the mark. To set the calendar right without ascending the terrace to measure shadows is mere empty talk and private fancy. He asked leave to attend court for a time, to oversee the senior director Zhou Lian and his colleagues, and before the winter solstice to take up station at the Observatory for day-and-night measurement, recording sun and moon day by day until the next winter solstice—there to test the twenty-four qi, the equinoxes and solstices and their new moons, solar lodge motion and lunar distance, the yellow and red paths, the stars culminating at dusk and dawn, and the degrees of the seven luminaries and four residuals; to compare these with the Yuan xinsi measurements, note agreement and divergence, and report each discrepancy in order. Further, let the Ministry of Rites seek out masters of principle and number and summon them to the capital to fix the annual difference in full, and so complete a calendar for the age." The memorial went to the Ministry of Rites for joint deliberation; Yue Bin held that the calendar must not be altered—a view at odds with Xiang's. The Ministry of Rites replied, "Xiang's wish to conduct observations himself is not without insight. Let the two ministers each lay out their views to the limit, exhaust every point of agreement and difference, and so bring the work into harmony with Heaven's way. The throne approved.

七年，欽天監奏：「閏十月朔，《回回曆》推日食二分四十七秒，《大統曆》推不食。」已而不食。十九年三月癸巳朔，臺官言日當食，已而不食。帝喜，以為天眷，然實由推步之疏也。隆慶三年，掌監事順天府丞周相刊《大統曆法》，其曆原曆敘古今諸曆異同。萬曆十二年十一有癸酉朔《大統曆》推日食九十二秒，《回回曆》推不食，已而《回回曆》驗。禮科給事中侯先春因言：「邇年月食在酉而曰戌，月食將既而曰未九分，差舛甚矣。《回回曆》科推算日月交食，五星淩犯，最為精密，何妨纂人《大統曆》中，以備考驗。」詔可。二十年五月戌夜月食，監官推算差一日。

The seventh year: the Directorate of Astronomy reported, "On the new moon of the intercalary tenth month, the Hui Calendar computed a solar eclipse of two parts and forty-seven seconds; the Grand Concordance computed none. In the event, there was no eclipse. The nineteenth year, on the guisi new moon of the third month, the observatory officers declared that the sun would be eclipsed; it was not. The emperor rejoiced, taking this as Heaven's favor; in truth it owed to the looseness of the computation. Longqing 3: Zhou Xiang, Shuntian vice prefect in charge of the directorate, published the Methods of the Grand Concordance Calendar, with an origin-treatise setting forth how calendars past and present differ. Wanli 12, on the guiyou new moon of the eleventh month: the Grand Concordance predicted a solar eclipse of ninety-two seconds; the Hui Calendar predicted none—and the Hui Calendar was borne out. Supervising secretary Hou Xianchun thereupon said, "Of late, lunar eclipses have fallen in the you hour yet been announced for xu; eclipses on the verge of totality have been called 'not yet nine-tenths'—the errors are grave. The Hui Calendar desk's reckonings of solar and lunar eclipses and the five planets' encroachments are the most exact; why not weave them into the Grand Concordance for reference and proof? An edict approved. Wanli 20, fifth month: on the xu night a lunar eclipse occurred, and the observatory officers' computation missed by a full day.

二十三年，鄭世子載堉進《聖壽萬年曆》、《律曆融通》二書。疏略曰：「高皇帝革命時，元曆未久，氣朔未差，故不改作，但討論潤色而已。積年既久，氣朔漸差。《後漢志》言『三百年鬥曆改憲』。今以萬曆為元，而九年辛巳歲適當『鬥曆改憲』之期，又協『乾元用九』之義，曆元正在是矣。臣嘗取《大統》與《授時》二曆較之，考古則氣差三日，推今則時差九刻。夫差雖九刻，處夜半之際，所差便隔一日。節氣差天一日，則置閏差一月。閏差一月，則時差一季。時差一季，則歲差一年。其失豈小小哉？蓋因《授時》減分太峻，失之先天；《大統》不減，失之後天。因和會兩家，酌取中數，立為新率，編撰成書，大旨出於許衡，而與曆不同。黃鐘乃律曆本原，而舊曆罕言之。新法則以步律呂爻象為首。堯時冬至日躔宿次，何承天推在須、女十度左右，一行推在女、虛間，元人曆議亦云在女、虛之交。而《授時曆》考之，乃在牛宿二度。《大統曆》考之，乃在危宿一度。相差二十六度，皆不與《堯典》合。新法上考堯元年甲辰歲，夏至午中，日在柳宿十二度左右，冬至午中，日在女宿十度左右，心昴昏中，各去午正不逾半次，與承天、一行二家之說合。此皆與舊曆不同大者，其餘詳見《曆議》。望敕大臣名儒參訂採用。」

Wanli 23: the Prince of Zheng, Zhu Zaiyu, presented two works—the Sacred Longevity Perpetual Calendar and the Concordance of Pitch Standards and Calendar. His memorial in brief said, "When the Founding Emperor seized the mandate, the Yuan calendar had not long been in use; qi and new moon had not yet drifted—so he did not undertake a new creation but only revised and polished. As the years piled up, qi and new moon gradually fell away. The Treatise of Later Han says, 'Every three hundred years the Dipper Calendar must revise the constitution.' Let Wanli serve as the origin: the xinsi year of the ninth reign-year falls exactly in the season when 'the Dipper Calendar revises the constitution,' and it accords with the principle that 'the Qian Origin employs nine'—the calendar origin lies here. Your servant has compared the Grand Concordance and Season Granting calendars: tested against antiquity, the qi differ by three days; projected to the present, the time differs by nine quarters. Though the discrepancy is but nine quarters, at the midnight turning it spans a full day. If the solar terms slip one day, intercalation slips one month. If intercalation slips one month, the seasons slip one quarter. If the seasons slip one quarter, the year slips one year. Can such error be called small? This is because the Season Granting pared its fractions too sharply, erring on the side of prematurity; the Grand Concordance did not pare them, erring on the side of lateness. Harmonizing the two schools and taking the middle number for a new ratio, I have compiled a book whose main lines follow Xu Heng, yet diverge from the received calendar. The yellow bell is the root of pitch-pipes and calendar alike, yet the old calendars seldom treat it. The new method places the stepping of pitch-pipes and hexagram images foremost. At Yao's era the winter-solstice solar lodge: He Chengtian placed it some ten degrees into Xu and Nu; Yi Xing between Nu and Xu; the Yuan calendar deliberations likewise at the juncture of Nu and Xu. Yet the Season Granting, tested against the record, places it at two degrees in the Ox. The Grand Concordance, tested likewise, places it at one degree in the Rooftop. They differ by twenty-six degrees—none accord with the Canon of Yao. The new method, projected to Yao's first year jiachen: at summer solstice, sun south at noon near twelve degrees in the Willow; at winter solstice, sun south at noon near ten degrees in the Woman; Heart and Mao culminating at dusk—each no more than half a lodge from the noon meridian—agreeing with He Chengtian and Yi Xing. These are the chief points on which it departs from the old calendar; the rest are set out in full in the Calendar Deliberations. I beg that the throne command great ministers and eminent scholars to collate and adopt it."

其法首曰步發斂。取嘉靖甲寅歲為曆元，元紀四千五百六十，期實千四百六十一，節氣歲差一秒七十五忽，歲周氣策無定率，各隨歲差求而用之。律應即氣應五十五日六十刻八十九分，律總旬週六十日。次曰步朔閏。朔望弦策與《授時》同，閏應十九日三十六刻十九分。次曰步日躔。日平行一度，躔周即天週三百六十五度二十五分，躔中半之，象策又半之，辰策十二分躔周之一。黃、赤道歲差，盈初縮末限，縮初盈末限，俱與《授時》同，周應二百三十八度二十二分三十九秒。按《授時》求日度法，以周應加積度，命起虛七，其周應為自虛七度至箕寸十度之數。《萬年曆法》以周應減積度，命起角初，其周應為箕十度至角初度之數，當為二百八十六度四十五分。今數不合，似誤。次曰步晷漏。北極出地度分，冬、夏至中晷恆數，並二至晝夜長短刻數，俱以京師為準。參以嶽臺，以見隨處裡差之數。次曰步月離。月平行、轉周、轉中，與《授時》同。離周即遲疾限三百三十六限十六分六十秒，離中半之，離象又半之。轉差一日九十刻六十分。轉應七日五十刻三十四分。次曰步交道。正交、中交與《授時》同。距交十四度六十六分六十六秒。交周、交中、交差，與《授時》同。交應二十日四十七刻三十四分。次曰步交食。日食交外限六度，定法六十一，交內限八度定法八十一。月食限定法與《授時》同。次曰步五緯。合應：土星二百六十二日三千二十六分，木星三百一十一千八百三十七分，火星三百四十三日五千一百七十六分，金星二百三十八千三百四十七分，水星九十一日七千六百二十八分。曆應：土星八千六百四日五千三百三十八分，木星四千一十八日六千七十三分，火星三百一十四日四十九分，金星六十日一千九百七十五分，水星二百五十三日七千四百九十七分。周率、度率及晨夕伏見度，俱與《授時》同。

Its method opens with a section called Stepping Accumulation and Contraction. It takes the jiayin year of Jiajing as calendar origin, with era count 4560, period solid 1461, and solar-term precession of one second and seventy-five huz; the year's circuit and qi policy have no fixed ratio, each being sought according to the annual difference. The pitch-pipes' response is the qi's response: fifty-five days, sixty quarters, and eighty-nine parts; the pitch-pipes' total cycle equals the ten-day week of sixty days. Next: Stepping New Moon and Intercalation. The policies for new moon, full moon, and quarters match the Season Granting; the intercalation response is nineteen days, thirty-six quarters, and nineteen parts. Next: Stepping Solar Lodge Motion. The sun's daily parallel is one degree; the lodge circuit is the celestial circuit of 365.25 degrees; the lodge center is half of this, the image policy half again, and the chen policy one-twelfth of the lodge circuit. For the yellow and red paths' annual difference—the fullness-at-start, contraction-at-end limit and the contraction-at-start, fullness-at-end limit—all match the Season Granting; the cycle response is 238 degrees, 22 minutes, and 39 seconds. By the Season Granting's method for solar degree: add the cycle response to accumulated degrees and command from seven degrees in the Void; its cycle response is the count from seven degrees in the Void to ten degrees in the Winnowing Basket. The Perpetual Calendar method subtracts the cycle response from accumulated degrees and commands from the start of the Horn; its cycle response is the count from ten degrees in the Winnowing Basket to the start of the Horn—which should be 286 degrees and 45 minutes. The figures now given do not agree—an apparent error. Next: Stepping Gnomon and Clepsydra. The north pole's altitude in degrees, the constant noon gnomon lengths at winter and summer solstice, and the day-and-night lengths in quarters at both solstices—all take the capital as standard. These are checked against the Yue Terrace to show the li-difference for each locality. Next: Stepping Lunar Distance. The moon's daily parallel, rotation circuit, and rotation center match the Season Granting. The distance circuit is the slow-and-fast limit of 336 limits, 16 parts, and 60 seconds; the distance center halves it, and the distance image halves again. The rotation difference is one day, ninety quarters, and sixty parts. The rotation response is seven days, fifty quarters, and thirty-four parts. Next: Stepping Crossing Paths. The true cross and central cross match the Season Granting. The distance from the crossing is 14 degrees, 66 minutes, and 66 seconds. The crossing circuit, crossing center, and crossing difference match the Season Granting. The crossing response is twenty days, forty-seven quarters, and thirty-four parts. Next: Stepping Eclipse Crossing. For solar eclipse: outside the crossing, the limit is six degrees and the fixed method sixty-one; inside the crossing, the limit is eight degrees and the fixed method eighty-one. For lunar eclipse, the limits and fixed methods match the Season Granting. Next: Stepping the Five Luminaries. Conjunction responses: Saturn 262 days and 3026 parts; Jupiter 311 days and 1837 parts; Mars 343 days and 5176 parts; Venus 238 days and 8347 parts; Mercury 91 days and 7628 parts. Calendar responses: Saturn 8604 days and 5338 parts; Jupiter 4018 days and 6073 parts; Mars 314 days and 49 parts; Venus 60 days and 1975 parts; Mercury 253 days and 7497 parts. Cycle rates, degree rates, and the degrees of morning and evening visibility—all match the Season Granting.

其議歲餘也，曰：「陰陽消長之理，以漸而積，未有不從秒起。《授時》考古，於百年之際頓加一分，於理未安。假如魯隱公三年酉歲，下距至元辛巳二千年，以《授時》本法算之，於歲實當加二十分，得庚午日六刻，為其年天正冬至。次年壬戌歲，下距至元辛巳一千九百九十年，本法當加十九分，得乙亥日五十刻四十四分，為其年天正冬至。兩冬至相減，得相距三百六十五日四十四刻四十四分，則是歲餘九分日之四，非四分日之一也。曆法之廖，莫甚於此。新法酌量，設若每年增損二秒，推而上之，則失昭公己丑；增損一秒至一秒半，則失僖公辛亥。今約取中數，其法置定距自相乘，七因八歸，所得百，約之為分，得一秒七十五忽，則辛亥、己丑皆得矣。」

Its deliberation on the year remainder runs: "The principle of yin and yang's waxing and waning accumulates by fine steps—none that does not begin from seconds. The Season Granting, in examining antiquity, suddenly adds one part at the century mark—sound in form, unsound in reason. Suppose the third year of Duke Yin, a you year, two thousand years down to the Yuan xinsi: by the Season Granting's original method the year solid should gain twenty parts, yielding the gengwu day at six quarters as that year's celestial winter solstice. The next year, renxu, nineteen hundred and ninety years down to xinsi: the original method should add nineteen parts, yielding the yihai day at fifty quarters and forty-four parts as the celestial winter solstice. Subtracting the two winter solstices leaves an interval of 365 days, forty-four quarters, and forty-four parts—so the year remainder is nine-fourths of a day, not one-fourth. No flaw in calendar reckoning runs deeper than this. The new method weighs the matter thus: if each year gains or loses two seconds, projecting upward misses Duke Zhao's jichou; if it gains or loses one to one-and-a-half seconds, Duke Xi's xinhai is missed. Taking the middle figure, the method sets the fixed interval multiplied by itself, applies the seven-eights reduction, obtains one hundred, and reduces to parts to yield one second and seventy-five huz—then both xinhai and jichou are secured."

其議日躔也，曰：「古曆見於《六經》，灼然可考者莫如日躔及中星。而推步家鮮有達者，蓋由不知夏時、周正之異也。大抵夏曆以節氣為主，周曆以中氣為主。何承天以正月甲子夜半合朔雨水為上元，進乖夏朔，退非周正。故近代推《月令》、《小正》者、皆不與古合。嘗以新法歲差，上考《堯典》中星，則所謂四仲月，蓋自節氣之始至於中氣之終，三十日內之中星耳後世執者於二分二至，是亦誤矣。」

Its deliberation on solar lodge motion says, "Among ancient calendars attested in the Six Classics, none are more clearly verifiable than solar lodge motion and culminating stars. Yet few computists have mastered them—chiefly because they do not grasp the difference between Xia months and Zhou correct months. Broadly speaking, the Xia calendar takes solar terms as primary; the Zhou takes the major qi as primary. He Chengtian took the first month, jiazi, midnight conjunction at Rain Water as upper origin—too far forward for the Xia new moon, too far back for Zhou correctness. Hence the recent reconstructions of the Monthly Ordinance and the Lesser Correct Month all fail to agree with antiquity. When I apply the new method's precession and look up to the culminating stars of the Canon of Yao, the so-called four mid-season months are simply the stars culminating at midnight within the thirty days from a solar term's opening to a major qi's close; later ages that fix rigidly on the two equinoxes and two solstices err likewise."

其議侯極也，曰：「自漢至齊、梁，皆謂紐星即不動處。惟祖恆之測知紐星去極一度有餘。自唐至宋，又測紐星去極三度有餘。《元志》從三度，蓋未有說也。新法不測紐星，以日景驗之，於正方案上，周天度內權指一度為北極，自此度右旋，數至六十七度四十一分，為夏至日躔所在。復至一百一十五度二十一分，為冬至日躔所在。左旋，數亦如之。四處並中心五處，各識一針。於二至日午中，將案直立向南取景，使三針景合，然後縣繩界取中綿，又取方十字界之，視橫界上距極出地度分也，即極出地度分也。」

Its deliberation on polar culminating stars says, "From Han through Qi and Liang, all held that the pivot star was the fixed point. Only Zu Chongzhi's measurements showed the pivot star standing more than one degree from the pole. From Tang through Song, measurement again found the pivot star more than three degrees from the pole. The Yuan History adopts a figure of three degrees, for which there is likely no adequate explanation. The new method does not measure the pivot star directly; it checks by the sun's shadow. On the upright azimuth plate, within the full circuit of degrees one provisionally designates one degree as the pole, then counts rightward to 67°41′ for the sun's position at the summer solstice. Counting onward to 115°21′ gives the sun's position at the winter solstice. Rotating leftward, one counts in the same way. At the four corners and the center—five points in all—each is marked with a needle. On the solstice days at noon, stand the plate upright facing south to catch the shadow, align the three needle-shadows, then suspend cords to find the center line and mark a square cross; read on the horizontal line the degrees and minutes of polar altitude above the horizon—that is the polar altitude.

其議晷景也，曰：「何承天立表測景，始知自漢以來，冬至皆後天三日。然則推步晷景，乃治曆之耍也。《授時曆》亦憑晷景為本，而《曆經》不載推步晷景之術，是為缺略，今用北極出地度數，兼弧矢二術以求之，庶盡其原。又隨地形高下，立差以盡變，前此所未有也。」又曰：「《授時曆》議據《前漢志》魯獻公十五年戊寅歲正月甲寅朔旦冬至，引用為首。夫獻公十五年下距隱西元年己未，歲百六十一年，其非春秋時明矣。而《元志》乃云『自春秋獻公以來』，又云『昭公冬至，乃日度失行之驗』，誤矣。夫獻公甲寅冬至，別無所據，惟劉歆《三統曆》言之。豈左傳不足信，而歆乃可信乎？太初元年冬至在辛酉，歆乃以為甲子，差天三日，尚不能知，而能逆知上下數百年乎？故凡春秋前後千載之間，氣朔交食，《長曆》、《大衍》所推近是，劉歆、班固所說全非也。」又曰：「《大衍曆》議謂宋元嘉十三年一月甲戌，景長為日度變行，《授時曆》議亦云，竊以為過矣。苟日度失行，當如歲差，漸漸而移。今歲既已不合，來歲豈能復合耶？蓋前人所測，或未密耳。夫冬至之景一丈有餘，表高晷長，則景虛而淡，或設望筒、副表、景符之類以求實景。然望筒或一低昂，副表、景符或一前卻，所據之表或稍有傾欹，圭面或稍有斜側，二至前後數日之景，進退只在毫釐之間，耍亦難辨。況委託之人，未智當否。九服之遠，既非自摯，所報晷景，寧足信乎？」

In his discussion of gnomon shadows he says: "He Chengtian set up a gnomon to measure shadows and first discovered that since Han times the winter solstice had consistently fallen three days behind Heaven. Computing and advancing gnomon shadows is therefore the essential task of calendar-making. The Shoushi calendar likewise rests on gnomon shadows, yet the Calendar Classic does not record how to compute them—a serious omission. The new method uses polar altitude together with arc and versed-sine techniques to derive them fully. It also sets corrections according to local elevation, accounting for every variation—something no earlier method had done." He also writes: "The Shoushi calendar's discussion takes as its starting point the Former Han Treatise entry for Duke Xian of Lu's fifteenth year (wuyin): new moon on jiayin at dawn on the winter solstice. Duke Xian's fifteenth year is 161 years before the first year of Duke Yin (jiwei) in the western chronology—clearly not the Spring and Autumn era at all. Yet the Yuan History says 'since Duke Xian of the Spring and Autumn period' and also that 'Duke Zhao's winter solstice proves the sun's motion had gone astray'—both are mistaken. The jiayin winter solstice of Duke Xian's year rests on no other authority than Liu Xin's Santong calendar. Is the Zuo Commentary untrustworthy while Liu Xin alone is to be believed? In the first year of the Taichu era the winter solstice fell in xinyou, yet Liu Xin placed it in jiazi—three days off Heaven. If he could not get that right, how could he know events centuries before or after? For the millennium surrounding the Spring and Autumn period, solar terms, new moons, and eclipses computed in the Chang calendar and Dayan system are roughly correct; what Liu Xin and Ban Gu asserted is wholly wrong." He also writes: "The Dayan calendar's discussion claims that in Yuanjia 13, on the first month, day jiaxu, an unusually long shadow proved the sun's motion had shifted; the Shoushi discussion repeats this—I believe both go too far. If the sun's motion truly went astray, it ought to shift gradually, as with precession. If it fails to match this year, how could it match again the next? Presumably the earlier measurements were simply not precise enough. At winter solstice the shadow exceeds one zhang; with a tall gnomon the shadow grows long, faint, and diffuse, so observers use sighting tubes, auxiliary tables, shadow markers, and similar devices to capture the true shadow. Yet a sighting tube may sit slightly high or low; auxiliary tables and markers may be placed slightly forward or back; the gnomon itself may lean; the measuring surface may tilt. Shadows a few days before and after the solstices differ by only hairs and millimeters—the decisive point is nearly impossible to tell apart. Moreover, the officials entrusted with the work may not even know whether their readings are right. Across the nine domains, since no one measured in person everywhere, how can reported gnomon shadows be fully trusted?

其議漏刻也，曰：「日月帶食出入，五星晨昏伏見，曆家設法悉因晷漏為準。而晷漏則隨地勢南北，辰極高下為異焉。元人都燕，其《授時曆》七曜出沒之早晏，四時晝夜之永短，皆準大都晷漏。國初都金陵，《大統曆》晷漏改徒南京，冬夏至相差三刻有奇。今推交食分秒，南北東西等差及五星定伏定見，皆因元人舊法，而獨改其漏刻，是以互神舛誤也。故新法晷漏，照依元舊。」

In his discussion of clepsydra gradations he says: "Whether the sun and moon rise or set near an eclipse, or the five planets appear, hide, or culminate at dawn or dusk—calendar-makers always base their methods on gnomon shadows and water-clock readings. Yet gnomon and clepsydra readings vary with latitude and with the altitude of the celestial pole. The Yuan capital was at Yan; the Shoushi calendar's times of rising and setting for the seven luminaries and the lengths of day and night through the four seasons all followed the gnomon and clepsydra of Dadu. When the dynasty was founded the capital was Jinling; the Datong calendar shifted the gnomon and clepsydra standard to Nanjing, so winter and summer solstice times differ by more than three ke. Today eclipse minutes, north-south and east-west corrections, and the fixed times of planetary appearance and hiding all still follow Yuan methods—yet the clepsydra gradations alone were changed, producing inconsistent errors throughout. The new method therefore restores gnomon and clepsydra standards to the Yuan originals.

其議日食也，曰：「日道與月道相交處有二，若正會於交，則食既，若但在交前後相近者，則食而不既。此天之交限也。又有人之交限，假令中國食既，戴日之下，所虧才半，化外之地，則交而不食。易地反觀，亦如之。何則？日如大赤丸，月如小黑丸，共縣一綿，日上而下，即其下正望之，黑丸必掩赤丸，似食之既；及旁觀有遠近之差，則食數有多寡矣。春分已後，日行赤道北畔，交外偏多，交內偏少。秋分已後，日行赤道南畔，交外偏少，交內偏多。是故有南北差。冬至已後，日行黃道東畔，午前偏多，午後偏少。夏至已後，日行黃道西畔，午前偏少，午後偏多。是故有東西差。日中仰視則高，旦暮平視則低。是有距午差。食於中前見早，食於中後見遲。是故有時差，凡此諸差，唯日有之，月則無也。故推交食，惟日頗難。欲推九服之變，必各據其處，考晷景之短長，揆辰極之高下，庶幾得之。《曆經》推定之數，徒以燕都所見者言之耳。舊云：『月行內道，食多有驗。月行外道，食多不驗。』又云：『天之交限，雖系內道，若在人之交限之外，類同外道，日亦不食。』此說似矣，而未盡也。假若夏到前後，日食於寅卯酉戌之間，人向東北、西北觀之，則外道食分反多於內道矣。日體大於月，月不能盡掩之，或遇食既，而日光四溢，形如金環，故日無食十分之理。雖既，亦止九分八十秒。《授時曆》日食，陽曆限六度，定法六十，陰曆限八度，定法八十。各置其限度，如其定法而一，皆得十分。今於其定法下，各加一數以除限度，則得九分八十餘秒也。」

In his discussion of solar eclipses he says: "The sun's path and the moon's path cross at two nodes. If they meet exactly at the node, the eclipse is total; if they only approach the node, there is a partial eclipse but not totality. This is the celestial eclipse limit. There is also a terrestrial eclipse limit: suppose the eclipse is total in China, yet beneath the subsolar point only half the disk is obscured, while beyond the frontier the conjunction occurs but no eclipse is seen. Change vantage point and the same principle applies in reverse. Why is this so? Imagine the sun as a large red sphere and the moon as a small black sphere on the same string. Viewed straight up from below when the sun is overhead, the black sphere fully covers the red one, like a total eclipse; but viewed obliquely from a distance, the eclipsed fraction varies. After the spring equinox the sun runs along the northern side of the equator: eclipses outside the node predominate, inside the node are fewer. After the autumn equinox it runs along the southern side: outside the node are fewer, inside more. Hence the north-south correction arises. After the winter solstice the sun travels the eastern side of the ecliptic: eclipses before noon are more frequent, after noon fewer. After the summer solstice it travels the western side: before noon fewer, after noon more. Hence the east-west correction arises. At noon one looks upward and the path is high; at dawn and dusk one looks level and it appears low. Hence the correction for distance from noon. An eclipse before noon is seen early; after noon, late. Hence the time correction. All these corrections apply only to the sun; the moon has none. Computing eclipses is therefore especially difficult for the sun. To calculate variations across the nine domains, one must work from each locality, measuring gnomon shadows and determining polar altitude—only then can one approximate the truth. The fixed figures in the Calendar Classic describe only what was observed at the Yan capital. An old rule says: "When the moon travels the inner path, eclipses usually prove accurate. When it travels the outer path, eclipses usually fail to match. Another old rule says: "Even when the celestial limit lies on the inner path, if it falls outside the terrestrial limit—like the outer path—the sun may not eclipse at all." This view sounds plausible but does not go far enough. Suppose an eclipse around the summer solstice falls between the yin, mao, you, and xu hours; an observer facing northeast or northwest may see a greater obscuration on the outer path than on the inner. The sun is larger than the moon and cannot be fully covered; even at totality light may spill around the rim in a golden ring—so a solar eclipse can never reach ten-tenths obscuration. Even at totality the maximum is nine parts and eighty seconds. In the Shoushi calendar for solar eclipses, the yang limit is six degrees with a fixed divisor of sixty, and the yin limit eight degrees with a fixed divisor of eighty. Each applied its limit and divisor to unity, yielding ten-tenths in every case. The new method adds one to each divisor before dividing the limit, producing nine parts and eighty-odd seconds.

其議月食也，曰：「暗虛者，景也。景之蔽月，無早晚高卑之異，四時九服其之殊。譬如縣一黑丸於暗室，其左燃燭，其右縣一白丸，若燭光為黑丸所蔽，則白丸不受其光矣。人在四旁觀之，所見無不同也。故月食無時差之說。自《紀元曆》妄立時差，《授時》因之，誤矣。」

In his discussion of lunar eclipses he says: "The dark void is simply the earth's shadow. When shadow covers the moon there is no difference of morning or evening, height or depth, season or region. It is like hanging a black sphere in a dark room with a candle on the left and a white sphere on the right: if the black sphere blocks the candle, the white sphere receives no light. Observers on any side see the same thing. Lunar eclipses therefore admit no time correction. The Jiyuan calendar wrongly introduced a time correction, which the Shoushi calendar adopted in turn—both were mistaken.

其議五緯也，曰：「古法推步五緯，不如變數之加減。北齊張子信仰觀歲久，知五緯有盈縮之變，當加減以求逐日之躔。蓋五緯出入黃道內外，各自有其道，視日遠近為遲疾，其變數之加減，如裡路之徑直斜曲也。宋人有言曰：『五星行度，惟留退之際最多差。自內而進者，其退必向外，自外而進者，其退必由內。其跡臺循柳葉，兩末銳於中間，往還之道相去甚遠。故星行兩末度稍遲，以其斜行故也。中間行度稍速，以其徑捷故也。』前代修曆，止增損舊法而已，未嘗實考天度。其法須測驗每夜昏曉夜半，月及五星所在度秒，置簿錄之。滿五年，其間去陰雲晝見日數外，可行三年實行，然後可以算術綴之也。」

In his discussion of the five planets he says: "Ancient methods for advancing the five planets are inferior to variable increment and decrement. Zhang Zixin of Northern Qi, observing over many years, saw that the five planets undergo accelerating and slowing variation and must be corrected day by day to track their true positions. The five planets move inside and outside the ecliptic, each on its own path, appearing faster or slower as they approach or recede from the sun; their corrective increments resemble the difference between a straight inner road and a winding outer one. A Song scholar wrote: "Among the five planets' motions, the greatest discrepancies occur at stations and retrogradations. A planet advancing from inside the path must retreat outward; one advancing from outside must retreat inward. Its track loops like a willow leaf—pointed at both ends, broad in the middle—with the outbound and return paths far apart. Hence at both ends the planet moves somewhat slowly because the path is oblique. In the middle it moves somewhat faster because the path is direct and short. Earlier dynasties, when revising calendars, only adjusted old formulas and never truly measured celestial positions. The proper method is to observe each night at dusk, dawn, and midnight the degrees and seconds of the moon and five planets and record them in a ledger. After five years—excluding cloudy nights and daytime observations—three years of reliable data will suffice to fit them by calculation.

書上，禮部尚書范謙奏：「歲差之法，自虞喜以來，代有差法之議，竟無晝一之規。所以求之者，大約有三：考月令之中星，測二至之日景，驗交食之分秒。考以衡管，測以臬表，驗以漏刻，斯亦危得之矣。曆家以周天三百六十五度四分度之一，紀七政之行，又析度為百分，分為百秒，可謂密矣。然渾象之體，徑僅數尺，布周天度，每度不及指許，安所置分秒哉？至於臬表之樹不過數尺，刻漏之籌不越數寸。以天之高且廣也，而以寸之物求之，欲其纖微不爽，不亦難乎？故方其差在公秒之間，無可驗者，至蹬逾一度，乃可以管窺耳。此所以窮古今之智七巧，不能盡其變歟？即如世子言，以《大統》、《授時》二曆相較，考古則氣差三日，推今則時差必刻。夫時差九刻，在亥子之間則移一日，在晦朔之交則移一月，此可驗之於近也。設移而前，則生明在二日之昏，設移而後，則生明在四日之夕矣。今似未至此也。其書應發欽天監參訂測驗。世子留心曆學，博通今古，宜賜獎諭。」從之。

When the work was submitted, Fan Qian, Minister of Rites, memorialized: "The method of precession has been debated since Yu Xi, yet no generation has settled on a single uniform rule. Approaches to determining it fall roughly into three: checking the meridian stars in the Monthly Ordinances, measuring solstice shadows, and verifying eclipse times to the second. One examines with pitch-pipes and sighting tubes, measures with gnomons, and verifies with clepsydras—none of which yields secure results easily. Calendar-makers reckon the full circuit at 365¼ degrees for the seven luminaries, subdividing each degree into a hundred parts and each part into a hundred seconds—precision on paper, at least. Yet an armillary sphere is only a few feet across; spread the full circuit upon it and each degree is narrower than a finger—where can seconds be marked? Gnomons stand no taller than a few feet; clepsydra rods measure no more than a few inches. Heaven is immeasurably high and wide, yet we seek it with inch-scale instruments—can hair-fine accuracy be anything but elusive? When the error lies within fractions of an arcsecond it cannot be confirmed; only when it exceeds a full degree can it be glimpsed through a sighting tube. Is this why all the ingenuity ever brought to the problem still cannot exhaust its variations? As the Prince of Zheng argues, comparing the Datong and Shoushi calendars: looking backward, solar terms differ by three days; projecting forward, the time correction would have to be several ke. A nine-ke time shift would move the calendar one day if it fell between hai and zi, or one month if it crossed a new-moon boundary—effects one could verify immediately. Shifted forward, the first crescent would appear on the evening of the second day; shifted back, on the evening of the fourth. At present we do not seem to have reached that stage. His book should be forwarded to the Directorate of Astronomy for collation and empirical testing. The Prince is deeply versed in calendrical learning and broadly learned in past and present methods; he should receive an imperial commendation. The emperor approved.

河南僉事刑雲路上書言：「治曆之耍治曆之耍，無逾觀象、測景、候時、籌策四事。今丙申年日至，臣測得乙未日未正一刻，而《大統》推在申正二刻，相差九刻。且今年立春、夏至、立冬皆適直子半之交。臣推立春乙亥，而《大統》推丙子；夏至壬辰，而《大統》推癸巳；立冬巳酉，而《大統》推庚戌。相隔皆一日。若或直元日於子半，則當退履端於月窮，而朝賀大禮在月正二日矣。豈細故耶？閏八月朔，日食，《大統》推初虧巳正二刻，食幾既，而臣候初虧巳正一刻，食止七分餘。《大統》實後天幾二刻，則閏應及轉應、交應，各宜增損之矣。」欽天監見雲路疏，甚惡之。監正張應候奏詆，謂其僭妄惑世。禮部尚書范謙乃言：「曆為國家大事，士夫所當講求，非曆士之所得私。律例所禁，乃妄言妖祥者耳。監官拘守成法，不能修改合天。幸有其人，所當和衷共事，不宜妒忌。乞以雲路提叔欽天監事，督率官屬，精心測候，以成鉅典。」議上，不報。

Xing Yunlu, Vice Commissioner of the Henan Branch, memorialized: "In calendar-making, nothing surpasses four tasks: observing phenomena, measuring shadows, awaiting the right moment, and computing with counting rods. At this year's solstice (bingchen), I measured the moment on yiwei day at the first ke of wei zheng, whereas the Datong calendar places it at the second ke of shen zheng—a discrepancy of nine ke. Moreover, this year's beginning of spring, summer solstice, and beginning of winter all fall precisely at the midnight jiaozi boundary. I compute the beginning of spring as yi hai; the Datong calendar gives bing zi; summer solstice as ren chen; the Datong gives gui si; beginning of winter as si you; the Datong gives geng xu. In each case the discrepancy was a full day. If New Year's Day were placed exactly at the midnight boundary, the year's beginning would have to be pushed back to month's end and the great New Year audience would fall on the second day of the first month. Could that be dismissed as a trifling matter? On the new moon of the intercalary eighth month there was a solar eclipse. The Grand Concordance predicted first contact in the second quarter of the si double-hour, with the eclipse nearly total; yet when I observed, first contact came in the first quarter of si and the eclipse reached only a little over seven-tenths. The Grand Concordance was in fact nearly two quarters behind heaven; the intercalation, rotation, and nodal correspondences should each be adjusted accordingly. When the Directorate of Astronomy saw Yunlu's memorial, it detested it deeply. Director Zhang Yinghou memorialized in denunciation, calling him presumptuous, reckless, and a deceiver of the world. Minister of Rites Fan Qian then said, "The calendar is a great affair of state, which scholars ought to pursue; it is not something calendar officers may monopolize. What the statutes forbid is only reckless talk of omens and prodigies. The observatory officers cling to established methods and cannot revise them to match heaven. Fortunately there is such a man; they ought to work together in good faith and not be jealous. I beg that Yunlu be put in charge of Directorate of Astronomy affairs, to lead the officials in careful observation and complete the great enterprise. The proposal went up but received no response.

三十八年，監推十一月壬寅朔日食分秒及虧圓之候，職方郎範守己疏駁其誤。禮官因請博求知曆學者，令與監官晝夜推測，庶幾曆法靡差。於是五官正周子愚言：「大西洋歸化遠臣龐迪峨、熊三撥等，攜有彼國曆法，多中國典籍所未備者。乞視洪中譯西域曆法例，取知曆儒臣率同監官，將諸書盡譯，以補典籍之缺。」先是，大西洋人利瑪竇進貢土物，而迪峨、三撥及能華同、鄧玉函、湯若望等先後至，俱精究天文曆法。禮部因奏：「精通曆法，如雲路、守己為時所推，請改授京卿，共理曆事。翰林院檢討徐光啟、南京工部員外郎李之藻亦皆精心曆理，可與迪峨、三撥等同譯西洋法，俾雲路等參訂修改。然曆法疏密，莫顯於交食，欲議修曆，必重測驗。乞敕所司修治儀器，以便從事。」疏入，留中。未幾雲路、之藻皆召至京，參預曆事。雲路據其所學，之藻則以西法為宗。

In year thirty-eight the Directorate predicted the magnitude in degrees and seconds and the phases of obscuration and fullness for the solar eclipse on the renyin new moon of the eleventh month; Bureau of Appointments director Fan Shouji memorialized to refute its errors. The ritual officials therefore asked to seek broadly those versed in calendrical learning and have them compute and observe day and night with the observatory officers, so that the calendar might be without error. Thereupon Five Offices Director Zhou Ziyu said, "The distant subjects from the Great Western Ocean who have submitted, Pang Di'e, Xiong Sanbo, and others, have brought their country's calendar methods, many of which are not found in Chinese classics. I beg that, following the precedent of Hong Zhong's translation of Western Region calendar methods, calendar-learned scholar-officials be taken to lead the observatory officers in translating all these books fully, to supply what the classics lack. Earlier the man of the Great Western Ocean Matteo Ricci had presented native products as tribute, and Di'e, Sanbo, Neng Huatong, Deng Yuhan, Adam Schall, and others had arrived in succession, all deeply versed in astronomy and the calendar. The Ministry of Rites therefore memorialized, "Those expert in the calendar, such as Yunlu and Shouji, are acclaimed by the age; I beg that they be transferred to capital ministerial posts to administer calendar affairs jointly. Hanlin Academician Xu Guangqi and Nanjing Works Bureau vice director Li Zhizao are also both deeply versed in calendrical principles and may translate the Western methods together with Di'e and Sanbo, so that Yunlu and others may consult, revise, and amend. Yet whether a calendar is coarse or fine is nowhere clearer than in eclipses and planetary encroachments; if one wishes to discuss revising the calendar, testing must be heavily emphasized. I beg that the responsible offices be ordered to repair and prepare instruments for convenient use in the work. The memorial entered but was held at court. Before long Yunlu and Zhizao were both summoned to the capital to take part in calendar affairs. Yunlu relied on his own learning; Zhizao took Western methods as his school.

四十一年，之藻已改銜南京太僕少卿，奏上西洋曆法，略言臺監推算日月交食時刻虧分之謬。而力薦迪峨、三撥及華民、陽瑪諾等，言：「其所論天文曆數，有中國昔賢所未及者，不徒論其數，又能明其所以然之理。其所制窺天、窺日之器，種種精絕。今迪峨等年齡向衰，乞敕禮部開局，取其曆法，譯出成書。」禮科姚永濟亦以為言。時庶務因循，未暇開局也。

In the forty-first year Zhizao, having changed his title to Nanjing Court of the Imperial Stud vice minister, memorialized presenting Western calendar methods, briefly stating the errors of the observatory and directorate in calculating the times, degrees of obscuration, and magnitudes of solar and lunar eclipses. He strongly recommended Di'e, Sanbo, Alphonse Vagnoni, and others, saying, "What they discuss in astronomy and calendrical numbers reaches what China's ancient worthies never attained; they do not merely discuss the numbers but can also clarify the reasons why they are so. The instruments they have made for viewing heaven and viewing the sun are in every kind exquisitely perfected. Now Di'e and the others are growing old in years; I beg that the Ministry of Rites be ordered to open a bureau, take their calendar methods, and translate them into completed books. The Ministry of Rites censor Yao Yongji also spoke to the same effect. At the time routine affairs dragged on and there was no leisure to open a bureau.

四十四年，雲路獻《七政真數》，言：「步曆之法，必以兩交相對。兩交正，而中間時刻分秒之度數，一一可按。日月之交食，五星之淩犯，皆日月五星之相交也。兩交相對，互相發明，七政之能事畢矣。」天啟元年春，雲路復詳述古今時刻，與欽天監所推互異。症新法至密，章下禮部。四月壬申朔日食，雲路所推食分時刻，與欽天監所推互異。自言新法至密，至期考驗，皆與天下不合。雲路又嘗論《大統》宮度交界，當以歲差考定，不當仍用《授時》三百年前所測之數。又月建月關半杓所指，鬥杓有歲差，而月建無改移。皆篤論也。

In the forty-fourth year Yunlu presented the True Numbers of the Seven Regulators, saying, "The method of pacing the calendar must take the two nodes in mutual opposition. When the two nodes are correct, the degrees of the intermediate times, moments, and seconds can each be checked off one by one. Solar and lunar eclipses and the five planets' encroachments are all intersections of sun, moon, and the five planets. The two nodes in mutual opposition illuminate each other; the full capability of the seven regulators is then complete. In the spring of Tianqi 1, Yunlu again set forth ancient and modern times in detail, which differed from what the Directorate of Astronomy calculated. He proved the new method most precise; the memorial was sent down to the Ministry of Rites. On the renshen new moon of the fourth month there was a solar eclipse; the eclipse magnitude and time Yunlu calculated differed from what the Directorate of Astronomy calculated. He himself said the new method was most precise, but when the day came for testing, none matched what was seen under heaven. Yunlu also once argued that the Grand Concordance's lodge-degree boundaries ought to be fixed by precession and ought not still to use numbers measured three hundred years earlier in the Season Granting Calendar. He also held that the monthly establishment and monthly barrier indicated by half the Dipper's handle show precession in the Dipper's handle but no shift in the monthly establishment—sound doctrines all. All were sound doctrines.

崇禎二年五月乙酉朔日食，禮部侍郎徐光啟依西法預推，順天府見食二分有奇，瓊州食既，大寧以北不食。《大統》、《回回》所推，順天食分時刻，與光啟妻異。已而光啟法驗，餘皆疏。帝切責監官。時五官正戈豐年等言：「《大統》乃國初所定，寮即郭守敬《授時曆》也，二百六十年毫未增損。自至元十八年造曆，越十八年為大德三年八月，已當食不食，六年六月又食而失推。是時守敬方知院事，亦付之無可奈佑，況斤斤守法者哉？今若循舊，向後不能無差。」於是禮部奏開局修改。乃以光啟督修曆法。光啟言：近世言曆諸家，大都宗郭守敬法，至若歲差環轉，歲實參差，天有緯度，地有經度，列宿有本行，月五星有本輪，日月有真會、視會，皆古所未聞，惟西曆有之。而舍此數法，則交食淩犯，終無密合理。宜取其法參互考訂，使與《大統》法會同歸一。」

On the yiyou new moon of the fifth month in Chongzhen 2 there was a solar eclipse. Vice Minister of Rites Xu Guangqi predicted by Western methods in advance: Shuntian Prefecture would see a little over two-tenths obscured, Qiongzhou total eclipse, and north of Daining no eclipse. What the Grand Concordance and Huihui calendars predicted for Shuntian's magnitude and time differed from Guangqi's. Afterward Guangqi's method was verified; the rest were all coarse. The emperor sharply rebuked the observatory officers. At the time Five Offices Director Ge Fengnian and others said, "The Grand Concordance was fixed at the dynasty's founding; it is namely Guo Shoujing's Season Granting Calendar, and for two hundred sixty years not a hair's breadth has been added or subtracted. From the eighteenth year of Zhiyuan when the calendar was made, after eighteen years came the eighth month of the third year of Dade, when an eclipse ought to have occurred but did not; in the sixth month of the sixth year there was an eclipse but it was not predicted. At that time Shoujing had only just learned of the Academy's affairs and likewise had no remedy—how much less those who fuss over guarding the law? If we now follow the old ways, hereafter there cannot fail to be discrepancies. Thereupon the Ministry of Rites memorialized to open a bureau for revision. Guangqi was then put in charge of supervising revision of the calendar law. Guangqi said: Recent schools that discuss the calendar mostly take Guo Shoujing's method as their ancestor; but as for precession's cyclical rotation, the true year's fractional discrepancies, heaven having latitude, earth having longitude, lodges having proper motion, moon and five planets having deferent circles, sun and moon having true conjunction and apparent conjunction—all were unheard of in antiquity and exist only in the Western calendar. Yet if one sets aside these several methods, eclipses and encroachments will in the end have no close-fitting principle. One ought to take their methods, consult and cross-check them, and make them meet and return to one with the Grand Concordance method."

已而光啟上曆法修正十事：其一，議歲差，每歲東行漸長短之數，以正古來百年、五十年、六十年多寡互異之說。其二，議歲實小餘，昔多今少，漸次改易，及日景長短歲歲不同之因，以定冬至，以正氣明朔。其三，每日測驗日行經度，以定盈縮加減真率，東西南北高下之差，以步月離。其四，夜測月行經緯度數，以定交轉遲疾真率，東西北高下之差，以步月離。其五，密測列宿以緯行度，以定七政盈縮、遲疾、順逆、違離、遠近之數。其六，密測五星經緯行度，以定小輪行度遲疾、留逆、伏見之數，東西南北高下之差，以推步淩犯。其七，推變黃道、赤道廣狹度數，密測二道距度，及月五星各道與黃道相距之度，以定交轉。其八，議日月去交遠近及真會、視會之因，以定距午時差之真率，以正交食。其九，測日行，考知二極出入地度數，以定周天緯度，以齊七政。因月食考知東西相距地輪經度，以定交食時刻。其十，依唐、元法，隨地測驗二極出入地度數，地輪經緯，以求晝夜晨昏永短，以正交食有無、先後、多寡之數。因舉南京太僕少卿李之藻、西洋人能華民、鄧玉涵。報可。九月癸卯開曆局。三年，玉函卒，又徵西洋人湯若望、羅雅穀譯書演算。光啟進本部尚書，仍督修曆法。

Shortly afterward Guangqi submitted ten items for calendar revision: first, discuss precession—the yearly eastward motion and the numbers of gradual lengthening and shortening—to correct the ancient divergent theories of one hundred, fifty, and sixty years. Second, discuss the true year's small remainder—more in the past, less now, gradually changing, and the reason the sun's shadow length differs year by year—to fix the winter solstice and correct the qi and clarify the new moon. Third, test daily the sun's motion in longitude to fix the true rates of addition and subtraction for surplus and deficit, and the differences of east, west, south, north, high, and low, to pace lunar departure. Fourth, at night measure the moon's motion in longitude and latitude to fix the true rates of nodal rotation's slow and fast phases, and the differences of east, west, north, high, and low, to pace lunar departure. Fifth, closely measure the lodges to fix latitude motion in degrees, to fix the seven regulators' numbers of surplus, deficit, slow, fast, direct, retrograde, departure, and approach. Sixth, closely measure the five planets' motion in longitude and latitude to fix the epicycle's degrees of slow and fast motion, station, retrogradation, and hiding and appearance, and the differences of east, west, south, north, high, and low, to compute and pace encroachments. Seventh, deduce changes in the ecliptic's and equator's breadth and narrowness in degrees, closely measure the two paths' separation in degrees and each of the moon and five planets' paths' distance from the ecliptic in degrees, to fix nodal rotation. Eighth, discuss the reasons for sun and moon's distance from the node and for true and apparent conjunction, to fix the true rate of the difference from noon, to correct eclipses. Ninth, measure the sun's motion and examine the degrees by which the two poles enter and leave the earth, to fix the degrees of the circuit of heaven's latitude and align the seven regulators. From lunar eclipses examine the terrestrial longitude separating east and west, to fix eclipse times. Tenth, following Tang and Yuan methods, test locally the degrees by which the two poles enter and leave the earth and the earth's longitude and latitude, to seek the lengths of day, night, dawn, and dusk, and correct the numbers of whether eclipses occur, their priority, lateness, and magnitude. He also recommended Nanjing Court of the Imperial Stud vice minister Li Zhizao and Westerners Alphonse Vagnoni and Deng Yuhan. Approved. In the ninth month on guimao the Calendar Bureau was opened. In the third year Yuhan died; Westerners Adam Schall and Giacomo Rho were again summoned to translate books and perform calculations. Guangqi was promoted to minister of his department and still supervised revision of the calendar law.

時巡按四御史馬如蚊薦資縣諸生冷守中精曆學以所呈曆書送局。光啟力駁其謬，並預推次年四月川食時刻，令其臨時比測。四年正月，光啟進《曆書》二十四卷。夏四月戊午，夜望月食，光啟預推分秒時刻方位。奏言：「日食隨地不同，則用地緯度算其食分多少，用地經度算其加時早晏。月食分秒，海內並同，止用地經度推求先後時刻。臣從輿地圖約略推步，開載各布政司月食初虧度分，蓋食分多少既天下皆同，則餘率可以類推，不若日食之經緯各殊，心須詳備也。又月體一十五分，則盡入暗虛亦十五分止耳。今推二十六分六十六十秒者，蓋暗虛體大於月，若食時去交稍遠，即月體不能全入暗虛，止從月體論其分數。是夕之食，極近於交，故月入暗虛十五分方為食既，更進一十一分有奇，乃得生光，故為二十六分有奇。如《回回曆》推十八分四十七秒，略同此法也。」已四川報次序守中所推月食實差二時，而新法密合。

At the time touring censor of the four circuits Ma Ruwen recommended Zi county scholar Leng Shouzhong, expert in calendrical learning, and sent the calendar book he presented to the bureau. Guangqi forcefully refuted its errors and also predicted in advance the time of the Sichuan eclipse in the fourth month of the following year, ordering him to compare and test on the spot when the time came. In the first month of the fourth year Guangqi presented twenty-four juan of the Calendar Book. In summer, on the night of wuwu in the fourth month, there was a lunar eclipse; Guangqi predicted in advance the degrees, seconds, time, and direction. He memorialized, saying, "Solar eclipses differ by place; therefore use terrestrial latitude to calculate how great the obscuration is, and use terrestrial longitude to calculate whether the additive time is early or late. Lunar eclipse degrees and seconds are the same throughout the realm; only terrestrial longitude is used to seek the prior and posterior times. Your subject roughly computed from the terrestrial map and set out the initial obscuration degrees for each provincial administration commission's lunar eclipse; since the magnitude is the same everywhere under heaven, the remaining rates can be inferred by analogy, unlike solar eclipses where longitude and latitude each differ and all must be set out in detail. Again, the moon's body is fifteen parts; fully entering the earth's shadow is also only fifteen parts. What is now computed as twenty-six parts, six tens, and ten seconds is because the earth's shadow is larger than the moon; if at eclipse the distance from the node is slightly far, the moon's body cannot fully enter the earth's shadow, and one counts only from the moon's body for the fraction. That night's eclipse was extremely close to the node; therefore the moon had to enter the earth's shadow fifteen parts before total eclipse, and advancing a further eleven-odd parts before light reappeared—hence a little over twenty-six parts. As the Huihui Calendar predicts eighteen parts, four tens, and forty-seven seconds, it is roughly the same as this method. Already Sichuan had reported; by comparison Shouzhong's predicted lunar eclipse was actually two hours off, while the new method matched closely.

光啟又進《曆書》二十一卷。冬十月辛丑朔日食，新法預順天見食二分一十二秒，應天以南下食，大漢以北食既，例以京師見食不及三分，不救護。光啟言：

Guangqi again presented twenty-one juan of the Calendar Book. In winter, on the xinchou new moon of the tenth month there was a solar eclipse. The new method predicted in advance that Shuntian would see two parts and twelve seconds obscured; south of Yingtian the eclipse would be below, north of Dahan total eclipse. By precedent, because the capital's eclipse was less than three-tenths, no protective rites were performed. Guangqi said:

月食在夜，加時早晚，若無定據。惟日食按晷定時，無可遷就。故曆法疏密，此為的症。臣等纂輯新法，漸次就緒，而向生交食為期尚遠，此時不與監臣共見，至成曆後，將何徵信？且是食之必當測俟，更有說焉。

Lunar eclipses occur at night; whether the additive time is early or late has no fixed basis. Only solar eclipses fix time by the gnomon and cannot be accommodated. Therefore whether a calendar is coarse or fine—this is the decisive test. Your subjects are compiling the new method, gradually nearing completion, yet the next eclipses are still far off and we will not be with the observatory officers to witness them; when the calendar is finished, what shall we take as proof? Moreover this eclipse must be observed and tested—there is more to say.

舊法食在正中，則無時差。今此食既在日中，而新法仍有時差者，蓋以七政運行皆依黃道，不由赤道。舊法所謂中乃道之午中，非黃道之正中也。黃赤道二道之中，獨冬夏至加時正午，乃得同度。今十月朔去冬至度數尚遠，兩中之差，二下三度有奇，豈可因加時近午，不加不減乎？適際此日，又值此時，足可驗時差之正術，二也。

In the old method, when the eclipse was at the center there was no time difference. Now this eclipse is at midday, yet the new method still has a time difference—because the seven regulators' motion all follows the ecliptic and not the equator. What the old method calls "center" is noon on the path, not true ecliptic center. Only at winter and summer solstice do the centers of the ecliptic and equator coincide at noon and attain the same degree. Now the first day of the tenth month is still far in degrees from the winter solstice; the difference between the two centers is a little over two or three degrees—how can one, because the additive time is near noon, add or subtract nothing? To meet this day and this time is enough to verify the correct method of time difference—second point.

本方之地經度，未得真率，則加時難定，其法心從交食時測驗數次，乃可較勘晝一。今此食依新術測候，其加時刻分，或後未合，當取從前所記地經度分，斟酌改定，此可以求裡差之真率，二也。

This region's terrestrial longitude has not obtained its true rate, so the additive time is hard to fix; the method must rely on testing at several eclipses before one can compare them fully. For this eclipse, if tested by the new method the additive time's degrees and minutes are afterward not in agreement, one should take the terrestrial longitude degrees recorded before and adjust them by deliberation—this can seek the true rate of local difference—second point.

時差一法，但知中無加減，而不知中分黃赤，今一經目見，人人知加時之因黃道，因此推彼，他術皆然，足以知學習之甚易，三也。

The method of time difference only knew that at center there is no addition or subtraction, but did not know that center divides yellow and red; now once seen with the eyes, everyone knows the additive time comes from the ecliptic; extending this to other methods, all are likewise easy to learn—third point.

即分數甚少，宜詳加測候，以求顯驗。帝是其言。至期，光啟率監臣預點日晷，調壺漏，用測高儀器測食甚日晷高度。又於密室中斜開一隙，置窺筒、遠鏡以測虧圓，晝日體分板分數圖板以定食分，其時刻、高度悉合，惟食甚分數未及二分。於是光啟言：「今食甚之度分密合，則經度裡差已無煩更定矣。獨食分未合，原推者蓋因太陽光大，能減月魄，必食及四五分以上，乃得與原推相合，然此測，用密室窺筒，故能得此分數，倘止憑目力，或水盆照映，則眩耀不定，恐少尚不止此也。」

Even when the eclipse magnitude is very small, one should observe and measure it carefully to obtain clear verification. The emperor approved his proposal. When the day arrived, Guangqi led the supervising officials to mark the sundial, adjust the clepsydra, and use altitude instruments to measure the gnomon height at greatest eclipse. In a darkened room he also cut an oblique slit and set up a sighting tube and telescope to measure the crescent of deficiency, using solar fraction boards and charts to fix the eclipse magnitude; the time and altitude all agreed, but the magnitude at greatest eclipse fell short of two-tenths. Guangqi then said, "Now the degrees and minutes at greatest eclipse match closely, so there is no need to adjust terrestrial longitude further. Only the eclipse magnitude does not agree. The original calculator probably assumed that the sun's great light can diminish the moon's disk, so only when the eclipse reached four or five tenths or more would it match his computation—yet this measurement used a sighting tube in a dark room and therefore obtained this fraction; if one relied only on the naked eye or a basin reflection, the glare would be unstable and the shortfall might be even greater."

時有滿城布衣魏文魁，著《曆元》、《曆測》二書，令其子象乾進《曆元》於朝，通政司送局考驗。光啟摘當極論者七事：其一，歲實自漢以來，代有減差，到《授時》減為二十四分二十五秒。依郭法百年消一，今當為二十一秒有奇。而《曆元》用趙知微三十六秒，翻覆驟加。其一，弧背求弦矢，宜用密率。今《曆測》中猶用徑一圍三之法，不合弧矢真數。其一，盈縮之限，不在冬夏至，宜在冬夏至後六度。今考日躔，春分迄夏至，夏至迄秋分，此兩限中，日時刻分不等。又立春迄立夏，立秋迄立冬，此兩限中，日時刻分亦不等。測量可見。其一，言太陰最高得疾，最低得遲，且以圭表測而得之，非也。太陰遲疾是入轉內事，表測高下是入交內事，豈容混推。而月行轉周之上，又復左旋，所以最高向西行極遲，最低向東行乃極疾，舊法正相反。其一，言日食正午無時差，非也。時差言距，非距赤道之午中，乃距黃道限東西各九十度之中也。黃道限之中，有距午前後二十餘度者，但依午正加減，焉能必合。其一，言交食定限，陰曆八度，陽曆六度，非也。日食，陰曆當十七度，陽曆當八度。月食則陰陽曆俱十二度。其一，《曆測》云：「宋文帝元嘉六年十一月己丑朔，日食不盡如鉤，晝星見。今以《授時》推之，止食六分九十六秒，郭曆舛矣。」夫月食天下皆同，日食九服各異。南宋都於金陵，郭曆造於燕地，北極出地差八度，時在十一月則食差當得二分弱，其云「不盡如鉤」，當在九分左右。郭曆推得七分弱，乃密合，非舛也。本局今定日食分數，首言交，次言地，次言時，一不可闕。已而文魁反覆論難，光啟更申前說，著為《學曆濁辨》。

At the time the commoner Wei Wenkuai of Mancheng had written two books, Calendar Origin and Calendar Measurement, and had his son Xiangqian present Calendar Origin at court; the Secretariat of Transmission sent it to the Bureau for examination. Guangqi selected seven points for decisive debate. First: since Han times the tropical year's surplus has been progressively reduced, until in the Season Granting it stood at twenty-four parts and twenty-five seconds. By Guo Shoujing's method of one part consumed per century, it should now be twenty-one-odd seconds. Yet Calendar Origin uses Zhao Zhiwei's thirty-six seconds and suddenly adds to it again and again. Second: in seeking chord and sagitta from arc, one should use a fine ratio. Calendar Measurement still uses the ratio diameter one to circumference three, which does not match the true arc-sagitta numbers. Third: the limits of the equation of center are not at the winter and summer solstices but six degrees after them. Examining solar motion now, from spring equinox to summer solstice and from summer solstice to autumn equinox—in these two limits the sun's hours, minutes, and parts are unequal. From beginning of spring to beginning of summer, and from beginning of autumn to beginning of winter—in these two limits too the sun's hours, minutes, and parts are unequal. Measurement makes this plain. Fourth: the claim that the moon is fastest at highest altitude and slowest at lowest, and that this was obtained by gnomon measurement, is wrong. Lunar anomaly belongs within the revolution cycle; gnomon measurement of altitude belongs within nodal passage—how can the two be mixed in one computation? Moreover, above the revolution cycle the moon again rotates leftward, so at highest altitude it moves westward and is extremely slow, while at lowest it moves eastward and is extremely fast—the old method has this exactly reversed. Fifth: the claim that a solar eclipse at noon on the central meridian has no time difference is wrong. Time difference concerns distance—not distance from the equator's noon meridian, but from the center of the ecliptic's eastern and western limits, each ninety degrees from the node. Within the ecliptic limits there are cases more than twenty degrees before or after noon; merely adding and subtracting by true noon, how can the result necessarily agree? Sixth: the claim that the fixed eclipse limit is eight degrees on the lunar path and six on the solar path is wrong. For a solar eclipse the lunar path should allow seventeen degrees and the solar eight. For a lunar eclipse both paths should allow twelve degrees. Seventh: Calendar Measurement says, "In the sixth year of Emperor Wen's Yuanjia reign, on the jichou new moon of the eleventh month, the solar eclipse was incomplete like a hook and daytime stars were visible. Computed now by the Season Granting, the eclipse would be only six parts ninety-six seconds—Guo's calendar is in error. Yet lunar eclipses are the same throughout the realm, while solar eclipses differ in each of the nine domains. The Southern Song capital was at Jinling, while Guo's calendar was made at the Yan capital; polar altitude there differs by eight degrees; in the eleventh month the eclipse difference ought to be about two-tenths less, so what they call "incomplete like a hook" should be about nine-tenths. Guo's calendar computed seven-tenths weak—that is a close match, not an error. This Bureau now fixes solar-eclipse magnitude by speaking first of nodal passage, next of terrestrial position, next of time—none of these may be omitted. Before long Wenkuai repeatedly debated in refutation; Guangqi restated his earlier views and set them down in Clarifying Muddled Calendar Learning.

其論歲實小餘及日食變差尤明晰。曰：「歲實小餘，自漢迄元漸次消減。今新法定用歲實，更減於元。不知者必謂不惟先天，更先《大統》。乃以推壬申冬至，《大統》得已亥寅正一刻，而新法得辰初一刻十八分。何也？蓋正歲年與步月離相似，冬至無定率，與定朔、定望無定率一也。朔望無定率，宜以平朔望加減之，冬至無定率，宜以平年加減之。故新法之平冬至，雖在《大統》前，而定冬至恆在《大統》後也。」又曰：「宋仁宗天聖二年甲子歲，五月丁亥朔，曆官推當食不食，諸曆推算皆云當食。夫於法則實當食，而於時則實不食。今當何以解之？蓋日食有變差一法，月在陰曆，距交十度強，於法當食。而獨此日此之南北差，變為東西差，故論天行，則地心與日月相參直，實不失食。而從人目所見，則日月相距近變為遠，實不得食。顧獨汴京為然，若從汴以東數千里，則漸見食，至東北萬餘裡外，則全見食也。夫變差時不同，或多變為少，或少變為多，或有變為無，或無變為有。推曆之難，全在此等。」未幾，光啟入愉閣。

His discussion of the tropical year's small remainder and of parallax variation in solar eclipses was especially clear. He said, "The tropical year's small remainder from Han through Yuan was gradually reduced. The new method now fixes the tropical year, reducing it still further from the Yuan figure. The uninformed will surely say it is not only ahead of Heaven but even ahead of the Grand Concordance. Yet computing the renshen winter solstice—the Grand Concordance gave hai hour, yin quarter, first quarter, while the new method gave chen, first quarter, eighteen parts. How can this be? Because the proper year's length resembles pacing lunar departure: the winter solstice has no fixed rate, just as mean new and full moons have no fixed rate. When new and full moons have no fixed rate, one should add and subtract from mean conjunction and opposition; when the winter solstice has no fixed rate, one should add and subtract from the mean year. Thus the new method's mean winter solstice, though it falls before the Grand Concordance, the true winter solstice always falls after it. He also said, "In the second year of Emperor Renzong's Tiansheng reign, a jiazi year, on the dinghai new moon of the fifth month, the calendar officers computed an eclipse but there was none; every calendar computed that there should be an eclipse. By the method there should indeed have been an eclipse; by observation there was indeed none. How is this to be explained? Because solar eclipses have a parallax-variation method: on that day the moon on the lunar path was a little over ten degrees from the node—by method an eclipse should occur. But on this day alone the north-south parallax became an east-west parallax; in terms of celestial motion, the earth's center and the sun and moon were truly aligned—there was indeed no failure of eclipse. Yet as seen by human eyes, what had been a near alignment became a distant one—no eclipse was truly visible. Only at Bianjing was this so; from Bian eastward for several thousand li the eclipse would gradually appear, and ten-thousand-odd li to the northeast it would be total. Parallax variation differs by time—sometimes much becomes little, or little much; sometimes eclipse becomes none, or none eclipse. The whole difficulty of calendar computation lies in matters such as these. Before long Guangqi entered the Yuyin Pavilion.

五年九月十五日，月食，監推初虧在卯初一刻，光啟等推在卯初三刻，回回科推在辰初初刻。三法異同，致奉詰問。至期測候，陰雲不見，無可徵驗。光啟具陳三法不同之故，言：

In year five, on the fifteenth day of the ninth month, there was a lunar eclipse. The Directorate computed first obscuration at mao, first quarter; Guangqi and his colleagues at mao, third quarter; the Huihui section at chen, first quarter. The three methods disagreed, and the throne sent an inquiry. When the day came, clouds hid the sky and nothing could be verified. Guangqi fully explained why the three methods differed, saying:

時刻之加減，由於盈縮、遲疾兩差。而盈縮差，舊法起冬夏至，新法起最高，最高有行分，惟宋紹興間與夏至同度。郭守敬後此百年，去離一度有奇，故未覺。今最高在夏至後六度。此兩法之盈縮差所不同也。遲疾差，舊法只用一轉周，新法謂之自行輪。自行之外，又有兩次輪。此兩法之遲疾差所以不同也。至於《回回曆》又異者，或由於四應，或由於裡差，臣實未曉其故。總之，三傢俱依本法推步，不能變法遷就也。

The addition and subtraction of time come from the equation of center and from the lunar anomaly's fast-slow difference. As for the equation difference, the old method starts at the winter and summer solstices, the new at the apogee; the apogee has motion in degrees—it coincided with the summer solstice only during the Shaoxing period of Song. A century after Guo Shoujing it had moved more than one degree from that point, and so the change went unnoticed. Now the apogee lies six degrees after the summer solstice. This is how the equation differences of the two methods diverge. As for the anomaly difference, the old method used only one revolution cycle; the new calls it the self-motion wheel. Outside the self-motion wheel there are again two secondary wheels. This is why the anomaly differences of the two methods differ. As for the Huihui calendar differing again, it may be from the four responses or from local difference; your subject truly does not know the reason. In sum, all three schools compute by their own methods and cannot alter their methods to accommodate one another.

將來有宜講求者二端：一曰食分多寡。日食時，陽晶晃耀，每先食而後見。月食時，遊氣紛侵，每先見而後食。其差至一分以上。今欲灼見實分，有近造窺筒，日食時，於密室中取其光景，映照尺素之上，初虧至復圓。分數真確，書然不爽。月食用以仰觀二體離合之際，鄞鄂著明。與目測迥異。此定分法也。一曰加時早晚。定時之術，壺漏為古法，輪鐘為新法，然不若求端於日星，晝則用日，夜則任用一星。皆以儀器測取經緯度數，推算得之。此定時法也。二法既立，則諸術之疏密，毫末莫遁矣。

There are two matters worth pursuing in future: first, whether the eclipse magnitude is greater or smaller. At a solar eclipse the sun's bright disk dazzles the eye—one always perceives obscuration too early and the return of light too late. At a lunar eclipse wandering vapors intrude—one always perceives brightness too early and obscuration too late. The discrepancy can exceed one part. To see the true magnitude clearly, a newly made sighting tube is used: at a solar eclipse, in a dark room one takes its light and projects it onto silk from first obscuration to full return of light. The fraction is exact and can be recorded without error. At a lunar eclipse one uses it to observe overhead when the two bodies join or part—the boundary stands out clearly. This is far unlike naked-eye observation. This is the method of fixing magnitude. Second, whether the additive time is early or late. For fixing time, the clepsydra is the ancient method and the wheel-clock the new—yet it is better to seek the reference in the sun and stars: by day use the sun, by night any one star. In every case one uses instruments to measure longitude and latitude and computes the time thereby. This is the method of fixing time. Once these two methods are established, whether any technique is coarse or fine—nothing can escape detection.

古今月食，諸史不載。日食，自漢至隋，凡二百九十三，而食於晦者七十七，晦前一日者三，初二日者三，其疏如此。唐至五代凡一百一十，而食於晦者一，初二日者一，初三日者一，稍密矣。宋凡一百四十八人，無晦食者，更密矣。猶有推食而不食者一。至加時差至四五刻者，當其時已然。可知高速無窮之事，必積時累世，仍稍見其端兒。故漢至今千七百歲，立法者十有三家，而守敬為最優，尚不能無刻之差，而況於沿習舊法者，何能現其精密哉？

Lunar eclipses ancient and modern—the various histories do not record them. Solar eclipses from Han through Sui number 293 in all—seventy-seven on the last day of the month, three on the day before last, three on the second day—so coarse. From Tang through the Five Dynasties there are 110—one on the last day, one on the second, one on the third—somewhat finer. Under Song there are 148, with none on the last day—finer still. Yet there is still one case where an eclipse was computed but none occurred. Where the additive-time error reached four or five quarters, that was already evident at the time. From this one may know that matters of boundless extreme speed must accumulate through ages and generations before one can slightly glimpse their beginning. Hence from Han to the present, across seventeen hundred years and thirteen schools of calendar-makers, Guo Shoujing was finest—yet even he was not without quarter error; how much less those who merely follow old methods—how could they display precision now?

是年，光啟又進《曆書》三千卷。明年冬十月，光啟以病辭曆務，以山東參政李天經代之。逾月而光啟卒。七年，魏文魁上言，曆官所推交食節氣皆非是。於是命魁入京測驗。是時言曆者四家，《大統》、《回回》外、別立西洋為西局，文魁為炙局。言人人殊，紛若聚訟焉。

That year Guangqi again presented three thousand juan of the Calendar Book. The next year, in the tenth month of winter, Guangqi resigned his calendar duties on grounds of illness, and Li Tianjing, surveillance vice censor of Shandong, replaced him. A month later Guangqi died. In year seven Wei Wenkuai memorialized that the calendar officers' computations of eclipses and solar terms were all wrong. Thereupon Wenkuai was ordered to the capital for testing. At the time four schools of calendrics were contending: the Grand Concordance and the Huihui, and besides these the Western method was set up as the Western Bureau while Wenkuai headed the Wei Bureau. Each man had his own view, noisy as a crowd at lawsuit.

天經繕進《曆書》凡二十九卷，並星屏一具，俱故輔光啟督率西人所造也。天經預推五星淩犯會合行度，言：「閏八月二十四，木犯積履屍氣。九月初四昏初，火土同度。初七卯正，金土同度。十一昏初，金火同度。舊法推火土同度，在初七，是後天三日。金火同度在初三，是先天八日。」而文魁則言，天經所報，木星犯積屍不合。天經又言：「臣於閏八月二十五日夜及九月初一日夜，同體臣陳六韋等，用窺管測，見積屍為數十小星圍聚，木與積屍，共納管中。蓋窺圓徑寸許，兩星相距三十分內者，方得同見。如觜宿三星相距二十七分，則不能同見。而文魁但據臆算，未經實測。據雲初二日木星已在柳前，則前此豈能越鬼宿而飛渡乎？」天經又推木星退行、順行，兩經鬼宿，其度分晷刻，已而皆驗，於是文魁說絀。

Tianjing edited and presented twenty-nine juan of the Calendar Book in all, together with one star screen—all formerly supervised by Guangqi with the Westerners. Tianjing pre-computed the five planets' conjunctions, violations, and motion in degrees, saying, "On the twenty-fourth day of the intercalary eighth month, Jupiter will violate the Accumulated Corpse star. On the fourth day of the ninth month, at the beginning of dusk, Mars and Saturn will share the same degree. On the seventh day, at mao exact, Venus and Saturn will share the same degree. On the eleventh, at the beginning of dusk, Venus and Mars will share the same degree. The old method places the Mars–Saturn conjunction on the seventh day—that is three days after heaven. Venus and Mars in conjunction fell on the third day—eight days ahead of heaven. But Wen Kui said that Tianjing's report of Jupiter encroaching on Piled Corpse did not agree. Tianjing also said, "On the night of the twenty-fifth of the intercalary eighth month and the night of the first day of the ninth month, together with colleagues including Chen Liuwei, I used a sighting tube to observe and saw Piled Corpse as several dozen small stars clustered together; Jupiter and Piled Corpse both entered the tube at once. The tube's circular aperture is about an inch across; only when two stars lie within thirty minutes of arc of each other can both be seen together. As the three stars of the Turtle Beak lodge are twenty-seven minutes apart, they cannot be seen together. Yet Wen Kui relied only on speculative calculation and had not carried out actual observation. According to him, on the second day of the ninth month Jupiter was already before Willow—how then could it earlier have flown across Ghost in a single leap? Tianjing further computed Jupiter's retrograde and direct motion, twice crossing Ghost; its degrees, minutes, and gnomon marks were afterward all verified, and Wen Kui's theory was defeated.

天經又進《曆書》三十二卷，並日晷、星晷、窺筒諸儀器。八年四月，又上《乙亥丙子七政行度曆》及《參訂曆法條議》二十六則。

Tianjing further presented thirty-two juan of the Calendar Book, together with sundials, star dials, sighting tubes, and other instruments. In the fourth month of year eight he again submitted the Calendar of the Seven Regulators' Motion in the yihai and bingzi cycles and twenty-six articles for revising calendar methods.

某七政公說之議七：一曰諸曜之應宜改。蓋日月五星平行起算之根則為應，乃某曜某日某時躔某宮次之數。今新法改定諸應，悉從崇禎元年戊辰前，冬至後，己卯日子正為始。二曰測諸曜行度，應用黃道儀。蓋太陽由黃道行，月星各有本道，出入黃道內外，不行赤道。若用赤道儀測之，所得經緯度分，須通以黃、赤通率表，不如用黃道儀，即得七政之本度為便也。三曰諸方七政行度，隨地不等。蓋日月東西見食，其時各有先後，既無庸疑矣。則太陽之躔二十四節氣，與月五星之掩食淩犯，安得不與交食同一理乎？故新法水成諸表，雖以順天府為主，而推算諸方行度亦皆各有本法。四曰諸曜加減分，用平、立、定三差法，尚不足。蓋加減平行以求自行，乃曆家耍務。第天實圓體，與平行異類，舊所用三差法，俱從句股平行定者，於天體未合。即扣盈縮損益之數，未得其真。今新法加減諸表，乃以圓齊圓，始可合天。五曰隨時隨地可求諸曜之經度。舊法欲得某日曜經度，必先推各曜冬至日所行宮度宿次，後乃以各段日度比算始得。今法不拘時日方所，只簡本表推步即是。六曰徑一圍三，非弧矢真法。蓋古曆家以直綿測圓形，名曰弧矢法，而算用徑一圍三，廖也。今立割圓八綿表，其用簡而大。弧矢等綿，但乘除一次，使能得之。七曰球上三角三弧形，非句股可盡。蓋古法測天以句股為本，然句股能禦直角，不能禦斜角。且天為圓球，其面上與諸道相割生多三弧形，句股不足以盡之。

A certain essay on the public discussion of the seven regulators sets forth seven points: first, the correspondences of the luminaries ought to be revised. For the root of parallel motion from which sun, moon, and five planets begin calculation is the correspondence—namely the number of a given luminary on a given day and hour occupying a given palace and lodge. Now the new method revises all correspondences, all taking as start the moment of midnight on the jimao day after the winter solstice before the wuchen year, first year of Chongzhen. Second, in measuring the luminaries' motion in degrees, one ought to use the ecliptic instrument. For the sun moves along the ecliptic; moon and stars each have their own paths, entering and leaving within and without the ecliptic, and do not move on the equator. If one measures with the equatorial instrument, the longitude and latitude degrees obtained must be converted through the yellow-red conversion table—not as convenient as using the ecliptic instrument, which immediately gives the seven regulators' fundamental degrees. Third, the seven regulators' motion in degrees differs by place. For east and west see eclipses at different times, each with its own prior and posterior sequence—this is beyond doubt. Then the sun's passage through the twenty-four qi and the moon and five planets' occultations, encroachments, and violations—how can they fail to follow the same principle as eclipses? Therefore the new method's completed tables, though taking Shuntian Prefecture as primary, also each have their own methods for computing motion in degrees for every region. Fourth, the luminaries' additive-subtractive parts using the three difference methods of mean, stepped, and fixed are still insufficient. For adding and subtracting parallel motion to seek true motion is the calendar expert's essential task. But heaven is truly a round body, unlike parallel motion in kind; the old three-difference methods, all fixed from right triangle parallel motion, do not fit the celestial body. Even when one applies the numbers of surplus, deficit, increase, and decrease, one does not obtain the truth. Now the new method's additive-subtractive tables use circle to match circle—only then can one fit heaven. Fifth, at any time and place one can seek the luminaries' longitude. The old method, to obtain a given day's luminary longitude, had first to compute each luminary's palace degree and lodge at the winter solstice day, then compare by each segment's daily degrees before obtaining it. The present method is not bound by time, day, or place; one need only consult the basic tables and compute by steps—that is all. Sixth, diameter one circumference three is not the true arc-sagitta method. For ancient calendar experts measured circles with straight chords, calling it the arc-sagitta method, yet in calculation used diameter one circumference three—crude indeed. Now we establish the eight-chord table for circle division; its use is simple yet great. Arc-sagitta and similar chords require only one multiplication and division to obtain them. Seventh, the three arc forms of spherical triangles are not exhaustible by right triangles. For the old method measured heaven with right triangles as foundation; yet right triangles can handle right angles, not oblique angles. Moreover heaven is a round sphere; on its surface where paths intersect many three-arc forms arise—right triangles are insufficient to exhaust them.

恆星之議四：一曰恆星本行，即所謂歲差，從黃道極起算。蓋各星距赤極度分，古今不同。其距赤道內外地也，亦古今不同。而距黃極或距黃道內外，則皆終古如一，所以知日月五星俱依黃道行。其恆星本行，應從黃極起算，以為歲差之率。二曰古今各宿度不同。蓋恆星以黃道極為極，故各宿距星行度，與赤道極時近時遠。行漸近極，即赤道所出過距星綿漸密，其本宿赤道弧則較小。漸遠極，即過距星綿漸疏，其本宿赤道弧則較大。此緣二道二極不同，非距星有異行，亦非距星有易位也。如觜宿距星，漢測距參二度，唐測一度，宋崇寧測半度，元郭守敬五分。今測之，不啻無分，且侵入參宿二十四分，非一症乎？三曰夜中測星定時。蓋太陽依赤道左行，每十五度為一小時。今任測一星距子午圈前後度分，又以本星經行與太陽經行查加減，得太陽距子午圈度分，因以變為真時刻。四曰宋時所定十二宮次，在某宿度，今不能定於某宿度。蓋因恆星有本行，宿度已右移故也。

On fixed stars there are four points: first, fixed stars' proper motion, namely what is called precession, is to be computed from the ecliptic pole. For each star's degrees from the red pole differ between antiquity and the present. Its distance within or without the equator likewise differs between antiquity and the present. But distance from the yellow pole or within or without the ecliptic is the same through all ages—whence we know sun, moon, and five planets all follow the ecliptic. Fixed stars' proper motion ought to be computed from the yellow pole as the rate of precession. Second, lodge degrees of each age differ between antiquity and the present. For fixed stars take the ecliptic pole as pole; therefore each lodge's distance-star motion in degrees is now near, now far from the equatorial pole. As motion gradually nears the pole, the equator's lines crossing the distance star grow denser, and that lodge's equatorial arc is comparatively smaller. As it gradually departs the pole, the lines crossing the distance star grow sparser, and that lodge's equatorial arc is comparatively larger. This is because the two paths and two poles differ—not that distance stars have different motion, nor that distance stars change position. As the Turtle Beak lodge distance star: Han measurement placed it two degrees from Shen; Tang one degree; Song Chongning half a degree; Yuan Guo Shoujing five minutes. Today's measurement shows not merely no minutes, but encroachment into the Shen lodge twenty-four minutes—is this not one proof? Third, determine time by measuring stars at midnight. For the sun moves leftward along the equator, fifteen degrees per small hour. Now measure at will one star's degrees before and after the meridian circle, then check addition and subtraction against that star's longitude motion and the sun's longitude motion, obtaining the sun's degrees from the meridian circle, and thereby convert to true time. Fourth, the twelve palace stations fixed in Song times at certain lodge degrees cannot now be fixed at those lodge degrees. This is because fixed stars have proper motion and lodge degrees have shifted rightward.

太陽之議四：一太陽盈縮之限，非冬、夏二至，所謂最高及最高沖出也。此限年年右行，今已過二至後六度有奇。二曰以圭表測冬夏二至，非法之善。蓋二至前後，太陽南北之行度甚微，計一丈之表，其一日之影差不過一分三十秒，則一秒得六刻有奇，若測差二三秒，即差幾二十刻，安所得準乎？今法獨用春、秋二分，蓋以此時太陽一日南北行二十四分，一日之景差一寸二分，即測差一二秒，算不得滿一刻，較二至為最密。三曰日出入分，應從順天府起算。蓋諸方北極出地不同，晨昏時刻亦因以異。《大統》依應天府算，上以晝夜長短，日月東刃西帶食，所推不準。今依天罕改定。四曰平節氣，非上天真節氣。蓋舊法氣策，乃歲週二十四分之一。然太陽之行有盈有縮，不得平分。如以平分，則春分後天二日，秋分先天二日矣。今悉改定。

On the sun there are four points: first, the sun's limits of surplus and deficit are not the winter and summer solstices—namely what are called apogee and apogee opposition. This limit shifts rightward year by year; now it has passed more than six degrees beyond the solstices. Second, using the gnomon table to measure winter and summer solstice is not a good method. For before and after the solstices the sun's north-south motion in degrees is extremely slight; reckoning a ten-chi table, one day's shadow difference is no more than one minute thirty seconds—then one second equals more than six quarters; if measurement errs two or three seconds, the error is nearly twenty quarters—how can one obtain accuracy? The present method uses only spring and autumn equinoxes; for at these times the sun moves twenty-four minutes north-south in one day, and one day's shadow difference is one cun two fen—so even an error of one or two seconds in measurement computes to less than one quarter, tighter than at the solstices. Third, sunrise and sunset parts ought to be computed from Shuntian Prefecture. For every region's north pole elevation above the earth differs, and dawn and dusk times likewise differ accordingly. The Grand Concordance computed from Yingtian Prefecture; for day-night length above, and sun and moon girdling east and west with eclipses, what was computed was inaccurate. Now it is revised according to the Tianhan [standard]. Fourth, mean qi are not heaven's true qi. For the old method's qi policy was one twenty-fourth of the year's circuit. Yet the sun's motion has surplus and deficit and cannot be evenly divided. If evenly divided, then after spring equinox heaven is two days late, and before autumn equinox heaven is two days early. Now all are revised and fixed.

太陰之議四：一曰朔望之外，別有損益分，一加減不足以盡之。蓋舊定太陰平行，算朔望加減，大率五度有奇，然兩弦時多寡不一，即《授時》亦言朔望外，平行數不定，明其理未著其法。今於加減外，再用一加減，名為二三均數。二曰緯度不能定於五度，時多時寡。古今曆家以交食分數及交泛等，測量定黃白二道相距約五度。然朔望外兩道距度，有損有益，大距計五度三公度之一。若一月有兩食，其弦時用儀求距黃道度五度，未能合天。三曰交行有損益分。蓋羅喉、計都即正交、中交行度，古今為平行。今細測之，月有時在交上，以平求之，必不合算。因設一加減，為交行均數。四曰天行無紫氣。舊謂生於閏餘，又為木之餘氣。今細考諸曜，無象可明，知為妄增。

On the moon there are four points: first, beyond new and full moon there are separate surplus-deficit parts; one addition-subtraction is insufficient to exhaust them. For the old fixed lunar parallel motion, computing new and full moon addition-subtraction, was mostly a little over five degrees; yet at the two quadratures the amounts differ—indeed the Season Granting also says beyond new and full moon the parallel numbers are not fixed, showing the principle but not setting forth the method. Now beyond addition-subtraction we apply another addition-subtraction, called the two-three mean numbers. Second, latitude cannot be fixed at five degrees—it is sometimes more, sometimes less. Calendar experts of antiquity and the present, from eclipse fractions and nodal breadth and the like, measured and fixed the yellow and white paths' mutual distance at about five degrees. Yet beyond new and full moon the two paths' separating distance has deficit and surplus; the greatest distance totals five degrees and three and one-third degrees. If in one month there are two eclipses, at quadrature using instruments to seek five degrees from the ecliptic cannot fit heaven. Third, nodal motion has surplus-deficit parts. For Rahu and Ketu are the motion in degrees of true node and mean node; antiquity and the present treat them as parallel motion. Now on fine measurement, sometimes the moon is on the node; seeking by parallel motion, the computation necessarily fails to agree. Therefore one addition-subtraction is set up as the nodal motion mean number. Fourth, celestial motion has no purple qi. The old said it arose from intercalary remainder and was also Jupiter's residual qi. Now on fine examination of the luminaries, no phenomenon can clarify it—knowing it was falsely added.

交食之議四：一曰日月景徑分恆不一。蓋日月時行最高，有時行最高，有時行最卑，因相距有遠近，見有大小。又因遠近竿太陰過景，時有厚薄，所以徑分不能為一。二曰日食午正非中限，乃以黃道九十度限為中限。蓋南北東西差俱依黃道，則時差安得不從黃道道論其初末以求中限乎？且黃道出地平上，兩象限自有其高，亦自有其中。此理未明，或宜加反減，宜減反加，凡加進不合者由此也。三曰日食初虧復圓，時刻多寡恆不等，非二時折半之說。蓋視差能變實行為視行，則以視差較食甚前後，鮮有不參差者。夫視差既食甚前後不一，又安能令視行前後一乎？今以視行推變時刻，則初虧復圓，其不能相等也明矣。四曰諸方各依地經推算時刻及日食分。蓋地面上東西見日月出沒，各有前後不同即所得時刻亦不同。故見食雖一而時刻異，此日月食皆一理。若日食則因視差隨地不一，即太陰視距不一，所見食分亦異焉。

On eclipses there are four points: first, sun and moon shadow diameters in parts are constantly not one. For sun and moon sometimes move at apogee and sometimes at perigee; because of distance near and far, they appear large or small. Also because of distance, pole, and the moon passing the shadow, sometimes thick, sometimes thin—therefore diameter parts cannot be one. Second, solar eclipse at noon is not the center limit; rather the ecliptic's ninety-degree limit is the center limit. For north-south and east-west differences all follow the ecliptic—then how can time difference fail to follow the ecliptic path in discussing beginning and end to seek the center limit? Moreover when the ecliptic rises above the horizon, the two quadrants each have their high point and also their center. This principle not understood, sometimes one ought to add yet subtracts, ought to subtract yet adds—all cases where addition and advance fail to agree arise from this. Third, solar eclipse initial obscuration and final fullness—moments and amounts are constantly unequal; it is not the theory of halving the two times. For parallax can change true motion into apparent motion; comparing parallax before and after greatest eclipse, rarely is there no discrepancy. If parallax before and after greatest eclipse is not one, how can one make apparent motion before and after one? Now using apparent motion to compute changing moments, then initial obscuration and final fullness clearly cannot be equal. Fourth, each region according to terrestrial longitude computes moments and solar eclipse parts. For on the earth's surface east and west see sun and moon rise and set each with different prior and posterior sequence—therefore the moments obtained also differ. Therefore though the eclipse seen is one, the moments differ—this is one principle for both solar and lunar eclipses. For solar eclipse, because parallax differs by place, the moon's apparent distance differs, and the eclipse fraction seen also differs.

五緯之議三：一曰五星應用太陽視行，不得以段目定之。蓋五星皆以太陽為主，與太陽合則疾行，沖則退行。且太陽之行有遲疾，則五星合伏日數，時寡時多，自不可以段目定其度分。二曰五星應加緯行。蓋五星出入黃道，各有定距度。又木、土、火三星沖太陽緯大，合太陽緯小。金、水二星順伏緯小，逆伏緯大。三曰測五星，當用恆星為準則。蓋測星用黃道儀外，宜用弧矢等儀。以所測緯星視距二恆星若干度分，依法布算，方得本星真經緯度分。或繪圖亦可免算。

On the five planets there are three points: first, the five planets ought to use the sun's apparent motion and cannot be fixed by segment entries. For the five planets all take the sun as master; in conjunction they move fast, in opposition they retrograde. Moreover the sun's motion has slow and fast phases; then the five planets' days of conjunction and hiding are sometimes few, sometimes many—by nature one cannot fix their degrees and minutes by segment entries. Second, the five planets ought to add latitude motion. For the five planets entering and leaving the ecliptic each have fixed separating degrees. Wood, Earth, and Fire likewise have large ecliptic latitude when in opposition to the Sun and small latitude when in conjunction. Venus and Mercury show small latitude during prograde stations and greater latitude during retrograde stations. Third, when measuring the five planets one should take fixed stars as the standard of reference. Besides the ecliptic instrument, arc-and-chord instruments and the like should also be employed in stellar measurement. One measures the planet's apparent distance from two fixed stars in degrees and minutes, applies the prescribed computation, and thereby obtains the planet's true longitude and latitude. A plotted diagram can serve the same purpose and avoid lengthy calculation.

是時新法書器俱完，屢測交食淩犯俱密合，但魏文魁等多方陰撓，內官實左右之。以故帝意不能決，諭天經同監局虛心詳究，務祈書一。是年，天經推水星伏見及木星所在之度，皆與《大統》各殊，而新法為合。又推八月二十七日寅正二刻，木、火、月三曜同在張六度，而《大統》推木在張四度，火、月張三度。至期，果同在張六度。九年正月十五日辛酉，曉望月食。天經及《大統》、《回回》、東局，各頂推虧圓食甚分秒時刻。天經恐至期雲掩難見，乃按裡差，推河南、山西所見時刻，奏遣官分行測驗。其日，天經與羅雅谷、湯若望、大理評事王應遴、禮臣李焻及監局守登、文魁等赴臺測驗，惟天經所推獨合。已而，河南所報盡合原推，山西則食時雲掩無從考驗。

By then the new method's treatises and instruments were complete, and repeated eclipse and occultation tests agreed closely with observation; yet Wei Wenku and others obstructed the reform at every turn, with eunuchs actively siding with them. The emperor therefore remained undecided and ordered Tianjing and the observatory officials to examine the matter impartially and produce a single authoritative calendar text. That year Tianjing's predictions for Mercury's visibility cycles and Jupiter's position all disagreed with the Grand Concordance but matched the new method. He further predicted that at the second quarter of the yin double-hour on the twenty-seventh of the eighth month, Jupiter, Mars, and the Moon would stand together at six degrees in Zhang, whereas the Grand Concordance had Jupiter at four degrees and Mars and the Moon at three. When the day arrived, all three bodies were indeed at six degrees in Zhang. In year nine, on the fifteenth of the first month (xinyou), a lunar eclipse occurred near dawn. Tianjing, the Grand Concordance calendar, the Huihui calendar, and the Eastern Bureau each computed in advance the phases of first contact, totality, and last contact, together with magnitude, seconds, and times. Fearing clouds at the capital, Tianjing applied terrestrial longitude correction, calculated visibility times for Henan and Shanxi, and memorialized to dispatch observers to test in both provinces. On the day of the eclipse Tianjing went to the observatory with Rho, Schall, Wang Yinglin of the Court of Judicial Review, ritual official Li Huang, and observatory officers including Shoudeng and Wenku; only Tianjing's figures proved correct. Reports from Henan soon confirmed the original predictions in full, while Shanxi was overcast at eclipse and could not be checked.

帝以測驗月食，新法為近，但十五日雨水，而天經以十三日為雨水，令再奏明。天經覆言：

The emperor held that the eclipse test favored the new method, but noted that the old calendar placed Rain Water on the fifteenth while Tianjing placed it on the thirteenth, and ordered him to explain this discrepancy. Tianjing replied:

諭節氣有二法：一為平節氣，一為定節氣。平節氣者，以一歲之實，二十四平分之，每得一十五日有奇，為一節氣。故從歲前冬至起算，必越六十日八十七刻有奇為雨水。舊法所推十五日子正一刻者此也，定節氣者，以三百六十為周天度，而亦以二十四平分之，每得一十五度為一節氣。從歲前冬至起算，曆五十九日二刻有奇，而太陽行滿六十度為雨水。新法所推十三日卯初二刻八分者此也。太陽之行胡盈有縮，非用法加減之，必不合天，安得平分歲實為節氣乎？以春分症之，其理更明。分者，黃赤相交之點，太陽行至此，乃晝夜平分。舊法於二月十四日下，注晝五十刻、夜五十刻是也。夫十四日書夜已平分，則新法推十四日春分者為合天，而舊法推十六日者，後天二日矣。知春分，則秋分及各節氣可知，而無疑於雨水矣。

He explained that solar terms are reckoned in two ways: mean solar terms and true solar terms. Mean solar terms take the true length of the year, divide it into twenty-four equal parts, and treat each interval of slightly more than fifteen days as one term. Counting from the winter solstice before New Year, Rain Water on this scheme falls only after sixty days and eighty-seven-odd quarters. The Grand Concordance date of the fifteenth at the first quarter of zi belongs to this mean-term method. True solar terms treat the celestial circuit as 360° and likewise divide it into twenty-four terms of fifteen degrees each. From the pre-year winter solstice, once fifty-nine days and two-odd quarters have passed and the Sun has moved sixty degrees, Rain Water has been reached. The new method's Rain Water on the thirteenth at the second quarter of mao and eight fen follows this true-term reckoning. The Sun's motion is sometimes fast and sometimes slow; without the method's corrective additions and subtractions it cannot match heaven. How, then, can one simply divide the year's true length into equal terms? The vernal equinox makes the point still clearer. The equinox is the intersection of the ecliptic and equator; when the Sun reaches that point, day and night are equal. The old calendar annotates the fourteenth of the second month with fifty day-quarters and fifty night-quarters, showing that day and night were already equal on that date. Since equality of day and night had already been reached on the fourteenth, the new method's equinox on that day accords with heaven, whereas the old method's equinox on the sixteenth is two days late. Once the vernal equinox is fixed, the autumnal equinox and the other terms follow, and Rain Water need no longer be disputed.

已而天經於春分屈期，每午赴臺測午正太陽高度。二月十四日高五十度八分，十五日高五十度三十分。末經乃言：

As the equinox approached, Tianjing went to the observatory each day at noon to measure the Sun's meridian altitude. On the fourteenth of the second month the altitude was 50°8′; on the fifteenth, 50°30′. He then memorialized further:

京師北極出地三十九度五十五分，則赤道應高五十度五分，春分日太陽正當赤道上，其午正高度與赤道高度等，過此則太陽高度必漸多，今置十四日所測高度，加以地半經差二分，較赤道已多五分。蓋原推春分在卯正二五分弱，是時每日緯行二十四分弱，時差二十一刻五分，則緯行應加五分強。至十五日，並地半徑較赤道高度已多至三十分，況十六日乎？是春分當在十四，不當在十六也。秋風京然。又出《節氣圖》曰：

At Beijing the north celestial pole stands 39°55′ above the horizon, so the equator at meridian passage should be 50°5′ high. On the equinox the Sun lies on the equator, and its noon altitude should equal that of the equator; thereafter the Sun's noon altitude must rise. Yet the altitude observed on the fourteenth, even after adding the two-fen semidiameter correction, already exceeds the equatorial altitude by five fen. The earlier computation had placed the equinox at the second quarter of mao and slightly under 5¼ fen; with the Sun then moving about 24 fen of latitude per day and a time difference of 21¼ quarters, the latitude correction should add slightly more than 5¼ fen. By the fifteenth, even after the semidiameter correction, the noon altitude exceeded the equator by thirty fen; on the sixteenth the discrepancy would be still greater. The equinox therefore belongs on the fourteenth, not the sixteenth. The throne accepted this and assented accordingly. He also submitted a Solar Term Diagram, explaining:

內規分三百六十五度四分度之一者，日度也。外規公三百六十度者，天度也。自冬至起算，越九十一日三十一刻六分，而始曆春分者，日為之限敢，乃在天則已逾二度餘矣。又越二百七十三日九十三刻，一十九分，而即交秋分者，亦日為之限也，乃在天不及二度餘。豈非舊法春分每後天二日，秋分先天二日耶？

The inner ring, graduated in 365¼°, marks solar motion. The outer ring, graduated in 360°, marks celestial motion. From the winter solstice, the mean solar calendar must pass 91 days, 31¼ quarters, and 6 fen before it registers the vernal equinox; by then the Sun in the sky has already advanced more than 2° beyond that point. Likewise, after another 273 days, 93¼ quarters, and 19 fen the mean calendar reaches the autumnal equinox, while in the sky the Sun still falls more than 2° short. Is this not proof that the old method places the vernal equinox two days late and the autumnal equinox two days early?

十年正月辛丑朔，日食，天經等預推京師師見食一分一十秒，應天及各省分秒各殊，惟云南、太原則不見食。其初虧、食甚、復圓時刻亦各異。《大統》推食一分六十三秒，《回回》推食三分七十秒，東局所推止遊氣侵光三十餘秒。而食時推驗，惟天經為密。時將廢《大統》，用新法，於上管理另局曆務代州知州郭正中言：「中曆必不可盡廢，西曆必不可專行。四曆各有短長，當參合諸家，兼收西法。」十一年正月，乃詔仍行《大統曆》，如交食經緯，晦朔弦望，因年遠有差者，旁求參考新法與回回科並存。上年，進天經光祿寺卿，仍管曆務，十四年十二月，天經言：「《大統》置閏，但論月無中氣，新法尤視合朔後先。今所進十五年新曆，其十月、十二月中氣，適交次月合朔時刻之前，所以月內雖無中氣，而實非閏月。蓋氣在朔前，則此氣尚屬前月之晦也。至十六年第二月止有驚蟄一節，而春分中氣，交第三月合朔之後，則第二月為閏正月，第三月為第二月無疑。」時帝已深知西法之密。迨十六年三月乙丑朔日食，測又獨驗。八月，詔西法果密，即改為《大統曆法》，通行天下。未幾國變，竟未施行。本朝用為憲曆。

In year ten, on the xinchou new moon of the first month, a solar eclipse occurred. Tianjing predicted one tenth of the Sun's disk obscured at the capital; Yingtian and the provinces differed in magnitude, and Yunnan and Taiyuan would see no eclipse. The times of first contact, maximum eclipse, and last contact likewise varied by location. The Grand Concordance predicted 1.63 tenths obscured, the Huihui calendar 3.70 tenths, and the Eastern Bureau only about three tenths of atmospheric haze dimming the Sun. At the eclipse, observation showed only Tianjing's prediction to be accurate. As the court was preparing to abandon the Grand Concordance for the new method, Guo Zhengzhong of Daizhou, who headed a separate calendar bureau, argued: "The Chinese calendar cannot be wholly discarded, nor should the Western calendar be enforced to the exclusion of all others. Each of the four systems has its strengths and weaknesses; the proper course is to synthesize them and incorporate Western methods alongside the rest. In the first month of year eleven an edict restored the Grand Concordance Calendar, allowing the new method and Huihui section to be consulted where long use had introduced error in eclipses, planetary positions, and lunation phases. The year before, Tianjing had been promoted to Director of the Court of Imperial Entertainments while retaining charge of the calendar. In the twelfth month of year fourteen he explained: "The Grand Concordance declares an intercalary month whenever a month lacks a mid-term qi, whereas the new method also weighs whether the conjunction precedes or follows that qi. In the fifteenth-year calendar he now submitted, the mid-term qi of the tenth and twelfth months fall just before the next month's conjunction; those months lack a mid-term qi yet are not truly intercalary. When a qi falls before the conjunction, it still belongs to the last day of the preceding month. In year sixteen the second month contains only the Awakening of Insects node, while the vernal-equinox mid-term qi falls after the third month's conjunction; the second month must therefore be an intercalary first month and the third month the true second month. By then the emperor was already convinced of the Western method's precision. When a solar eclipse on the yichou new moon of the third month in year sixteen was tested, the new method alone proved correct. In the eighth month an edict declared the Western method accurate, renamed it the Grand Concordance Calendar Method, and ordered its use throughout the empire. Soon afterward the dynasty fell, and the reform was never fully implemented. The present dynasty has adopted it as the official calendar.

按明制，曆官皆世業，成、弘間尚能建修改之議，萬曆以後則皆專己守殘而已。其非曆官而知曆者，鄭世子而外，唐順之、周述學、陳壤、袁黃、雷宗皆有著述。唐順之未有成書，其議論散見周述學之《曆宗通議》、《曆宗中經》。袁黃著《曆法新書》，其天地人三元，則本之陳壤。而雷宗亦著《合壁連珠曆法》皆會通回回曆以入《授時》，雖不能如鄭世子之精微，其於中西曆理，亦有所發明。邢雲路《古今律曆考》，或言本出魏文魁手，文魁學本慮淺，無怪其所疏《授時》，皆不得其旨也。

Under Ming institutions calendar officers held hereditary posts; during Chenghua and Hongzhi some still urged reform, but after Wanli they merely defended their own methods and resisted change. Outside the hereditary corps, Tang Shunzhi, Zhou Shuxue, Chen Rang, Yuan Huang, and Lei Zong also wrote on calendrics, besides the Prince of Zheng. Tang Shunzhi produced no finished treatise; his views appear scattered in Zhou Shuxue's Calendar School General Discourse and Calendar School Central Classic. Yuan Huang's New Book of Calendar Methods, especially its doctrine of the three origins of heaven, earth, and man, derives from Chen Rang. Lei Zong's Combined Luminaries and Linked Pearls Calendar Method likewise seeks to reconcile the Huihui calendar with the Season Granting system. None equals the Prince of Zheng in refinement, yet each advanced understanding of Chinese and Western calendrics. Some attribute Xing Yunlu's Investigation of Calendrics and Pitch Pipes Through the Ages to Wei Wenku; Wenku's learning was shallow, which explains why his commentary on the Season Granting system so often misses the point.

西洋人之來中土者，皆自稱甌羅巴人。其曆法與回回同，而加精密。嘗考前代，遠國之人言曆法者多在西域，而東南北無聞。唐之《九執律》，元之《萬年曆》，及洪武間所譯《回回曆》，皆西域也。蓋堯命義、和仲叔分宅四方，義仲、義叔、和叔則以隅夷、南交、朔方為限，獨和仲但曰：「宅西」，而不限以地，豈非當時聲教之西被者遠哉。至於週末，疇人子弟分散。西域、天方諸國，接壤西陲，百若東南有大海之阻，又無極北嚴寒之畏，則抱書器而西征，勢固便也。甌羅巴在回回西，其風俗相類，而好奇喜新競勝之習過之。故則曆法與回回同源，而世世增修，遂非回回所及，亦其好勝之慾為之也。義、和既失其守，古籍之可見者，僅有《周髀》範圍，亦可知其源流之所自矣。夫旁搜採以續千百年之墜緒，亦禮秀求野之意也，故備論也。

Westerners who came to China called themselves men of Europa. Their calendrical methods share a common origin with the Huihui system, but with greater refinement. Looking back through history, experts in calendrics from distant lands appear chiefly in the Western Regions; the east, south, and north produced nothing comparable. The Tang Nine Executions pitch standards, the Yuan Ten-Thousand-Years Calendar, and the Huihui Calendar translated in the Hongwu reign all came from the Western Regions. Yao charged Yi, He, Zhong, and Shu to establish observatories in the four quarters; Yi Zhong, Yi Shu, and He Shu were assigned to Yuyi, Nanjiao, and Shuofang, but He Zhong was told only to "dwell in the west" without a territorial limit—surely because the civilizing influence had already extended far to the west. By the late Zhou period the hereditary astronomers had scattered. The Western Regions and the lands of Tianfang adjoin the western frontier; had the southeast not been blocked by ocean and the far north not been forbiddingly cold, carrying astronomical learning westward would have been the easier path. Europa lies west of the Islamic lands; the peoples resemble one another, but the appetite for novelty and competitive mastery runs even stronger there. Their calendars therefore share the Huihui origin yet were refined generation after generation until they surpassed it—another fruit of that competitive spirit. After Yi and He lost their posts, the Zhou Gnomon alone survives among early texts to show how the tradition began. To gather widely and revive a thousand years of broken transmission is the spirit of "seeking knowledge in the villages"—hence this full account.

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