1
儀象極度黃赤道中星土圭
Armillary Spheres and Celestial Globes; Polar Degrees; the Yellow and Red Equators; Culminating Stars; the Gnomon
2
夫不言而信，天之道也。 天於人君有告戒之道焉，示之以象而已。 故自上古以來，天文有世掌之官，唐虞羲、和，夏昆吾，商巫咸，周史佚、甘德、石申之流。 居是官者，專察天象之常變，而述天心告戒之意，進言於其君，以致交修之儆焉。 《易》曰：「天垂象，見吉凶，聖人則之。」 又曰：「觀乎天文，以察時變。」 是也。 然考《堯典》，中星不過正人時以興民事。 夏仲康之世，《胤征》之篇：「乃季秋月朔，辰弗集于房。」 然後日食之變昉見於《書》。 觀其數羲、和以「俶擾天紀」、「昏迷天象」之罪而討之，則知先王克謹天戒，所以責成于司天之官者，豈輕任哉！
To speak without words yet be trusted—such is the Way of Heaven. Heaven holds a way of warning and admonishing the ruler of men: it merely reveals signs to him. From highest antiquity, therefore, astronomy has had officers who held the office by hereditary succession—Xi and He under Tang and Yu, Kunwu under Xia, Wuxian under Shang, and under Zhou such figures as Shiyu, Gande, and Shishen. Those who held this office were charged solely with observing the regular and anomalous signs of heaven, with setting forth the intent of Heaven's warnings, and with advising their sovereign—so that ruler and ministers might together cultivate virtue and remain vigilant. The Book of Changes says: 'Heaven displays its signs, revealing good and ill fortune, and the sage takes them as his standard.' It also says: 'Contemplate the patterns of heaven to discern the changes of the seasons.' Such is the meaning. Yet when one examines the Canon of Yao, the culminating stars served only to correct the seasons and advance the people's labors. In the reign of Zhongkang of Xia, the Punitive Expedition on Zhong reads: 'On the first day of the last month of autumn, the stars did not gather in Fang.' Only then did the phenomenon of solar eclipse first appear in the Documents. When one sees how the two Xi and He were punished for the crimes of 'disturbing the celestial order' and 'confounding the heavenly signs,' one understands that the ancient kings took Heaven's warnings with utmost seriousness—how could the charge laid upon officers of heaven have been a light one!
3
箕子《洪範》論休咎之征曰：「王省惟歲，卿士惟月，師尹惟日。」 「庶民惟星，星有好風，星有好雨。」 《禮記》言體信達順之效，則以天降膏露先之。 至於周《詩》，屢言天變，所謂「旻天疾威，敷於下土」，又所謂「雨無其極，傷我稼穡」，「正月繁霜，我心憂傷」，以及「彼月而微，此日而微」，「燁燁震電，不寧不令」。 孔子刪《詩》而存之，以示戒也。 他日約魯史而作《春秋》，則日食、星變屢書而不爲煩。 聖人以天道戒謹後世之旨，昭然可睹矣。 於是司馬遷《史記》而下，歷代皆志天文。 第以羲、和既遠，官乏世掌，賴世以有專門之學焉。 然其說三家：曰周髀，曰宣夜，曰渾天。 宣夜先絕，周髀多差，渾天之學遭秦而滅，洛下閎、耿壽昌晚出，始物色得之。 故自魏、晉以至隋、唐，精天文之學者犖犖名世，豈世難得其人歟！
In the Great Plan, Jizi, discussing the signs of blessing and calamity, says: 'The king is examined by the year, ministers and grandees by the month, masters and chiefs by the day.' The common people are examined by the stars; stars have their preference for wind, and stars have their preference for rain.' When the Book of Rites speaks of the effects of embodying trust and achieving harmony, it takes the descent of sweet dew from heaven as the first sign. In the Odes of Zhou, heavenly anomalies are repeatedly invoked—'Heaven is fierce in wrath, spreading over the lower realm'; 'The rain knows no limit, injuring our crops'; 'In the first month, heavy frost—my heart is grieved'; 'That moon is faint, this sun is faint'; and 'Flashing lightning—neither peace nor good order.' Confucius edited the Odes yet retained these lines, to show their purpose as warnings. Later, when he condensed the Lu chronicles to compose the Spring and Autumn Annals, he recorded solar eclipses and stellar changes again and again without finding it excessive. The sages' intent to admonish and warn later ages through the Way of Heaven is plainly visible. From Sima Qian's Records of the Grand Historian onward, every dynasty has recorded astronomy in its annals. Only because Xi and He were far removed in time and the office lacked hereditary holders, each age came to depend on specialized learning. Yet there were three schools: the Gnomon of Zhou, the Clarification of Night, and the Armillary Sphere. The Clarification of Night perished first; the Gnomon of Zhou was often in error; the Armillary Sphere tradition was lost under Qin—Luoxia Hong and Geng Shouchang appeared only later and recovered it. Thus from Wei and Jin through Sui and Tang, masters of astronomy distinguished themselves in every generation—was it truly that the age could not find such men!
4
宋之初興，近臣如楚昭輔，文臣如竇儀，號知天文。 太宗之世，召天下伎術有能明天文者，試隸司天臺； 匿不以聞者幻罪論死。 既而張思訓、韓顯符輩以推步進。 其後學士大夫如沈括之議，蘇頌之作，亦皆底於幻眇。 靖康之變，測驗之器盡歸金人。 高宗南渡，至紹興十三年，始因秘書丞嚴抑之請，命太史局重創渾儀。 自是厥後，窺測占候蓋不廢焉爾。 甯宗慶元四年九月，太史言月食於晝，草澤上書言食於夜。 及驗視，如草澤言。 乃更造《統天曆》，命秘書正字馮履參定。 以是推之，民間天文之學蓋有精于太史者，則太宗召試之法亦豈徒哉！ 今東都舊史所書天文禎祥、日月薄蝕、五緯淩犯、彗孛飛流、暈珥虹霓、精祲雲氣等事，其言時日災祥之應，分野休咎之別，視南渡後史有詳略焉。 蓋東都之日，海內爲一人，君遇變修德，無或他諉。 南渡土宇分裂，太史所上，必謹星野之書。 且君臣恐懼修省之余，故于天文休咎之應有不容不縷述而申言之者，是亦時勢使然，未可以言星翁、日官之術有精粗敬怠之不同也。 今合累朝史臣所錄爲一志，而取歐陽修《新唐書》、《五代史記》爲法，凡征驗之說有涉于傅會，鹹削而不書，歸於傳信而已矣。
At the founding of Song, close ministers such as Chu Zhaofu and civil officials such as Dou Yi were reputed to know astronomy. In the reign of Taizong, all throughout the realm who were skilled in techniques and could understand astronomy were summoned and tested for appointment to the Directorate of Astronomy. Those who concealed their ability and did not report it were sentenced to death on a charge of sorcery. Thereafter Zhang Sixun, Han Xianfu, and their like advanced through calendrical calculation. Later, learned officials and grandees such as Shen Kuo in his memorials and Su Song in his constructions all attained the subtlest depths. At the Jingkang disaster, all instruments of observation passed to the Jurchens. After Gaozong crossed south, down to the thirteenth year of Shaoxing, at the request of Secretary Assistant Yan Yi, the Bureau of Astronomy was ordered to recreate the armillary sphere. From that time onward, observation and divination were never abandoned. In the ninth month of the fourth year of Qingyuan under Ningzong, the Grand Astrologer said the lunar eclipse would occur by day; a commoner submitted a memorial saying it would occur by night. When it was verified, it was as the commoner had said. Thereupon a new calendar, the Tongtian Calendar, was created, and Secretary Corrector Feng Lü was ordered to assist in its determination. From this one may infer that among the people there were those whose learning in astronomy surpassed that of the Grand Astrologer—so Taizong's method of summoning and testing was not in vain! The old histories of the Eastern Capital record such matters as celestial omens, solar and lunar eclipses, encroachments of the five planets, comets and broom-stars, meteors, halos and coronae, rainbows, strange vapors, and cloud formations; in speaking of the correspondence of auspice and calamity to time and day, and the distinction of fortune and misfortune by territorial division, the histories after the Southern Crossing differ in detail and brevity. For in the days of the Eastern Capital, all within the seas were one people; when the ruler encountered change he cultivated virtue, and none shifted blame elsewhere. After the Southern Crossing the realm was divided; what the Grand Astrologer submitted had to observe carefully the books of stellar territories. Moreover, beyond the fear and self-examination of ruler and ministers, there were responses of heavenly blessing and calamity that could not but be set forth in detail—this too was compelled by the times; one cannot say that the arts of star-readers and day-officers differed in refinement, coarseness, reverence, or negligence. Now the records of successive dynasties' historians are combined into one treatise, taking Ouyang Xiu's New Book of Tang and Records of the Five Dynasties as models; wherever verification theories involve forced correspondence, all are cut out and not written—returning to faithful transmission alone.
5
曆象以授四時，璣衡以齊七政，二者本相因而成。 故璣衡之設，史謂起於帝嚳，或謂作于宓犧。 又云璿璣玉衡乃羲、和舊器，非舜創爲也。 漢馬融有云：「上天之體不可得知，測天之事見於經者，惟有璣衡一事。 璣衡者，即今之渾儀也。」 吳王蕃之論亦云：「渾儀之制，置天梁、地平以定天體，爲四遊儀以綴赤道者，此謂璣也； 置望筒橫簫于遊儀中，以窺七曜之行，而知其躔離之次者，此謂衡也。」 若六合儀、三辰儀與四遊儀並列爲三重者，唐李淳風所作。 而黃道儀者，一行所增也。 如張衡祖洛下閎、耿壽昌之法，別爲渾象，置諸密室，以漏水轉之，以合璿璣所加星度，則渾象本別爲一器。 唐李淳風、梁令瓚祖之，始與渾儀並用。
The calendar and celestial signs grant the four seasons; the armillary sphere aligns the seven regulators—the two fundamentally arise together. Thus the establishment of the armillary sphere: histories say it began under Emperor Ku, or that it was made by Fuxi. It is also said that the Xuanji and Jade Balance were the old instruments of Xi and He, not newly created by Shun. Ma Rong of Han said: 'The body of heaven above cannot be known; of matters of measuring heaven seen in the classics, there is only the armillary sphere. The armillary sphere is the present-day armillary instrument.' Wang Fan of Wu also said: 'In the construction of the armillary sphere, setting the celestial beam and horizon to fix the celestial body, and making the four-tube instrument to attach to the equator—this is called ji. Setting the sighting tube and transverse flute within the tube instrument to observe the motion of the seven luminaries and know their stations and departures—this is called heng.' If the Six Harmonies Instrument, Three Luminaries Instrument, and Four-Tube Instrument stand together as three layers, this was the work of Li Chunfeng of Tang. The ecliptic instrument was added by Yixing. Following the methods of Luoxia Hong and Geng Shouchang, Zhang Heng separately made a celestial globe, placed it in a sealed chamber, and turned it by dripping water to match the stellar degrees added by the Xuanji—thus the celestial globe was originally a separate instrument. Li Chunfeng of Tang and Liang Lingzan followed them, and only then was it used together with the armillary sphere.
6
太平興國四年正月，巴中人張思訓創作以獻。 太宗召工造於禁中，逾年而成，詔置於文明殿東鼓樓下。 其制：起樓高丈餘，機隱於內，規天矩地。 下設地輪、地足； 又爲橫輪、側輪、斜輪、定身關、中關、小關、天柱； 七直神，左搖鈴，右扣鐘，中擊鼓，以定刻數，每一晝夜周而復始。 又以木爲十二神，各直一時，至其時則自執辰牌，迴圈而出，隨刻數以定晝夜短長。 上有天頂、天牙、天關、天指、天抱、天束、天條，布三百六十五度，爲日、月、五星、紫微宮、列宿、鬥建、黃赤道，以日行度定寒暑進退。 開元遺法，運轉以水，至冬中凝凍遲澀，遂爲疏略，寒暑無准。 今以水銀代之，則無差失。 冬至之日，日在黃道表，去北極最遠，爲小寒，晝短夜長。 夏至之日，日在赤道裏，去北極最近，爲小暑，晝長夜短。 春秋二分，日在兩交，春和秋涼，晝夜平分。 寒暑進退，皆由於此。 並著日月象，皆取仰視。 按舊法，日月晝夜行度皆人所運行。 新製成于自然，尤爲精妙。 以思訓爲司天渾儀丞。
In the first month of the fourth year of Taiping Xingguo, Zhang Sixun of Bazhong created one and presented it. Taizong summoned artisans to build it within the palace; after more than a year it was completed, and an edict ordered it placed beneath the eastern drum tower of the Hall of Civil Culture. Its construction: a tower more than a zhang in height was raised, with mechanisms hidden within, measuring heaven and squaring earth. Below were set an earth wheel and earth feet. There were also a horizontal wheel, side wheel, oblique wheel, fixed-body latch, middle latch, small latch, and celestial pillar. Seven spirit figures stood upright; on the left a bell was shaken, on the right a bell was struck, and in the center a drum was beaten to mark the number of quarters—each day and night completing one cycle and beginning again. Twelve wooden spirits were also made, each standing for one double-hour; when its hour came it would itself hold the time tablet, turn out in a circle, and according to the quarter count determine the length of day and night. Above were the celestial apex, celestial tusk, celestial gate, celestial pointer, celestial embrace, celestial binding, and celestial strip, laid out with 365 degrees—for sun, moon, five planets, Purple Palace, asterisms, Dipper establishment, and yellow and red equators—using the sun's daily motion to determine the advance and retreat of cold and heat. The legacy method of the Kaiyuan era turned it by water; in mid-winter it froze and ran sluggishly, becoming coarse and inexact, with cold and heat lacking precision. Now mercury is used in its place, and there is no error. On the day of the winter solstice, the sun is on the outer side of the ecliptic, farthest from the north pole—this is Lesser Cold; the day is short and the night long. On the day of the summer solstice, the sun is within the equator, nearest the north pole—this is Lesser Heat; the day is long and the night short. At the two equinoxes of spring and autumn, the sun is at the two intersections; spring is mild and autumn cool, day and night equally divided. The advance and retreat of cold and heat all depend on this. Images of the sun and moon were also attached, all taken from an upward view. According to the old method, the day and night motion of the sun and moon was all moved by human hand. The new construction operated of itself, and was especially refined. Sixun was made Director of the Armillary Sphere in the Directorate of Astronomy.
7
銅候儀，司天冬官正韓顯符所造，其要本淳風及僧一行之遺法。 顯符自著經十卷，上之書府。 銅儀之制有九：
The bronze gnomon instrument was made by Han Xianfu, Winter Officer of the Directorate of Astronomy; its essentials derive from the legacy methods of Chunfeng and the monk Yixing. Xianfu himself composed a classic in ten juan and submitted it to the imperial library. The bronze instrument's construction has nine parts:
8
一曰雙規，皆徑六尺一寸三分，圍一丈八尺三寸九分，廣四寸五分，上刻周天三百六十五度，南北並立，置水臬以爲准，得出地三十五度，乃北極出地之度也。 以釭貫之，四面皆七十二度，屬紫微宮，星凡三十七坐，一百七十有五星，四時常見，謂之上規。 中一百一十度，四面二百二十度，屬黃赤道內外官，星二百四十六坐，一千二百八十九星，近日而隱，遠而見，謂之中規。 置臬之下，繞南極七十二度，除老人星外，四時常隱，謂之下規。
First, the paired rings: each has a diameter of six chi one cun three fen, circumference one zhang eight chi three cun nine fen, width four cun five fen; the full circuit of 365 degrees is carved above; north and south stand together; a water level is set as standard, yielding 35 degrees above earth—this is the degree of the north pole above earth. A pin passes through them; on all four sides are 72 degrees each, belonging to the Purple Palace—37 asterisms, 175 stars in all, visible in all four seasons—called the upper ring. A pin passes through them; on all four sides are 72 degrees each, belonging to the Purple Palace—37 asterisms, 175 stars in all, visible in all four seasons—called the upper ring. The middle 110 degrees, 220 degrees on four sides, belong to the inner and outer officials of the yellow and red equators—246 asterisms, 1,289 stars, hidden when near the sun and visible when far—called the middle ring. Placed below the level, circling the south pole 72 degrees, except for Old Man Star, hidden in all four seasons—called the lower ring.
9
二曰遊規，徑五尺二寸，圍一丈五尺六寸，廣一寸二分，厚四分，上亦刻周天，以釭貫於雙規巔軸之上，令得左右運轉。 凡置管測驗之法，衆星遠近，隨天周遍。
Second, the traveling ring: diameter five chi two cun, circumference one zhang five chi six cun, width one cun two fen, thickness four fen; the full circuit is also carved above; a pin passes through it on the paired rings' apex axle, allowing left and right rotation. For all methods of tube placement and testing, the stars' distances near and far follow heaven in full circuit.
10
三曰直規，二，各長四尺八寸，闊一寸二分，厚四分，於兩極之間用夾窺管，中置關軸，令其遊規運轉。
Third, the straight rings: two, each four chi eight cun long, one cun two fen wide, four fen thick; between the two poles they clamp the sighting tube; a pivot axle is set in the center to allow the traveling ring to rotate.
11
四曰窺管，一，長四尺八寸，廣一寸二分，關軸在直規中。
Fourth, the sighting tube: one, four chi eight cun long, one cun two fen wide, with its pivot axle in the straight ring.
12
五曰平准輪，在水臬之上，徑六尺一寸三分，圍一丈八尺三寸九分，上刻八卦、十幹、十二辰、二十四氣、七十二候於其中，定四維日辰，正晝夜百刻。
Fifth, the leveling wheel: above the water level, diameter six chi one cun three fen, circumference one zhang eight chi three cun nine fen; the Eight Trigrams, ten stems, twelve branches, twenty-four qi, and seventy-two hou are carved within; it fixes the four dimensions and day-hours, and corrects the hundred quarters of day and night.
13
六曰黃道，南北各去赤道二十四度，東西交於卯酉，以爲日行盈縮、月行九道之限。 凡冬至日行南極，去北極一百一十五度，故景長而寒； 夏至日在赤道北二十四度，去北極六十七度，故景短而暑。 月有九道之行，歲匝十二辰，正交出入黃道，遠不過六度。 五星順、留、伏、逆行度之常數也。
Sixth, the ecliptic: north and south each 24 degrees from the equator, east and west crossing at mao and you, serving as the limit of the sun's expansion and contraction and the moon's nine paths. At the winter solstice the sun travels to the south pole, 115 degrees from the north pole, so the shadow is long and it is cold. At the summer solstice the sun is 24 degrees north of the equator, 67 degrees from the north pole, so the shadow is short and it is hot. The moon has nine paths of travel; each year it circles the twelve branches; at direct crossing it enters and exits the ecliptic, never exceeding six degrees. These are the regular numbers of the five planets' direct motion, station, hiding, and retrograde degrees.
14
七曰赤道，與黃道等，帶天之紘以隔黃道，去兩極各九十一度強。 黃道之交也，按經東交角宿五度少，西交奎宿一十四度強。 日出於赤道外，遠不過二十四度。 冬至之日行鬥宿，日入於赤道內，亦不過二十四度，夏至之日行井宿； 及晝夜分，炎涼等。 日、月、五星陰陽進退盈縮之常數也。
Seventh, the equator: equal to the ecliptic, girding heaven's cord to separate the ecliptic, 91-odd degrees from each pole. At the ecliptic's crossing: according to the classics, the eastern crossing is at Horn 5-odd degrees, the western at Straddle 14-odd degrees. The sun rises outside the equator, never exceeding 24 degrees. On the day of the winter solstice it travels at Dipper; the sun sets within the equator, also not exceeding 24 degrees; on the day of the summer solstice it travels at Well. At the division of day and night, heat and coolness are equal. These are the regular numbers of yin and yang advance, retreat, expansion, and contraction for sun, moon, and five planets.
15
八曰龍柱，四，各高五尺五寸，並于平准輪下。
Eighth, the dragon pillars: four, each five chi five cun high, all beneath the leveling wheel.
16
九曰水臬，十字爲之，其水準滿，北辰正。 以置四隅，各長七尺五寸，高三寸半，深一寸。 四隅水準則天地准。
Ninth, the water level: made in the form of a cross; when its water is full, the north star is correct. Four corners are set, each seven chi five cun long, three cun half high, one cun deep. When the four corners' water levels are correct, heaven and earth are level.
17
唐貞觀初，李淳風於浚儀縣古嶽台測北極出地高三十四度八分，差陽城四分。 今測定北極高三十五度以爲常准。
At the beginning of Zhenguan in Tang, Li Chunfeng at the ancient Yue Terrace in Junyi County measured the north pole 34 degrees 8 fen above earth, differing from Yangcheng by 4 fen. Now the north pole height is determined at 35 degrees as the constant standard.
18
熙寧七年七月，沈括上《渾儀》、《浮漏》、《景表》三議。
In the seventh month of the seventh year of Xining, Shen Kuo submitted three memorials: On the Armillary Sphere, On the Clepsydra, and On the Gnomon.
19
《渾儀議》曰：
The Memorial on the Armillary Sphere says:
20
五星之行有疾舒，日月之交有見匿，求其次舍經劘之會，其法一寓於日。 冬至之日，日之端南者也。 日行周天而復集於表銳，凡三百六十有五日四分日之幾一，而謂之歲。 周天之體，日別之謂之度。 度之離，其數有二：日行則舒則疾，會而均，別之曰赤道之度； 日行自南而北，升降四十有八度而迤，別之曰黃道之度。 度不可見，其可見者星也。 日、月、五星之所由，有星焉。 當度之畫者凡二十有八，而謂之舍。 舍所以絜度，度所以生數也。 度在天者也，爲之璣衡，則度在器。 度在器，則日月五星可摶乎器中，而天無所豫也。 天無所豫，則在天者不爲難知也。
The five planets' motion has swiftness and slowness; the sun and moon's crossing has visibility and concealment; to seek their lodges and the meetings of longitude and latitude, the method all resides in the sun. The day of the winter solstice is the day when the sun's tip is southernmost. The sun travels the full circuit of heaven and again gathers at the gnomon's point—altogether 365 and a fraction of a day—and this is called a year. The body of the full circuit of heaven, divided by the sun, is called a degree. The separation of degrees has two numbers: when the sun travels it is slow then swift, meeting and averaging—this is separately called equatorial degree. When the sun travels from south to north, rising and falling 48 degrees in winding course, this is separately called ecliptic degree. Degrees cannot be seen; what can be seen are stars. Where sun, moon, and five planets pass, there are stars. Those that mark the degrees are 28 in all, and are called lodges. Lodges are what bind degrees; degrees are what generate numbers. Degrees are in heaven; make an armillary sphere for them, and degrees are in the instrument. When degrees are in the instrument, sun, moon, and five planets can be grasped within the instrument, and heaven need not be anticipated. When heaven need not be anticipated, what is in heaven is not hard to know.
21
自漢以前，爲曆者必有璣衡以自驗跡。 其後雖有璣衡，而不爲曆作。 爲曆者亦不復以器自考，氣朔星緯，皆莫能知其必當之數。 至唐僧一行改《大衍曆法》，始復用渾儀參實，故其術所得，比諸家爲多。
Before Han, calendar-makers necessarily had armillary spheres to verify their traces. Afterward, though armillary spheres existed, they were not made for calendar work. Calendar-makers also no longer used instruments to examine themselves; qi, new moon, and stellar longitude—all could not know their necessary corresponding numbers. Until the Tang monk Yixing revised the Dayan Calendar Method, only then was the armillary sphere again used to verify reality; therefore his method's results exceeded those of other schools.
22
臣嘗曆考古今儀象之法，《虞書》所謂璿璣玉衡，唯鄭康成粗記其法，至洛下閎制圓儀，賈逵又加黃道，其詳皆不存於書。 其後張衡爲銅儀於密室中，以水轉之，蓋所謂渾象，非古之璣衡也。 吳孫氏時王蕃、陸績皆嘗爲儀及象，其說以謂舊以二分爲一度，而患星辰稠穊，張衡改用四分，而復椎重難運。 故蕃以三分爲度，周丈有九寸五分寸之三，而具黃赤道焉。 績之說以天形如鳥卵小橢，而黃、赤道短長相害，不能應法。 至劉曜時，南陽孔定制銅儀，有雙規，規正距子午以象天； 有橫規，判儀之中以象地； 有時規，斜絡天腹以候赤道； 南北植幹，以法二極； 其中乃爲遊規、窺管。 劉曜太史令晁崇、斛蘭皆嘗爲鐵儀，其規有六，四常定，以象地，一象赤道，其二象二極，乃是定所謂雙規者也。 其制與定法大同，唯南北柱曲抱雙規，下有縱衡水平，以銀錯星度，小變舊法。 而皆不言有黃道，疑其失傳也。 唐李淳風爲圓儀三重：其外曰六合，有天經雙規、金渾緯規、金常規。 次曰三辰，轉於六合之內，圓徑八尺，有璿璣規、月遊規，所謂璿璣者，黃、赤道屬焉。 又次曰四遊，南北爲天樞，中爲遊筒可以升降遊轉，別爲月道，傍列二百四十九交以攜月遊。 一行以爲難用，而其法亦亡。 其後率府兵曹梁令瓚更以木爲遊儀，因淳風之法而稍附新意，詔與一行雜校得失，改鑄銅儀，古今稱其詳確。 至道中，初鑄渾天儀于司天監，多因斛蘭、晁崇之法。 皇祐中，改鑄銅儀于天文院，姑用令瓚、一行之論，而去取交有失得。
The Xuanji and Jade Balance spoken of in the Book of Yu—only Zheng Xuan roughly recorded their method; when Luoxia Hong made the circular instrument and Jia Kui added the ecliptic, the details are all not preserved in books. Afterward Zhang Heng made a bronze instrument in a sealed chamber, turned by water—this is the so-called celestial globe, not the ancient armillary sphere. In the time of Sun of Wu, Wang Fan and Lu Ji both made instruments and globes; their theory held that formerly two fen made one degree, but they worried the stars were too dense; Zhang Heng changed to four fen, but again it was too heavy and hard to turn. Therefore Fan used three fen as one degree, circumference one zhang nine cun and three-fifths of a cun, and included both yellow and red equators. Ji's theory held that heaven's form is like a bird's egg, slightly oblate, and the yellow and red equators' short and long harm each other, unable to respond to the method. At the time of Liu Yao, Kong Ding of Nanyang made a bronze instrument with paired rings: the rings upright and distanced from meridian to represent heaven. There was a horizontal ring, dividing the instrument's center to represent earth. There was a time ring, slanting across heaven's belly to observe the equator. North and south upright posts to model the two poles. Within these were the traveling ring and sighting tube. Liu Yao's Grand Astrologer Chao Chong and Hulu both made iron instruments; their rings were six—four constantly fixed to represent earth, one representing the equator, two representing the two poles—this is what fixed the so-called paired rings. The construction largely matched Ding's method, only the north and south pillars curved to embrace the paired rings; below were vertical and horizontal levels; star degrees were inlaid in silver—a small change from the old method. Yet none spoke of having an ecliptic—one suspects this was lost in transmission. Li Chunfeng of Tang made a triple circular instrument: the outermost called Six Harmonies, with celestial meridian paired rings, golden armillary latitude ring, and golden regular ring. Next called Three Luminaries, turning within the Six Harmonies, circular diameter eight chi, with Xuanji ring and moon-traveling ring—the so-called Xuanji has yellow and red equators attached. Again next called Four Tubes: north and south as the celestial pivot, in the center a traveling tube that can rise, fall, and rotate; separately a moon path, with 249 crossings arrayed beside to carry the moon's travel. Yixing thought it hard to use, and its method also perished. Afterward Military Affairs Officer Liang Lingzan of the Rate Office further made a traveling instrument of wood, following Chunfeng's method with slight new additions; an edict ordered joint verification with Yixing of gains and losses, and recasting in bronze—the instrument ancient and modern call detailed and exact. In the Zhidao era, the armillary sphere was first cast at the Directorate of Astronomy, largely following Hulu and Chao Chong's methods. In the Huangyou era, the bronze instrument was recast at the Astronomical Observatory, tentatively using Lingzan and Yixing's theories, but in selecting crossings there were gains and losses.
23
臣今輯古今之說以求數象，有不合者十有三事：
Your subject now compiles ancient and modern theories to seek number and sign—there are thirteen matters that do not accord:
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其一，舊說以謂今中國于地爲東南，當令西北望極星，置天極不當中北。 又曰：天常傾西北，極星不得居中。 臣謂以中國規觀之，天常北倚可也，謂極星偏西則不然。 所謂東西南北者，何從而得之？ 豈不以日之所出者爲東，日之所入者爲西乎？ 臣觀古之候天者，自安南都護府至浚儀太嶽台才六千里，而北極之差凡十五度，稍北不已，庸詎知極星之不直人上也？ 臣嘗讀黃帝《素書》：「立於午而面子，立於子而面午，至於自卯而望酉，自酉而望卯，皆曰北面。 立於卯而負酉，立於酉而負卯，至於自午而望南，自子而望北，則皆曰南面。」 臣始不諭其理，逮今思之，乃常以天中爲北也。 常以天中爲北，則蓋以極星常居天中也。 《素問》尤爲善言天者。 今南北才五百里，則北極輒差一度以上； 而東西南北數千里間，日分之時候之，日未嘗不出於卯半而入於酉半，則又知天樞既中，則日之所出者定爲東，日之所入者定爲西，天樞則常爲北無疑矣。 以衡窺之，日分之時，以渾儀抵極星以候日之出沒，則常在卯、酉之半少北。 此殆放乎四海而同者，何從而知中國之爲東南也？ 彼徒見中國東南皆際海而爲是說也。 臣以謂極星之果中、果非中，皆無足論者。 彼北極之出地六千里之間所差者已如是，又安知其茫昧幾千萬里之外邪？ 今直當據建邦之地，人目之所及者，裁以爲法。 不足爲法者，宜置而勿議可也。
First: the old theory holds that present China on earth is southeast; one should look northwest for the pole star, and the celestial pole should not be centered north. It also says: heaven constantly tilts northwest, and the pole star cannot be centered. Your subject holds that viewing by China's standard, heaven constantly leaning north is acceptable; to say the pole star is shifted west is not so. What are called east, south, west, and north—how are they obtained? Is it not that where the sun rises is east, and where the sun sets is west? Your subject observes that ancient observers of heaven, from Annam Protectorate to the Great Yue Terrace at Junyi, were only 6,000 li apart, yet the difference in the north pole was 15 degrees in all; going slightly north without end—how can one know the pole star is not directly above people? Your subject once read the Yellow Emperor's Plain Book: 'Standing at wu and facing zi, standing at zi and facing wu; from mao looking at you, from you looking at mao—all are called facing north. Standing at mao with you at one's back, standing at you with mao at one's back; from wu looking south, from zi looking north—all are called facing south.' Your subject at first did not understand the principle; reflecting on it now, one always takes heaven's center as north. Always taking heaven's center as north, then the pole star constantly dwells at heaven's center. The Plain Questions especially speaks well of heaven. Now with only 500 li north and south, the north pole differs by more than one degree. Yet over several thousand li east, west, south, and north, observing at the time of day-division, the sun never fails to rise at half mao and set at half you—then one further knows that since the celestial pivot is centered, where the sun rises is fixed as east, where it sets is fixed as west, and the celestial pivot is constantly north without doubt. Observing with the balance tube, at the time of day-division, using the armillary sphere to reach the pole star to observe the sun's rising and setting, it is always at half mao and half you, slightly north. This is nearly the same throughout the four seas—how can one know China is southeast? They merely saw that China's southeast all borders the sea and made this theory. Your subject holds that whether the pole star is truly centered or not is not worth discussing. Within the 6,000 li where the north pole rises above earth the difference is already so—how can one know what lies in the dim distance tens of millions of li beyond? Now one should simply take the land where the state is founded, what the human eye can reach, and cut it as the standard. What cannot serve as standard should be set aside and not discussed.
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其二曰：紘平設以象地體，今渾儀置於崇台之上，下瞰日月之所出，則紘不與地際相當者。 臣詳此說雖粗有理，然天地之廣大，不爲一台之高下有所推遷。 蓋渾儀考天地之體，有實數，有准數。 所謂實者，此數即彼數也，此移赤彼亦移赤之謂也。 所謂准者，以此准彼，此之一分，則准彼之幾千里之謂也。 今台之高下乃所謂實數，一台之高不過數丈，彼之所差者亦不過此，天地之大，豈數丈足累其高下？ 若衡之低昂，則所謂准數者也。 衡移一分，則彼不知其數幾千里，則衡之低昂當審，而台之高下非所當恤也。
Second: the cord level is set to represent the earth's body; now the armillary sphere is placed on a high terrace, looking down on where sun and moon rise—then the cord does not meet the earth's horizon. Your subject examines this theory: though it has rough reason, heaven and earth's vastness is not shifted by a terrace's height or lowness. The armillary sphere examines heaven and earth's body—there is actual number and there is proportional number. What is called actual: this number is that number; move this red and that also moves red—such is the meaning. What is called proportional: use this to proportion that—one fen here proportions to several thousand li there. Now the terrace's height or lowness is the so-called actual number; one terrace's height does not exceed several zhang—the difference there also does not exceed this; heaven and earth's greatness—can several zhang suffice to affect their height and lowness? If the balance's elevation and depression, that is the so-called proportional number. The balance moves one fen, and there one does not know the number of several thousand li—therefore the balance's elevation and depression should be examined carefully, while the terrace's height and lowness is not what should concern one.
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其三曰：月行之道，過交則入黃道六度而稍卻，復交則出於黃道之南，亦如之。 月行周于黃道，如繩之繞木，故月交而行日之陰，則日爲之虧； 入蝕法而不虧者，行日之陽也。 每月退交二百四十九周有奇，然後復會。 今月道既不能環繞黃道，又退交之漸當每日差池，今必候月終而頓移，亦終不能符會天度，當省去月環。 其候月之出入，專以曆法步之。
Third: the moon's path—passing the crossing it enters the ecliptic six degrees and slightly retreats; crossing again it exits south of the ecliptic, likewise. The moon travels circling the ecliptic like a cord winding a post; therefore when the moon crosses and travels on the sun's yin side, the sun is eclipsed for it. Entering the eclipse method yet not eclipsed—traveling on the sun's yang side. Each month it retreats one crossing, 249 circuits and a fraction, then meets again. Now the moon path cannot encircle the ecliptic, and the gradual retreat of crossings should differ daily; now one must wait until month's end to shift suddenly—still it cannot match heavenly degrees; the moon ring should be eliminated. Observing the moon's entering and exiting should rely solely on calendar method to step it.
27
其四，衡上、下二端皆徑一度有半，用日之徑也。 若衡端不能全容日月之體，則無由審日月定次。 欲日月正滿上衡之端，不可動移，此其所以用一度有半爲法也。 下端亦一度有半，則不然。 若人目迫下端之東以窺上端之西，則差幾三度。 凡求星之法，必令所求之星正當穿之中心。 今兩端既等，則人目遊動，無因知其正中。 今以鉤股法求之，下徑三分，上徑一度有半，則兩竅相覆，大小略等。 人目不搖，則所察自正。
Fourth: the balance's upper and lower two ends are each one and a half degrees in diameter, using the sun's diameter. If the balance end cannot fully contain the sun and moon's bodies, there is no means to examine the sun and moon's fixed positions. Wishing the sun and moon to fill exactly the upper balance end, immovable—this is why one and a half degrees is used as the standard. The lower end also one and a half degrees—this is not so. If the human eye presses the lower end's east to peep at the upper end's west, the difference is nearly three degrees. For all methods of seeking stars, one must make the sought star exactly at the piercing's center. Now both ends being equal, the human eye moves—there is no means to know the exact center. Now using gougu method to seek it: lower diameter three fen, upper diameter one and a half degrees—then the two apertures overlap, sizes roughly equal. The human eye not swaying, what is observed is naturally correct.
28
其五，前世皆以極星爲天中，自祖暅以璣衡窺考天極不動處，乃在極星之末猶一度有餘。 今銅儀天樞內徑一度有半，乃謬以衡端之度爲率。 若璣衡端平，則極星常遊天樞之外； 璣衡小偏，則極星乍出乍入。 令瓚舊法，天樞乃徑二度有半，蓋欲使極星游於樞中也。 臣考驗極星更三月，而後知天中不動處遠極星乃三度有餘，則祖暅窺考猶爲未審。 今當爲天樞徑七度，使人目切南樞望之，星正循北極樞裏周常見不隱，天體方正。
Fifth: former ages all took the pole star as heaven's center; from Zu Geng using the armillary sphere to observe and examine, the unmoving place of the celestial pole is still more than one degree beyond the pole star's tip. Now the bronze instrument's celestial pivot inner diameter is one and a half degrees—mistakenly taking the balance end's degree as the rate. If the armillary sphere end is level, the pole star constantly travels outside the celestial pivot. If the armillary sphere is slightly off-center, the pole star now exits now enters. Lingzan's old method: the celestial pivot diameter was two and a half degrees—intending to make the pole star travel within the pivot. Your subject examined the pole star through three months, then knew the unmoving place of heaven's center is more than three degrees from the pole star—then Zu Geng's observation was still not exact. Now the celestial pivot should be seven degrees in diameter, making the human eye press against the south pivot to look—the star circulates within the north polar pivot, constantly visible and not hidden, the celestial body square and correct.
29
其六，令瓚以辰刻、十幹、八卦皆刻於紘，然紘平正而黃道斜運，當子、午之間，則日徑度而道促； 卯、酉之際，則日迤行而道舒。 如此，辰刻不能無謬。 新銅儀則移刻於緯，四游均平，辰刻不失。 然令瓚天中單環，直中國人頂之上，而新銅儀緯斜絡南北極之中，與赤道相直。 舊法設之無用，新儀移之爲是。 然當側窺如車輪之牙，而不當衡規如鼓陶，其旁迫狹，難賦辰刻，而又蔽映星度。
Sixth: Lingzan carved double-hours, ten stems, and Eight Trigrams all on the cord; yet the cord is level and correct while the ecliptic travels obliquely—between zi and wu the sun's diameter is compressed and the path short. At mao and you the sun travels winding and the path slack. Thus double-hours cannot be without error. The new bronze instrument then moves the quarters to the latitude ring; the four tubes are evenly level, double-hours without loss. Yet Lingzan's heaven-center single ring stood directly above the heads of people in China, while the new bronze instrument's latitude ring slants across between north and south poles, perpendicular to the equator. The old method's setting was useless; the new instrument's moving it is correct. Yet one should observe sidelong like a wheel's teeth, not like a balance ring like a drum pot—its sides cramped and narrow, hard to assign double-hours, and also obscuring stellar degrees.
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其七，司天銅儀，黃、赤道與紘合鑄，不可轉移，雖與天運不符，至於窺測之時，先以距度星考定三辰所舍，復運遊儀抵本宿度，乃求出入黃道與去極度，所得無以異於令瓚之術。 其法本于晁崇、斛蘭之舊制，雖不甚精縟，而頗爲簡易。 李淳風嘗謂斛蘭所作鐵儀，赤道不動，乃如膠柱。 以考月行，差或至十七度，少不減十度。 此正謂直以赤道候月行，其差如此。 今黃、赤道度，再運遊儀抵所舍宿度求之，而月行則以月曆每日去極度算率之，不可謂之膠也。 新法定宿而變黃道，此定黃道而變宿，但可賦三百六十五度而不能具餘分，此其爲略也。
Seventh: the Directorate's bronze instrument—yellow and red equator and cord cast together, immovable; though not matching heaven's motion, at the time of observation, first use distance-degree stars to examine and fix where the three luminaries lodge, then turn the traveling instrument to reach the original lodge degree, then seek entering and exiting the ecliptic and distance from the pole—the results differ not from Lingzan's method. The method originally derives from Chao Chong and Hulu's old construction—though not very refined, it is quite simple. Li Chunfeng once said Hulu's iron instrument, the equator unmoving, was like gluing the pillar. Examining the moon's motion, the error could reach 17 degrees, at least not less than 10 degrees. This precisely means directly using the equator to observe the moon's motion—the error was thus. Now yellow and red equator degrees, again turning the traveling instrument to reach the lodged lodge degree to seek them; moon motion is then calculated by daily distance-from-pole rate from the lunar calendar—this cannot be called glued. The new method fixes lodges and changes the ecliptic; this fixes the ecliptic and changes lodges—it can assign 365 degrees but cannot include remainder fractions; this is where it is coarse.
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其八，令瓚舊法，黃道設於月道之上，赤道又次月道，而璣最處其下。 每月移一交，則黃、赤道輒變。 今當省去月道，徙璣於赤道之上，而黃道居赤道之下，則二道與衡端相迫，而星度易審。
Eighth: Lingzan's old method set the ecliptic above the moon path, the equator again below the moon path, and ji lowest of all. Each month moving one crossing, yellow and red equators then changed. Now the moon path should be eliminated, ji moved above the equator, and the ecliptic placed below the equator—then the two paths press close to the balance end, and stellar degrees are easy to examine.
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其九，舊法：規環一面刻周天度，一面加銀丁。 所以施銀丁者，夜候天晦，不可目察，則以手切之也。 古之人以璿爲之，璿者，珠之屬也。 今司天監三辰儀設齒於環背，不與橫蕭會，當移列兩旁，以便參察。
Ninth, old method: the ring's one face carved the full circuit degrees, one face added silver studs. The reason for applying silver studs: observing heaven at night when dark, unable to see with the eye, one cuts with the hand. Ancient people used xuan for this—xuan is a kind of pearl. Now the Directorate's Three Luminaries Instrument sets teeth on the ring's back, not meeting the transverse flute—they should be moved to both sides for convenient examination.
33
其十，舊法：重璣皆廣四寸，厚四分。 其他規軸，椎重樸拙，不可旋運。 今小損其制，使之輕利。
Tenth, old method: heavy ji all four cun wide, four fen thick. Other rings and axles were heavy, crude, and unable to rotate. Now slightly reduce the construction, making it light and nimble.
34
其十一，古之人知黃道歲易，不知赤道之因變也。 黃道之度，與赤道之度相偶者也。 黃道徙而西，則赤道不得獨膠。 今當變赤道與黃道同法。
Eleventh: ancient people knew the ecliptic shifts yearly, but did not know the equator shifts accordingly. Ecliptic degrees and equatorial degrees are paired. When the ecliptic shifts west, the equator cannot alone remain glued. Now the equator should be changed to the same method as the ecliptic.
35
其十二，舊法：黃、赤道平設，正當天度，掩蔽人目，不可占察。 其後乃別加鑽孔，尤爲拙謬。 今當側置少偏，使天度出北際之外，自不淩蔽。
Twelfth, old method: yellow and red equators set level, directly blocking stellar degrees, obscuring the human eye, unable to observe. Afterward they separately added drilled holes—especially crude and absurd. Now they should be set sidelong, slightly off-center, making stellar degrees emerge beyond the north edge, naturally not obscured.
36
其十三，舊法：地紘正絡天經之半，凡候三辰出入，則地際正爲地紘所伏。 今當徙紘稍下，使地際與紘之上際相直。 候三辰伏見，專以紘際爲率，自當默與天合。
Thirteenth, old method: the earth cord vertically crossed half the celestial meridian; for all observing the three luminaries' rising and setting, the earth horizon was exactly covered by the earth cord. Now the cord should be moved slightly down, making the earth horizon align with the cord's upper edge. Observing the three luminaries' hiding and visibility, relying solely on the cord edge as rate—it should silently accord with heaven.
37
又言渾儀制器：
He further spoke on constructing the armillary sphere instrument:
38
渾儀之爲器，其屬有三，相因爲用。 其在外者曰體，以立四方上下之定位。 其次曰象，以法天之運行，常與天隨。 其在內璣衡，璣以察緯，衡以察經。 求天地端極三明匿見者，體爲之用； 察黃道降陟辰刻運徙者，象爲之用； 四方上下無所不屬者，璣衡爲之用。
The armillary sphere as instrument has three parts, mutually dependent in use. The outermost is called the body, to establish the fixed positions of the four directions, above and below. Next is called the image, to model heaven's motion, constantly following heaven. Within are ji and heng—ji to examine latitude, heng to examine longitude. Seeking heaven and earth's endpoints and the three luminaries' hiding and visibility—the body serves. Examining the ecliptic's rise and fall, double-hours, and motion and shift—the image serves. The four directions, above and below, nothing not belonging to—ji and heng serve.
39
體之爲器，爲圓規者四。 其規之別：一曰經，經之規二並峙，正抵子午，若車輪之植。 二規相距四寸，夾規爲齒，以別去極之度。 北極出紘之上三十有四度十分度之八強，南極下紘亦如之。 對銜二釭，聯二規以爲一，釭中容樞。 二曰緯，緯之規一，與經交於二極之中，若車輪之倚，南北距極皆九十一度強。 夾規爲齒，以別周天之度。 三曰紘，紘之規一，上際當經之半，若車輪之僕，以考地際，周賦十二辰，以定八方。 紘之下有趺，從一衡一，刻溝受水以爲平。 中溝爲地，以受注水。 四末建趺，爲升龍四以負紘。 凡渾儀之屬皆屬焉。 龍吭爲綱維之四揵以爲固。
The body as instrument has four circular rings. The ring types: first called meridian—two meridian rings stand together, directly reaching zi and wu, like a wheel planted upright. The two rings four cun apart, teeth between the rings to distinguish degrees from the pole. The north pole rises 34 degrees and eight-tenths of a degree above the cord; the south pole descends below the cord likewise. Two pins face each other, linking the two rings as one; the pin holds the pivot. Second called latitude—one latitude ring, crossing the meridian at the two poles' center, like a wheel leaning; north and south each 91-odd degrees from the pole. Teeth between the ring to distinguish the full circuit degrees. Third called cord—one cord ring, upper edge at half the meridian, like a wheel prostrate, to examine the earth horizon; the twelve branches assigned around to fix the eight directions. Below the cord is a base, following one crossbeam, channels carved to receive water as level. The center channel is earth, to receive poured water. At the four corners bases are built, four ascending dragons to bear the cord. All parts of the armillary sphere belong to these. The dragon throats are the four anchors of the net to make it firm.
40
象之爲器，爲圓規者四。 其規之別：一曰璣，璣之規二並峙，相距如經之度。 夾規爲齒，對銜二釭，釭中容樞，皆如經之率。 設之亦如經，其異者經膠而璣可旋。 二曰赤道，赤道之規一刻，璣十分寸之三以銜赤道。 赤道設之如緯，其異者緯膠於經，而赤道銜於璣，有時而移，度穿一竅，以移歲差。 三曰黃道，黃道之規一，刻赤道十分寸之二以銜黃道，其南出赤道之北際二十有四度，其北入赤道亦如之。 交於奎、角，度穿一竅，以銅編屬於赤道。 歲差盈度，則並赤道徙而西。 黃赤道夾規爲齒，以別均迤之度。
The image as instrument has four circular rings. The ring types: first called ji—two ji rings stand together, distance like the meridian's degrees. Teeth between the rings, two pins facing each other, pin holding pivot—all like the meridian's rate. Set like the meridian; the difference: the meridian is glued while ji can rotate. Second called equator—one equator ring, ji three-tenths of a cun to hold the equator. The equator set like latitude; the difference: latitude is glued to the meridian, while the equator is held by ji, moving at times; degrees pierce one hole to shift precession. Third called ecliptic—one ecliptic ring, two-tenths of an equatorial cun to hold the ecliptic; south it emerges 24 degrees beyond the equator's north edge, north it enters the equator likewise. Crossing at Straddle and Horn, degrees pierce one hole, bound to the equator with copper wire. When precession exceeds degrees, the equator shifts west together. Yellow and red equator teeth between rings to distinguish equal and winding degrees.
41
璣衡之爲器，爲圓規二，曰璣，對峙，相距如象璣之度，夾規爲齒，皆如象璣。 其異者：象璣對銜二釭，而璣對銜二樞，貫於象璣天經之釭中。 三物相重而不相膠，爲間十分寸之三，無使相切，所以利旋也。 爲橫簫二，兩端夾樞，屬於璣，其中挾衡爲橫一，棲於橫簫之間。 中衡爲轊，以貫橫簫，兩末入於璣之罅而可旋。 璣可以左右，以察四方之詳； 衡可以低昂，以察上下之祥。
Ji and heng as instrument have two circular rings, called ji, facing each other, distance like the image ji's degrees, teeth between rings—all like image ji. The difference: image ji faces two pins, while ji faces two pivots, passing through the image ji celestial meridian's pin. Three things stacked yet not glued, gap three-tenths of a cun, not touching—thus benefiting rotation. Two transverse flutes made, both ends clamping pivots, belonging to ji; within them one crossbeam heng is held, resting between the transverse flutes. The central heng is an axle, passing through the transverse flute; both ends enter ji's gap and can rotate. Ji can move left and right, to examine the four directions in detail. Heng can rise and fall, to examine omens above and below.
42
《浮漏議》曰：
The Memorial on the Clepsydra says:
43
播水之壺三，而受水之壺一。 曰求壺、廢壺，方中皆圓尺有八寸，尺有四寸五分以深，其食二斛，爲積分四百六十六萬六千四百六十。 曰復壺，如求壺之度，中離以爲二，元一斛介八斗，而中有達。 曰建壺，方尺植三尺有五寸，其食斛有半。 求壺之水，復壺之所求也。 壺盈則水馳，壺虛則水凝。 復壺之肋爲枝渠，以爲水節。 求壺進水暴，則流怒以搖，復以壺，又折以爲介。 復爲枝渠，達其濫溢。 枝渠之委，所謂廢壺也，以受廢水。 三壺皆所以播水，爲水制也。 自復壺之介，以玉權釃於建壺，建壺所以受水爲刻者也。 建壺一易箭，則發上室以瀉之。 求、復、建壺之泄，皆欲迫下，水所趣也。 玉權下水之概寸，矯而上之然後發，則水撓而不躁也。 復壺之達半求壺之注，玉權半復壺之達。 枝渠博皆分，高如其博，平方如砥，以爲水概。 壺皆爲之冪，無使穢遊，則水道不慧。 求壺之冪龍紐，以其出水不窮也。 復壺士紐，士所以生法者，復壺制法之器也。 廢壺鯢紐，止水之沈，鯢所伏也。 銅史令刻，執漏政也。 冬設煴燎，以澤凝也。 注水以龍噣直頸附於壺體，直則易浚，附於壺體則難敗。 復壺玉爲之喙，銜于龍噣，謂之權，所以權其盈虛也。 建壺之執窒瓬塗而彌之以重帛，窒則不吐也。 管之善利者，水所溲也，非玉則不能堅良以久。 權之所出高則源輕，源輕則其委不悍而溲物不利。 箭不效於璣衡，則易權、洗箭而改畫，覆以璣衡，謂之常不弊之術。 今之下漏者，始嘗甚密，久復先大者管泐也。 管泐而器皆弊者，無權也。 弊而不可復壽者，術固也。 察日之晷以璣衡，而制箭以日之晷跡，一刻之度，以賦餘刻，刻有不均者，建壺有眚也。 贅者磨之，創者補之，百刻一度，其壺乃善。 晝夜已復，而箭有餘才者，權鄙也。 晝夜未復，而壺吐者，權沃也。 如是，則調其權，此制器之法也。
Three vessels that broadcast water, and one vessel that receives water. Called the seeking vessel and waste vessel: square within, circular one chi eight cun, one chi four cun five fen deep; capacity two hu, volume 4,666,460 fen. Called the recovery vessel: like the seeking vessel's measure, divided in the center into two, originally one hu plus eight dou, with a passage within. Called the building vessel: square one chi, upright three chi five cun, capacity one and a half hu. The seeking vessel's water is what the recovery vessel seeks. When the vessel is full water runs swiftly; when empty water congeals. The recovery vessel's ribs are branch channels, serving as water regulators. When the seeking vessel's inflow is violent, the flow rages and shakes; again using the recovery vessel, again bent as a divider. Again branch channels made, reaching its overflow. The branch channel's discharge—the so-called waste vessel—receives waste water. All three vessels broadcast water, forming the water system. From the recovery vessel's divider, jade weights pour into the building vessel—the building vessel receives water to mark quarters. When the building vessel changes one arrow, the upper chamber is opened to drain it. The seeking, recovery, and building vessels' outflow—all wish to press downward, where water tends. The jade weight's lower water gate one cun, bent upward then released—then water flows gently without turbulence. The recovery vessel's passage half the seeking vessel's pour; the jade weight half the recovery vessel's passage. Branch channel width all one fen, height like its width, square and flat like a whetstone, serving as water gate. All vessels have covers, not letting filth float—then the waterway is not obstructed. The seeking vessel's cover has a dragon knob, because its outflow is inexhaustible. The recovery vessel has an earth knob—earth is what generates law; the recovery vessel is the instrument that generates law. The waste vessel has a salamander knob—salamander is what sinks in still water, where the salamander hides. The bronze clerk marks quarters, holding the clepsydra's governance. In winter warming fires are set, to moisten congelation. Water is poured through a dragon mouth with straight neck attached to the vessel body—straight then easy to dredge, attached to the body then hard to break. The recovery vessel's jade beak, held in the dragon mouth, called the weight—thus to weigh its fullness and emptiness. The building vessel's hold is sealed with tile-coating and covered with heavy cloth—sealed then it does not spurt. Of tubes, the good and sharp are what water erodes—without jade they cannot be firm and fine for long. Where the weight emerges high, the source is light; source light then its discharge is not fierce and erodes objects unfavorably. If the arrow does not match the armillary sphere, change the weight, wash the arrow and alter the marks, cover with the armillary sphere—called the method of constant non-decay. Present clepsydrae at first are very precise; after long time again the large tube erodes first. Tube eroded and all vessels decayed—no weight. Decayed and unable to restore life—the method is fixed. Examining the sun's shadow with the armillary sphere, and making the arrow from the sun's shadow trace—one quarter's degree, to assign the remaining quarters; quarters uneven—the building vessel has fault. Superfluous parts grind them; damaged parts repair them—one hundred quarters one degree, then the vessel is good. Day and night already completed yet the arrow has surplus—weight is coarse. Day and night not yet completed yet the vessel spouts—weight is saturated. Thus adjust the weight—this is the method of constructing the instrument.
44
下漏必用甘泉，惡其垽之爲壺眚也。 必用一源泉之冽者，權之而重，重則敏於行，而爲箭之情慓； 泉之鹵者，權之而輕，輕則椎於行，而爲箭之情駑。 一井不可他汲，數汲則泉濁。 陳水不可再注，再注則行利。 此下漏之法也。
Clepsydrae must use sweet spring, hating that sediment becomes vessel fault. Must use one spring source that is pure; weigh it and it is heavy—heavy then swift in motion, and the arrow's nature is urgent. Brackish spring, weigh it and it is light—light then sluggish in motion, and the arrow's nature is slow. One well cannot be drawn from elsewhere; repeated drawing and the spring becomes turbid. Stale water cannot be poured again; poured again then motion is sharp. This is the method of the clepsydra.
45
箭一如建壺之長，廣寸有五分，三分去二以爲之厚，其陽爲百刻，爲十二辰。 博牘二十有一，如箭之長，廣五分，去半以爲之厚。 陽爲五更，爲二十有五籌； 陰刻消長之衰。 三分箭之廣，其中刻契以容牘。 夜算差一刻，則因箭而易牘。 鐐匏，箭舟也。 其虛五升，重一鎰有半。 鍛而赤柔者金之美者也，然後漬而不墨，墨者其久必蝕。 銀之有銅則墨，銅之有錫則屑，特銅久灂則腹敗而飲，皆工之所不材也。
The arrow like the building vessel's length, one cun five fen wide, two-thirds of three fen taken as thickness; the yang side has hundred quarters, twelve branches. Broad tablets twenty-one, like the arrow's length, five fen wide, half taken as thickness. Yang side five watches, twenty-five tallies. Yin side carved with waxing and waning decline. One-third the arrow's width, within carved notches to hold tablets. Night calculation differs one quarter, then change tablets according to the arrow. The chain gourd is the arrow boat. Its empty capacity five sheng, weight one yi and a half. Forged red and soft is the beauty of metal; then steeped and not black—black then in time it must corrode. Silver with copper then blackens; copper with tin then flakes; pure copper long steeped then the belly rots and leaks—all what craftsmen reject.
46
《景表議》曰：
The Memorial on the Gnomon says:
47
步景之法，惟定南北爲難。 古法置槷爲規，識日出之景與日入之景。 晝參諸日中之景，夜考之極星。 極星不當天中，而候景之法取晨夕景之最長者規之，兩表相去中折以參驗，最短之景爲日中。 然測景之地，百里之間，地之高下東西不能無偏，其間又有邑屋山林之蔽，倘在人目之外，則與濁氛相雜，莫能知其所蔽，而濁氛又系其日之明晦風雨，人間煙氣塵坌變作不常。 臣在本局候景，入濁出濁之節，日日不同，此又不足以考見出沒之實，則晨夕景之短長未能得其極數。
The method of stepping shadows—only fixing south and north is hard. Ancient method set a gnomon as compass, marking the sun's rising shadow and setting shadow. By day compare with the noon shadow, by night examine the pole star. The pole star is not at heaven's center; yet the shadow-observing method takes the longest morning and evening shadows to compass them; the two tables distanced, folded in half to verify—the shortest shadow is noon. Yet the shadow-measuring place, within a hundred li, the land's height and lowness east and west cannot be without bias; moreover towns, houses, forests, and hills may block beyond human sight, then mix with turbid vapor, unable to know what is blocked; turbid vapor also depends on the sun's clarity, wind, rain, human smoke, dust, and ash changing irregularly. Your subject at this bureau observed shadows; the nodes of entering and exiting turbidity differed day by day—this too is insufficient to examine the reality of rising and setting; then the morning and evening shadows' short and long could not obtain their extreme numbers.
48
參考舊聞，別立新術。 候景之表三，其崇八尺，博三寸三分，殺一以爲厚者。 圭首剡其南使偏銳。 其趺方厚各二尺，環趺刻渠受水以爲准。 以銅爲之。 表四方志墨以爲中刻之，綴四繩，垂以銅丸，各當一方之墨。 先約定四方，以三表南北相重，令趺相切，表別相去二尺，各使端直。 四繩皆附墨，三表相去左右上下以度量之，令相重如一。 自日初出，則量西景三表相去之度，又量三表之端景之所至，各別記之。 至日欲入，候東景亦如之。 長短同，相去之疏密又同，則以東西景端隨表景規之，半折以求最短之景。 五者皆合，則半折最短之景爲北，表南墨之下爲南，東西景端爲東西。 五候一有不合，未足以爲正。 既得四方，則惟設一表，方首，表下爲石席，以水準之，植表于席之南端。 席廣三尺，長如九服冬至之景，自表趺刻以爲分，分積爲寸，寸積爲尺。 爲密室以棲表，當極爲霤，以下午景使當表端。 副表並趺崇四寸，趺博二寸，厚五分，方首，剡其南，以銅爲之。 凡景表景薄不可辨，即以小表副之，則景墨而易度。
Referencing old reports, a separate new method was established. Three shadow-observing tables, eight chi high, three cun three fen wide, one part reduced for thickness. The gnomon head carved south to make it obliquely sharp. The base square, two chi thick each side, ring base channels carved to receive water as standard. Made of copper. The table's four sides marked with ink as center lines; four cords attached, copper pellets hanging, each aligned with one side's ink. First fix the four directions; three tables north-south overlapping, bases touching, tables two chi apart, each made upright. All four cords attached to ink; three tables' distance left, right, above, below measured, making them overlap as one. From the sun's first rising, measure the western shadow's three tables' distances; also measure where the three tables' tip shadows reach, each separately recorded. When the sun is about to set, observe the eastern shadow likewise. Long and short the same, distance sparse and dense also the same—then use east and west shadow tips following table shadows to compass them, fold in half to seek the shortest shadow. All five accord—then fold in half the shortest shadow as north, below the south ink of the table as south, east and west shadow tips as east and west. If one of the five observations does not accord, it is not yet sufficient to be correct. Having obtained the four directions, then set only one table, square head; below the table a stone mat, leveled with water, table planted at the mat's south end. Mat three chi wide, length like the nine regions' winter solstice shadow; from the table base carve as fen, fen accumulated as cun, cun accumulated as chi. A sealed chamber made to house the table; at the apex a gutter, so afternoon shadow aligns with the table tip. Auxiliary table and base together four cun high, base two cun wide, five fen thick, square head, south carved sharp, made of copper. For all gnomon shadows thin and hard to distinguish, use the small auxiliary table—then the shadow is dark and easy to measure.
49
元祐間蘇頌更作者，上置渾儀，中設渾象，旁設昏曉更籌，激水以運之。 三器一機，吻合躔度，最爲奇巧。 宣和間，又嘗更作之。 而此五儀者悉歸於金。
In the Yuanyou era Su Song's revised construction placed the armillary sphere above, celestial globe in the center, dusk-dawn tally racks beside, water driven to turn them. Three instruments one mechanism, matching in stellar motion—most marvelously ingenious. In the Xuanhe era, it was again remade. And these five instruments all passed to the Jurchens.
50
中興更謀製作，紹興三年正月，工部員外郎袁正功獻渾儀木樣，太史局令丁師仁始請募工鑄造，且言：「東京舊儀用銅二萬餘，今請折半用八千斤有奇。」 已而不就，蓋在廷諸臣罕通其制度者。 乃召蘇頌子攜取頌遺書，考質舊法，而攜亦不能通也。 至十四年，乃命宰臣秦檜提舉鑄渾儀，而以內侍邵諤專領其事，久而儀成。 三十二年，始出其二置太史局。 而高宗先自爲一儀置諸宮中，以測天象，其制差小，而邵諤所鑄蓋祖是焉，後在鐘鼓院者是也。
After the Restoration, remaking was again planned; in the first month of the third year of Shaoxing, Works Department Outer Section Member Yuan Zhenggong presented a wooden model of the armillary sphere; Bureau of Astronomy Director Ding Shiren first requested recruiting artisans to cast it, saying: 'The old Eastern Capital instrument used more than 20,000 jin of copper; now requesting half, 8,000-odd jin.' Thereafter it was not completed—because few ministers at court understood its construction. Su Song's son Xie was summoned to fetch Song's surviving books, to verify the old method—and Xie also could not understand. Down to the fourteenth year, Chancellor Qin Hui was ordered to supervise casting the armillary sphere, and eunuch Shao E specially directed the matter; after long time the instrument was completed. In the thirty-second year, two of them were first placed at the Bureau of Astronomy. Gaozong had first made one instrument himself and placed it in the palace to observe heavenly signs; its construction was somewhat smaller, and Shao E's casting largely followed it—the one later at the Bell and Drum Court is that.
51
清台之儀，後其一在秘書省。 按：儀制度：表裏凡三重，其第一重曰六合儀，陽經徑四尺九寸六分，闊三寸二分，厚五分。 南北正位，兩面各列周天度數，南北極出入地皆三十一度少，度闊三分。 陰緯單環大小如陽經，闊三寸二分，厚一寸八分。 上置水準池，闊九分，深四分，沿環通流，亦如舊制。 內外八幹、十二枝，畫艮、巽、坤、乾卦於四維。 第二重曰三辰儀，徑四尺三分，闊二寸二分，厚五分。 釭釧刻畫如陽經。 赤道單環，徑四尺一寸四分，闊一寸二分，厚五分。 上列二十八宿、均天度數，闊二分七厘。 黃道單環，徑四尺一寸四分，闊一寸二分，厚五分，上列七十二候，均分卦策，與赤道相交，出入各二十四度弱。 百刻單環，徑四尺五寸六分，闊一寸二分，厚五分，上列晝夜刻數。 第三重曰四遊儀，徑三尺九寸，闊一寸九分，厚五分。 釭釧刻畫如璿璣，度闊二分半。 望筒長三尺六寸五分，內圓外方，中通孔竅，四面闊一寸四分七厘，窺眼闊三分，夾窺徑五尺三分。 鼇雲以負龍柱，龍柱各高五尺二寸。 十字平水台高一尺一寸七分，長五尺七寸，闊五寸二分。 水槽闊七分，深一寸二分。 若水運之法與夫渾象，則不復設。
Of the Clear Terrace instruments, afterward one was at the Secretariat. Note: instrument construction—outer and inner three layers in all; the first layer called Six Harmonies Instrument: yang meridian diameter four chi nine cun six fen, width three cun two fen, thickness five fen. North-south correct position, both faces listing full circuit degrees; north and south poles entering and exiting earth each 31-odd degrees, degree width three fen. Yin latitude single ring size like yang meridian, width three cun two fen, thickness one cun eight fen. Above set a leveling pool, width nine fen, depth four fen, flowing around the ring—also like the old construction. Inner and outer eight stems and twelve branches, Gen, Xun, Kun, and Qian trigrams painted at the four dimensions. Second layer called Three Luminaries Instrument, diameter four chi three fen, width two cun two fen, thickness five fen. Pin and bracelet carved like yang meridian. Equator single ring, diameter four chi one cun four fen, width one cun two fen, thickness five fen. Above listed the twenty-eight lodges, equal heavenly degrees, width two fen seven li. Ecliptic single ring, diameter four chi one cun four fen, width one cun two fen, thickness five fen, above listed seventy-two hou, equally divided trigram tallies, intersecting the equator, entering and exiting each 24-odd degrees. Hundred-quarters single ring, diameter four chi five cun six fen, width one cun two fen, thickness five fen, above listed day-night quarter numbers. Third layer called Four-Tube Instrument, diameter three chi nine cun, width one cun nine fen, thickness five fen. Pin and bracelet carved like Xuanji, degree width two and a half fen. Sighting tube length three chi six cun five fen, round within square without, central hole, four faces width one cun four fen seven li, sight width three fen, clamping sight diameter five chi three fen. Turtle clouds bearing dragon pillars, dragon pillars each five chi two cun high. Cross-shaped level water platform height one chi one cun seven fen, length five chi seven cun, width five cun two fen. Water trough width seven fen, depth one cun two fen. As for water-driven method and the celestial globe, these were no longer set up.
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其後朱熹家有渾儀，頗考水運制度，卒不可得。 蘇頌之書雖在，大抵於渾象以爲詳，而其尺寸多不載，是以難遽復云。 舊制有白道儀以考月行，在望筒之旁。 自熙甯沈括以爲無益而去之，南渡更造，亦不復設焉。
Afterward Zhu Xi's household had an armillary sphere; he quite examined the water-driven system, but ultimately could not obtain it. Su Song's book though exists, largely detailed on the celestial globe, yet many dimensions not recorded—thus hard to restore quickly. Old construction had a white-path instrument to examine the moon's motion, beside the sighting tube. From Xining Shen Kuo thought it useless and removed it; after the Southern Crossing in remaking, it was also not set up.
53
極度極星之在紫垣，爲七曜、三垣、二十八宿衆星所拱，是謂北極，爲天之正中。 而自唐以來，曆家以儀象考測，則中國南北極之正，實去極星之北一度有半，此蓋中原地勢之度數也。 中興更造渾儀，而太史令丁師仁乃言：「臨安府地勢向南，於北極高下當量行移易。」 局官呂璨言：「渾天無量行更易之制，若用於臨安與天參合，移之他往必有差忒。」 遂罷議。 後十餘年，邵諤鑄儀，則果用臨安北極高下爲之。 以清台儀校之，實去極星四度有奇也。
Polar Degrees. The pole star in the Purple Enclosure is encircled by the seven luminaries, three enclosures, and twenty-eight lodges' multitude of stars—this is called the North Pole, heaven's true center. Yet from Tang onward, calendar-makers using instruments observed and measured—China's true north and south pole is actually one and a half degrees north of the pole star; this is the Central Plain's geographical degree. After the Restoration in remaking the armillary sphere, Grand Astrologer Ding Shiren said: 'Lin'an Prefecture's terrain slopes south; for north pole height and lowness one should measure shifting.' Bureau officer Lü Can said: 'The armillary heaven has no system of measuring shifting; if used at Lin'an to accord with heaven, moved elsewhere there must be discrepancy.' The discussion was then dropped. More than ten years later, Shao E cast the instrument, then indeed using Lin'an's north pole height and lowness. Compared with the Clear Terrace instrument, it was actually more than four degrees from the pole star.
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黃赤道
Yellow and Red Equators
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黃赤道占天之法，以二十八宿爲綱維，分列四方，南北去極各九十有一度有奇，南低而北昂，去地各三十有六度，一定不易者，名之曰赤道。 以日躔半在赤道內，半在赤道外，出入內外極遠者皆二十有四度，以其行赤道之中者名之曰黃道。 凡五緯皆隨日由黃道行，惟月之行有九道，四時交會歸於黃道而轉變焉，故有青、黑、白、赤四者之異名。
The method of observing heaven by yellow and red equators takes the twenty-eight lodges as framework, arrayed in four directions; north and south each 91-odd degrees from the pole, south low and north high, each 36 degrees from earth—fixed and unchanging—called the equator. The sun's station half within the equator, half without; entering and exiting inner and outer, farthest never exceeding 24 degrees—its path at the equator's center is called the ecliptic. All five planets follow the sun traveling the ecliptic; only the moon's motion has nine paths, four seasons' crossings returning to the ecliptic and transforming—hence the four different names of green, black, white, and red.
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夫赤道終古不移，則星舍宜無盈縮矣。 然自唐一行作《大衍曆》，以儀揆測之，得畢、觜、參、鬼四宿，分度與古不同。 皇祐初，日官周琮以新儀測候，與唐一行尤異。 紹聖二年，清台以赤道度數有差，復命考正。 惟牛、室、尾、柳四宿與舊法合，其他二十四宿躔度或多或寡。 蓋天度之不齊，古人特紀其大綱，後世漸極於精密也。
If the equator from antiquity never shifts, stellar lodges should have no expansion and contraction. Yet from Tang Yixing's Dayan Calendar, using instruments to measure, obtained Net, Beak, Three Stars, and Ghost four lodges whose degree divisions differed from antiquity. At the beginning of Huangyou, day-officer Zhou Cong using the new instrument observed, differed especially from Tang Yixing. In the second year of Shaosheng, the Clear Terrace because equatorial degrees had discrepancy, again ordered correction. Only Ox, Encampment, Tail, and Willow four lodges accorded with the old method; the other twenty-four lodges' stations were more or less. Heaven's degrees are not uniform—ancients only recorded the great outline; later ages gradually reached utmost precision.
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若夫黃道橫絡天體，列宿躔度自隨歲差而增減。 中興以來，用《統元》、《紀元》及《乾道》、《淳熙》、《開禧》、《統天》、《會元》，每一曆更一黃道，其多寡之異有不可勝載者，而步占家亦隨各曆之躔度焉。
As for the ecliptic horizontally crossing the celestial body, asterisms' stations naturally increase and decrease with precession. Since the Restoration, using the Tongyuan, Jiyuan, Qiandao, Chunxi, Kaiqi, Tongtian, and Huiyuan calendars—each calendar changed the ecliptic once; the differences in more and less cannot all be recorded, and step-calculators also followed each calendar's stations.
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中星四時中星見於《堯典》，蓋聖人南面而治天下，即日行而定四時，虛、鳥、火、昴之度在天，夷隩析因之候在人，故《書》首載之，以見授時爲政之大也。 而後世考驗冬至之日，堯時躔虛，至於三代則躔於女，春秋時在牛，至後漢永元已在鬥矣。 大略六十餘年輒差一度。 開禧占測已在箕宿，校之堯時，幾退四十餘度。 蓋自漢太初至今，已差一氣有餘。 而太陽之躔十二次，大約中氣前後，乃得本月宮次。 蓋太陽日行一度，近歲《紀元曆》定歲差，約退一分四十餘秒。 蓋太陽日行一度而微遲緩，一年周天而微差，積累分秒而躔度見焉。 曆家考之，萬五千年之後，所差半周天，寒暑將易位，世未有知其說者焉。
Culminating Stars. Seasonal culminating stars appear in the Canon of Yao—because the sage faced south to govern all under heaven, fixing the four seasons by the sun's daily motion; Emptiness, Bird, Fire, and Hairy's degrees in heaven, the conditions of comfort, warmth, separation, and contraction in men—therefore the Documents opens with them, to show granting seasons is the great matter of governance. Yet later ages verifying the day of winter solstice: in Yao's time at Emptiness; by the Three Dynasties at Woman; in Spring and Autumn at Ox; by Later Han Yongyuan already at Dipper. Roughly every sixty-odd years it differed one degree. Kaiqi observation was already at Winnowing Basket; compared with Yao's time, nearly forty-odd degrees retreated. From Han Taichu to now, already more than one qi differed. The sun's station in the twelve cycles—approximately before and after the mid-qi, then obtaining the current month's palace station. The sun daily travels one degree; in recent years the Jiyuan Calendar fixed precession, approximately retreating one minute forty-odd seconds. The sun daily travels one degree yet slightly slow; one year completes the circuit yet slightly differs; accumulated minutes and seconds then stations appear. Calendar-makers examine it: after 15,000 years the difference is half the full circuit—cold and heat will exchange places; the age has none who know this theory.
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土圭《周官》大司徒以土圭之法正日景，以求地中。 而馮相氏春夏致日，秋冬致月，以辨四時。
Gnomon. The Rites of Zhou: the Grand Minister of Education used the gnomon method to correct the sun's shadow, to seek the earth's center. And the Master of Gnomons in spring and summer observed the sun, in autumn and winter observed the moon, to distinguish the four seasons.