Volume 506 Biographies 293: Astronomers and Mathematicians 1

推步之學，由疏漸密。泰西新法，晚明始入中國，至清而中、西薈萃，遂集大成。聖祖聰明天亶，研究曆算，妙契精微。一時承學之士，蒸蒸鄉化，肩背相望。二百年來，推步之學，日臻邃密，匪特闢古學之榛蕪，抑且補西人之罅漏。嘉慶初，阮元撰疇人傳，後學一再續之，唐、宋以來，於斯為盛。今甄其卓然名家者著於篇，其政事、文學登於列傳及儒林、文苑者；西人官欽天監，廁於卿貳，各自有傳者：不具列焉。

The science of calendrical astronomy grew ever more precise, advancing step by step from rough methods to refined ones. Western astronomical techniques reached China only in the late Ming, but under the Qing Chinese and European scholarship converged until the tradition reached its full flowering. The Kangxi Emperor, endowed with brilliant native genius, studied calendrical astronomy until he mastered its finest subtleties. Scholars who studied under him turned to the discipline in ever-growing numbers, crowding the field shoulder to shoulder. Over two centuries calendrical astronomy grew daily more rigorous—clearing away the tangle of obsolete Chinese methods while also filling gaps in Western theory. Early in the Jiaqing era Ruan Yuan compiled the Biographies of Astronomers and Mathematicians, and later scholars added to it again and again; never since the Tang and Song had the field been so richly documented. Here we record those who stood out as masters of the discipline; those whose careers in government or letters are treated in other biographical chapters, in the Confucian scholars section, or in the literary garden; and Westerners who served in the Directorate of Astronomy at vice-ministerial rank, each with a biography of their own, are not listed in full here.

薛鳳祚，字儀甫，淄川人。少習算，從魏文魁遊，主持舊法。順治中，與法人穆尼閣談算，始改從西學，盡傳其術，因著算學會通正集十二卷，考驗二十八卷，致用十六卷。其曰對數比例者，乃西算以假數求真數之便法也；曰中法四線，以西法六十分為度，不便以十進位，改從古法，以百分為度，所列止正弦、餘弦、正切、餘切，故曰四線。其推步諸書：曰太陽太陰諸行法原，曰木火土三星經行法原，曰交食法原，曰歷年甲子，曰求歲實，曰五星高行，曰交食表，曰經星中星，曰西域回回術，曰西域表，曰今西法選要，曰今法表，皆會中、西以立法。以順治十二年乙未天正冬至為元，諸應皆從以起算。以三百六十五日二十三刻三分五十七秒五微為歲實，黃、赤道交度有加減，恆星歲行五十二秒，與天步真原法同。梅文鼎謂其書詳於法，而無快論以發其趣，蓋其時新法初行，中、西文字輾轉相通，故詞旨未能盡暢。然貫通其中、西，要不愧為一代疇人之功首-{云}-。

Xue Fengzuo, whose courtesy name was Yifu, came from Zichuan. As a youth he studied mathematics under Wei Wenkui and championed the traditional Chinese methods. During the Shunzhi era he discussed mathematics with the French Jesuit Ferdinand Verbiest, converted to Western methods, and mastered them fully. He then produced three works: the twelve-juan Main Collection of the Comprehensive Mastery of Mathematical Learning, the twenty-eight-juan Examination and Verification, and the sixteen-juan Practical Application. His section on logarithmic proportion explains the Western shortcut of using artificial numbers to obtain true values; and his "Chinese method of four lines" reverts from the Western sexagesimal degree to the ancient centesimal division, listing only sine, cosine, tangent, and cotangent—hence the name "four lines." His calendrical works include treatises on solar and lunar motion, on Jupiter, Mars, and Saturn, on eclipses, on sexagenary cycles, on determining the tropical year, on planetary motion, eclipse tables, fixed and culminating stars, Islamic astronomy, and selected Western methods—each synthesizing Chinese and European principles into a unified system. He took the winter solstice of Shunzhi 12 (1655) as his epoch, and derived all calendrical correspondences from that starting point. He set the tropical year at 365 days, 23 ke, 3 fen, 57 miao, and 5 wei, with adjustments for the obliquity of the ecliptic and a precession of 52 seconds per year—matching the True Origins of Celestial Paces method. Mei Wending observed that Xue's works are meticulous in technique but lack the lively exposition that would reveal their deeper interest—understandable, since the new Western methods had only just arrived and terms were still passing awkwardly between Chinese and European usage. Yet in uniting Chinese and Western learning he remains without question the leading figure of his generation among astronomers and mathematicians.

鳳祚定歲實秒數為五十七，與奈端合，與穆尼閣以為四十五秒者不同，則其學非墨守穆氏可知。或譏其謹守穆尼閣成法，依數推衍，非篤論也。

Fengzuo set the fractional seconds of the tropical year at fifty-seven, agreeing with Newton rather than Verbiest's figure of forty-five—showing that he did not merely parrot Verbiest's teachings. Critics who charge him with merely following Verbiest's formulas and pushing numbers without insight are not offering fair judgment.

杜知耕，字端甫，號伯瞿，柘城舉人。精研幾何，以利瑪竇、徐光啟所譯幾何原本復加刪削，作幾何論約七卷，後附十條，則知耕所作也。言其法似為本書所無，其理實涵各題之內，非能於本書之外別生新義也。稱後附者，以別於丁氏、利氏之增題也。又雜取諸家算學，參以西人之說，依古九章為目，作《數學鑰》六卷。言數非圖不明，圖非指不明，圖中用甲乙等字作志者，代指也，故其書於圖解尤詳。梅文鼎稱其圖註九章，頗中肯綮雲。

Du Zhigeng, whose courtesy name was Duanfu and style name Boqu, was a provincial graduate of Zhecheng. He specialized in geometry and produced a seven-juan Concise Discourse on Geometry by further abridging Ricci and Xu Guangqi's translation of Euclid; the ten supplementary propositions appended at the end are his own work. Although his methods appear to go beyond the original text, their principles are already implicit in Euclid's propositions; he did not invent doctrines outside the book itself. They are called "appendices" to distinguish them from the supplementary propositions added by Ding and Ricci. He also compiled the six-juan Key to Mathematics, drawing on various schools of mathematics and Western explanations, organized according to the topics of the ancient Nine Chapters. He held that numbers require diagrams to be understood and diagrams require labels—and that the jia, yi, and other symbols in his figures serve as pointers. For this reason his book is especially rich in illustrated explanations. Mei Wending praised his illustrated commentary on the Nine Chapters as striking right at the heart of the matter.

龔士燕，字武任，武進人。少穎異能文，講求性理，旁通算術，發明蔡氏律呂新書，推演黃鍾圜徑、開方密率諸法，而於元太史郭守敬授時術尤得其秘。如求冬至時刻，上推百年加一算，以為歲週三百六十五日二十四刻二十五分之內，滿百年消長一分。核之春秋日食三十七事，多與符合。又如推晦、朔、弦、望，以太陽之盈與太陰之遲，以太陰之疾與太陽之縮皆相並，為同名相從；以太陽之盈與太陰之疾，以太陰之遲與太陽之縮皆相減，為異名相消：乃得盈縮遲疾化為加減時刻之差。以此加減朔望之大、小餘分，得定朔弦望諸時刻。至盈、縮、遲、疾，郭守敬創平、立、定三差，理隱數繁，能審其機括，繪圖以明之。

Gong Shiyan, whose courtesy name was Wuren, came from Wujin. As a youth he was precocious in letters and devoted to Neo-Confucian philosophy while also mastering mathematics. He elaborated Cai Yuan's treatise on pitch pipes, worked out methods for pipe diameters and root-extraction ratios, and above all penetrated the secrets of Guo Shoujing's Season-Granting calendar. To find the moment of the winter solstice, for example, he projected backward century by century, holding that within a tropical year of 365 days, 24 ke, and 25 parts, one part accumulates or diminishes over each hundred years. Checked against the thirty-seven solar eclipses recorded in the Spring and Autumn Annals, his results agreed in most cases. In calculating new moons, first and last quarters, and full moons, he combined the sun's excess with the moon's slowness and the moon's speed with the sun's contraction—adding terms of the same sign; while subtracting the sun's excess from the moon's speed and the moon's slowness from the sun's contraction—canceling unlike terms—thereby converting excess, contraction, speed, and slowness into corrections to the calculated times. Applying these corrections to the remainder terms for new and full moons yielded the definitive times of new moons, quarters, and full moons. Guo Shoujing's three-tier interpolation for excess, contraction, speed, and slowness was conceptually obscure and numerically intricate; Gong Shiyan grasped its inner logic and illustrated it with diagrams.

又如赤道變黃道之法，謂在二至後者，以度率一零八六五除赤道積度變為黃道宿度；在二分後者，以度率一零八六五乘赤道積度變為黃道宿度。凡此授時之術，引伸益明。其餘月離五星等法，與回回、西洋諸算，遇有疑難，無不洞悉。至日、月體徑有大小，交食限數有淺深，具見其奧。且悟唐順之弧容直闊之法，以推求太陰出入黃道，在內在外，不離乎六度。自是一應七政、氣朔、交食諸端，按法而推，百不失一。

In converting equatorial to ecliptic coordinates, after the solstices he divided accumulated equatorial degrees by the ratio 10865 to obtain ecliptic lodge degrees; and after the equinoxes he multiplied accumulated equatorial degrees by the same ratio. In all such techniques of the Season-Granting calendar he extended and clarified Guo's methods. He likewise mastered lunar theory, planetary methods, and Islamic and Western calculations, leaving no difficulty unresolved. He fully grasped the varying apparent diameters of the sun and moon and the depth of eclipse limits. He also applied Tang Shunzhi's method of arc, chord, and width to calculate the moon's latitude north and south of the ecliptic, which never exceeds six degrees. Thereafter, in all calculations involving the seven luminaries, seasonal qi, new moons, and eclipses, his predictions scarcely missed once in a hundred.

康熙六年，詔募天下知算之士，於是入都。其時欽天監用大統算七政多不合天，奉旨在觀像台每日測驗，而金星比算差至十度。因修改古法，乃據七年所測表景推測太盈縮，又據日測五星行度，考其遲疾。彼此推求加減，氣、閏、轉、交諸應，測驗皆與天合。蓋其法亦本郭守敬，太陽為氣應，推冬至日躔用之；太陰週天為轉應，朔望用之；日月地球之運，同在一直線，視點上為交應，推日月食用之；合氣盈、朔虛之奇零為閏應，推閏月用之；此外又有金、木、水、火、土同聚一宿為合應，推五星用之。

In 1667 an edict called for mathematicians from across the empire, and he went to the capital. The Directorate was then using the Datong calendar, whose planetary positions often failed to match observation; ordered to test daily at the observatory, he found Venus off by as much as ten degrees. He revised the old methods accordingly, using seven years of gnomon measurements to determine solar excess and contraction and daily planetary observations to establish their varying speeds. Cross-checking these results, he found that all correspondences—for seasonal qi, intercalation, lunar rotation, and eclipses—matched observation. His system, like Guo Shoujing's, used the sun's motion as the qi correspondence for calculating the winter solstice; the moon's circuit as the rotation correspondence for new and full moons; the alignment of sun, moon, and earth as the crossing correspondence for eclipses; and the fractional remainders of qi excess and new-moon deficit as the intercalation correspondence for leap months; with the conjunction of all five planets in one lodge as a further correspondence for planetary calculations.

修改諸應，取順治元年甲申為元，以應世祖章皇帝撫有中夏之祥，欽天監名為『改應法』。既改氣、閏、轉、交諸應，復改遲、疾限及求差諸法，又改冬至黃道日出分依步中星內法。又盈縮遲疾無積度，日食無時差，皆與天合。台官交章保薦。八年，曆書告成，奏對武英殿，授歷科博士。時有薦西人南懷仁等於朝，及其實測諸術，驗且捷，遂定用西法，而古歷卒不行。

He revised all these correspondences, taking Shunzhi 1 (1644) as the epoch to mark the Shunzhi Emperor's pacification of China. The Directorate named the system the Revised Correspondences Method. Having revised the qi, intercalation, rotation, and crossing correspondences, he also revised the limits for planetary speed and slowness, the methods for calculating differences, and the winter-solstice ecliptic sunrise fraction using the inner method of pacing culminating stars. Excess, contraction, speed, and slowness required no accumulated degrees, and solar eclipses showed no time discrepancy—all matching observation. Observatory officials submitted memorial after memorial recommending him. In 1669 the calendrical treatise was finished; he was received in audience at the Hall of Martial Eminence and appointed Doctor of the Calendar Section. Westerners including Ferdinand Verbiest were then recommended at court; their observational methods proved both accurate and efficient, and the Western calendar was adopted—Gong's revised traditional system never took effect.

十年，以疾歸，著有像緯考一卷、歷言大略一卷。其天體論一卷及闇虛、中星、交食、定朔、五星諸論俱佚。

In 1671 he returned home on account of illness. He left behind the one-juan Examination of Astral Images and Constellations and the one-juan General Outline of Calendrical Discourse. His Treatise on Celestial Bodies and his essays on the dark void, culminating stars, eclipses, fixed new moons, and the five planets are all lost.

王錫闡，字曉菴，吳江人。兼通中、西之學，自立新法，用以測日、月食不爽秒忽。每遇天晴霽，輒登屋臥鴟吻察星象，竟夕不寐。著曉庵新法六卷，序曰；『炎帝八節，曆之始也，而其書不傳。黃帝、虞、夏、殷、周、魯七曆，先儒謂系偽作。今七曆俱存，大指與漢曆相似，而章蔀氣朔，未睹其真，為漢人所託無疑。太初、三統，法雖疏遠，而創始之功，不可泯也。劉洪、姜岌，次第闡明，何、祖專力表、圭，益稱精切。自此南、北曆象，率能好學深思，多所推論，皆非淺近所及。唐曆大衍稍密，然開元甲子當食不食，一行乃為諛詞以自解，何如因差以求合乎？』

Wang Xichan, whose courtesy name was Xiao'an, came from Wujiang. He mastered both Chinese and Western astronomy, devised his own methods, and predicted solar and lunar eclipses to the nearest second. Whenever the sky cleared, he would climb onto the roof ridge to watch the stars and stay awake through the night. He wrote the six-juan New Methods of Xiao'an, whose preface reads: 'The Flame Emperor's eight seasonal divisions marked the origin of calendrics, but no text survives. The seven calendars attributed to the Yellow Emperor, Yu, Xia, Shang, Zhou, and Lu were dismissed by earlier scholars as forgeries. All seven texts survive today and broadly resemble Han calendrics, but their cycle, obscuration, qi, and new-moon systems are clearly Han fabrications. The Taichu and Santong calendars, though crude in method, deserve credit as pioneering efforts. Liu Hong and Jiang Kui developed them further; He Chengtian and Zu Chongzhi concentrated on gnomon tables and measuring rods with ever greater precision. Thereafter calendrical astronomers north and south studied deeply and advanced many theories beyond the reach of superficial minds. The Tang Dayan calendar was somewhat more precise, yet it failed to predict the Kaiyuan eclipse; Yixing offered flattery instead of explanation—why not have adjusted the constants to match observation?'

又曰：『明初元統造大統曆，因郭守敬遺法，增損不及百一，豈以守敬之術果能度越前人乎？守敬治曆，首重測日，余嘗取其表景，反覆布算，前後牴牾。餘所創改，多非密率。在當日已有失食失推之咎，況乎遺籍散亡，法意無徵。兼之年遠數盈，違天漸遠，安可因循不變耶？元氏藝不逮郭，在廷諸臣，又不逮元，卒使昭代大典，踵陋襲偽。雖有李德芳苦爭之，然德芳不能推理，而株守陳言，無以相勝，誠可嘆也！』

He also wrote: 'At the founding of the Ming, Yu Tong produced the Datong calendar from Guo Shoujing's legacy with changes of less than one part in a hundred—did Guo's methods truly surpass all who came before? Guo prioritized solar measurement in calendar-making; I have recalculated from his gnomon data repeatedly and found internal contradictions. Many of the constants I devised are not precise ratios. Even in Guo's own day the calendar missed eclipses and failed predictions; how much worse now that his texts are lost and their rationale cannot be verified? Moreover, as the centuries pass the constants drift further from observation—how can one cling to the old system unchanged? Yuan officials fell short of Guo; Ming ministers fell short of the Yuan—so the calendrical canon of a glorious age descended into mediocrity and inherited error. Li Defang fought hard against this, but he could not argue from principle and merely repeated old formulas, unable to prevail—a truly lamentable outcome!'

又曰：『萬曆季年，西人利氏來華，頗工曆算。崇禎初，命禮臣徐光啟譯其書，有曆指為法原，曆表為法數，書百餘卷，數年而成，遂盛行於世。言曆者莫不奉為俎豆。吾謂西曆善矣，然以為測候精詳可也，以為深知法意未可也。循其理而求通，可也；安其誤而不辨，不可也。姑舉其概：二分者，春、秋平氣之中；二至者，日道南、北之中也。大統以平氣授人時，以盈縮定日躔。西人既用定氣，則分、正為一，因譏中歷節氣差至二日。夫中曆歲差數強，盈縮過多，惡得無差？然二日之異，乃分、正殊科，非不知日行之朓朒而致誤也。歷指直以怫己而譏之，不知法意一也。諸家造曆，必有積年日法，多寡任意，牽合由人。守敬去積年而起自辛巳，屏日法而斷以萬分，識誠卓也。西曆命日之時以二十四，命時之分以六十，通計一日為分一千四百四十，是復用日法矣。至於刻法，彼所無也。近始每時四分之，為一日之刻九十六。彼先求度而後日，尚未覺其繁，施之中曆則窒矣。乃執西法反謂中曆百刻不適於用，何也？且日食時差法之九十有六，與日刻之九十六何與乎？而援以為據，不知法意二也。天體渾淪，初無度分可指，昔人因一日日躔命為一度，日有疾徐，斷以平行，數本順天，不可損益。西人去週天五度有奇，斂為三百六十，不過取便割圜，豈真天道固然？而黨同伐異，必曰日度為非，詎知三百六十尚非天真有此度數乎？不知法意三也。上古置閏，忄互於歲終，蓋歷術疏闊，計歲以置閏也。中古法日趨密，始計月以置閏，而閏於積終，故舉中氣以定月，而月無中氣者即為閏。大統專用平氣，置閏必得其月，新法改用定氣，致一月有兩中氣之時，一歲有兩可閏之月，若辛丑西曆者，不亦盭乎！夫月無平中氣者，乃為積餘之終，無定中氣者，非其月也。不能虛衷深考，而以鹵莽之習，侈支離之學，是以歸餘之後，氣尚在晦；季冬中氣，已入仲冬；首春中氣，將歸臘杪。不得已而退朔一日以塞人望，亦見其技之窮矣，不知法意四也。天正日躔，本起子半，後因歲差，自丑及寅。若夫合神之說，乃星命家猥言，明理者所不道。西人自命歷宗，何至反為所惑，謂天正日躔定起丑初乎？況十二次命名，悉依星象，如隨節氣遞遷，雖子午不妨異地，豈玄枵、鳥咮亦無定位耶？不知法意五也。歲實消長，昉於統天，郭氏用之，而未知所以當用；元氏去之，而未知所以當去。西人知以日行最高求之。而未知以二道遠近求之，得其一而遺其一。當辨者一也。歲差不齊，必緣天運緩疾，今欲歸之偶差，豈前此諸家皆妄作乎？黃、白異距，生交行之進退；黃、赤異距，生歲差之屈伸；其理一也。曆指已明於月，何蔽於日？當辨者二也。日躔盈縮最高，斡運古今不同，揆之臆見，必有定數。不惟日月星應同，但行遲差微，非畢生歲月所可測度耳。西人每詡數千年傳人不乏，何以亦無定論？當辨者三也。日月去人時分遠近，兒徑因分大小，則遠近大小，宜為相似之比例。西法日則遠近差多，而兒徑差少；月則遠近差少，而兒徑差多。因數求理，難會其通。當辨者四也。日食變差，機在交分，日軌交分，與月高交分不同；月高交於本道，與交於黃道者又不同。曆指不詳其理，歷表不著其數，豈黃道一術足窮日食之變乎？當辨者五也。中限左右，日月兒差，時或一東一西。交、廣以南，日月兒差，時或一南一北。此為兒差異向與兒差同向者加減迥別，歷指豈以非所常遇，故置不講耶？萬一遇之，則學者何以立算？當辨者六也。日光射物，必有虛景，虛景者，光徑與實徑之所生也。闇虛恆縮，理不出此。西人不知日有光徑，僅以實徑求闇虛。及至推步不符，复酌損徑分以希偶合。當辨者七也。月食定望，惟食甚為然，虧复四限，距望有差。日食稍離中限，即食甚已非定朔。至於虧复，相去尤遠。西曆乃言交食必在朔、望，不用朓朒次差。當辨者八也。』

He also wrote: 'In the late Wanli era the Jesuit Matteo Ricci came to China, well versed in calendrical astronomy. Early in the Chongzhen reign Xu Guangqi was ordered to translate his works—the Calendar Guide as theoretical foundation and Calendar Tables as numerical methods—more than a hundred juan completed over several years and soon dominant everywhere. Every calendrical scholar treated them as canonical scripture. I grant that the Western calendar is excellent for precise observation, but I do not grant that its authors fully understood the principles behind their methods. One may follow their reasoning to reach understanding; but one must not accept their errors without question. Consider one example: the equinoxes mark the midpoints of mean spring and autumn qi; the solstices mark the sun's southern and northern limits. The Datong calendar used mean qi for seasonal division and excess and contraction for the sun's daily position. The Westerners used fixed qi, conflating equinoxes with solstices, and mocked the Chinese calendar for solar terms off by as much as two days. The Chinese precession constant was too large and the excess-contraction values excessive—some discrepancy was inevitable. Yet the two-day gap arose because equinoxes and solstices belong to different categories—not from ignorance of the sun's varying speed. The Calendar Guide criticized out of partisan spirit without grasping the underlying rationale—its first error. Every traditional calendar relied on accumulated epoch years and day divisors, arbitrarily chosen and fitted by hand. Guo Shoujing discarded epoch years, beginning from the xinsi year, and replaced day divisors with decimal parts—a truly remarkable insight. The Western calendar divides the day into twenty-four hours and each hour into sixty minutes, totaling 1,440 parts per day—a return to the old day-divisor system. They had no equivalent of the Chinese ke division. Only recently did they quarter each hour to produce ninety-six ke per day. They calculate degrees before days—a procedure they find manageable but that would cripple the Chinese system. Yet they insist on Western methods and declare the Chinese hundred-ke system useless—on what grounds? And what connection does the ninety-six in the eclipse time-difference formula bear to the ninety-six ke of the day? Yet they cite it as proof without grasping the underlying rationale—its second error. Heaven is a single sphere, originally without fixed degrees to mark; the ancients therefore defined one day's solar travel as one degree. Because the sun moves unevenly, they used mean motion—the reckoning follows heaven itself and cannot be arbitrarily adjusted. The Westerners trimmed a little more than five degrees from the celestial circuit and rounded it to 360—merely a convenient way to divide the circle. How could that truly be heaven's own measure? Yet in partisan spirit they attack rivals and declare solar degrees wrong—do they not realize that even 360 is not a number heaven truly ordained? Failure to grasp the underlying rationale—its third error. In earliest times intercalary months were placed mutually at year's end, because calendar methods were still crude and intercalation was reckoned by the year. In middle antiquity methods grew steadily finer; intercalation was reckoned by month and placed at the end of accumulated surplus. Months were fixed by median qi, and any month lacking median qi was declared intercalary. The Great Uniformity calendar used mean qi exclusively and always placed intercalation in the correct month. The new methods switched to true qi, producing months with two median qi and years with two eligible intercalary months—as in the xinchou Western calendar. Is that not perverse? A month without mean median qi marks the end of accumulated surplus; one without true median qi is not a proper month at all. Unable to investigate with open-minded rigor, they fall back on crude habits and indulge in fragmented learning—so that after returning the remainder, the qi still falls on the new moon; the midwinter median qi has already shifted into mid-winter's second month; and the first-spring median qi is about to slip back to the end of the twelfth month. Forced to retreat the new moon by one day to satisfy public expectation, they revealed the limits of their art—failure to grasp the underlying rationale, its fourth error. The sun's position at celestial New Year originally began at half zi; later, through precession, it shifted from chou toward yin. The doctrine of 'harmonizing with the spirit' is vulgar astrological chatter—serious thinkers do not entertain it. Westerners style themselves masters of calendrical astronomy—how could they be taken in and insist that the celestial New Year's solar position must begin at the start of chou? Moreover the twelve celestial stations are all named from star patterns—if they shifted with the solar terms, zi and wu might tolerate different locations, but would Xuanxiao and Niaozhui also lose fixed positions? Failure to grasp the underlying rationale—its fifth error. Variation in the true length of the year began with the Tongtian calendar; Guo Shoujing adopted it without knowing why it should be used; the Yuan calendar makers discarded it without knowing why it should be discarded. Westerners know to derive it from the sun's apogee. Yet they do not know to derive it from the differing distances of the two paths—they grasp one factor and neglect the other. This is the first matter requiring clarification. Precession is not uniform; it must arise from the varying speed of celestial motion. To reduce it to mere accidental error—is every earlier school to be dismissed as fabricating nonsense? The differing distances between the yellow and white paths produce the advance and retreat of syzygies; the differing distances between the yellow and red paths produce the flexing and extension of precession; the underlying principle is the same. The Calendar Guide already explains this for the moon—why remain blind to it for the sun? This is the second matter requiring clarification. The sun's excess-contraction and apogee have revolved differently through past and present ages; by reason there must be a fixed value. Not the sun and moon alone but the stars as well should follow the same law—the differences in speed are slight and cannot be measured within a single lifetime. Westerners boast that their tradition has never lacked successors for thousands of years—why then do they still have no settled conclusion? This is the third matter requiring clarification. As the sun and moon vary in fractional distance from the observer, their apparent diameters vary in size—distance and size ought therefore to stand in similar proportion. In Western methods the sun shows a large difference in distance but a small difference in apparent diameter; for the moon, a small difference in distance but a large difference in apparent diameter. Seeking principle from the numbers alone, one can hardly grasp the underlying coherence. This is the fourth matter requiring clarification. Parallax variation in solar eclipses turns on node distance—the sun's orbital node distance differs from the moon's altitude node distance; and the moon's altitude node on its own path differs again from that at the ecliptic intersection. The Calendar Guide does not explain this principle and the Calendar Tables do not give its numbers—can ecliptic methods alone exhaust the variations of solar eclipses? This is the fifth matter requiring clarification. To the left and right of the central limit, solar and lunar parallax may at times run one east and one west. South of the intersection breadth, solar and lunar parallax may at times run one south and one north. Because parallax in opposite directions and in the same direction require very different additions and subtractions—does the Calendar Guide pass over them simply because they are rarely encountered? If such a case occurs once in ten thousand, how are students to establish their calculations? This is the sixth matter requiring clarification. When sunlight strikes an object it always casts a virtual image, produced by the light radius together with the real radius. The dark shadow cone is always contracted—the principle admits no exception. Westerners do not recognize that the sun has a light radius and compute the shadow cone from the real radius alone. When their calculations fail to match observation, they trim the radius parts hoping for a chance fit. This is the seventh matter requiring clarification. A lunar eclipse falls at exact full moon only at greatest eclipse; the four limits of beginning and end of eclipse stand at varying distances from full moon. For solar eclipses, once the path departs even slightly from the central limit, greatest eclipse no longer falls at exact new moon. At the limits of beginning and end, the distance is especially great. The Western calendar nevertheless insists that eclipses must fall at new and full moon and makes no use of the secondary waxing-and-waning correction. This is the eighth matter requiring clarification.'

又曰：『語云：「步歷甚難，辨歷甚易。」蓋言象緯森羅，得失無所遁也。據彼所說，亦未嘗自信無差。五星經度，或失二十餘分，躔離表驗，或失數分，交食值此，所失當以刻計；凌犯值此，所失當以日計矣。故立法不久，違錯頗多，餘於歷說已辨一二。乃癸卯七月望食當既不既，與夫失食失推者何異乎？且譯書之初，本言取西曆之材質，歸大統之型範，不謂盡隳成憲，而專用西法，如今日者也。餘故兼採中、西，去其疵類，參以己意，著曆法六篇，會通若干事，改正若干事，表明若干事，增輯若干事，立法若干事。舊法雖舛，而未遽廢者，兩存之；理雖可知，而上下千年不得其數者，缺之；雖得其數，而遠引古測，未經目信者，別見補遺，而正文仍襲其故。為日一百幾十有幾，為文萬有千言，非敢妄雲窺其堂奧，庶幾初學之津梁也。』

He also said: 'As the saying goes: "Computing a calendar is very hard; judging one is very easy." ' It means that the stars stand densely arrayed in the sky—right and wrong cannot escape them. Judging by their own statements, they have never been fully confident of their own accuracy. The five planets' ecliptic longitudes may err by more than twenty minutes, and transit tables by several minutes—when eclipses are affected, the error should be reckoned in ke; when occultations are affected, the error should be reckoned in days. The system has not long been established and errors are numerous—I have already addressed one or two of them in my Discourse on Calendars. Yet the full-moon eclipse of the seventh month of guimao should not have been total—how does that differ from missing an eclipse or botching a prediction? Moreover, when the books were first translated the aim was to take the substance of the Western calendar and fit it to the Great Uniformity model—not to destroy established institutions entirely and rely exclusively on Western methods, as is done today. I therefore drew on both Chinese and Western sources, removed their defects, added my own views, and wrote six treatises on calendar methods—harmonizing, correcting, clarifying, supplementing, and establishing new rules in various matters. Where old methods are wrong but not yet ready for abolition, both versions are preserved; where the principle is clear but the numbers cannot be obtained across a thousand years, gaps are left; where numbers are obtained but rest on ancient measurements at far remove and not personally verified, supplements are given separately while the main text retains the old reckoning. The work took more than a hundred days and runs to more than ten thousand words. I do not dare claim to have glimpsed the inner sanctum; I hope only to serve as a bridge for beginners.'

其法：度法百分，日法百刻，週天三百六十五度二十五分六十五秒五十九微三十二纖，內外準分三十九分九十一秒四十九微，次準九十一分六十八秒八十六微，黃道歲差一分四十三秒七十三微二十六纖。列宿經緯：角一十度七十三分七十九秒，南二度一分二十三秒，亢一十度八十二分二十四秒，北三度一分一秒，氐一十八度一十六分一十四秒，北四十三分九十六秒，房四度八十三分六十三秒，南五度四十六分一十九秒，心七度六十六分二秒，南三度九十七分三十八秒，尾一十五度八十二分七十八秒，南一十五度二十一分九十秒，箕九度四十六分九十六秒，南六度五十九分四十九秒，南斗二十四度一十九分八十二秒，南三度八十八分九十三秒，牽牛七度七十九分五十五秒，北四度七十五分一十七秒，婺女一十一度八十二分二秒，北八度二十分五十九秒，虛一十度一十二分九十一秒，北八度八十二分七十秒，危二十度四十一分四秒，北一十度八十五分六十二秒，營室一十五度九十二分二十秒，北一十度七十一分七十一秒。

His system used a degree divisor of one hundred parts and a day divisor of one hundred ke; the celestial circuit was 365° 25′ 65″ 59 micro-units 32 fibrils; inner and outer standard parts 39′ 91″ 49 micro-units; secondary standard 91′ 68″ 86 micro-units; and ecliptic precession 1′ 43″ 73 micro-units 26 fibrils. Fixed-star coordinates: Horn—10° 73′ 79″ longitude, 2° 01′ 23″ south latitude; Neck—10° 82′ 24″, 3° 01′ 01″ north; Root—18° 16′ 14″, 43′ 96″ north; Room—4° 83′ 63″, 5° 46′ 19″ south; Heart—7° 66′ 02″, 3° 97′ 38″ south; Tail—15° 82′ 78″, 15° 21′ 90″ south; Winnowing Basket—9° 46′ 96″, 6° 59′ 49″ south; Southern Dipper—24° 19′ 82″, 3° 88′ 93″ south; Leading Ox—7° 79′ 55″, 4° 75′ 17″ north; Maid—11° 82′ 02″, 8° 20′ 59″ north; Emptiness—10° 12′ 91″, 8° 82′ 70″ north; Rooftop—20° 41′ 04″, 10° 85′ 62″ north; Encampment—15° 92′ 20″, 10° 71′ 71″ north.

先是曉菴新法未成，作歷說六篇，歷策一篇，其說精核，與新法互有詳略。又隱括中、西步術，作大統西曆啟蒙。丁未歲，因推步大統法作丁未歷稿。辛酉八月朔日食，以中、西法及己法豫定時刻分秒，至期，與徐發等以五家法同測，己法獨合，作推步交朔測小記。又以治歷首重割圜，作圜解。測天當據儀晷，造三晷，兼測日、月、星，因作三辰晷志。俱能究術數之微奧，補西人所不逮。與同時青州薛鳳祚齊名，稱『南王北薛』雲。歷策有云：『每遇交會，必以所步、所測課較疏密，疾病寒暑無間，變週、改應、增損、經緯、遲疾諸率，於茲三十年所。』亦可以想見作者實測之詣力矣。

Earlier, before Xiao'an's new methods were complete, he wrote six essays in the Discourse on Calendars and one in Calendar Strategy—arguments precise and penetrating, each complementing the new methods with greater or lesser detail. He also compactly summarized Chinese and Western computational methods in Introduction to the Great Uniformity and Western Calendar. In the dingwei year he applied the Great Uniformity step methods to write the Dingwei Calendar Draft. At the new-moon solar eclipse of the eighth month of xinyou he predetermined the hour, minute, and second using Chinese, Western, and his own methods; when the day came he observed with Xu Fa and others using five schools' methods—his alone matched—and he wrote a brief record of computational conjunction and new-moon observation. Because calendar-making above all depends on dividing the circle, he also wrote Circle Solution. Heavenly observation should rely on instruments and gnomons; he built three gnomons for joint observation of sun, moon, and stars and wrote the Record of the Three Luminaries Gnomon. All these works plumb the subtle depths of calendrical arithmetic and remedy what Westerners had not reached. He shared equal fame with his contemporary Xue Fengzuo of Qingzhou, and the pair were known as 'Wang in the south, Xue in the north.' Calendar Strategy says: 'At every conjunction he compared computed and observed results for accuracy; through sickness, cold, and heat without interruption he revised cycle periods, epochs, increments and decrements, longitudes and latitudes, and slow-and-fast ratios—for some thirty years.' One can thus imagine the author's mastery and dedication in actual observation.

潘檉樟，字力田。與王錫闡同邑友善。錫闡嘗館其家，講論算法，常窮日夜。檉樟著辛丑歷辨曰：『昔堯命羲和，曰以閏月定四時成歲，蓋曆法首重置閏。而春秋傳曰：「先王之正時也，履端於始，舉正於中，歸餘於終。」所謂始者，取氣朔分齊為曆元也；所謂中者，月以中氣為定，無中氣者則為閏也；所謂終者，積氣盈、朔虛之數而閏生焉也。自漢以降，歷術雖屢變，未有能易此者。唯西域諸歷則不然，其法有閏年、有閏日，而無閏月。蓋中歷主日，而西曆主度，不可強同也。今之為西曆者，乃以日躔求定氣、求閏月，不惟盡廢中國之成憲，而亦自悖西域之本法矣。故十餘年來，宮度既紊，氣序亦訛。如戊子之閏三月也，而置在四月；庚寅之閏十一月也，而置在明年之二月；癸巳之閏七月也，而置在六月；己亥之閏正月也，而置在三月。其為舛誤，何可勝言！然非深於歷者，未易指摘。至於辛丑之閏月，則其失顯然無以自解矣。何也？閏法論平氣而不當論定氣，若以平氣，則是年小雪在十月晦，冬至在十一月朔，而閏在兩月之間。所謂閏前之月中氣在晦，閏後之月中氣在朔者也。今以定氣，則秋分居九月朔，故預於七月朔置閏，然後秋分仍在八月，而霜降、小雪各歸其月。無如大寒定氣乃在十一月朔，而十二月又無中氣，既不可再置一閏，則是同一無中氣之月，而或閏或否。彼所云太陽不及交宮即置為閏者，何獨於此而自背其法乎？蓋孟秋非歸餘之終，故天正不能履端於始，地正不能舉正於中也。如此，則四時不定，歲功不成，而閏法又安用之？且壬寅正月，定朔舊法在丙子丑初，即彼法亦在丙子子正，則辛丑之季冬當為大盡，而明年正月中氣复移於今歲之秒。彼亦自覺其未安，故進歲朔於乙亥，而季冬為小盡之月，皆所謂欲蓋彌彰者耳。即辛丑歲朔，以彼法推，當會於亥正，而今在戌正，差至六刻，其他牴牾，更難枚舉。噫！作法如是，而猶自以為盡善，可乎？蓋其說以日行盈縮為節氣短長，每遇日行最盈，則一月可置一氣，是古有氣盈、朔虛，而今更有氣虛、朔盈矣。然或晦朔兩節而中氣介其間。如丙戌仲冬，去閏稍遠，猶可不論；獨辛丑仲冬，冬至、大寒俱在晦朔，去閏最近，進退無據。苟且遷就，有不勝其弊者。夫閏法之主平氣，行之已數千年矣，今一變其術，未久而輒窮，至於無可如何，則又安取紛更為也！』檉樟後坐法死。弟耒，亦學曆算，見文苑傳。

Pan Chengzhang, whose courtesy name was Litian. He was on friendly terms with Wang Xichan, being from the same district. Xichan once lodged at his home and discussed calculation methods, often through entire days and nights. Chengzhang wrote the Xinchou Calendar Discourse, saying: 'In antiquity Yao charged Xi and He to fix the four seasons and complete the year by intercalary months—calendar methods from the outset give foremost weight to intercalation. Yet the Zuo Commentary says: "When the former kings corrected the seasons, they tread the beginning at the start, raise correctness at the middle, and return the remainder at the end." ' What is called 'start' means taking the alignment of qi and new moon as the calendar epoch; what is called 'middle' means fixing the month by median qi—a month without median qi is intercalary; what is called 'end' means that when accumulated qi surplus and new-moon deficit reach their limit, intercalation arises. From the Han dynasty onward, though calendar methods changed repeatedly, none could alter this principle. Only the various Western-region calendars differ—their systems have intercalary years and intercalary days, but no intercalary months. The Chinese calendar is day-centered and the Western calendar degree-centered—they cannot be forcibly equated. Those who now compile the Western calendar seek true qi and intercalary months from solar motion—not only abolishing China's established institutions but contradicting the Western region's own original methods. For more than ten years lodge degrees have fallen into disorder and the sequence of qi has gone wrong. In wuzi, intercalation should have fallen in the third month but was placed in the fourth; in gengyin, intercalation should have fallen in the eleventh month but was placed in the second month of the following year; in guisi, intercalation should have fallen in the seventh month but was placed in the sixth. in jihai, intercalation should have fallen in the first month but was placed in the third. such errors are beyond reckoning! yet only those deeply versed in calendrics can readily expose them. but in xinchou, the error in placing the intercalary month is plainly indefensible. Why is this? Intercalation law treats mean qi, not true qi; under mean qi, that year Minor Snow would fall on the last day of the tenth month, the winter solstice on the first of the eleventh, and intercalation would fall between the two months. This is the principle that before intercalation the month's median qi falls on the last day, and after intercalation on the first. Under true qi, the autumn equinox falls on the first of the ninth month, so they place intercalation early, at the first of the seventh, to keep the equinox in the eighth month and restore Frost Descent and Minor Snow to their proper months. Yet Major Cold falls on the first of the eleventh month, and the twelfth month again lacks median qi; unable to add another intercalation, they leave the same kind of month sometimes intercalary and sometimes not. They say that when the sun fails to reach the palace of intersection, intercalation is required—why do they alone abandon that rule here? The first month of autumn does not mark the end of the accumulated remainder, so the heavenly year cannot begin at the proper start, nor the earthly year hold its proper middle. Then the four seasons lose their order, the year's work cannot be completed, and what use remains for intercalation law? In renyin, the old fixed-new-moon method puts the first month at the beginning of the second quarter of bingzi; even by their own reckoning it falls at the first quarter of bingzi, so xinchou's last winter month should be a full month—yet next year's first-month median qi shifts back into the final moments of this year. Finding their own method unsatisfactory, they advanced the year's new moon to yihai and made the last winter month short—a classic case of trying to cover up only to reveal more. For xinchou's year new moon, their method would place it at the first quarter of hai, yet it falls at the first quarter of xu—a six-quarter error; other contradictions are beyond counting. Alas! To craft a method like this and still call it perfect—is that acceptable? Their doctrine treats the sun's varying speed as the shortening and lengthening of qi; when the sun moves fastest, one month can hold only one qi—in antiquity there were qi surplus and new-moon deficit, but now qi deficit and new-moon surplus appear as well. Yet sometimes two qi fall on the last and first days with median qi sandwiched between. In bingxu's mid-winter, far from intercalation, one might overlook the point; but in xinchou's mid-winter, the winter solstice and Major Cold both fall on the last and first days, closest to intercalation, leaving no firm ground for either course. Patchwork compromises produce evils that cannot be borne. Intercalation law has rested on mean qi for thousands of years; change the method and it is soon exhausted, until nothing can be done—why then pursue endless revisions!' Chengzhang was later executed under the law. His younger brother Lei also studied calendrics and calculation; see the Literary Worthies biography.

方中通，字位伯，桐城人。集諸家之說，著數度衍二十四卷，附錄一卷。言：『九章皆出於句股，環矩以為圓，合矩以為方，方數為典。以方出圓，句股之所生也；少廣，方圓所出也。方田、商功，皆少廣所出。一方一圓，其間不齊，始出差分，而均輸對差分之數，盈朒借差求均。又差分、均輸所出，而以方程濟其窮。度量衡原出黃鍾，粟佈出焉，黃鍾出於方圓者也。』又言：『古法用竹徑一寸長六分二百七十一而成六觚為一握，後世有珠算而古法亡矣。泰西之筆算、籌算，皆出九九。尺算即比例規，出三角。乘莫善於籌，除莫善於筆，加減莫善於珠，比例莫善於尺。』其珠算歸法，三一三十一，四一二十二之類，『十』字俱作『餘』字。其尺算以三尺交加，取數祗用平分一線。時廣昌揭暄亦明算術，與中通論難日輪大小，得光肥影瘦之故，及古今歲差之不同，須測算消長以齊之。一晝夜人一萬三千五百息，每息宗動天行十萬里有奇。別錄為一書，曰揭方問答。

Fang Zhongtong, whose courtesy name was Weibo, was a native of Tongcheng. He gathered the theories of various schools and wrote Shudu Yan in twenty-four juan, with one supplementary juan. He wrote: 'The Nine Chapters all derive from right triangles; circling the square yields a circle, joining squares yields a square, and square numbers are the standard. deriving the circle from the square is what right triangles produce; Finding side and breadth derives from square and circle. Field measurement and construction works both derive from finding side and breadth. Where square and circle do not align, proportional division first arises; fair distribution counters proportional division, and surplus and deficit borrow differences to seek balance. These in turn derive from proportional division and fair distribution, while simultaneous equations resolve what they cannot. Weights and measures originally derived from the Yellow Bell, and millet spread out from it—the Yellow Bell itself comes from square and circle.' He also wrote: 'The ancient method used bamboo one inch in diameter and two hundred seventy-one six-tenths long, shaped into a hexagonal grip; later generations had the abacus, and the ancient method was lost. Western pen calculation and tally calculation both derive from the nine-nine multiplication table. Ruler calculation is the proportional compass, derived from trigonometry. For multiplication nothing beats tallies, for division nothing beats the pen, for addition and subtraction nothing beats the abacus, and for proportion nothing beats the ruler.' In his abacus division mnemonics—such as 'three-one thirty-one, four-one twenty-two'—the character for 'ten' is always written as 'remainder.' In his ruler calculation three rulers are joined crosswise, and numbers are taken using only a single bisecting line. At the time Jie Xuan of Guangchang also mastered calculation; he and Zhongtong debated the size of the solar wheel, explaining why light appears broad and shadow narrow, and that ancient and modern precession differ and must be reconciled by measurement and calculation of their waxing and waning. In one day and night a person takes thirteen thousand five hundred breaths; with each breath the Primum Mobile travels more than one hundred thousand li. This was separately recorded as a book called Questions and Answers with Jie on Surveying.

揭暄，字子宣，廣昌人。著璇璣遺述七卷，一名寫天新語。論日月東行如槽之滾丸，而月質不變。又謂七政之小輪。皆出自然，如盤水之運旋而周遭，以行疾而成旋渦，遂成留逆。於五星西行，日月盈縮，皆設譬多方，言之近理。康熙己巳，以草稿寄梅文鼎，抄其精語為一卷，稱其『深明西術，而又別有悟入，其言多古今所未發』。卒年逾八十。

Jie Xuan, whose courtesy name was Zixuan, was a native of Guangchang. He wrote Xuanji Yishu in seven juan, also known as Xietian Xinyu. He argued that the sun and moon move eastward like a ball rolling in a trough, while the moon's substance remains unchanged. He also spoke of the minor epicycles of the seven luminaries. all arise naturally, like water swirling in a basin and circling about—as swift motion forms a vortex, station and retrograde motion result. On the five planets' westward motion and the sun and moon's varying speed, he offered many analogies that sound plausible. In the jisi year of Kangxi he sent a draft to Mei Wending, who copied its finest passages into one juan and praised him as 'deeply versed in Western methods yet with insight of his own; his remarks mostly say what antiquity and the present had never articulated.' He died aged over eighty.

梅文鼎，字定九，號勿庵，宣城人。兒時侍父士昌及塾師羅王賓仰觀星象，輒了然於次舍運轉大意。年二十七，師事竹冠道士倪觀湖，受麻孟旋所藏台官交食法，與弟文鼐、文鼏共習之。稍稍發明其立法之故，補其遺缺，著歷學駢枝二卷，後增為四卷，倪為首肯。

Mei Wending, whose courtesy name was Dingjiu and style name Wu'an, was a native of Xuancheng. As a child, while attending his father Shichang and tutor Luo Wangbin, he would gaze at the stars and immediately grasp the broad pattern of lodges and their motion. At twenty-seven he took the Bamboo-Crowned Daoist Ni Guanhu as his master, received the Bureau Methods for Conjunctions and Eclipses kept by Ma Mengxuan, and studied them with his brothers Wenlin and Wenji. He gradually explained the reasons behind the methods, filled their gaps, and wrote Calendar Learning Parallels in two juan, later expanded to four; Ni gave his approval.

值書之難讀者，必欲求得其說，往往廢寢忘食。殘編散帖，手自抄集，一字異同，不敢忽過。疇人子弟及西域官生，皆折節造訪，有問者，亦詳告之無隱，期與斯世共明之。所著曆算之書凡八十餘種。

Whenever he encountered a difficult book, he had to grasp its meaning, often neglecting sleep and food. From scattered fragments and loose sheets he copied and assembled by hand, not daring to overlook even a single character's difference. Disciples of calendrical families and Western-region official students all humbled themselves to visit him; to any who asked, he explained fully without concealment, hoping to clarify matters together with the age. The books on calendrics and calculation that he wrote numbered more than eighty kinds.

讀元史授時曆經，歎其法之善，作元史歷經補註二卷。又以授時集古法大成，因參校古術七十餘家，著古今曆法通考七十餘卷。授時以六術考古今冬至，取魯獻公冬至證統天術之疏，然依其本法步算，與授時所得正同，作春秋以來冬至考一卷。元史西征庚午元術，西征者，謂太祖庚辰；庚午元者，上元起算之端也。曆志訛太祖庚辰為太宗，不知太宗無庚辰也。又訛上元為庚子，則於積年不合。考而正之，作庚午元算考一卷。授時非諸古術所能方，郭守敬所著歷草，乃歷經立法之根，拈其義之精微者，為郭太史歷草補註二卷。立成傳寫魯魚，不得其說，不敢妄用，作大統立成註二卷。授時術於日躔盈縮、月離遲疾，並以垛積招差立算，而九章諸書無此術，從未有能言其故者，作平立定三差詳說一卷，此發明古法者也。唐九執術為西法之權輿，其後有婆羅門十一曜經及都聿利斯經，皆九執之屬。在元則有札馬魯丁西域萬年術，在明則馬沙亦黑、馬哈麻之回回術、西域天文書，天順時具琳所刻天文實用，即本此書，作回回曆補註三卷，西域天文書補註二卷，三十雜星考一卷。表景生於日軌之高下，日軌又因裡差變移，作四省表景立成一卷。周髀所言裡差之法，即西人之說所自出，作周髀算經補註一卷。渾蓋之器，最便行測，作渾蓋通測憲圖說訂補一卷。西國以太陽行黃道三十度為一月，作西國日月考一卷。西術中有細草，猶授時之有通軌也，以歷指大意隱括而注之，作七政細草補註三卷。新法有交食蒙求、七政蒙引二書，並逸，作交食蒙求訂補二卷、附說二卷。監正楊光先不得已日食圖，以金環食與食甚分為二圖，而各有時刻，其誤非小，作交食作圖法訂誤一卷。新法以黃道求赤道交食，細草用儀象志表，不如弧三角之親切，作求赤道宿度法一卷。謂中、西兩家之法，求交食起復方位，皆以東西南北為言。然東西南北惟日月行至午規而又近天頂，則四方各正其位。非然，則黃道有斜正之殊，而自虧至复，經歷時刻，輾轉遷移，弧度之勢，頃刻易向。且北極有高下，而隨處所見必皆不同，勢難施諸測驗。今別立新法，不用東西南北之號，惟人所見日月員體，分為八向，以正對天頂處為上，對地平處為下，上下聯爲直線，作十字橫線，命之曰左、曰右，此四正向也；曰上左、上右，曰下左、下右，則四隅向也。乃以定其受蝕之所在，則舉目可見，作交食管見一卷。太陽之有日差，猶月離交食之有加減時，因表說含糊有誤，作日差原理一卷。火星最為難算，至地谷而始密，解其立法之根，作火緯圖法一卷。訂火緯表記，因及七政，作七政前均簡法一卷。天問略取緯不真，而列表從之誤，作黃赤距緯圖辨一卷。新法帝星、句陳經緯刊本互異，作帝星句陳經緯考異一卷。測帝星、句陳二星為定夜時之簡法，作星軌真度一卷。以上皆以發明新法算書，或正其誤，或補其缺也。

Reading the Shoushi Calendar Classic in the Yuan History, he admired the excellence of its methods and wrote Supplementary Notes to the Yuan History Calendar Classic in two juan. Seeing that Shoushi gathered ancient methods into a great synthesis, he collated more than seventy ancient methods and wrote Comprehensive Inquiry into Ancient and Modern Calendar Methods in more than seventy juan. Shoushi used six methods to investigate ancient and modern winter solstices; taking Duke Xian of Lu's winter solstice to prove the deficiencies of the Tongtian method, yet following its original method in step calculation, the result matched Shoushi exactly—he wrote Investigation of Winter Solstices since the Spring and Autumn Annals in one juan. The Yuan History's Western Campaign Gengwu Origin method—'Western Campaign' refers to Taizu's gengchen year; 'Gengwu Origin' is the starting point for upper-origin reckoning. The Calendar Treatise wrongly makes Taizu's gengchen into Taizong's, not knowing Taizong had no gengchen year. It also wrongly makes the upper origin gengzi, which then does not agree with the accumulated years. He investigated and corrected this, writing Investigation of the Gengwu Origin Calculation in one juan. Shoushi cannot be matched by ancient methods; Guo Shoujing's Calendar Draft was the root of the calendar classic's methods—selecting the subtle points of its meaning, he wrote Supplementary Notes to Grand Astrologer Guo's Calendar Draft in two juan. The ready tables in transmission had scribal errors and their meaning could not be recovered—he dared not use them arbitrarily, and wrote Notes on the Datong Ready Tables in two juan. Shoushi methods for solar motion excess and deficit and lunar motion slow and fast both use stacked accumulation and interpolation by differences; the Nine Chapters and other books have no such art, and none had ever been able to say why—he wrote Detailed Explanation of the Three Differences (Level, Upright, Fixed) in one juan—this elucidates ancient methods. Tang's Nine Executions method is the source of Western methods; afterward came the Brahmin Eleven Luminaries Sutra and the Zoroastrian Sutra, all belonging to the Nine Executions. In the Yuan there was Jamal al-Din's Western-region perpetual-year method; in the Ming, Mashayihei and Mahama's Huihui methods and Western-region astronomical books; the Tianwen Shiyong engraved by Ju Lin in the Tianshun era was based on this book—he wrote Huihui Calendar Supplementary Notes in three juan, Western-region Astronomical Book Supplementary Notes in two juan, and Investigation of Thirty Miscellaneous Stars in one juan. Shadow length arises from the height and low of the sun's path, and the sun's path in turn shifts with parallax—he wrote Ready Tables for Shadow Length of the Four Provinces in one juan. The parallax method spoken of in Zhou Bi is what Western doctrine derives from—he wrote Supplementary Notes to the Zhou Bi Suan Jing in one juan. The armillary and celestial-globe instruments are most convenient for field measurement—he wrote Revised and Supplemented Illustrated Explanation of the Armillary and Celestial Globe for General Measurement in one juan. Western countries take the sun's thirty degrees along the ecliptic as one month—he wrote Investigation of Western Sun and Moon in one juan. Western methods have detailed procedures, as Shoushi has the general track—summarizing the Calendar Directive's main idea in condensed notes, he wrote Supplementary Notes to the Detailed Procedures of the Seven Luminaries in three juan. The New Methods had Mengqiu and Mengyin on eclipses and the seven luminaries, both lost—he wrote Revised and Supplemented Eclipse Mengqiu in two juan with two juan of appended explanations. Supervisor Yang Guangxian, in his makeshift solar-eclipse diagrams, split the annular eclipse and greatest eclipse into two diagrams, each with its own time—the error is no small one—he wrote Correcting Errors in the Method of Drawing Eclipses in one juan. The New Methods seek ecliptic-to-equator eclipses; the detailed procedures use tables from the Instrument Record, inferior to spherical trigonometry—he wrote Method for Finding Equatorial Lodge Degrees in one juan. He held that Chinese and Western methods alike, in seeking the directions of eclipse beginning and ending, speak only in terms of east, west, south, and north. Yet east, west, south, and north are correct in their positions only when the sun and moon reach the meridian and are near the zenith. Otherwise the ecliptic has variations of obliquity and rectitude, and from first contact to totality the elapsed times shift and turn—the arc's momentum changes direction in an instant. Moreover the north pole has varying altitude, and what is seen everywhere must differ—it is hard to apply to measurement. He now set forth a new method, not using the names east, west, south, and north, but only the round bodies of sun and moon as seen, divided into eight directions—taking the point directly opposite the zenith as 'above' and opposite the horizon as 'below,' linking above and below in a straight line and drawing a cross line, naming them 'left' and 'right'—these four are the cardinal directions; 'Upper-left, upper-right,' 'lower-left, lower-right' are the four diagonal directions. By this he fixed where the eclipse is received—visible at a glance—and wrote Eclipse Tube Observation in one juan. The sun's solar parallax is like the added and subtracted times in lunar motion and eclipses; because the gnomon explanation was vague and wrong, he wrote Principles of Solar Parallax in one juan. Mars is hardest to calculate; only with Tycho did it become precise—explaining the root of its methods, he wrote Diagram Method for Martian Longitude in one juan. Revising the notes on the Martian table and extending to the seven luminaries, he wrote Simplified Method for the Seven Luminaries' Prior Mean in one juan. Tianwen Lüe takes incorrect declination but follows it in tables—he wrote Diagram Discourse on Right Ascension and Declination Distance in one juan. In the New Methods, printed editions of the Imperial Star and Gouchen longitudes and latitudes differ—he wrote Investigation of Variants in the Imperial Star and Gouchen Longitudes and Latitudes in one juan. Measuring the Imperial Star and Gouchen as a simplified method for fixing nighttime hours, he wrote True Degrees of Star Tracks in one juan. All the above elucidate the New Methods' computational books, correcting errors or filling gaps.

康熙己未，明史開局，曆志為錢塘吳任臣分修，經嘉禾徐善、北平劉獻廷、毗陵楊文言，各有增定，最後以屬黃宗羲，又以屬文鼎，摘其訛誤五十餘處，以算草、通軌補之，作明史曆志擬稿一卷。雖為大統而作，實以闡明授時之奧，補元史之缺略也。其總目凡三：曰法原，曰立成，曰推步。而法原之目七：曰句股測望，曰弧天割圜，曰黃赤道差，曰黃赤道內外度，曰白道交週，曰日月五星平立定三差，曰裡差刻漏。立成之目凡四：曰太陽盈縮，曰太陰遲疾，曰晝夜刻，曰五星盈縮。推步之目凡六：曰氣朔，曰日躔，曰月離，曰中星，曰交食，曰五星。

In the jiwei year of Kangxi the Ming History project opened; the Calendar Treatise was assigned to Wu Renchen of Qian Tang, with additions from Xu Shan of Jiahe, Liu Xianting of Beiping, and Yang Wenyan of Piling; lastly it was entrusted to Huang Zongxi, then again to Wending, who extracted more than fifty errors and supplemented them with calculation drafts and general tracks, writing Draft of the Ming History Calendar Treatise in one juan. Though written for the Datong system, it in fact elucidates the profundities of Shoushi and supplements the Yuan History's omissions. Its general headings number three: Origins of Methods, Ready Tables, and Step Calculation. Under Origins of Methods there are seven items: right-triangle surveying, arc-of-heaven circle-cutting, ecliptic-equator difference, inner and outer ecliptic-equator degrees, white-path cycle intersection, level-upright-fixed three differences of sun, moon, and five planets, and parallax clepsydra. Ready Tables has four items: solar excess and deficit, lunar slow and fast, day-night quarters, and five-planet excess and deficit. Step Calculation has six items: qi and new moon, solar motion, lunar motion, culminating stars, eclipses, and five planets.

又作曆志贅言一卷，大意言：『明用大統，實即授時，宜詳元史缺載之事，以補其未備。又回回曆承用三百年，法宜備書。又鄭世子歷學已經進呈，宜詳述。他如袁黃之曆法新書，唐順之、周學述之會通回曆，以庚午元曆之例例之，皆得附錄。其西洋歷方今現行，然崇禎朝徐、李諸公測驗改憲之功，不可沒也，亦宜備載緣起。』

He also wrote one juan of Appended Remarks on the Calendar Treatise, arguing in essence that although the Ming used the Datong calendar, it was really the Shoushi system in practice, and that the Yuan History's omissions should be fully set forth to fill the gaps. The Islamic calendar had been used for three centuries as well, and its methods deserved full treatment. Prince Zheng's calendar studies had already been presented to the throne and ought to be described in detail. Other works, such as Yuan Huang's New Book of Calendrical Methods and the Unified Hui Calendar by Tang Shunzhi and Zhou Xueshu, could be appended following the precedent set for the Gengwu Epoch Calendar. The Western calendar was then in current use, yet the work of Xu, Li, and their colleagues during the Chongzhen reign in observation and calendar reform must not be forgotten; the origins of that reform also deserved full account.'

己巳，至京師，謁李光地於邸第，謂曰；『曆法至本朝大備矣，而經生家猶若望洋者，無快論以發其趣也。宜略仿元趙友欽革象新書體例，作簡要之書，俾人人得其門戶，則從事者多，此學庶將大顯。』因作歷學疑問三卷。

On a jisi day he reached the capital and called on Li Guangdi at his residence, saying: 'Calendrical science is now thoroughly developed in our dynasty, yet classical scholars still stand before it like men gazing at the sea—there is no engaging work to stir their interest. He suggested writing a concise book modeled on the format of Zhao Youqin's New Book of Remodeling the Image, so that anyone could find a doorway into the subject—then more people would take it up, and the discipline might at last flourish widely.' On this basis he wrote Questions on Calendar Studies in three juan.

光地扈駕南巡，駐蹕德州，有旨取所刻書籍回奏，光地匆遽未及攜帶，遂以所辢刻歷學疑問謹呈。奉旨：『朕留心曆算多年，此事朕能決其是非，將書留覽再發。』二日後，召見光地，上云：『昨所呈書甚細心，且議論亦公平，此人用力深矣，朕帶回宮中仔細看閱。』光地因求皇上親加御筆，批駁改定，上肯之。

When Guangdi accompanied the emperor on a southern tour and the court halted at Dezhou, an edict called for printed books to be presented. Guangdi had not brought his in the rush, so he respectfully submitted a freshly printed copy of Questions on Calendar Studies instead. The response came: 'I have devoted myself to calendrics and calculation for many years and can judge such matters myself. Keep the book for my review and submit it again later.' Two days later Guangdi was summoned. The emperor said, 'Yesterday's book shows great care, and the argument is fair. This author has worked deeply. I shall take it back to the palace and read it through carefully.' Guangdi then asked the emperor to annotate and correct it in his own hand, and the emperor agreed.

明年癸未春，駕复南巡，於行在發回原書，面諭光地：『朕已細細看過。』中間圈點塗抹及簽貼批語，皆上手筆也。光地復請此書疵繆所在，上云：『無疵繆，但算法未備。』蓋其書本未完成，故聖諭及之。

The following spring, in the guiwei year, the emperor toured south again. At the traveling palace he returned the original book and told Guangdi in person, 'I have read it carefully.' Every circle, deletion, and pasted marginal note in it was in the emperor's own hand. Guangdi asked again where the book erred. The emperor said, 'There are no errors—but the computational methods are incomplete.' The imperial remark reflected the fact that the book was still unfinished.

未幾，聖祖西巡，問隱淪之士，光地以關中李顒、河南張沐及文鼎三人對。上亦夙知顒及文鼎，乙酉二月，南巡狩，光地以撫臣扈從，上問：『宣城處士梅文鼎焉在？』光地以『尚在臣署』對。上曰：『朕歸時，汝與偕來，朕將面見。』四月十九日，光地與文鼎伏迎河干，越晨，俱召對御舟中，從容垂問，至於移時，如是者三日。上謂光地曰：『歷象算法，朕最留心，此學今鮮知者，如文鼎，真僅見也。其人亦雅士，惜乎老矣！』連日賜御書扇幅，頒賜珍饌。臨辭，特賜『績學參微』四大字。越明年，又命其孫瑴成內廷學習。

Before long, on a western tour the Sage Emperor asked about scholars living in retirement, and Guangdi named three: Li Yong of Guanzhong, Zhang Mu of Henan, and Wending. The emperor already knew Yong and Wending well. In the second month of the yiyou year, during a southern tour, Guangdi attended as provincial governor. The emperor asked, 'Where is the Xuancheng recluse Mei Wending?' Guangdi replied, 'He is still at my office.' The emperor said, 'When I return, bring him with me—I shall see him in person.' On the nineteenth day of the fourth month, Guangdi and Wending waited by the river to receive the emperor. The next morning both were summoned to audience aboard the imperial boat and questioned at length, leisurely and without haste—and so for three days running. The emperor told Guangdi, 'Astronomy and calculation are what I follow most closely. Few people today know this field—someone like Wending is truly a rare find. 'He is a cultivated man as well—alas, he is old!' For several days running the emperor bestowed fans bearing his own calligraphy and sent him choice delicacies. At their parting he was specially granted four characters in the emperor's hand: 'Penetrating Subtlety through Sustained Learning.' The next year he also ordered Wending's grandson Jucheng to study at the Inner Court.

五十三年，瑴成奉上諭：『汝祖留心律曆多年，可將律呂正義寄一部去，令看，或有錯處，指出甚好。夫古帝王有「都俞籲咈」四字，後來遂止有「都俞」，即朋友之間，亦不喜人規勸，此皆是私意。汝等須竭力克去，則學問長進。可並將此意寫與汝祖知之。』恩寵為古所未有。

In the fifty-third year Jucheng received an imperial message: 'Your grandfather has devoted himself to pitch-pipes and calendrics for many years. Send him a copy of Correct Meaning of Pitch Pipes to examine. If he finds mistakes, it would be excellent for him to point them out. Ancient rulers honored the four words 'assent, approve, call out, and dissent'; later only 'assent and approve' remained. Even among friends, people no longer welcome correction—all of this springs from private feeling. You must strive with all your strength to overcome this, and your learning will advance. Write this as well to your grandfather so that he may know my meaning.' Such imperial favor was without precedent.

文鼎圖注各直省及蒙古各地南北東西之差，為書一卷，名分天度裡。地既渾員，則所云二百五十里一度，緯度則然，若經度離赤道遠，則里數漸狹。故惟路正東西行，自有一定算法；路或斜行，則其法不可用為立法。若兩地各有北極高度，又有相距之經度，而無相距里數，是有兩邊一角，而求餘一邊，即可以知斜距之裡。若先有斜距之里數而求經度，是為三邊求角，亦可以知相距之經度。其法並用斜弧三角形立算，可與月食求經度之法相參，而且簡易的確。

Wending mapped and annotated the north-south and east-west differences among the provinces and Mongol regions, producing one juan titled Dividing Celestial Degrees into Li. Because the earth is a sphere, the figure of two hundred fifty li per degree applies to latitude; the farther one moves from the equator in longitude, the narrower each degree becomes in li. Only for routes running due east or west is there a fixed method of calculation; if the route runs obliquely, that method cannot serve as a general law. Given the polar altitude at two places and the longitudinal difference between them, but not the distance in li, one has two sides and one angle and can solve for the remaining side—thus obtaining the oblique distance in li. If one begins with the oblique distance in li and seeks the longitude, it becomes a three-sides-to-angle problem, and the longitudinal difference can be found the same way. The method relies throughout on spherical trigonometry; it compares well with finding longitude from lunar eclipses, and is at once simple and exact.

文鼎於測算之圖與器，一見即得要領，古六合、三辰、四遊之儀，以意約為小制，皆合。又自制為月道儀，揆日測高諸器，皆自出新意。嘗登觀像台，流覽新制六儀，及元郭守敬簡儀、明初渾球，指數其中利病，皆如素習。其書有測器考二卷，又自鳴鐘說一卷，壺漏考一卷，日晷備考一卷，赤道提晷一卷，勿菴揆日器一卷，加時日軌高度表一卷，揆日測說一卷，璇璣尺解一卷，測量定時簡法一卷，勿庵測望儀式一卷，勿庵仰觀儀式一卷，月道儀式一卷。

In surveying diagrams and instruments, Wending grasped essentials at a glance. He reduced the ancient Six Harmonies, Three Luminaires, and Four Wanders instruments to small-scale models by his own design, and all worked correctly. He also devised a lunar-path instrument and various sun-gauging and altitude-measuring devices, all original inventions of his own. Once on the observatory tower he examined the newly made Six Instruments, Guo Shoujing's simplified armillary from the Yuan, and the early Ming celestial globe, and identified their strengths and flaws as if he had long been familiar with them. His writings include Examination of Surveying Instruments in two juan, along with one juan each on self-sounding bells, clepsydra vessels, sundials, the ecliptic gnomon, the Wuan sun-gauging instrument, a table of solar-track altitudes by added hours, a discourse on sun-gauging measurement, an explanation of the armillary scale, a simple method of survey-based timekeeping, the forms of the Wuan surveying and upward-gazing instruments, and the lunar-path instrument.

其說曰：『月道出入於黃道，猶黃道之出入於赤道也。自古及今，未有為之儀器者。今依渾蓋北密南疏之度，以黃極為樞，而月道半在其內，半出其外，則月緯大小之理，及正交、中交、交前、交後之法，可以眾著。儀以銅為之，略如渾蓋，其上盤為月道，亦如渾蓋天盤之黃道圈；其下盤黃道經緯，分宮分度，並以黃極為心，而侭邊以黃緯九十五度少半為限。出黃道南五度少半，月道所到也。』

He explained: 'The Moon's path crosses in and out of the ecliptic just as the ecliptic crosses in and out of the equator. From antiquity until his day, no instrument had been built for it. Following the armillary sphere's denser graduations toward the north and sparser toward the south, with the ecliptic pole as pivot and the lunar path half inside and half outside it, the logic of lunar latitude and the methods for direct, middle, pre-nodal, and post-nodal positions could all be made plainly visible. The instrument was cast in bronze, roughly like an armillary cover; its upper plate represents the lunar path, corresponding to the ecliptic ring on the armillary's upper plate; the lower plate shows ecliptic longitude and latitude by mansion and degree, all centered on the ecliptic pole, with the rim set at a limit of ninety-five and a half degrees of ecliptic latitude. Five and a half degrees south of the ecliptic marks the limit reached by the lunar path.'

禮部郎中李煥鬥嘗從文鼎問曆法，作答李祠部問歷一卷。滄州老儒劉介錫同客天津，問曆法，作答劉文學問天象一卷。又言生平於難讀之書，每手疏而攜諸篋，以待明者問之，於歷學尤多，作思問編一卷。緯度以測日高，因知北極為用甚博，古用二至二分，今則逐日可測，承友人之問，作七十二候太陽緯度一卷。潘天成從文鼎學歷，而苦於布算，作寫歷步曆法一卷授之。又授時步交食式一卷，文鼎季弟文鼏之稿也。步五星式六卷，文鼎與其仲弟文鼐共成之者也。

Li Huandou, a director in the Ministry of Rites, once studied calendrics with Wending; Wending answered his questions in one juan titled Reply to Director Li's Questions on the Calendar. Liu Jiexi, an elder scholar of Cangzhou, while staying with Wending as guests in Tianjin, asked about calendrical methods; Wending answered in one juan titled Literary Liu's Questions on Celestial Phenomena. He also said that throughout his life, whenever he encountered a difficult book he would make handwritten notes and keep them in his satchel against the day a knowledgeable person might ask about them; he had especially many such notes on calendrics, which he compiled as one juan titled Reflections on Questions. Latitude is found by measuring the sun's altitude, from which one sees how widely the north celestial pole is applicable. Antiquity used only the two solstices and two equinoxes; now measurement can be taken day by day. At a friend's request he wrote one juan on Solar Latitude for the Seventy-two Pentades. Pan Tiancheng studied calendrics with Wending but struggled with rod calculation; Wending wrote one juan on Written Calendar Step Methods and gave it to him. There was also one juan of Shoushi Step Methods for Eclipses—a draft by Wending's youngest brother Wenmi. Step Methods for the Five Planets in six juan was completed jointly by Wending and his second brother Wenai.

文鼎每得一書，皆為正其訛闕，指其得失，又古歷列星距度考一卷，從殘壞之本，尋其普天星宿，入宿去極度分，中缺二星，又從閩中林侗寫本補完之，而斷以為授時之法。萬曆中利瑪竇入中國，始倡幾何之學，以點線面體為測量之資，制器作圖，頗為精密。學者張皇過甚，未暇深考，輒薄古法為不足觀；而株守舊法者，又斥西人為異學：兩家之說，遂成隔礙。文鼎集其書而為之說，用籌、用尺、用筆，稍稍變從我法。若三角、比例等，原非中法可賅，特為表出。古法方程，亦非西法所有，則專著論，以明古人之精意不可湮沒。又為九數存古，以著其概。總為中西算學通例一卷。

Whenever Wending acquired a book, he corrected its textual errors, identified its gaps, and assessed its merits and flaws. He also wrote one juan titled Examination of Fixed Star Distances in the Ancient Calendar, working from a damaged original to recover the whole-sky star positions, their mansion entries, and polar distances in degrees and minutes. Two stars were missing; he restored them from a copy made by Lin Tong in Fujian and adjudicated the result by Shoushi methods. During the Wanli reign Matteo Ricci came to China and first promoted the study of geometry, using points, lines, surfaces, and solids as the basis of measurement; the instruments he made and diagrams he drew were notably precise. Scholars exaggerated its claims, without taking time for thorough study, and readily dismissed traditional methods as beneath notice; while those who clung to the old methods denounced Western learning as alien doctrine, so the two schools became walled off from each other. Wending collected their books and wrote commentaries, adapting rod, ruler, and brush calculation gradually to Chinese methods. Trigonometry, proportion, and the like had no native Chinese equivalent; he set them forth separately. Traditional simultaneous equations, which Western methods lacked, he treated in dedicated essays to show that the ancients' fine insights must not be lost. He also wrote Preserving the Ancients in the Nine Numbers to set forth the general picture. These were combined as one juan titled General Principles of Chinese and Western Mathematics.

餘分九種：一，勿庵籌算七卷。二，筆算五卷。皆易橫為直，以便中文。三，度算一卷，原無算例，其弟文鼏補之，而參以嘉禾陳藎謨尺算用法。又有矩算，用一尺一方板，則文鼎所創。四，比例數解四卷。釋穆尼閣所譯之對數。五，三角法舉要五卷。其目有五：曰測量名義，曰算例，曰內容外切，曰或問，曰測量。六，方程論六卷，安溪李鼎徵為刻於泉州。七，幾何摘要三卷，就原本刪繁補遺。八，句股測量二卷，就周髀、海島諸術，錄要以存古意。九，九九數存古十卷，九數即九章隸首之法，僅存者九章之目耳。後有作者，莫能出其範圍。

The remainder comprises nine works: first, Wuan Counting-rods in seven juan. Second, Brush Calculation in five juan. Both convert horizontal notation to vertical, to suit Chinese writing. Third, Degree Calculation in one juan—originally without worked examples, which his brother Wenmi supplied, incorporating Chen Jingmo of Jiahe's ruler-calculation methods. There is also rectangle calculation, using a one-foot square board—an invention of Wending's own. Fourth, Explanation of Proportional Numbers in four juan. It explains the logarithms in Verbiest's translation. Fifth, Essentials of Trigonometry in five juan. It has five sections: Surveying Terminology, Worked Examples, Inscribed and Circumscribed Figures, Questions, and Surveying. Sixth, Treatise on Simultaneous Equations in six juan, printed at Quanzhou by Li Dingzheng of Anxi. Seventh, Summary of Geometry in three juan, trimming the original's excess and repairing its omissions. Eighth, Right-triangle Surveying in two juan, gathering essentials from the Zhou Bi and Sea Island methods to preserve the spirit of the ancients. Ninth, Preserving the Ancients in the Nine Numbers in ten juan—the Nine Numbers being the methods of the first chapter of the Nine Chapters; only the chapter titles of the Nine Chapters now survive. Later writers could not surpass its range.

外有書一十七種為續編：一，少廣拾遺一卷。古有一乘方至九乘方相生之圖，而莫詳所用。後或增之至十乘，惟四乘方與十乘方不可藉用他法，因為推演至十二乘方，有條不紊。二，方田通法一卷，算家有捷田二十三法，廣之為百二十有四。三，幾何補編四卷。幾何原本六卷，止於測面，七卷以後，未經譯出，取測量全義量體諸率，實考其作法根源，以補原書之未備。而原書二十等面體之說，向固疑其有誤者，今乃得其實數。又原本理分中末線，但有求作之法，而莫知所用。今依法求得十二等面及二十等面之體積，因得其各體中棱線及輳心對角諸線之比例。又兩體互相容及兩體與立方、立員諸體相容各比例，並以理分中末線為法，乃知此線不為徒設。四，西鏡錄訂註一卷。五，權度通幾一卷。重學為西術一種，載於比例規解者多譌誤，今以南勳卿儀象志互相訂補，其數始真。六，奇器補註二卷。關中王公徵奇器圖說所述引重轉木諸制，並有裨於民生日用，而又本於西人重學，以明其意。嘗以書史所傳，如漢杜詩作水《廠義》以便民，及王氏農書諸水器之類，睹記所及，如劉繼莊詩集載筒車灌田法，稍為輯錄，以補其所遺，而圖與說不相應者正之，以西字為識者易之。七，正弦簡法補一卷。大測諸書，言作八線表之法詳矣，薛鳳祚書有用矢線求度法，為之作圖，以明其意。因得兩法，在六宗、三要之外，而為用加捷。兩法者，一曰正弦方冪倍而退位得倍弧之矢，一曰正矢進位折半得半弧正弦上方冪。八，弧三角舉要五卷。曆書皆三角法也，內分二支：一曰平三角，一曰弧三角。凡曆法所測，皆弧度也，弧線與直線不能為比例，則剖析渾員之體，而各於弧線中得其相當直線。即於無句股中尋出句股，此法之最奇而確者。弧三角之用法雖多，而其最著明者，為黃赤交變一圖。反覆推論，瞭如列眉，熟此一端，則其餘不難推及矣。測量全義第七、第八、第九卷專明此理，而舉例不全，且多錯謬。其散見諸歷指者，僅存用數，無從得其端倪。天學會通圈線三角法，作圖草率，往往不與法相應。一以正弧三角為綱，仍用渾儀解之。正弧三角之理，盡歸句股。參伍其變，斜弧三角之理，亦歸句股矣。其目：曰弧三角體式，曰正弧句股，曰求餘角法，曰弧角比例，曰垂線，曰次形，曰垂弧捷法，曰八線相當。九，環中黍尺五卷。舉要中弧度之法已詳，然更有簡妙之用宜知。測量全義原有斜弧兩矢較之例，所立圖姑為斜望之形，而無實度可言。今一以平儀正形為主，凡可以算得者，即可以器量。渾儀真像，呈諸片楮，而經緯歷然，無絲毫隱伏假借。至於加減代乘除之用，曆書舉其名不詳其說，疑之數十年，而後得其條貫，即初數次數甲數乙數諸法。其目：曰總論，曰先數後數，曰平儀論，曰三極通幾，曰初數次數，曰加減法，曰甲數乙數，曰加減捷法，曰加減又法，曰加減通法。十，巉堵測量二卷。古法斜剖立方，成兩巉堵形，巉堵又剖為二，成立三角，立三角為量體所必需，然此義皆未發。今以渾儀黃赤道之割切二線成立三角形，立三角本實形，今諸線相遇成虛形，與實形等，而四面皆句股，西法通於古法矣。又於餘弧取赤道及大距弧之割切線，成句股方錐形，亦四面皆句股，即弧度可相求，亦不言角，古法通於西法矣。二者並可以堅楮為儀象之，則八線相為比例之理，瞭如掌紋。而郭守敬員容方直矢接句股之法，不煩言說而解。其目：曰總論，曰立三角摘要，曰渾員內容立三角，曰句股錐，曰句股方錐，曰方巉堵容員巉堵，曰員容方直儀簡法，曰郭太史本法，曰角即弧解。十一，用句股解幾何原本之根一卷。幾何不言句股，而其理莫能外。故其最難通者，以句股釋之則明。惟理分中末線似與句股異源，今為遊心於立法之初，仍不外乎句股，益信古句股義包舉無遺。徐光啟譯大測表，名之曰割圜句股八線表，其知之矣。十二，幾何增解數則。其目有四：曰以方斜較求斜方，曰切線角與員內角交互相應，曰量無法四邊形捷法，曰取平行線簡法。並就幾何各題而增，不入補編，附前條共卷。十三，仰觀覆矩二卷。一查地平經度為日出入方位，一查赤道經度為日出入時刻，並依裡差，用弧三角立算，與曆書法微別。十四，方員冪積二卷。曆書周徑率至二十位，然其入算，仍用古率十一與十四之比例，豈非以乘除之際難用多位歟？今以表列之，取數殊易，乃為之約法，則徑與週之比例即方、員二冪之比例，亦即為立方、立員之比例，殊為簡易直捷。十五，麗澤珠璣一卷。友朋之益，取其有關算學者。十六，算器考一卷。十七，數學星槎一卷。

Beyond these are seventeen works forming a sequel: first, Supplementary Recovery of Lesser Breadth in one juan. Antiquity had a diagram showing mutual generation from the first through the ninth power, but no one explained how it was used. Later writers extended it to the tenth power; only the fourth and tenth powers could not be handled by other methods, so he worked them out systematically up to the twelfth power. Second, General Method for Rectangular Fields in one juan—earlier calculators knew twenty-three shortcut field formulas; he expanded them to one hundred twenty-four. Third, Supplement to Geometry in four juan. The first six books of Euclid's Elements stop at plane measurement; the seventh and following books were never translated. He took the solid ratios from Complete Meaning of Measurement, verified their construction in practice, and used them to fill what the original left incomplete. He had long suspected the original's treatment of twenty-sided solids was wrong; now he obtained their true values. The original also gave methods for constructing mean and extreme ratio lines, but no one knew what they were for. Applying those methods, he found the volumes of twelve- and twenty-sided solids and thereby the ratios among their edges, center-to-vertex lines, and diagonals. He also worked out the ratios by which two solids inscribe each other, and by which two solids inscribe cubes, cylinders, and other figures—all by the mean and extreme ratio line, showing that it had not been introduced without purpose. Fourth, Revised Annotation to the Western Mirror in one juan. Fifth, Weights and Measures Applied to Geometry in one juan. Statics is one branch of Western mathematics; the account in Explanation of the Proportional Compass contains many errors. Cross-checking against Nan Xunqing's Treatise on Instruments and Images, he corrected the figures. Sixth, Supplementary Annotation to Marvelous Devices in two juan. The lifting and turning mechanisms described in Wang Zheng's Diagrams and Explanations of Marvelous Devices from Guanzhong all serve daily life; he also drew on Western statics to explain their principles. He also collected from histories and records such examples as Han Du Shi's water-powered bellows built for the people's benefit and the various water devices in Wang Zhen's Book of Agriculture; from notes he happened upon, such as the bucket-wheel irrigation method in Liu Jizhuang's collected poems, he compiled what was missing; where diagrams and text did not agree he corrected them, and replaced Western characters with more familiar ones. Seventh, Supplement to the Simplified Sine Method in one juan. Books on grand surveying explain in detail how to construct the eight-line table; Xue Fengzuo's book gives a method of finding degrees by versed sine lines—for this Wending made diagrams to clarify the idea. In this way he obtained two methods beyond the Six Divisions and Three Essentials, and more convenient to apply. The two methods are these: one doubles the sine squared and shifts the decimal to obtain the versed sine of a double arc; the other advances the versed sine's place value and halves it to obtain the squared term above the sine of a half arc. Eighth, Essentials of Spherical Trigonometry in five juan. Calendar books all use trigonometry, divided into two branches: plane trigonometry and spherical trigonometry. Everything calendrical methods measure is arc measure; since arcs and straight lines cannot be proportional, one dissects the sphere and from each arc derives its corresponding straight line. It finds right triangles where none seem to exist—the most marvelous and exact of methods. Spherical trigonometry has many uses, but the clearest is the diagram of variation between the ecliptic and the equator. Repeated reasoning makes it as plain as the brows on one's face; master this one point and the rest follows easily. Volumes seven through nine of Complete Meaning of Measurement treat this principle, but their examples are incomplete and riddled with errors. What appears scattered in various calendar guides gives only the working numbers, offering no clue to the underlying principle. The circular-line trigonometry in Compendium of Astronomical Learning is diagrammed carelessly, often failing to match the methods. He takes right spherical triangles as the framework and still explains them with the armillary sphere. The principles of right spherical triangles all reduce to right triangles. By combining and varying them, the principles of oblique spherical triangles also reduce to right triangles. Its sections are: Forms of Spherical Triangles, Right Spherical Triangles, Methods of Finding Remaining Angles, Ratios of Arcs and Angles, Perpendicular Lines, Secondary Figures, Shortcut Methods for Perpendicular Arcs, and Equivalence of the Eight Lines. Ninth, Millet Measures Within the Sphere in five juan. Essentials already explains arc methods in detail, yet simpler and more elegant applications remain worth knowing. Complete Meaning of Measurement originally included an example comparing two versed sines of oblique arcs, but its diagram was only an oblique view with no real measurable values. Now he takes the true form on the plane instrument as primary: whatever can be calculated can also be measured with instruments. The true image of the armillary sphere is laid out on paper, with celestial coordinates clearly marked and without the slightest concealment or artifice. As for using addition and subtraction in place of multiplication and division, calendar books name the methods but do not explain them; he puzzled over them for decades before grasping their logic—the methods of initial numbers, secondary numbers, A-numbers, and B-numbers. Its sections are: General Discussion, First Numbers and Later Numbers, Discussion of the Plane Instrument, Geometry of the Three Poles, Initial and Secondary Numbers, Addition-Subtraction Methods, A-Numbers and B-Numbers, Shortcut Addition-Subtraction Methods, Alternative Addition-Subtraction Methods, and General Addition-Subtraction Methods. Tenth, Chan-du Measurement in two juan. The ancient method of obliquely sectioning a cube yields two chan-du forms; each chan-du is halved to form triangular prisms, which are essential for measuring solids—yet none of this had been explained. Now, using the secant and tangent lines of the ecliptic and equator on the armillary sphere to form a triangular prism—a solid in principle, but here the intersecting lines form a virtual figure equal to the solid, with all four faces right triangles—thus linking Western methods to ancient ones. Further, from the remaining arc he takes the secant and tangent lines of the equator and great-distance arc to form a right-triangular square pyramid, also with four right-triangle faces, so arc degrees can be found mutually without speaking of angles—thus linking ancient methods to Western ones. Both can be modeled in stiff paper as instrument figures, making the proportional relations among the eight lines as clear as the lines on one's palm. Guo Shoujing's methods of circular containment, square containment, straight lines, versed sines, and joined right triangles become clear without much explanation. Its sections are: General Discussion, Summary of Triangular Prisms, Triangular Prisms Within the Sphere, Right-Triangle Pyramids, Right-Triangle Square Pyramids, Square Chan-du Containing Circular Chan-du, Simplified Methods for Circular and Square Containment Instruments, Grand Astrologer Guo's Original Methods, and Explanation That Angles Are Arcs. Eleventh, Using Right Triangles to Explain the Foundations of Euclid's Elements in one juan. Geometry does not speak of right triangles, yet its principles cannot go beyond them. What is hardest to grasp becomes clear when explained through right triangles. Only the mean and extreme ratio line seems to spring from a different source, yet when one reflects on how the methods were first established, they still do not go beyond right triangles—confirming that the ancient doctrine of right triangles embraces everything. When Xu Guangqi translated the grand surveying table, he named it the Table of Eight Lines for Circle-Division Right Triangles—he understood this. Twelfth, Additional Geometric Solutions and Rules. It has four sections: Finding the Slant and Square by Comparing Square and Slant, Correspondence Between Tangent-Line Angles and Inscribed Circular Angles, Shortcut Method for Measuring Irregular Quadrilaterals, and Simplified Method for Taking Parallel Lines. All are additions to various geometric problems, not entered in the supplement, appended to the preceding entry in the same juan. Thirteenth, Gazing Upward and the Inverted Gnomon in two juan. One investigates horizon longitude for the direction of sunrise and sunset, and equatorial longitude for the times of sunrise and sunset; both follow local correction and are calculated by spherical trigonometry, differing slightly from calendar-book methods. Fourteenth, Square and Circle Powers and Volumes in two juan. Calendar books give the ratio of circumference to diameter to twenty places, yet in calculation they still use the ancient ratio of eleven to fourteen—is this not because many places are awkward in multiplication and division? Tabulating the values makes them easy to use; he devised a simplified method whereby the ratio of diameter to circumference is also the ratio of square and circular areas, and likewise of cubes and cylinders—a simple and direct approach. Fifteenth, Pearls from the Beautiful Marsh in one juan. It collects the benefits of friendship, selecting what bears on mathematics. Sixteenth, Examination of Calculating Instruments in one juan. Seventeenth, Mathematical Star Raft in one juan.

文鼎歷學疑問，曾呈御覽，後又引申其說，作歷學疑問補二卷，皆平正通達，可為步算家準則。

Wending's Questions on Calendrical Learning was once presented for imperial review; later he extended his views in a two-juan Supplement to Questions on Calendrical Learning—all even-handed and penetrating, and fit to serve as standards for computational mathematicians.

文鼎為學甚勤，劉輝祖同舍館，告桐城方苞曰：『吾每寐覺，漏鼓四五下，梅君猶構燈夜誦，乃今知吾之玩日而愒時也。』居京師時，裕親王以禮延致朱邸，稱梅先生而不名。李文貞公命子鍾倫從學，介弟鼎徵及群從皆執弟子之禮。宿遷徐用錫，晉江陳萬策，景州魏廷珍，河間王之銳，交河王蘭生，皆以得與參校為榮。家多藏書，頻年遊歷，手抄雜帙不下數万卷。歲在辛丑，卒，年八十有九。上聞，特命有地治者經紀其喪，士論榮之。

Wending studied with great diligence. Liu Huizu, who shared his lodgings, told Fang Bao of Tongcheng: "Whenever I wake at the fourth or fifth watch, Master Mei is still reading by lamplight—and only now do I see how I have squandered my days and wasted my time." While in the capital, Prince Yu courteously invited him to his vermilion residence and addressed him as Master Mei without using his given name. Li Wenzhong ordered his son Zhonglun to study under him; his younger brother Dingzheng and all their cousins observed the rites of disciples. Xu Yongxi of Suqian, Chen Wance of Jinjiang, Wei Tingzhen of Jingzhou, Wang Zhirui of Hejian, and Wang Lansheng of Jiaohe all counted it an honor to join in collating. His family owned many books, and through years of travel he hand-copied miscellaneous works numbering no less than tens of thousands of juan. In the xinchou year he died, aged eighty-nine. When the Emperor heard, he specially ordered the local magistrate to manage the funeral; scholars regarded this as an honor.

子以燕，字正謀。康熙癸酉舉人。於算學頗有悟入，有法與加減同理，而取徑特殊，能於恆星曆指中摘出致問，文鼎所謂『能助餘之思』也。早卒。

His son was Yiyan, courtesy name Zhengmou. A metropolitan graduate of the guiyou year of the Kangxi reign. He had considerable insight in mathematics; he devised a method analogous to addition-subtraction but taking a special approach, and could extract questions from the Fixed Star Calendar Guide—what Wending called "able to assist my thinking." He died young.

瑴成，字玉汝，以燕子。文鼎疑日差既有二根，即宜列二表，瑴成以為：『定朔時既有高卑盈縮之加減矣，復用於此，豈非複乎？』文鼎因其說，然後悟交食之非缺，比之童烏九歲能與太玄。康熙乙未進士，改編修，與修國史。瑴成肄業蒙養齋，以故數學日進。御製數理精蘊、歷象考成諸書，皆與分纂。所著增刪算法統宗十一卷，赤水遺珍一卷，操縵卮言一卷。

Juecheng, courtesy name Yuru, was Yiyan's son. Wending doubted that since the solar equation has two roots, two tables should be listed. Juecheng argued: "When fixing the new moon, elevation and contraction have already been added and subtracted—using them again here, would that not be redundant?" Through his son's argument Wending came to see that eclipses were not incomplete—comparable to Tong Xuan, who at nine could discuss the Supreme Mystery. A metropolitan graduate of the yiwu year of the Kangxi reign, he was appointed Hanlin compiler and helped compile the National History. Juecheng studied at the Mengyang Studio, and for this reason his mathematics improved daily. He shared in compiling such works as the Imperially Produced Essentials of Mathematical Principles and Examination of Calendars and Astronomy. His works include Revised and Expanded Comprehensive Collection of Computational Methods in eleven juan, Treasures Left at Red Water in one juan, and Random Words at the Loom in one juan.

明代算家，不解立天元術，瑴成謂立天元一即西法之借根方，其說曰；『嘗讀授時曆草求弦矢之法，先立天元一為矢，而元學士李冶所著測圜海鏡，亦用天元一立算。傳寫魯魚，算式訛舛，殊不易讀。明唐荊川、顧箬溪兩公互相推重，自謂得此中三昧。荊川之說曰：「藝士著書，往往以秘其機為奇，所謂天元一云爾，如積求之云爾，漫不省其為何語。」而箬溪則言：「細考測圜海鏡，如求城徑，即以二百四十為天元，半徑即以一百二十為天元，即知其數，何用算為？似不必立可也。」二公之言如此，餘於顧說頗不謂然，而無以解也。後供奉內廷，蒙聖祖仁皇帝授以藉根之法，且諭曰：「西人名此書為阿爾熱八達，譯言東來法也。」敬受而讀之，其法神妙，誠算法之指南，而竊疑天元一之術頗與相似。復取授時曆草觀之，乃煥然冰釋，殆名異而實同，非徒似之而已。夫元時學士著書，台官治歷，莫非此物。乃歷久失傳，猶幸遠人慕化，復得故物。東來之名，彼尚不忘所自，而明人視若贅疣而欲棄之。噫！好學深思如唐、顧二公，尚不能知其意，而淺見寡聞者，又何足道哉？』

Ming mathematicians did not understand the Celestial Element method; Juecheng held that setting Celestial Element Unity is the Western borrowed-root method. His explanation says: "I once read in the Draft of the Shoushi Calendar the method of finding chord and versed sine, first setting Celestial Element Unity as the versed sine; and Yuan scholar Li Ye's Sea Mirror of Circle Measurement also uses Celestial Element Unity to set up calculations." Copyists introduced fish-and-deer errors, and the calculation formats are corrupt—very hard to read. The Ming scholars Tang Jingchuan and Gu Ruxi held each other in esteem and claimed to have mastered the essence of this art. Jingchuan said: "Craftsmen who write books often treat concealing their method as a marvel—what they call 'Celestial Element Unity' and such, or 'product accumulation' and such—people casually fail to understand what these terms mean." Ruxi said: "Examining Sea Mirror of Circle Measurement closely—as when finding a city wall's diameter, one sets two hundred forty as the Celestial Element, or for the radius sets one hundred twenty as the Celestial Element; once you know the number, what need for calculation? It seems setting it up may be unnecessary." Such were the two men's words. I tended to disagree with Gu's view but had no way to explain why. Later, serving in the inner court, I received from the benevolent Emperor Shengzu the borrowed-root method, and was instructed: "Westerners name this book Algebra, translated as the Method from the East." I respectfully received and read it; the method is marvelous, truly a guide to calculation—and I privately suspected the Celestial Element Unity technique was quite similar. Taking up the Draft of the Shoushi Calendar again, I saw it clearly—as though ice had melted; the names differ but the substance is the same, not merely similar. Yuan scholars who wrote books, calendar officials who regulated the calendar—all used this method. Yet after long loss of transmission, fortunately distant peoples admiring our civilization restored the old method. Its name 'from the East'—they still remembered its origin—yet Ming scholars treated it as a useless growth and wished to discard it. Alas! Even diligent and deep thinkers like Tang and Gu could not grasp its meaning—what then of shallow and narrow minds?"

明史館開，瑴成與修天文、曆志，呈總裁書曰：『一曆志半系先祖之藁，但屢經改竄，非復原本，其中訛舛甚多。凡有增刪改正之處，皆逐條簽出。一，天文志不宜入曆志，擬仍另編。蓋歷以欽若授時，置閏成歲，其術委曲繁重，其理精微，為說深長。且有明二百七十餘年沿革非一事，造歷者非一家，皆須入志。雖盡力刪削，卷帙猶繁。若加入天文誌之說，則恐冗雜不合史法。自司馬氏分歷與天官為二書，歷代因之，似不可易。一，天文志例載天體、星座、次舍、儀器、分野等事，遼史謂天象千古不變，歷代之志天文者近於衍，其說似是而非。蓋天象雖無古今之異，而古今之言天者，則有疏密之殊。況恆星去極，交宮中星，晨昏隱現，歲歲有差，安得謂千古不易？今擬取天文家精妙之說著於篇；其不足信者，擬削之。』

When the Ming History office opened, Juecheng participated in compiling the astronomical and calendar treatises and submitted a letter to the chief compiler: "First, the Calendar Treatise is half based on our ancestor's draft, but it was repeatedly altered and is no longer the original; there are many errors within." Wherever there are additions, deletions, or corrections, each has been flagged item by item. First, the Astronomical Treatise should not be placed in the Calendar Treatise; it is proposed to compile it separately. For the calendar sets seasons according to time, intercalates months to complete the year—its methods are intricate and weighty, its principles subtle, its explanations deep and long. Moreover, over the Ming's more than two hundred seventy years the changes were not of one kind, and calendar-makers were not of one school—all must enter the treatise. Even with every effort at reduction, the volume remains large. If astronomical treatise material were added, it would likely be redundant and violate historiographical form. Since Sima Qian separated calendar and celestial offices into two books, later dynasties followed this—it seems unchangeable. Second, the Astronomical Treatise customarily records celestial bodies, constellations, lodges, instruments, field-allotments, and such; the Liao History says celestial phenomena are unchanged through the ages and that dynastic astronomical treatises verge on redundancy—this view seems plausible but is wrong. Though celestial phenomena differ not between ancient and modern times, those who speak of heaven in ancient and modern times differ in precision and detail. Moreover, fixed stars' polar distance, the stars crossing palace and center, their morning and evening visibility—all vary year by year; how can one say they are unchanged through the ages? It is proposed to set down in the text the refined theories of astronomers; and to cut what is not trustworthy."

又時憲志用圖論曰：『客問於梅子曰：『史以紀事，因而不創。聞子之志時憲也用圖，此固廿一史所無，而子創為之，宜執事以為非體而欲去之也。而子固執己見，复呶呶上言，獨不記昌黎之自訟乎？吾竊為子危之！』梅子曰：『吾聞史之道貴信而直，餘本不原為史官，總裁謂時憲、天文兩志非專家不能辦，不以為固陋而委任之。余既不獲辭，不得不盡其職。今客謂舊史無圖而疑餘之創，竊謂史之記事，亦視其信否耳，因、創非所計也。夫後史之增於前者多矣，漢書十志，已不侔於八書，而後漢皇后本紀，與魏書之志釋老，唐書之傳公主，宋史之傳道學，皆前史所無，又何疑於國史用圖之為創哉？且客未讀明史耶？明史於割員弧矢、月道距差諸圖，備載曆志，何明史不疑為創，而顧疑餘乎？』客曰：『後史增於前者，必非無因，若明史之用圖，亦有說歟？』梅子曰：『疑以傳疑，信以傳信，春秋法也，作史者誰能易之？古之治歷者數十家，大率不過增損日法，益天週，減歲餘，以求合一時而已。即太初之起數鍾律，大衍之造端蓍策，亦皆牽合，並未能深探天行之故，而發明其所以然之理。本未嘗有圖，史臣何從取而載之？至元郭太史修授時，不用積年日法，全憑實測，用句股割員以求弦矢，於是有割圜諸圖載於歷草。作元史時，不知採摭，則宋、王諸公之疏也。明之大統，實即授時。本朝纂修明史諸公，以義非圖不明，遂採歷草入志，其識極超。復經聖君賢相鑑定，不以為非體而去之，俾精義傳於無窮，洵足開萬古作史者之心胸矣。至於時憲立法之妙，義蘊之奧，悉具於圖，更不可去。如必以去圖為合體，豈以明史為非體，而本朝之制不足法歟？且客亦知時憲之圖所自來乎？我聖祖仁皇帝憫絕學之失傳，留心探索四十餘年，見透底蘊，始親授儒臣，作圖立說，以闡明千古不傳之秘，即御製歷象考成是也。餘親承聖訓，實與彙編之列。彼前輩纂修明史，尚不忍沒古人之善，創例以傳之。而餘以承學之臣，恭紀御製，顧恐失執事之意，而遷就迎合，以致聖學不彰，貽誤後學，尚得謂之信史乎？不信之史，人可塞責，而何用餘越俎而代之？餘之呶呶，非沽直，不得已也。然則韓子之自訟，亦謂其言之可以已者耳。使韓子果務為容悅以求倖免，則諍臣之論，佛骨之表，又何為若是其侃侃哉？』客唯唯而退。』

Also, On Using Diagrams in the Shixian Treatise says: "A guest asked Master Mei: 'History records events and follows precedent without innovation.'" I hear that in your Shixian Treatise you use diagrams—this is indeed absent from all twenty-one standard histories, yet you have innovated; the chief compiler should consider this improper and wish to remove it." Yet you stubbornly hold your view and repeatedly petition above—do you not recall Han Yu's self-reproach?" I privately worry for you!" Master Mei replied: "I have heard that the way of history values trustworthiness and directness. I originally did not wish to be a historiographer; the chief compiler said the Shixian and Astronomical treatises required specialists and entrusted them to me without deeming me stubborn and narrow." Having been unable to refuse, I have no choice but to fulfill my office to the utmost. You say the earlier histories contained no diagrams and suspect I have invented something new. I would reply that when a history records events, what matters is whether the account is credible—not whether it follows precedent or breaks new ground. Later dynastic histories expanded upon their predecessors in countless ways—the Book of Han's ten treatises already exceed the Former Han's eight; yet the Later Han's annals of empresses, the Wei History's treatise on Buddhism and Daoism, the Tang History's biographies of princesses, and the Song History's biographies of Neo-Confucian scholars—all absent from earlier histories—were accepted without question. Why should using diagrams in our national history be singled out as an unwarranted innovation? Have you not read the History of Ming? The History of Ming fully records in its calendar treatise diagrams for arc-segments, chords and arcs, and lunar path deviations—why is the Ming History never accused of innovation, while I alone am suspected? The guest said: "When later histories expand upon earlier ones, they surely have their reasons. Did the History of Ming also have grounds for using diagrams?" Master Mei said: "Pass on what is doubtful as doubtful and what is credible as credible—that is the method of the Spring and Autumn Annals. What historiographer would presume to alter it?" Dozens of schools of calendrical astronomy existed in antiquity, but most merely adjusted the day-count divisor, increasing the celestial circumference and reducing the year's remainder—seeking only to fit the calendar to their own age. Even the Taichu calendar's derivation of numbers from pitch-pipes and bells, and the Dayan calendar's origin in yarrow-stalk divination, were strained expedients—none of them truly penetrated the causes of celestial motion or revealed the principles behind them. Since there were originally no diagrams to begin with, what could historiographers have drawn upon to include them? When Grand Astrologer Guo Shoujing revised the Shoushi calendar in the Yuan, he abandoned accumulated-year methods and day-count divisors, relying entirely on observation. Using right-triangle geometry and arc-segment calculations to derive chords and arcs, he produced circle-division diagrams recorded in the calendar draft. When the History of Yuan was compiled, the editors failed to gather and include these materials—a lapse on the part of Song Lian, Wang Xun, and their colleagues. The Ming dynasty's Datong calendar was in essence the Shoushi calendar. The scholars who compiled the History of Ming in our dynasty, recognizing that the meaning could not be conveyed without diagrams, incorporated the calendar draft into the treatise—an act of exceptional foresight. Furthermore, after review by the sage emperor and his wise ministers, the diagrams were not removed as violations of proper historiographical form, allowing profound principles to be transmitted for all time—a precedent that truly broadened the horizons of every historiographer since. As for the Shixian calendar, the elegance of its methods and the depth of its principles are fully embodied in its diagrams—they cannot be removed. If removing diagrams is required for proper historiographical form, would you declare the History of Ming improper—and our dynasty's precedent unworthy of following? Do you even know where the Shixian diagrams originated? Our Sacred Ancestor, the Benevolent Emperor Kangxi, grieved that this supreme learning had fallen out of transmission. For more than forty years he devoted himself to probing its depths until he grasped its foundations. Only then did he personally instruct his Confucian officials, devising diagrams and theories to illuminate secrets withheld for a thousand years—the Imperially Composed Essentials of Calendrical Astronomy. I personally received the emperor's instruction and took part in the compilation. The predecessors who compiled the History of Ming could not bear to suppress the achievements of the ancients—they established a new precedent to preserve them. As one who inherited this learning, I am charged with reverently recording the imperial work. If I fear displeasing my superiors and compromise to accommodate them, allowing the emperor's learning to go unrecorded and misleading future scholars—can such a history still be called trustworthy? An untrustworthy history lets everyone off the hook—why should I overstep my bounds to substitute for others? My persistent remonstrance is not self-righteous posturing—it is something I cannot avoid. Han Yu's self-reproach meant simply that his words might have stopped short of going so far. If Han Yu had truly sought favor and exemption, why were his remonstrances as a loyal critic and his memorial on the Buddha relics so bold and uncompromising? The guest murmured assent and withdrew.'

又儀象論曰：『齊政授時，儀象與算術並重。蓋非算術，無以預推節候以前民用；非儀象，無以測現在之行度，以驗推步之疏密，而為修改之端也。虞書『璇璣玉衡』，為儀象之權輿，其制不傳。漢人創造渾天儀，即璣衡遺制，唐、宋皆仿為之。至元始有簡儀、仰儀、闚幾、景符等器，視古加詳矣。明於齊化門南倚城築觀像台，仿元制作渾儀、簡儀、天體三儀，置於台上，台下有晷影堂，圭表壺漏，國初因之。康熙八年，命造新儀，十一年，告成，安置台上，其舊儀移藏他室。五十四年，西人紀理安欲炫其能而滅棄古法，復奏制象限儀，遂將所遺舊器用作廢銅，僅存明仿元渾儀、簡儀、天體三儀而已。所製象限儀成，亦置台上。按明史云：『嘉靖間修相風桿及簡、渾二儀，立四大表以測晷影，而立運儀、正方案、懸晷、偏晷，具備於觀像台，一以元法為斷。』餘於康熙五十二三年間，充蒙養齋彙編官，屢赴觀像台測驗。見台下所遺舊器甚多，而元制簡儀、仰儀諸器，俱有王珣、郭守敬監造姓名。雖不無殘缺，然睹其遺制，想見創造苦心，不覺肅然起敬也。乾隆年間，監臣受西人之愚，屢欲廢台下餘器作銅送製造局，賴廷臣奏請存留，禮部奉旨查檢，始知僅存三儀，殆紀理安之燼餘也。夫西人欲藉技術以行其教，故將盡滅古法，使後世無所考，彼益得以居奇，其心叵測。乃監臣無識，不思存什一於千百，而反助其為虐，何哉？乾隆九年冬，有旨移置三儀於紫微殿前，古人法物，庶幾可以永存矣。』

There is also a Treatise on Astronomical Instruments, which states: "In regulating the calendar and proclaiming the seasons, astronomical instruments and mathematical calculation are equally essential. Without mathematical calculation, seasonal markers cannot be computed in advance to guide the people's affairs; Without astronomical instruments, present celestial positions cannot be measured to test the accuracy of computational methods and provide grounds for revision. The "Armillary Sphere and Jade Transverse" in the Book of Yu represent the earliest astronomical instruments, though their construction has not survived. The Han created the armillary sphere, preserving the form of the ancient instruments; Tang and Song dynasties each reproduced them. Only in the Yuan did instruments such as the simplified armillary, altitude-measuring instrument, sighting frame, and shadow marker appear—considerably more refined than their ancient predecessors. In the Ming, an observatory platform was built against the city wall south of Qihua Gate, with armillary sphere, simplified armillary, and celestial globe modeled on Yuan designs placed atop it. Below stood the sundial shadow hall, gnomon, and water clock—the early Qing preserved this arrangement. In the eighth year of Kangxi, new instruments were commissioned; they were completed in the eleventh year and installed on the platform, while the old instruments were moved to storage elsewhere. In the fifty-fourth year of Kangxi, the Jesuit Kilian Stumpf, seeking to display his own abilities while discarding ancient methods, petitioned to have a quadrant instrument made and had the remaining old instruments melted down for scrap—leaving only the Ming replicas of the Yuan armillary sphere, simplified armillary, and celestial globe. When the new quadrant instrument was completed, it too was placed on the platform. The History of Ming records: "During the Jiajing reign, the wind-vane pole and the simplified and armillary instruments were restored; four great gnomons were erected to measure sundial shadows; motion, orientation, suspended, and inclined sundials were all installed on the observatory platform, following Yuan methods throughout. During the fifty-second and fifty-third years of Kangxi, while serving as a compiler in the Mengyang Studio, I made repeated visits to the observatory platform for measurements. I found many old instruments stored below the platform; the Yuan-era simplified armillary, altitude instrument, and other instruments all bore the names of Wang Xun and Guo Shoujing as supervising craftsmen. Though damaged in places, contemplating these surviving instruments and imagining the painstaking labor of their creators, one cannot but feel a solemn reverence. During the Qianlong reign, the observatory director, misled by Western missionaries, repeatedly sought to melt down the remaining instruments below the platform for the manufacturing bureau. Only when court officials memorialized for their preservation and the Ministry of Rites investigated by imperial order was it discovered that only the three instruments survived—likely all that escaped Kilian Stumpf's destruction. The Westerners seek to use technical learning to propagate their religion; by abolishing ancient methods they leave later generations with nothing to compare against, allowing themselves to hold a monopoly—their motives are not easily fathomed. Yet the observatory director, lacking discernment, thought nothing of preserving even a fraction of the heritage and instead aided in its destruction—why? In the winter of the ninth year of Qianlong, an imperial order relocated the three instruments to the forecourt of the Hall of Purple Tenuity—the ancient instruments might at last be preserved for posterity."

又論句股曰：『句股和較相求，言算學者莫不留心，其法可謂詳且備矣，未有以句股積與句弦和較為問者。元學士李冶著測圜海鏡，用餘句、餘股立算，神明變化，幾如五花八門，亦未及此。豈俱未計及耶？抑有其法而遺之耶？統宗少廣章內，雖有句股積及句弦較兩題，乃偶合於句三股四之數，非通法。昔待罪蒙養齋，彙編數理精蘊，意欲立法以補其缺。先用平方輾轉推求，皆不能禦，思之累日，而後得用帶縱立方求句股二法。』

There is also a Treatise on Right Triangles, which states: "Mathematicians everywhere study the methods for finding sums and differences of triangle sides—these techniques are thorough and well established. Yet no one has posed problems involving both the product of the legs and the sum or difference of a leg and the hypotenuse together. The Yuan scholar Li Ye's Sea Mirror of Circle Measurement, though ingenious in its use of surplus legs and sides and wondrously varied in its transformations, does not reach this case either. Did they all fail to consider this? Or did they possess the method but allow it to be lost? The Comprehensive Mastery's chapter on lesser breadth contains two problems on the product of legs and the leg-hypotenuse difference, but these happen to work only for the classic three-four-five triangle—not as a general method. When I served in the Mengyang Studio compiling the Essentials of Mathematical Principles, I intended to devise methods to fill this gap. I first tried successive applications of square-root extraction without success. After days of reflection, I finally derived two methods using mixed-dimension cubics to solve for the legs."

卒，年八十有三，諡文穆。

He died at the age of eighty-three and was granted the posthumous title Wenmu.

鈁，字導和，瑴成第四子也。瑴成纂叢書輯要六十餘卷，圖皆所繪。刪訂統宗圖，十之七八，皆出其手。年二十六，卒。

Fang, styled Daohe, was the fourth son of Juecheng. Juecheng compiled the Essentials of the Collectanea in more than sixty volumes; Fang drew all the illustrations. In revising the diagrams for the Comprehensive Mastery, seven or eight tenths were his own work. He died at the age of twenty-six.

文鼐，字和仲，文鼎從弟也。初學歷時，未有五星通軌，無從入算。與兄文鼎取元史歷經，以三差法佈為五星盈縮立成，然後算之，共成步五星式六卷。早卒。

Wen Nai, styled Hezhong, was Wen Ding's younger cousin. When he first studied calendrical astronomy, no comprehensive ephemeris for the five planets existed, and there was no basis for computation. Together with his elder brother Wen Ding, he drew on the calendar classic in the History of Yuan, using the three-difference method to compile tables of planetary elongation and contraction, then computed from them—producing the Six Fascicles of Methods for Computing the Five Planets. He died young.

文鼏，字爾素，文鼎季弟也。著中西經星同異考一卷。以三垣二十八宿星名，依步天歌次第，臚列其目，而以中、西有無多寡分注其下，載古歌、西歌於後。古歌即步天歌，西歌則利瑪竇所撰經天該也。其南極諸星，則據湯若望算書及南懷仁儀象志，為考證補歌，附之於末。其發凡略言：『齊七政，非先定恆星，則無從著手。故曰『七政如乘傳，恆星其地志也；七政如行棋，恆星其楸局也。』曰『恆』者，謂其終古不易；曰『經』者，謂其不同緯星南北行，『經』亦有『恆』之義焉。是編專以中、西兩家所傳之星歌星名考其多寡同異，故曰經星同異考。星官之書，自黃帝始，重黎、羲和，志天文者，紛糅不一。漢張衡云：『中外之官常明者百有二十四，可名者三百二十，為星二千五百，微星之數蓋萬一千五百二十。』至三國時，太史令陳卓始列甘、石、巫咸三家所著星，總二百八十三官，一千四百八十四星。自唐以來，以儀考測，迨宋兩朝志，始能言某星去極若干度，入某星若干度，為說較詳。此中國之言星學者。西儒星學遠有端緒，據其書所譯，周赧王丙寅古地末一測，漢永和戊寅多祿某一測，明嘉靖乙酉尼谷老一測，萬曆乙酉第谷一測，崇禎戊辰湯若望一測。國朝康熙壬子，南懷仁著儀象志，又依歲差改定黃經及赤經。今依南公志表，稽其大小，分為六等。一等大星一十有六，二等星六十有八，三等星二百有八，四等星五百一十有二，五等星三百四十有二，六等星七百三十有二，總計一千八百七十八。其微茫小星，則不能以數計。此泰西之學也。』

Wen Bi, styled Ersu, was Wen Ding's youngest brother. He wrote the Examination of the Same and Different among Chinese and Western Fixed Stars in one fascicle. He listed the star names of the Three Enclosures and Twenty-eight Lunar Mansions in the order of the Song of Pacing the Heavens, annotating below each entry whether it appeared in Chinese and Western catalogues and in what numbers, with the ancient and Western star songs appended at the end. The ancient song is the Song of Pacing the Heavens; the Western song is Matteo Ricci's Summary of the Heavens. For the stars of the South Pole, he drew on Adam Schall's computational works and Ferdinand Verbiest's Treatise on Astronomical Instruments to provide verification and supplementary songs, appended at the end. In his introductory overview he briefly states: "To regulate the seven luminaries, one must first establish the fixed stars—without them there is no place to begin. Thus it is said: 'The seven luminaries are like relay horses on a post road; the fixed stars are their route maps; The seven luminaries are like chess pieces in motion; the fixed stars are the board.' What is called 'fixed' means that they are unchanged from antiquity to the present; What is called 'longitude star' means that they differ from latitude stars in their north-south motion—'longitude' also carries the sense of 'fixed.' This work specifically compares the star songs and star names transmitted by Chinese and Western traditions for their numbers, correspondences, and differences—hence its title, Examination of the Same and Different among Fixed Stars. Star catalogues begin with the Yellow Emperor; Chongli, Xi and He, and later astronomers all recorded the heavens, but their accounts are inconsistent. Zhang Heng of the Han wrote: "Among the inner and outer asterisms, those constantly bright number one hundred twenty-four; those that can be named number three hundred twenty, comprising two thousand five hundred stars; faint stars probably exceed eleven thousand five hundred twenty." By the Three Kingdoms period, Grand Astrologer Chen Zhuo first compiled the stars recorded by the three schools of Gan, Shi, and Wuxian—altogether two hundred eighty-three asterisms and one thousand four hundred eighty-four stars. From the Tang onward, instruments were used for measurement; by the annals of the Northern and Southern Song, astronomers could finally specify how many degrees a given star stood from the pole and how many degrees it entered another star—the accounts became considerably more detailed. Such is the Chinese tradition of stellar astronomy. Western stellar astronomy has ancient origins; according to their translated works, observations were made in the bingyin year of King Nan of Zhou by Giacomo, in the wuyin year of the Han Yonghe era by Ptolemy, in the yiyou year of the Ming Jiajing era by Copernicus, in the yiyou year of Wanli by Tycho Brahe, and in the wuchen year of Chongzhen by Adam Schall. In our dynasty, in the renzi year of Kangxi, Ferdinand Verbiest wrote the Treatise on Astronomical Instruments, revising ecliptic and right ascension according to precession. Following Verbiest's treatise tables, stars are classified by magnitude into six grades. First-magnitude stars number sixteen; second-magnitude sixty-eight; third-magnitude two hundred eight; fourth-magnitude five hundred twelve; fifth-magnitude three hundred forty-two; sixth-magnitude seven hundred thirty-two—a total of one thousand eight hundred seventy-eight. Faint and tiny stars cannot be counted. Such is the Western tradition of stellar astronomy."

文鼏又有累年算稿，文鼎為錄存，名曰授時步交食式一卷。又有幾何類求新法，算書中比例規解，本無算例，文鼎作度算，用文鼏所補，而參之以陳藎謨尺算用法。

Wen Bi also left computational drafts accumulated over many years; Wen Ding preserved them under the title Shoushi Methods for Computing Eclipses in one fascicle. He also left a New Method of Geometric Analogical Seeking; the Explanation of the Proportional Compass in computational books originally lacked worked examples—Wen Ding wrote degree calculations using Wen Bi's supplements, cross-referenced with Chen Jingmo's Methods for Using the Calculating Ruler.

明安圖，字靜庵，蒙古正白旗人。官欽天監監正。受數學於聖祖，預修禦定歷象考成後編、禦定儀象考成。因西士杜德美用連比例演周徑密率及求正弦、正矢之法，知其理深奧，索解未易，因積思三十餘年，著割圜密率捷法四卷。一曰步法，於杜氏三法外，補創弧背求通弦、求矢法，仍杜氏原法，但通加一四除耳。又弦、矢求弧背，並通弦、矢求弧背，凡六法，合杜氏共成九法。其弦求弧背法，以弦為連比例二率，半徑為一率，求得二、四、六、八、十諸率，以一、三、五、七、九之五數各自乘，為累次乘數。二、三、四、五、六、七、八、九相挨，兩兩相乘，為累次除數，即用二率為第一得數。復置四率，以第一乘數乘之，第一除數除之，為第二得數。又置六率，以第一、第二乘數乘之，第一、第二除數除之，為第三得數。又置八率，以第一、第二、第三乘數乘之，第一、第二、第三除數除之，為第四得數。如是累求，至所得數祗一位止，乃★之，即所求之弧背也。矢求弧背法，倍正矢為連比例三率，亦以半徑為一率，求得五、七、九、十一諸率。以一、二、三、四、五之五數各自乘，為屢次乘數，三、四、五、六、七、八、九、十相挨，兩兩相乘，為屢次除數，即用三率為第一得數。復置五率，以第一乘數乘之，第一除數除之，為第二得數。又置七率，以第一、第二乘數乘之，第一、第二除數除之，為第三得數。又置九率，以第一、第二、第三乘數乘之，第一、第二、第三除數除之，為第四得數。如是累求，至所得數祗一位而止。開平方，即所求之弧背也，通弦求弧背，亦各加一四除。矢求弧背，則三率又多加一四。因更創餘弧求弦矢，餘弦矢求本弧，及借弧與正、餘弦互求四術。二曰用法，以角度求八線，及直線、弧線、三角形邊角相求，共設七題。謂今法所以密於古者，以用三角形也。然三角形非用八線表不能相求，惟用此法，以之立表則甚易，以之推三角形，則不用表而得數同。三、四兩卷曰法解，皆闡明弦、矢與弧背相求之根。其法先以一分弧通弦求二分弧通弧弦之數，次以一分、二分弧通弦求三分、四分全弧通弦之數，以一分三分弧通弦求五分全弧通弦之數。又因二分、五分相乘得十分，十分自乘得百分，十分、百分相乘得千分，十分、千分相乘得萬分。遂以半徑為一率，一分弧通弦為二率，各如相乘之率數，求得十、百、千、萬諸分弧率數。比例得弧背求通弦，應減四率二十四分之一，加六率八十分之一，減八率一百六十八分之一，加十率二百八十八分之一，減十二率四百四十分之一，加十四率六百二十四分之一，減十六率八百四十分之一。各四歸之，則二十四得六，為二三相乘數；八十得二十，為四五相乘數；一百六十八得四十二，為六七相乘數；二百八十八得七十二，為八九相乘數；四百四十得一百一十，為十與十一相乘數；六百二十四得一百五十六，為十二與十三相乘數；八百四十得二百一十，為十四與十五相乘數。故以二、三、四、五、六、七、八、九等數兩兩相乘，為屢次除數。又以通弦求得二率一分多，四率一分，六率九分，八率二百二十五分，十率一萬一千二十五分，十二率八十九萬三千二十五分，十四率一億八百五萬六千二十五分，得後率分數為實。各遞降二等，使二率降為四率，四率降為六率，得前率分數為法。以法除實，得四率一分，為一自乘數；六率九分，為三自乘數；八率二十五分，為五自乘數；十率四十九分，為七自乘數；十二率八十一分，為九自乘數；十四率一百二十一分，為十一自乘數；十六率一百六十九分，為十三自乘數：故以一、三、五、七、九等數各自乘為屢次乘數。次求通弦法，求得十、百、千、萬諸分弧正矢率數，比例得弧背求正矢，應減五率十二分之一，加七率三十分之一，減九率五十六分之一，加十一率九十分之一，減十三率一百三十二分之一，加十五率一百八十二分之一，減十七率二百四十分之一；而十二為三四相乘數，三十為五六相乘數，五十六為七八相乘數，九十為九與十相乘數，一百三十二為十一與十二相乘數，一百八十二為十三與十四相乘數，二百四十為十五與十六相乘數，故以三、四、五、六、七、八、九等數兩兩相乘，為屢次除數。又以正矢求得五率一分多，七率四分，九率三十六分，十一率五百七十六分，十三率一萬四千四百分，十五率五十一萬八千四百分，十七率二千五百四十萬一千六百分，為後率分數，各遞降二等為前率分數。如前通弦法，除得五率一分為一自乘數，七率四分為二自乘數，九率九分為三自乘數，十一率十六分為四自乘數，十三率二十五分為五自乘數，十五率三十六分為六自乘數，十七率四十九為七自乘數，故以一、二、三、四、五等數各自乘，為屢次乘數。書未成而卒，子新續之。

Ming Antu, styled Jing'an, was a Mongol of the Plain White Banner. He served as Director of the Directorate of Astronomy. He studied mathematics under the Sacred Ancestor Kangxi and participated in compiling the Later Compilation of the Imperially Fixed Essentials of Calendrical Astronomy and the Imperially Fixed Essentials of Astronomical Instruments. When the Jesuit Pierre Jartoux used continued proportions to derive the precise ratio of circumference to diameter and methods for computing sine and versed sine, Ming Antu recognized the profound difficulty of these principles. After more than thirty years of reflection, he wrote the Quick Methods of Circle Division and Precise Ratios in four fascicles. The first section covers step methods: beyond Jartoux's three methods, he devised methods for deriving full chord and versed sine from arc length, still following Jartoux's original approach but with one added and four divided throughout. He also devised six methods for deriving arc length from chord and versed sine, and from full chord and versed sine—nine methods in all, together with Jartoux's three. In his method for deriving arc length from chord, the chord serves as the second term of a continued proportion and the radius as the first; from these one obtains the second, fourth, sixth, eighth, and tenth terms. The odd numbers one, three, five, seven, and nine serve respectively as successive multipliers. The numbers two through nine in succession, multiplied in adjacent pairs, form successive divisors; the second term of the proportion yields the first result. Next, taking the fourth term, multiply by the first multiplier and divide by the first divisor to obtain the second result. Again taking the sixth term, multiply by the first and second multipliers and divide by the first and second divisors to obtain the third result. Again taking the eighth term, multiply by the first, second, and third multipliers and divide by the first, second, and third divisors to obtain the fourth result. Proceed by repeated iteration until the result has only one significant place left; combine the terms, and the sum is the arc length sought. In the method for deriving arc length from versed sine, the doubled versed sine serves as the third term of a continued proportion and the radius as the first; from these one obtains the fifth, seventh, ninth, and eleventh terms. The numbers one through five serve respectively as successive multipliers; three through ten in succession, multiplied in adjacent pairs, form successive divisors; the third term of the proportion yields the first result. Next, taking the fifth term, multiply by the first multiplier and divide by the first divisor to obtain the second result. Again taking the seventh term, multiply by the first and second multipliers and divide by the first and second divisors to obtain the third result. Again taking the ninth term, multiply by the first, second, and third multipliers and divide by the first, second, and third divisors to obtain the fourth result. Proceed by repeated iteration until the result has only one significant place left. Extract the square root, and the result is the arc length sought; for deriving arc length from full chord, likewise apply one added and four divided throughout. When deriving arc length from versed sine, one additionally adds one to the third term and divides by four. He further devised four methods: deriving chord and versed sine from a remainder arc, deriving the original arc from remainder chord and versed sine, and mutual conversion between a borrowed arc and sine and cosine. The second section covers application methods: deriving the eight trigonometric lines from an angle, and mutual solution of straight lines, arcs, and the sides and angles of triangles — seven worked problems in all. The present methods surpass the ancient ones precisely because they employ triangles. Triangles cannot be solved without the eight-line table — yet with this method alone, composing tables becomes very easy, and triangles can be solved to the same numerical results without any table at all. The third and fourth fascicles, titled Explication of Methods, explain the foundations of converting among chord, versed sine, and arc length. The method first derives the full chord of a two-part arc from that of a one-part arc; next derives the full chords of three- and four-part arcs from those of one- and two-part arcs; and derives the full chord of a five-part arc from those of one- and three-part arcs. Two-part and five-part arcs multiplied yield ten-part; ten-part squared yields hundred-part; ten-part and hundred-part multiplied yield thousand-part; ten-part and thousand-part multiplied yield ten-thousand-part. Taking the radius as the first term and the full chord of a one-part arc as the second, and applying the corresponding products of terms, one obtains the proportional terms for ten-, hundred-, thousand-, and ten-thousand-part arcs. By proportion, to derive full chord from arc length: subtract one twenty-fourth from the fourth term, add one eightieth to the sixth, subtract one one-hundred-sixty-eighth from the eighth, add one two-hundred-eighty-eighth to the tenth, subtract one four-hundred-fortieth from the twelfth, add one six-hundred-twenty-fourth to the fourteenth, and subtract one eight-hundred-fortieth from the sixteenth. Dividing each by four: twenty-four yields six, the product of two and three; eighty yields twenty, the product of four and five; one hundred sixty-eight yields forty-two, the product of six and seven; two hundred eighty-eight yields seventy-two, the product of eight and nine; four hundred forty yields one hundred ten, the product of ten and eleven; six hundred twenty-four yields one hundred fifty-six, the product of twelve and thirteen; eight hundred forty yields two hundred ten, the product of fourteen and fifteen. Hence the numbers two through nine, multiplied in adjacent pairs, form successive divisors. From the full chord one further obtains fractional parts at the second term (one part plus), fourth term (one part), sixth term (nine parts), eighth term (two hundred twenty-five parts), tenth term (eleven thousand twenty-five parts), twelfth term (eight hundred ninety-three thousand twenty-five parts), and fourteenth term (one hundred eight million fifty-six thousand twenty-five parts); these later-term fractions serve as dividends. Each descends two ranks in succession — the second term to the fourth, the fourth to the sixth — yielding the earlier-term fractions as divisors. Dividing dividend by divisor yields one part at the fourth term, the square of one; nine parts at the sixth term, the square of three; twenty-five parts at the eighth term, the square of five; forty-nine parts at the tenth term, the square of seven; eighty-one parts at the twelfth term, the square of nine; one hundred twenty-one parts at the fourteenth term, the square of eleven; one hundred sixty-nine parts at the sixteenth term, the square of thirteen; hence the odd numbers one, three, five, seven, and nine serve respectively as successive multipliers. Next, the method for deriving full chord: having obtained proportional terms for the versed sines of ten-, hundred-, thousand-, and ten-thousand-part arcs, to derive versed sine from arc length by proportion — subtract one twelfth from the fifth term, add one thirtieth to the seventh, subtract one fifty-sixth from the ninth, add one ninetieth to the eleventh, subtract one one-hundred-thirty-second from the thirteenth, add one one-hundred-eighty-second to the fifteenth, and subtract one two-hundred-fortieth from the seventeenth; Twelve is three times four, thirty is five times six, fifty-six is seven times eight, ninety is nine times ten, one hundred thirty-two is eleven times twelve, one hundred eighty-two is thirteen times fourteen, and two hundred forty is fifteen times sixteen; hence three through nine, multiplied in adjacent pairs, form successive divisors. From the versed sine one further obtains fractional parts at the fifth term (one part plus), seventh term (four parts), ninth term (thirty-six parts), eleventh term (five hundred seventy-six parts), thirteenth term (fourteen thousand four hundred parts), fifteenth term (five hundred eighteen thousand four hundred parts), and seventeenth term (twenty-five million four hundred one thousand six hundred parts) as later-term dividends, each paired by descending two ranks with an earlier-term divisor. As in the full-chord method above, division yields one part at the fifth term (the square of one), four parts at the seventh (the square of two), nine parts at the ninth (the square of three), sixteen parts at the eleventh (the square of four), twenty-five parts at the thirteenth (the square of five), thirty-six parts at the fifteenth (the square of six), and forty-nine at the seventeenth (the square of seven); hence one through five serve respectively as successive multipliers. He died before the book was finished; his son Xin completed it.

新，字景臻，安圖季子。充食俸生。安圖病且革，以所著捷法授之，新遵父命，與門下士陳際新、張肱共續成之。

Xin, styled Jingzhen, was Ming Antu's youngest son. He held stipend-student status at the Imperial Academy. When Ming Antu lay near death, he entrusted his manuscript of the Quick Methods to Xin; obeying his father's charge, Xin together with his disciples Chen Jixin and Zhang Gong brought the work to completion.

陳際新，字舜五，宛平諸生。官靈臺郎，為監正。續明安圖割圜密率捷法，尋緒推究，質以生前面授之言。至乾隆甲午，始克成書。

Chen Jixin, styled Shunwu, was a licentiate of Wanping. He served as Assistant Director of the Imperial Observatory and rose to Director. He continued Ming Antu's Quick Methods of Circle Division and Precise Ratios, following its thread of reasoning and checking it against what his teacher had taught him in person. In the jiawu year of the Qianlong reign, he at last finished the book.

劉湘煃，字允恭，江夏人。聞梅文鼎以曆算名當世，鬻產走千餘裡，受業其門，湛思積悟，多所創獲。文鼎得之甚喜，曰：『劉生好學精進，啟予不逮！』其與人書曰：『金、水二星，歷指所說未徹，得劉生說，而後二星之有歲輪，其理確不可易。』因以所著歷學疑問囑之討論，湘煃為著訂補三卷。又謂曆法自漢、唐以來，五星最疏，故其遲、留、伏、逆皆入於佔，至元郭守敬出，而五星始有推步經度之法，而緯則猶未備。西法舊亦未有緯度，至地谷而後有五星緯度，已在守敬後矣。曆書有法原、法數，並為曆法統宗。法原者，七政與交食之歷指也；法數者，七政與交食經緯之表也，故歷指實為造表之根本。今歷所載金、水，歷指如其法以造表，則與所步之表不合，如其表以推算測天，則又密合，是歷雖有表數，而猶未知立表之根也。』乃作五星法像五卷，文鼎深契其說，摘其要目為五星紀要。

Liu Xianghuang, styled Yunong, came from Jiangxia. Learning that Mei Wending was famed throughout the age for calendrical astronomy, he sold his estate and traveled more than a thousand li to become his pupil; long brooding ripened into insight, and he made many original discoveries. Mei Wending was delighted and declared: 'Student Liu is diligent and ever advancing — he has taught me what I lacked!' In a letter he wrote: 'The calendrical treatises' account of Mercury and Venus remains incomplete; only after Liu's explanation did the annual revolutions of these two planets become a principle beyond dispute.' He therefore entrusted his Questions in Calendar Learning to him for discussion, and Xianghuang wrote three fascicles of emendations and supplements. He also argued that since Han and Tang times the five planets had been treated most crudely in calendrical methods, so that their retrogradation, station, occultation, and opposition all devolved upon divination; only with Guo Shoujing in the Yuan did the five planets first receive methods for computing ecliptic longitude — yet latitude remained incomplete. Western methods too originally lacked planetary latitude; only after Tycho Brahe did latitude for the five planets appear — and that was already after Guo Shoujing. Calendrical treatises distinguish Origins of Methods and Tabular Numbers, both forming the comprehensive foundation of calendrical science. Origins of Methods are the calendrical treatises on the seven luminaries and eclipses; Tabular Numbers are the longitude and latitude tables for the seven luminaries and eclipses; the calendrical treatises are thus truly the root from which tables are composed. For Mercury and Venus in the present calendar, tables composed by the treatises' methods disagree with the computed tables — yet calculations from those tables match the heavens precisely. The calendar thus possesses tabular numbers without knowing the root from which the tables were established.' He therefore wrote Five Fascicles on the Images of Five-Planet Methods; Mei Wending deeply endorsed his views and extracted the main points under the title Essentials of the Five Planets.

湘煃又欲為渾蓋通憲天盤安星之用，以戊辰曆元加歲差，用弧三角法，作恆星經緯表根一卷，及月離交均表根、黃白距度表根各一卷，皆補新法所未及也。所著又有論日、月食算稿各一卷，各省北極出地圖說一卷，答全椒吳荀淑曆算十問書一卷。

Xianghuang also sought to serve the placement of stars on the armillary-sphere and astrolabe disks of the Comprehensive Ritual of the Armillary Sphere; taking the wuchen calendar epoch and adding precession, he applied spherical trigonometry to write one fascicle each on the foundations of the fixed-star longitude-latitude table, the lunar mean-motion and eclipse table, and the solar-lunar separation table — all filling gaps left by the new methods. He also left one fascicle each of draft calculations on solar and lunar eclipses, one fascicle on provincial maps of polar altitude, and one fascicle of replies to the ten questions on calendar and calculation posed by Wu Xunshu of Quanjiao.

王文啟，字宋賢，號惺齋，嘉興人。乾隆辛未進士，授將樂縣知縣。究心律曆句股之學，著書已刻者為惺齋雜著。內有史記、漢書正譌兩種，其正史記之譌者，為律書、曆書、天官書各一卷；正漢書之譌者，為律曆志上下二卷。未刻者為曆法記疑、句股衍、角度衍、九章雜論。而句股衍一書，因繁求簡，最為精晰。分甲、乙、丙三集，甲集術原三卷，乙集綱要二卷，丙集晰義四卷。甲集首卷通論術原，為句股因積求邊張本。二卷專論立方，因及平方法。三卷專論和數開立方，所以盡立方諸數之變。乙集兩卷，為相求法百二十三則之綱要。丙集四卷，即相求法，逐則分晰其義，專取發明立法之意。

Wang Wenqi, whose courtesy name was Songxian and style name Xingzhai, came from Jiaxing. A jinshi of the xinwei year of Qianlong, he was appointed magistrate of Jiangle County. He devoted himself to pitch-pipes, calendrical astronomy, and right-triangle mathematics; his published works appear in the Miscellaneous Writings of Xingzhai. These include two works emending the Records of the Grand Historian and the Book of Han — emending the Grand Historian's Treatises on Pitchpipes, Calendar, and the Celestial Offices, one fascicle each; and emending the Book of Han's Treatise on Pitchpipes and Calendar in two fascicles, upper and lower. Unpublished works include Notes on Doubts in Calendrical Methods, Development of Right Triangles, Development of Angles, and Miscellaneous Essays on the Nine Chapters. Of these, Development of Right Triangles — distilling simplicity from complexity — is the most lucid. It is divided into three collections — A, B, and C: Collection A, Origins of Methods, three fascicles; Collection B, Essentials, two fascicles; Collection C, Clarified Meanings, four fascicles. The first fascicle of Collection A surveys the origins of methods, laying the groundwork for deriving sides from areas in right triangles. The second fascicle treats cube extraction and extends to square extraction as well. The third fascicle treats cube extraction by sum-number, exhausting the variations of all cubic quantities. The two fascicles of Collection B summarize one hundred twenty-three rules of mutual solution. The four fascicles of Collection C present the mutual-solution methods themselves, analyzing each rule in detail to elucidate the intent behind each method's formulation.

其總序曰：『句股弦相求法，參以和較，凡得七十八則，求句股中函數。又有冪積求容員、容方、容縱方，及依弦作底求容方，與句股求外方、外員之數。又有積數與句股和較相求容方，與句股餘數相求之法。綜而計之，凡得二十九則。立表測量，得求高、求遠、求深三則，重表亦然。舊算書多簡略，詳者又苦錯出無緒。間嘗力為區別，使各以類從，先定相求法百十三則。甲申仲秋，复理前緒，逐一布算，捷於舊法，而舊法仍附見，以資參考。至以中函積與弦之所和、所較相求而得句、股、弦之正數，舊法罕見，今亦竊擬一法，以附於後。又別創截弦分兩，及補句求股、補股求句之法，分為六則，使不成句股之形，亦化為句股。並載不成句股求中函積二則，容方、容員四則，外切員徑一則，員內累求句股六則，凡又一十九則。以該西術三角之算，兼備割員之用。使學者知周髀一經，於術無所不該。後人不能觸類旁通，以盡其變，故使西術得出而爭勝，其實西術亦出周髀，不能出折句為股之外也。』

Its general preface reads: 'Methods for mutual solution among leg, base, and hypotenuse, incorporating sum and difference — seventy-eight rules in all, seeking quantities contained within the triangle. There are also methods for deriving inscribed circles, squares, and rectangles from area, for inscribing a square with the hypotenuse as base, and for outer squares and circles from the legs and base. There are also methods for inscribing a square from area together with the sum or difference of legs and base, and for solving from remainder quantities of legs and base. Altogether, twenty-nine rules. Surveying with erected tables yields three rules for height, distance, and depth; double tables add the same. Older computational books are mostly terse; detailed ones scatter their material without order. I have labored to sort them by kind, first establishing one hundred thirteen rules of mutual solution. In the mid-autumn of jiashen I resumed my earlier work, working through each calculation in turn — swifter than the old methods, though the old methods are still appended for reference. As for deriving the true leg, base, and hypotenuse from the contained area together with the sum or difference of the hypotenuse — a case rarely treated in old methods — I now venture to propose a method of my own, appended at the end. I also devised methods of bisecting the hypotenuse and of supplementing the leg to find the base or the base to find the leg — six rules in all — so that figures that are not right triangles may be reduced to right triangles. Also included are two rules for contained area in non-right triangles, four for inscribed squares and circles, one for the diameter of an externally tangent circle, and six for successive right triangles within a circle — nineteen rules in all. These cover Western trigonometry and serve circle division as well. Thus students may see that the Zhou Bi classic embraces every computational method without exception. Later scholars could not extend it by analogy to exhaust its variations, and so Western methods came forward to vie for supremacy — yet Western methods too derive from the Zhou Bi and cannot go beyond folding the leg to form the base.'

又略例引言曰：『算家句股一門，為術最繁，非鑿指一數以為布算之準，難以虛領其義。然如廣三修四見於經者，特其正例，正例外變例尤多。必欲正變兼呈，則一卷中彼此錯出，使閱者耳目數易，轉增煩憒。茲特標舉略例，併不成句股之形亦附見焉，以盡句股之變，而該三角之法。』

Again, the introduction to the brief examples reads: 'Among computational specialists, the branch of right triangles is the most elaborate; unless one fixes on concrete numbers as the standard for calculation, it is hard to grasp the meaning in the abstract. Cases like width three and height four in the classic are only standard examples; variant cases are far more numerous. To present standard and variant cases together in one fascicle is to jumble them endlessly, wearying the reader and adding confusion. Here I mark out brief examples only, also appending non-right-triangle cases, to exhaust the variations of right triangles while covering trigonometric methods.'

又答友問句股書曰：『欲求句股，先學開方，方有正方、縱方之異。縱方則以修廣之和、較數開之，其次則求四率比例，有三率求四率之法，有二率求三率之法，又有一率求三率之法。知此即可以知求句、股、弦各無零數法。以三率之中率為主，倍中率為股，首末二率相減為句，相加為弦。依此衍之，得句股略例十數則，然後以句、股、弦為正數，兩數相加為和，相減為較。又有句股三數相加減之和較數，弦與和，和弦與較和三數相加之和數也；弦與較，較弦與和較三數相減之較數也。三數相加減，今名之為兼三和較。凡正數和較之數各三，兼三和較各二，共十三數。十三數中，隨舉兩數，即可求句股弦全數。凡得相求法九十四則，而容方、容員、截股分兩、立表測量單表、重表之法，猶不與焉。其次則求截弦分兩之法，是為一句股分兩句股，即可以知不成句股亦可以分兩句股。不成句股分兩句股，即西法三角算之所由名，今則總以句股概之。其法取大小兩句股形，小股與大句同數者合為一形，即為不成句股之形。分之為兩，則所謂中垂線者，即小矩之股，大矩之句。以此衍之，又得不成句股略例二十餘則。依類推之，又得合形分兩、削形求全二法。合形分兩，則有正合形截偶分兩、反合形截中分兩、偏合形截邊分兩之法。削形求全，則有削去正矩、偏矩之殊，偏矩中又有淺削、深削之分。知此則句股之學盡矣。』元啟嘗曰：『我無他長，惟好學深思，心知其意而已。』然其句股術一書，幾欲駕梅文鼎而上之，為算術中不可少之書云。

Again, in Reply to a Friend's Questions on Right Triangles he writes: 'To work with right triangles, one must first learn root extraction; squares divide into true squares and rectangles. For rectangles, extract roots from the sum and difference of length and width; next apply four-term proportion — methods exist for finding the fourth term from three, three terms from two, and three terms from one. Master this, and one can derive legs, base, and hypotenuse each without fractional remainders. Take the middle term of a three-term proportion as primary: double it for the base, subtract the outer terms for the leg, add them for the hypotenuse. Extending this yields a dozen or so brief rules for right triangles; then treat leg, base, and hypotenuse as true numbers — their sum and difference follow from addition and subtraction. There are also composite sum-difference quantities formed by adding and subtracting the three right-triangle numbers; the sum of hypotenuse and sum, and the sum of hypotenuse and difference — three quantities whose sum forms the aggregate total; There are the hypotenuse and the difference, and the difference between the hypotenuse and the sum and difference — difference quantities obtained by subtracting among three numbers. When the three numbers are combined by addition and subtraction, the result is now called the composite triple sum-difference. There are three true numbers and three sum-difference numbers, two composite triple sum-differences — thirteen quantities in all. From any two of the thirteen quantities, one can determine the full leg, base, and hypotenuse. Ninety-four mutual-seeking rules are obtained in all, yet methods for inscribed squares and circles, cutting the base into two parts, and single- or double-staff gnomon measurement are still left out. Next come methods for cutting the hypotenuse into two parts — one right triangle split into two — from which one sees that even non-right triangles can be divided the same way. Splitting a non-right triangle into two right triangles is the very origin of Western trigonometry's name; here all such cases are treated under the right-triangle framework. The method takes two right triangles, one large and one small; when the small base equals the large leg and they are joined, the result is a non-right triangle. Split it in two, and the so-called median perpendicular is the base of the smaller rectangle and the leg of the larger one. Extending the method yields twenty-odd brief rules for non-right triangles. Pursuing the categories further, one obtains two more methods: composite-shape division and truncated-shape completion. Composite-shape division includes straight composite cutting at the even point, reverse composite cutting at the center, and oblique composite cutting at the edge. Truncated-shape completion distinguishes cutting away the true rectangle from the oblique rectangle; among oblique rectangles there are further shallow and deep cuts. Master this, and the study of right triangles is complete.' Yuan Qi once said: 'I have no other gift — only that I love learning and think deeply until I grasp the meaning in my heart.' Yet his Right-Triangle Methods nearly surpasses Mei Wending — a work indispensable in the mathematical arts.'

朱鴻，字云陸，秀水人。嘉慶七年進士，改翰林院庶吉士，散館授編修。擢御史，歷給事中，出官督理湖南糧儲道。研精算學。同郡錢儀吉譔三國會要，集乾象、景初二術成，嘗為作注。烏程陳杰時為台官博士，陽湖董祐誠亦客京邸，皆日從講數，各出所得相質問。舊無橢圓求週術，為祐誠言，圜柱斜剖，則成橢員，可以句股形求之。祐誠既發明其說，係以圖釋。初得杜德美割圜九術寫本，以示祐誠，創圖解三卷。既成，復得密率捷法於李潢家，則蒙古監正明安圖師弟續繹之書也，與傳寫本互異。鴻曾依杜法步算，徑一者，週三一四一五九二六五三五八九七九三二三八四六二六四三一八六三六七四七二二七九五一四，週十者，徑三一八三零九八八六一八三七九零六七一五三七七六七五四六六九六三八九零五六六六一。徐有玉採入務民義齋算學中。道光十年後，辭官仍居京師，譔考工記車製參解。又評程氏易疇考工創物小記，多所糾正云。

Zhu Hong, whose courtesy name was Yunlu, came from Xiushui. In the seventh year of Jiaqing he passed the metropolitan examination, entered the Hanlin Academy as a bachelor, and after completing his term was appointed a compiler. He was promoted to censor, served as supervising secretary, and then left the capital to supervise the Hunan grain and storage circuit. He devoted himself to the mastery of computational mathematics. Qian Yiji of the same prefecture compiled Essentials of the Three Kingdoms, assembling the Ganxiang and Jingchu calendar systems into one work, and Zhu once annotated it. Chen Jie of Wucheng was then doctor of the Directorate, and Dong Youcheng of Yanghu also lodged in the capital; they studied computation together daily, each presenting his findings for mutual critique. No method yet existed for finding the circumference of an ellipse; he told Youcheng that a cylinder cut obliquely forms an ellipse, which can be determined by right triangles. Youcheng worked out the theory and illustrated it with diagrams. When he first obtained a manuscript of Du Demei's Nine Methods for Circle Division, he showed it to Youcheng and produced three juan of illustrated explanations. After it was finished, he obtained Quick Methods for Precise Ratio at Li Huang's home — a work continued by the Mongolian director Mingantu and his disciples — which differed from circulated copies. Hong once worked through Du's method step by step: for a diameter of one, the circumference is 3.1415926535897932384626433186; for a circumference of ten, the diameter is 3.18309886183790671537766546696389056661. Xu Youyu included it in the mathematical collection of Wumin Yizhai. After the tenth year of Daoguang he resigned from office but remained in the capital, compiling a commentary on chariot construction in the Kaogongji. He also reviewed Cheng Yichou's Brief Notes on Creating Things in the Kaogongji and corrected many points.

博啟，字繪亭，滿洲正白旗人。乾隆中，官欽天監監副。嘗因句股和較之術，前人論之極詳，獨句股形中所容之方邊、員徑、垂線三事，尚缺而未備。爰以三事分配和較，創法六十。惜其書未刊，法不傳。今所傳者，惟有方邊及垂線求句、股、弦一題。法用平行線剖容方冪為四小句股形，借垂線為小句股和，借方邊為小弦，求小句小股。以小股與垂線比，若方邊與句比；以小句與垂線比，若方邊與弦比。道光初，方履亨官監正，每舉此題課士。其後得甘泉羅士琳力為表章，博術乃復明於世。

Bo Qi, whose courtesy name was Huiting, was a Manchu of the Plain White Banner. During the Qianlong reign he served as vice-director of the Directorate of Astronomy. Working from right-triangle sum-difference methods — though predecessors had treated them exhaustively — he found three matters still missing within inscribed right triangles: the square side, the circle diameter, and the altitude. He distributed the three matters among sum-differences and devised sixty methods. Regrettably his book was never published, and the methods were lost. What survives today is only a single problem seeking the leg, base, and hypotenuse from the square side and the altitude. The method uses parallel lines to divide the inscribed-square area into four small right triangles; taking the altitude as the small sum of leg and base and the square side as the small hypotenuse, one finds the small leg and base. As the small base is to the altitude, so the square side is to the leg; As the small leg is to the altitude, so the square side is to the hypotenuse. Early in the Daoguang era, when Fang Lüheng served as director, he set this problem for students every year. Later Luo Shilin of Ganquan championed the methods vigorously, and Bo's techniques were restored to the world.

羅論云：『曩者聞方慎菴監正言繪亭監副有是法，失傳。因仿監副遺法，用平行線剖半員冪為四小句股形，以半圓徑減垂線餘，借為小句股和，借半員為小弦，求得小句、小股。以小股比垂線，若半員徑比股；以小股比股，若半員徑比弦。又以半員徑減方邊，得較。用平行線剖較冪為四小句股形，借半員徑為小句股和，借較為小弦，求得小句、小股。以小股比半員徑，若方邊比句；以小句比半員徑，若方邊比股，以小股比股，若較比弦。用補副監之遺。復用天元術演得三事和較六十題，更立天、地兩元為廣例二十五術，撰句股容三事拾遺四卷。更試變通其術，禦以八線，取方邊用方斜率，得容方中之斜線。以垂線為一率，半徑為二率，斜線為三率，求得四率為正割。檢八線表得度用，與四十五度相加減，得垂線所分之大小兩弧，副以半徑為一率，垂線為二率，小弧正割為三率，求得四率為句。如以大弧正割為三率，求得四率為股，又如以大小兩弧之兩正切為三率，求得四率，為大小兩弧之兩分弦，相併得弦餘。二題倣此，其得數同，而尾數有奇零。以八線表所列之數至單位止，單位以下，棄其餘分，故不能如句股與天元所得之密合。或有妄詆天元術不能馭三角和較者，抑知天元創於宋、明之間，安能逆知西法之有三角而豫為立法？要在學者善為會通耳。試設平三角形，有一角而角在兩邊之中，有大邊與對邊和，有小邊與對邊和，求三道及垂線，此西人常法所不能御者。若立天元一術，則任求何邊或和數或較數，皆一平方即得。然則天元之與西法，其優劣可見矣。』

Luo wrote: 'Long ago I heard Director Fang Shen'an say that Vice-Director Huiting had these methods, but they had been lost. Following the vice-director's surviving method, I use parallel lines to divide the semicircle area into four small right triangles; subtract the altitude from the semicircle radius and take the remainder as the small sum of leg and base, the semicircle as the small hypotenuse, and thus find the small leg and base. As the small base is to the altitude, so the semicircle radius is to the base; As the small base is to the base, so the semicircle radius is to the hypotenuse. Subtract the square side from the semicircle radius to obtain the difference. Use parallel lines to divide the difference area into four small right triangles; take the semicircle radius as the small sum of leg and base and the difference as the small hypotenuse, and find the small leg and base. As the small base is to the semicircle radius, so the square side is to the leg; As the small leg is to the semicircle radius, so the square side is to the base; as the small base is to the base, so the difference is to the hypotenuse. In this way I supplement what the vice-director left incomplete. He further used the tianyuan method to derive sixty problems on the sum-differences of the three matters, set up heaven and earth as two unknowns for twenty-five extended methods, and compiled four juan titled Right-Triangle Inscribed Three Matters: Recovered Fragments. He also tried adapting the method through the eight lines, applying the square oblique ratio to the square side to obtain the oblique line within the inscribed square. With the altitude as the first ratio, the radius as the second, and the oblique line as the third, the fourth ratio obtained is the secant. Look up the degree value in the eight-line table; add and subtract it with forty-five degrees to obtain the large and small arcs into which the altitude divides the figure; then with radius as first ratio, altitude as second, and small-arc secant as third, the fourth ratio obtained is the leg. If the large-arc secant is taken as the third ratio, the fourth ratio obtained is the base; or if the tangents of the large and small arcs are taken as the third ratio, the fourth ratios obtained are the partial chords of the two arcs — added together they give the chord complement. The other two problems follow the same pattern; the results agree, but odd fractional remainders appear at the end. Because the eight-line table gives numbers only to the unit place and discards everything below, it cannot match the exact agreement obtained by right triangles and the tianyuan method. Some rashly denounce the tianyuan method as unable to handle triangular sum-differences — yet tianyuan arose between the Song and Ming; how could it have foreseen Western trigonometry and legislated for it in advance? What matters is that scholars know how to integrate the two traditions skillfully. Suppose a plane triangle with one angle between two sides, given the sum of a large side and its opposite and the sum of a small side and its opposite — to find the three sides and the altitude; ordinary Western methods cannot handle this. Set up a single tianyuan method, and whichever side, sum, or difference one seeks is obtained at once by one quadratic equation. From this the relative strengths of tianyuan and Western methods are plain.'

許如蘭，字芳谷，全椒人。乾隆三十年舉人，大挑知縣，分發福建。因親老改江西，歷任浮梁、新建等縣事。丁憂服闋，赴福建，題補侯官，未履任，會瘴氣發，病卒。

Xu Rulan, whose courtesy name was Fanggu, came from Quanjiao. In the thirtieth year of Qianlong he passed the provincial examination, was selected as a magistrate in the grand assignment, and was posted to Fujian. Because his parents were elderly he was transferred to Jiangxi, where he served successively in Fuliang, Xinjian, and other counties. After completing mourning for his parents, he went to Fujian and was appointed to Houguan, but before he could take office a miasma outbreak struck and he died of illness.

如蘭性敏，所讀書皆究心精妙，於曆算始習西法，通薛鳳祚所譯天步真原、天學會通。時同縣山西寧武同知吳烺受梅文鼎學於劉湘煃，如蘭因並習梅氏曆算。又於乾隆四十年夏，謁戴震於京都，受句股割圜記。四十四年，謁董化星于常州。戴傳緝古算經十書，而董則專業薛氏者也。由是兼通中、西之學。

Rulan was quick by nature and pursued every book he read to its finest subtleties. In calendrical astronomy he first studied Western methods and mastered Xue Fengzuo's translations True Principles of Celestial Steps and Comprehensive Astronomy. At the time Wu Liang of the same county, vice-prefect of Ningwu in Shanxi, had studied Mei Wending's learning under Liu Xiangkui, and Rulan therefore studied Mei's calendrical astronomy as well. In the summer of the fortieth year of Qianlong he visited Dai Zhen in the capital and received his Record of Right Triangles and Circle Division. In the forty-fourth year he visited Dong Huaxing in Changzhou. Dai transmitted the ten books of the Collected Ancient Mathematical Classics, while Dong specialized in the Xue school. Thus he mastered both Chinese and Western learning.

嘗謂其弟子胡早春曰：『古人以句股方程列於小學，童而習之，人人能曉，今則老宿不能通其義。一則時尚帖括，視句股為不急之務；再則習為風雅，不屑持籌握算，效疇人子弟所為。噫，過矣！』又謂：『士大夫不精弧矢之術，雖識天文，無益也。疇人算工不明像數之理，雖能步算，無益也。』著有乾象拾遺、春暉樓集諸書，今多散佚。

He once told his disciple Hu Zaochun: 'The ancients placed right triangles and equations in elementary studies; children learned them and everyone understood — now even venerable scholars cannot grasp their meaning. First, the fashion for examination essays makes right triangles seem unimportant; Second, people cultivate literary elegance and disdain holding counting rods and doing calculation, as astronomers' sons are expected to do. Alas, this goes too far!' He also said: 'If scholar-officials do not master arc-and-arrow methods, knowing astronomy does them no good. If astronomers and calculators do not understand the principles of image and number, being able to compute step by step does them no good.' He authored works including Recovered Fragments of the Ganxiang Calendar and the Chunhui Lou Collection; most are now scattered and lost.

其存者，有書梅氏月建非專言斗柄論後，略曰：『天氣渾淪，無可識認，古人不得已，即以恆星為天以識日躔。恆星積久而差，冬至日躔不在原宿，始立歲差之法。古謂恆星不動，而黃道西移。今測普天星座皆動，其經緯之度，不隨赤道運轉，而順黃道東移。故謂黃道不動，而恆星東行，與七政同一法。』又謂：『古人以中數為歲，朔數為年。上古氣朔同日，故月建起於節氣，而不起於中氣；日躔過宮，起於中氣，而不起於節氣。起於節氣，故曰冬至子之半；起於中氣，故曰冬至日躔星紀之次也。然則一歲十二建，乃天道經歷十二辰，故謂之月建，此萬古不易者也。斗柄所指分位不真，且恆星東移，積久有差，辨之誠是也。但古人云：「斗為帝車，斟酌元氣而布之四方」。又曰：「招搖柬指。」不過言天道無跡。可見順時布化，斗柄有像可徵耳。拘泥其詞，則惑矣。』其歲差說略曰：『恆星一年東行五十餘秒，又黃、赤二道斜交，並非平行，於左旋至速之中，微斜牽向右。日之於天，猶經緯之於日也。日行至黃道分至節氣之限，則春秋寒暑皆隨之而應。七政躔於各宮，遇各宮燥濕寒溫風雨，則隨恆星之性而應。然則冬、夏二至，乃黃道上子、午之位也。春、秋二分，乃黃道上卯、酉之位也。惟唐、虞時冬至日躔虛中，恆星之子中，正逢黃道之子中。嗣是漸差，而東周在女，漢在斗，今在箕。黃道之子，非恆星之子也。以丑宮初度為冬至者，因周時冬至恆星已差至丑，周人即以恆星為黃道之十二次，故命丑為星紀，言諸星以此紀也。其實丑乃周時恆星之宿度，並非恆星之子中。今並不在丑，又移至寅十餘度矣。由今箕一以上溯古虛五，歷年四千有餘，已差至五十八度，此恆星東行之明驗也。』其他著論無關曆算者不錄。

Among what survives is a postscript to Mei's essay that monthly establishment does not speak exclusively of the Dipper handle, which reads in part: 'Heavenly qi is undifferentiated chaos, with nothing by which to recognize it; the ancients, having no alternative, took the fixed stars as heaven to mark the sun's motion. Fixed stars drift over long ages; at winter solstice the sun no longer stands in its original lodge, and so the method of precession was established. The ancients held that fixed stars do not move while the ecliptic shifts westward. Modern measurement shows that all constellations move; their longitudes and latitudes do not rotate with the equator but shift eastward along the ecliptic. Hence one holds that the ecliptic is fixed while fixed stars move eastward — the same principle applied to the seven luminaries.' He also said: 'The ancients took the median count as the year and the new-moon count as the annual cycle. In high antiquity solar terms and new moons fell on the same day, so monthly establishment began from solar terms rather than median qi; The sun's passage through lodges began from median qi rather than solar terms. Because it began from solar terms, one says 'winter solstice at the midpoint of zi'; Because it began from median qi, one says 'at winter solstice the sun stands in the lodge Xingji.' Thus the twelve monthly establishments of a year mark heaven's passage through the twelve branches — hence the term monthly establishment; this is what never changes through the ages. The position indicated by the Dipper handle is inexact, and fixed stars shift eastward — over long ages drift accumulates; to distinguish this is indeed correct. But the ancients said: 'The Dipper is the emperor's chariot, measuring out primordial qi and distributing it to the four directions.' They also said: 'The Dipper indicator points east.' — this merely means the Way of Heaven leaves no visible trace. One can see that in distributing seasonal transformation, the Dipper handle provides a visible sign. Cling rigidly to the words and one is led astray.' His brief account of precession reads: 'Fixed stars move eastward more than fifty seconds a year; moreover, the ecliptic and equator cross obliquely and are not parallel — amid the swiftest leftward rotation, they are slightly pulled obliquely to the right. The sun's relation to heaven is like longitude and latitude in relation to the sun. When the sun reaches the equinoxes, solstices, and solar terms on the ecliptic, spring and autumn, cold and heat all respond in turn. As the seven luminaries pass through the zodiacal palaces, they respond to each palace's climate—dry or damp, cold or warm, windy or rainy—according to the character of the fixed stars there. The winter and summer solstices therefore mark the zi and wu positions on the ecliptic. The spring and autumn equinoxes mark the mao and you positions on the ecliptic. Only in the era of Yao and Shun did the winter solstice sun stand at the middle of the Emptiness lodge, with the fixed stars' zi midpoint coinciding exactly with the ecliptic's zi midpoint. Since then it has drifted steadily—in the Eastern Zhou to Maid, in the Han to Dipper, and today to Winnowing Basket. The ecliptic's zi point is not the fixed stars' zi point. Taking the first degree of the chou palace as winter solstice reflects the Zhou era, when the solstice among fixed stars had drifted to chou; the Zhou identified the fixed-star lodges with the ecliptic's twelve stations, naming chou Star Chronicle—the register by which all stars were reckoned. In fact chou marked only where the fixed stars stood at the solstice in Zhou times—not the true midpoint of zi among the fixed stars. Today the solstice stands nowhere near chou, having shifted another dozen degrees into yin. Tracing from today's Winnowing Basket 1° back to the Emptiness 5° of antiquity, more than four thousand years yield a displacement of fifty-eight degrees—clear proof of the fixed stars' eastward drift.' His other writings unrelated to calendrical astronomy are not included here.

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