1
《易》曰：「觀乎天文以察時變。」 是故古之哲王，法垂象以施化，考庶徵以致理，以授人時，以考物紀，修其德以順其度，改其過以慎其災，去危而就安，轉禍而為福者也。 夫其五緯七紀之名數，中官外官之位次，淩歷犯守之所主，飛流彗孛之所應，前史載之備矣。
The Changes says: "Observe the patterns of heaven to discern the changes of the seasons." Therefore the sage kings of antiquity took the celestial signs as their model to extend their transforming influence, examined the many portents to achieve good order, granted the seasons to the people, verified the records of things, cultivated their virtue to accord with the celestial measures, corrected their faults to guard against calamities—leaving danger for safety and turning misfortune into blessing. As for the names and numbers of the five planets and seven regulators, the positions of the inner and outer official stars, what transits, invasions, and stations govern, and what meteors, comets, and broom-stars portend—earlier histories record them fully.
2
武德年中，薛頤、庾儉等相次為太史令，雖各善於占候，而無所發明。 貞觀初，將仕郎直太史李淳風始上言靈臺候儀是後魏遺範，法制疏略，難為占步。 太宗因令淳風改造渾儀，鑄銅為之，至七年造成。 淳風因撰《法象志》七卷，以論前代渾儀得失之差，語在《淳風傳》。 其所造渾儀，太宗令置於凝暉閣以用測候，既在宮中，尋而失其所在。 玄宗，太史頻奏日蝕不效，詔沙門一行改造新曆。 一行奏云：「今欲創曆立元，須知黃道進退，請太史令測候星度。」 有司云：「承前唯依赤道推步，官無黃道遊儀，無由測候。」 時率府兵曹梁令瓚待制於麗正書院，因造遊儀木樣，甚為精密。 一行乃上言曰：「黃道游儀，古有其術而無其器。 以黃道隨天運動，難用常儀格之，故昔人潛思皆不能得。 今梁令瓚創造此圖，日道月交，莫不自然契合，既于推步尤要，望就書院更以銅鐵為之，庶得考驗星度，無有差舛。」 從之，至十三年造成。 又上疏曰：
During the Wude era, Xue Yi, Yu Jian, and others served in succession as Director of the Astronomy Bureau; though each was skilled at divination and observation, none made any innovations. At the beginning of the Zhenguan era, Li Chunfeng, Attendant Gentleman on probation attached to the Astronomy Bureau, first submitted that the Lingtai observatory instruments were a Northern Wei legacy—crude in regulation and difficult to use for astronomical computation. Emperor Taizong therefore ordered Chunfeng to redesign the armillary sphere, casting it in bronze; it was completed in the seventh year. Chunfeng thereupon compiled the Record of Celestial Forms in seven scrolls, discussing the gains, losses, and discrepancies of armillary spheres in earlier ages; the account is in his biography. The armillary sphere he made was ordered placed in the Ninghui Pavilion for measurement and observation; being within the palace, its whereabouts were soon lost. Under Emperor Xuanzong, the Astronomy Bureau repeatedly reported that solar eclipses did not match predictions, and an edict ordered the monk Yixing to revise the calendar anew. Yixing memorialized, saying: "We now wish to establish a new calendar and set the epoch, and must know the ecliptic's advance and retreat. I request the Director of the Astronomy Bureau measure and observe the star degrees." The responsible officials said: "Following precedent we rely only on the equator for computation. The office has no ecliptic traveling instrument, and there is no means to measure and observe." At that time Liang Lingzan, military staff officer of the Shuai Prefecture, was on special assignment at the Lizheng Academy; he therefore made a wooden model of the traveling instrument, which was extremely precise. Yixing then submitted, saying: "The ecliptic traveling instrument—antiquity had the method but not the device. Because the ecliptic moves with the turning of heaven, it is difficult to fix with a constant instrument; therefore men of old who pondered deeply could not obtain it. Now Liang Lingzan has created this diagram—the solar path and lunar crossing naturally accord without discrepancy; it is especially essential for computation. I hope it may be recast in bronze and iron at the academy, so that star degrees may be verified without error." The request was approved; it was completed in the thirteenth year. He also submitted a memorial, saying:
3
自周室衰微，疇人喪職，其制度遺象，莫有傳者。 漢興，丞相張蒼首創律曆之學。 至武帝詔司馬遷等更造漢曆，乃定東西、立晷儀、下漏刻，以追二十八宿相距星度，與古不同。 故唐都分天部，洛下閎運算轉曆，今赤道曆星度，則其遺法也。 後漢永元中，左中郎將賈逵奏言：「臣前上傅安等用黃道度日月，弦望多近。 史官壹以赤道度之，不與天合，至差一日以上。 願請太史官日月宿簿及星度課，與待詔星官考校。」 奏可。 「問典星待詔姚崇等十二人，皆曰：『星圖有規法，日月實從黃道，官無其器，不知施行。』， 大司農丞耿壽昌奏，以圓儀度日月行，考驗天運。 日月行赤道，至牽牛、東井，日行一度，月行十五度； 至婁、角，日行一度，月行十三度，此前代所共知也。」 是歲永元四載也。 明年，始詔太史造黃道銅儀。 冬至，日在斗十九度四分度之一，與赤道定差二度。 史官以校日月弦望，雖密近，而不為望日。 儀，黃道與度運轉，難候，是以少終其事。 其後劉洪因黃道渾儀，以考月行出入遲速。 而後代理曆者不遵其法，更從赤道命文，以驗賈逵所言，差謬益甚，此理曆者之大惑也。
From the decline of the Zhou house, the calendar officers lost their offices; their institutions and preserved instruments had no transmitters. When Han arose, Chancellor Zhang Cang first established the study of pitch-pipes and calendrics. At Emperor Wu's edict, Sima Qian and others further made the Han calendar, fixing east and west, setting up the gnomon instrument and lowering the clepsydra graduations, to pursue the star degrees between the twenty-eight lodges—different from antiquity. Therefore Tang Du divided the celestial departments; Luoxia Hong computed and turned the calendar—the present equatorial lodge star degrees are his transmitted method. In the Yongyuan era of Later Han, Left Palace Attendant Jia Kui memorialized, saying: "I previously submitted that Fu An and others, using the ecliptic to measure the Sun and Moon, found new and full moons mostly close. The astronomy officers uniformly measured by the equator; it did not accord with heaven, differing by more than a full day. I wish to request the astronomy office's ledgers of the Sun, Moon, and lodges and the star-degree assessments, to be examined and corrected with the star-official attendants awaiting edict." The memorial was approved. "When the canonical-star attendants awaiting edict Yao Chong and twelve others were questioned, all said: 'The star charts have regulated methods; the Sun and Moon truly follow the ecliptic. The office has no instrument, and does not know how to implement it.' Grand Agriculture Assistant Geng Shouchang memorialized, using a circular instrument to measure the Sun and Moon's motion, verifying the heavenly cycle. The Sun and Moon travel the equator; at Oxherd and Eastern Well, the Sun travels one degree per day and the Moon fifteen degrees; at Bond and Horn, the Sun travels one degree per day and the Moon thirteen degrees—this earlier generations jointly knew." This was the fourth year of Yongyuan. The next year, an edict first ordered the Astronomy Bureau to make a bronze ecliptic instrument. At the winter solstice, the Sun was at nineteen and one-quarter degrees of Dipper, with a fixed difference of two degrees from the equator. The astronomy officers used it to verify the Sun and Moon at new and full moon; though closely near, it was not at the day of full moon. The instrument—the ecliptic turning with the degree motion—was difficult to observe; therefore they scarcely completed the task. Afterward Liu Hong, by the ecliptic armillary sphere, examined the Moon's motion in entering and leaving and its slowness and speed. But later calendar-makers did not follow his method, again assigning lodges from the equator to verify what Jia Kui said—the errors grew ever greater; this is the great confusion of those who regulate calendrics.
4
今靈臺鐵儀，後魏明元時都匠解蘭所造，規制樸略，度刻不均，赤道不動，乃如膠柱，不置黃道，進退無准。 此據赤道月行以驗入曆遲速，多者或至十七度，少者僅出十度，不足以上稽天象，敬授人時。 近秘閣郎中李淳風著《法象志》，備載黃道渾儀法，以玉衡旋規，別帶日道，傍列二百四十九交，以攜月游，用法頗雜，其術竟寢。
The present Lingtai iron instrument was made by the master craftsman Xie Lan in the Mingyuan era of Northern Wei; its regulated form was crude and summary, its degree graduations uneven, the equator unmoving—like a glued pillar—without setting the ecliptic, advance and retreat without standard. This, relying on the equator to verify the Moon's entering the calendar for slowness and speed, in some cases reached seventeen degrees, in others barely exceeded ten degrees—insufficient to examine heavenly signs above and reverently grant the seasons to the people. Recently Secretariat Gentleman Li Chunfeng authored the Record of Celestial Forms, fully recording the method of the ecliptic armillary sphere—using the jade transversal revolving ring, separately carrying the solar path, with two hundred forty-nine crossings arrayed beside to carry the Moon's travel; the method was rather complex, and the technique in the end was abandoned.
5
臣伏承恩旨，更造遊儀，使黃道運行，以追列舍之變，因二分之中以立黃道，交於軫、奎之間，二至陟降各二十四度。 黃道之內，又施白道月環，用究陰陽朓朒之數，動合天運，簡而易從，足以製器垂象，永傳不朽。
Your subject, receiving the gracious directive, has further made the traveling instrument, causing the ecliptic to move, to pursue the changes of the lodges; at the two equinoxes the ecliptic is established, crossing between Chariot and Strider; at the two solstices ascent and descent are each twenty-four degrees. Within the ecliptic, a white-path lunar ring is also set, to investigate the numbers of yin and yang, waxing and waning; moving in accord with the heavenly cycle, simple and easy to follow—sufficient to make instruments and suspend signs, transmitted forever without perishing.
6
於是玄宗親為製銘，置之於靈臺以考星度。 其二十八宿及中外官與古經不同者，凡數十條。 又詔一行與梁令瓚及諸術士更造渾天儀，鑄銅為圓天之象，上具列宿赤道及周天度數。 注水激輪，令其自轉，一日一夜，天轉一周。 又別置二輪絡在天外，綴以日月，令得運行。 每天西轉一幣，日東行一度，月行十三度十九分度之七，凡二十九轉有餘而日月會，三百六十五轉而日行匝。 仍置木櫃以為地平，令儀半在地下，晦明朔望，遲速有准。 又立二木人于地平之上，前置鐘鼓以候辰刻，每一刻自然擊鼓，每辰則自然撞鐘。 皆于櫃中各施輪軸，鉤鍵交錯，關鎖相持。 既與天道合同，當時共稱其妙。 鑄成，命之曰「水運渾天俯視圖」，置於武成殿前以示百僚。 無幾而銅鐵漸澀，不能自轉，遂收置於集賢院，不復行用。
Thereupon Emperor Xuanzong personally composed an inscription and placed it at the Lingtai to examine star degrees. The twenty-eight lodges and inner and outer official stars that differ from the ancient canon number several tens of entries. An edict also ordered Yixing, Liang Lingzan, and various technicians to further make an armillary celestial globe, casting bronze as the image of the round heaven, above equipped with the lodges, equator, and degrees of the full circuit. Water was poured to drive the wheel, causing it to turn of itself; in one day and one night, heaven turns one circuit. Two wheels were separately set linked outside heaven, carrying the Sun and Moon, causing them to move. Each day heaven turns west one coin-width; the Sun travels east one degree, the Moon thirteen and seven-nineteenths degrees; in twenty-nine-odd turns the Sun and Moon meet; in three hundred sixty-five turns the Sun completes a circuit. A wooden cabinet was still set as the horizon, causing half the instrument to be below ground; dark moon, bright moon, new moon, and full moon, slowness and speed had standards. Two wooden figures were also set on the horizon, with bell and drum placed before to mark the double-hours and graduations; each graduation naturally struck the drum, each double-hour naturally struck the bell. All within the cabinet wheels and axles were set, hooks and keys interlaced, bolts and locks mutually holding. Being in accord with the heavenly Way, at the time all together praised its marvel. When casting was complete, it was named the "Water-Driven Armillary Celestial Globe Viewed from Above" and placed before the Wucheng Hall to show the hundred officials. Before long the bronze and iron gradually stiffened and could not turn of itself; it was then collected and stored in the Jixian Academy, no longer used.
7
今錄遊儀制度及所測星度異同，分遣使諸州所測日晷長短，李淳風、僧一行所定十二次分野，武德已來交蝕及五星祥變，著於篇。
Now recorded are the regulations of the traveling instrument and the star degrees measured, their similarities and differences; separately dispatched envoys to the various prefectures to measure the lengths of solar shadows; the twelve divisions and field-allotments fixed by Li Chunfeng and the monk Yixing; eclipses and the five planets' auspicious changes since Wude—set forth in this treatise.
8
旋樞雙環：外一丈四尺六寸一分，豎八分，厚三分，直徑四尺五寸九分，即古所謂旋儀也。 南北斜兩極，上下循規各三十四度，兩面各畫周天度數。 一面加釘，並用銀飾，使東西運轉如渾天遊儀。 中旋樞軸至兩極首內，孔徑大兩度半，長與旋環徑齊，並用古尺四分為度。
Rotating-pivot double ring: outer one zhang four chi six cun one fen, vertical eight fen, thick three fen, diameter four chi five cun nine fen—the ancient so-called rotating instrument. North and south oblique poles, above and below following the ring each thirty-four degrees; on both faces the degrees of heaven's full circuit are painted. On one face nails are added, all silver ornamented, causing east-west rotation like the armillary traveling instrument. The central rotating-pivot axle to within the two pole heads, bore diameter two and a half degrees, length equal to the rotating ring's diameter—all using the ancient chi with four fen as one degree.
9
玉衡望筒：長四尺五寸八分，廣一寸二分，厚一寸，孔徑六分，古用玉飾之。 玉衡，衡施於軸中，旋運持正，用窺七曜及列星之闊狹，外方內圓，孔徑一度半，周日輪也。
Jade-transversal sighting tube: length four chi five cun eight fen, width one cun two fen, thickness one cun, bore diameter six fen; antiquity used jade to ornament it. The jade transversal, the transversal set in the axle's center, rotating and holding true, used to sight the seven luminaries and the lodges' breadth and narrowness—outer square inner round, bore diameter one and a half degrees, the circuit of the daily wheel.
10
陽經雙環：外一丈七尺三寸，內一丈四尺六寸四分，廣四寸，厚四分，直徑五尺四寸四分，置於子午。 左右用八柱相固，兩面畫周天度數，一面加釘，並銀飾之。 半出地上，半入地下，雙間挾樞軸及玉衡望筒，旋環於中也。
Yang meridian double ring: outer one zhang seven chi three cun, inner one zhang four chi six cun four fen, width four cun, thickness four fen, diameter five chi four cun four fen, set at south and north noon. Left and right eight pillars mutually fixed; on both faces the degrees of heaven's full circuit are painted; on one face nails are added, all silver ornamented. Half emerges above ground, half enters below ground; the double interval clasps the pivot axle and jade-transversal sighting tube, the rotating ring in the center.
11
陰緯單環：外內廣厚周徑，皆准陽經，與陽經相銜各半，內外俱齊。 面平上為天，以下為地，橫周陽環，謂之陰渾也。 面上為兩界，內外為周天百刻。 平上御製銘序及書，並金為字。
Yin latitude single ring: outer and inner width, thickness, and full circuit diameter all follow the yang meridian; with the yang meridian each links half, inner and outer both level. The face above level is heaven; below is earth; horizontally circling the yang ring—this is called the yin armillary. On the face above are two boundaries; inner and outer are the hundred graduations of heaven's full circuit. On the level above are the imperially composed inscription preface and text, all characters in gold.
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天頂單環：外一丈七尺三寸，豎廣八分，厚三分，直徑五尺四寸四分。 當中國人頂之上，東西當卯酉之中，稍南，使見日出入，令與陽經、陰緯相固，如殼之裹黃。 南去赤道三十六度，去黃道十二度，去北極五十五度，去南北平各九十一度強。
Celestial apex single ring: outer one zhang seven chi three cun, vertical width eight fen, thickness three fen, diameter five chi four cun four fen. At the apex above the Central Land's people, east-west at the center of mao and you, slightly south, causing the Sun's emergence and setting to be seen; causing it to be fixed with the yang meridian and yin latitude, like a shell wrapping the yellow. South from the equator thirty-six degrees, from the ecliptic twelve degrees, from the north pole fifty-five degrees, from north and south level each ninety-one degrees strong.
13
赤道單環：外一丈四尺五寸九分，橫八分，厚三分，直徑四尺九寸。 赤道者，當天之中，二十八宿之列位也。 其本，後魏解蘭所造也。 因著雙規，不能運動。 臣今所造者，上列周天星度，使轉運隨天，仍度穿一穴，隨穴退交，不有差謬。 即知古者秋分，日在角五度，今在軫十三度； 冬至，日在牽牛初，今在斗十度。 擬隨差卻退，故置穴也。 傍在卯酉之南，上去天頂三十六度而橫置之。
Equator single ring: outer one zhang four chi five cun nine fen, horizontal eight fen, thickness three fen, diameter four chi nine cun. The equator is at heaven's center—the arrayed positions of the twenty-eight lodges. Its original was made by Xie Lan of Northern Wei. Because double rings were attached, it could not move. What your subject has now made lists the star degrees of heaven's full circuit above, causing rotation to follow heaven; still a hole is bored through the degree, following the hole retreating at crossing, without error. Thus one knows that in antiquity at the autumn equinox the Sun was at five degrees of Horn; now it is at thirteen degrees of Chariot; at the winter solstice the Sun was at the beginning of Oxherd; now it is at ten degrees of Dipper. Intending to follow the difference retreating backward, therefore the hole is set. Beside it south of mao and you, above going to the celestial apex thirty-six degrees and set horizontally.
14
黃道單環：外一丈五尺四寸一分，橫八分，厚四分，直徑四尺八寸四分。 日之所行，故名黃道。 古人知有其事，竟無其器，遂使太陽陟降，積歲有差。 月及五星，亦隨日度出入，規制不知准的，斟量為率，疏闊尤多。 臣今創置此環，置於赤道環內，仍開合使隨轉運，出入四十八度，而極畫兩方，東西列周天度數，南北列百刻，使見日知時，不有差謬。 上列三百六十策，與用卦相准，度穿一穴，與赤道相交。
Ecliptic single ring: outer one zhang five chi four cun one fen, horizontal eight fen, thickness four fen, diameter four chi eight cun four fen. Where the Sun travels—therefore named the ecliptic. Men of old knew the matter existed but ultimately had no instrument; thus the Sun's ascent and descent accumulated years of discrepancy. The Moon and five planets also follow the Sun's degrees in entering and leaving; regulated form did not know the mark, measuring and setting the rate—crude and broad, especially many. Your subject now newly sets this ring, placed within the equator ring, still opening and closing causing it to follow rotation, entering and leaving forty-eight degrees; at the poles both sides are painted, east-west listing heaven's full-circuit degrees, south-north listing the hundred graduations, causing the Sun to be seen and time known, without error. Above are listed three hundred sixty tally-rods, matching the used hexagrams; a hole is bored through the degree, intersecting with the equator.
15
白道月環：外一丈五尺一寸五分，橫度八分，厚三分，直徑四尺七寸六分。 月行有迂曲遲疾，與日行緩急相反。 古無其器，今創置於黃道環內，使就黃道為交合，出入六十度，以測每夜行度。 上畫周天度數，穿一穴，擬移交會，並用銅鐵為之。
White-path lunar ring: outer one zhang five chi one cun five fen, horizontal width eight fen, thickness three fen, diameter four chi seven cun six fen. The Moon's motion has winding curves, slowness and speed, opposite to the Sun's motion in ease and urgency. Antiquity had no instrument; now newly set within the ecliptic ring, causing it to take the ecliptic as crossing and union, entering and leaving sixty degrees, to measure each night's traveled degrees. Above are painted heaven's full-circuit degrees; a hole is bored, intending to follow the shifting crossing and union—all cast in bronze and iron.
16
李淳風《法象志》說有此日月兩環，在旋儀環上。 既用玉衡，不得遂於玉衡內別安一尺望筒。 運用既難，其器已澀。
Li Chunfeng's Record of Celestial Forms says these Sun and Moon two rings are on the rotating-instrument ring. Since the jade transversal is used, one cannot then separately install within the jade transversal another one-chi sighting tube. Application being difficult, the instrument already stiffened.
17
游儀四柱，龍各高四尺七寸。 水槽、山各高一尺七寸五分。 槽長六尺九寸，高廣各四寸。 水池深一寸，廣一寸五分。 龍者能興雲雨，故以飾柱。 柱在四維，龍下有山雲，俱在水準槽上，並銅為之。
Traveling instrument four pillars—the dragons each four chi seven cun high. Water trough and mountain each one chi seven cun five fen high. Trough length six chi nine cun, height and width each four cun. Water pool depth one cun, width one cun five fen. Dragons can raise cloud and rain—therefore used to ornament the pillars. The pillars at the four corners; below the dragons are mountain and cloud—all on the water-level trough, all cast in bronze.
18
游儀初成，太史所測二十八宿等與《經》同異狀：
When the traveling instrument was first completed, the Astronomy Bureau's measurements of the twenty-eight lodges and others, compared with the Canon, same and different in condition:
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角二星，十二度； 赤道黃道度與古同。 舊《經》去極九十一度，今則九十三度半。 《星經》云：「角去極九十一度，距星正當赤道，其黃道在赤道南，不經角中。」 今測角在赤道南二度半，黃道復經角中，即與天象符合。
Horn two stars, twelve degrees; equatorial and ecliptic degrees same as antiquity. The old Canon's departure from the pole ninety-one degrees; now ninety-three and a half degrees. The Star Canon says: "Horn's departure from the pole ninety-one degrees; the distance star is directly at the equator; its ecliptic is south of the equator, not passing through Horn's center." Now measured, Horn is two and a half degrees south of the equator; the ecliptic again passes through Horn's center—thus accord with the heavenly image.
20
亢四星，九度。 舊去極八十九度，今九十一度半。 氐四星，十六度。 舊去極九十四度，今九十八度。 房四星，五度。 舊去極一百八度，今一百一十度半。 心三星，五度。 舊去極一百八度，今一百一十一度。 尾九星，十八度。 舊去極一百二十度，一云一百四十一度，今一百二十四度。 箕四星，十一度。 舊去極一百一十八度，今一百二十度。 南斗六星，二十六度。 舊去極一百一十六度，今一百一十九度。 牽牛六星，八度。 舊去極一百六度，今一百四度。 須女四星，十二度。 舊去極一百度，今一百一度。 虛二星，十度。 舊去極一百四度，今一百一度。 北星舊圖入虛宿，今測在須女九度。 危三星，十七度。 舊去極九十七度，今九十七度。 北星舊圖入危宿，今測在虛六度半。 室二星，十六度。 舊去極八十五度，今八十三度。 東壁二星，九度。 舊去極八十六度，今八十四度。
Neck four stars, nine degrees. Old departure from pole eighty-nine degrees; now ninety-one and a half degrees. Root four stars, sixteen degrees. Old departure from pole ninety-four degrees; now ninety-eight degrees. Room four stars, five degrees. Old departure from pole one hundred eight degrees; now one hundred ten and a half degrees. Heart three stars, five degrees. Old departure from pole one hundred eight degrees; now one hundred eleven degrees. Tail nine stars, eighteen degrees. Old departure from pole one hundred twenty degrees; one account one hundred forty-one degrees; now one hundred twenty-four degrees. Winnowing Basket four stars, eleven degrees. Old departure from pole one hundred eighteen degrees; now one hundred twenty degrees. Southern Dipper six stars, twenty-six degrees. Old departure from pole one hundred sixteen degrees; now one hundred nineteen degrees. Oxherd six stars, eight degrees. Old departure from pole one hundred six degrees; now one hundred four degrees. Bond Maiden four stars, twelve degrees. Old departure from pole one hundred degrees; now one hundred one degrees. Emptiness two stars, ten degrees. Old departure from pole one hundred four degrees; now one hundred one degrees. The north star in old charts entered Emptiness lodge; now measured at nine degrees of Bond Maiden. Rooftop three stars, seventeen degrees. Old departure from pole ninety-seven degrees; now ninety-seven degrees. The north star in old charts entered Rooftop lodge; now measured at six and a half degrees of Emptiness. Encampment two stars, sixteen degrees. Old departure from pole eighty-five degrees; now eighty-three degrees. Eastern Wall two stars, nine degrees. Old departure from pole eighty-six degrees; now eighty-four degrees.
21
奎十六星，十六度。 舊去極七十六度，一云七十度，今七十三度。 東壁九度，奎十六度，此錯以奎西大星為距，即損壁二度，加奎二度，今取西南大星為距，即奎、壁各不失本度。 婁三星，十三度。 舊去極八十度，今七十七度。 胃三星，十四度。 昴七星，十一度。 舊去極七十四度，今七十二度。 畢八星，十七度。 舊去極七十八度，今七十六度。 觜觿三度，舊去極八十四度，今八十二度。 畢赤道與黃道度同。 觜赤道二度，黃道三度。 其二宿俱當黃道斜虛。 畢有十六度，尚與赤道度同。 觜總二度，黃道損加一度，此即承前有誤。 今測畢有十七度半，觜觿半度，並依天正。 參十星，舊去極九十四度，今九十二度。 東井八星，三十三度。 舊去極七十度，今六十八度。 輿鬼五星，舊去極六十八度，今古同也。 柳八星，十五度。 舊去極七十七度，一云七十九度，今八十度半。 柳，合用西頭第三星為距，比來錯取第四星，今依第三星為正。 七星十度，舊去極九十一度，一云九十三度，今九十三度半。 張六星，十八度。 舊去極九十七度，今一百度。 張六星，中央四星為硃鳥嗉，外二星為翼。 比來不取膺前為距，錯取翼星，即張加二度半，七星欠二度半。 今依本《經》為定。
Strider sixteen stars, sixteen degrees. Old departure from pole seventy-six degrees; one account seventy degrees; now seventy-three degrees. Eastern Wall nine degrees, Strider sixteen degrees—this wrongly took Strider's western great star as distance star, thus reducing Wall two degrees and adding Strider two degrees; now taking the southwestern great star as distance star, Strider and Wall each keep their original degrees. Bond three stars, thirteen degrees. Old departure from pole eighty degrees; now seventy-seven degrees. Stomach three stars, fourteen degrees. Hairy Head seven stars, eleven degrees. Old departure from pole seventy-four degrees; now seventy-two degrees. Net eight stars, seventeen degrees. Old departure from pole seventy-eight degrees; now seventy-six degrees. Beak three degrees; old departure from pole eighty-four degrees; now eighty-two degrees. Net's equatorial and ecliptic degrees are the same. Beak equator two degrees, ecliptic three degrees. These two lodges both lie where the ecliptic slants empty. Net has sixteen degrees, still the same as equatorial degrees. Beak totals two degrees; the ecliptic loses and gains one degree—this follows the earlier error. Now measured, Net has seventeen and a half degrees, Beak half a degree—both according to the heavenly standard. Three Stars ten stars; old departure from pole ninety-four degrees; now ninety-two degrees. Eastern Well eight stars, thirty-three degrees. Old departure from pole seventy degrees; now sixty-eight degrees. Chariot Ghost five stars; old departure from pole sixty-eight degrees; now same as antiquity. Willow eight stars, fifteen degrees. Old departure from pole seventy-seven degrees; one account seventy-nine degrees; now eighty and a half degrees. Willow should use the third star from the west head as distance star; previously the fourth star was wrongly taken; now the third star is taken as correct. Seven Stars ten degrees; old departure from pole ninety-one degrees; one account ninety-three degrees; now ninety-three and a half degrees. Extended Net six stars, eighteen degrees. Old departure from pole ninety-seven degrees; now one hundred degrees. Extended Net six stars—the central four stars are the Vermillion Bird's crop; the outer two stars are Wings. Previously the breast front was not taken as distance star; Wings stars were wrongly taken—thus Extended Net gained two and a half degrees, Seven Stars lost two and a half degrees. Now following the original Canon as fixed.
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翼二十二星，十八度。 舊去極九十七度，今一百三度。 軫四星，十七度。 舊去極九十八度，今一百度。 文昌，舊二星在鬼，四星在井； 今四星在柳，一星在鬼，一星在井。 北斗，魁第一星舊在七星一度，今在張十三度。 第二星舊在張二度，今在張十二度半。 第三星舊在翼二度，今在翼十三度。 第四星舊在翼八度，今在翼十七度太。 第五星舊在軫八度，今在軫十度半。 第六星舊在角七度，今在角四度少。 第七星舊在亢四度，今在角十二度少。 天關，舊在黃道南四度，今當黃道。 天江，舊在黃道外，今當黃道。 天囷，舊在赤道外，今當赤道。 三臺：上臺舊在井，今測在柳； 中臺舊在七星，今在張。 建星，舊去黃道北半度，今四度半。 天苑，舊在昴、畢，今在胃、昴。 王良，舊五星在壁，今四星在奎，一星在壁外。 屏，舊在觜，今在畢宿。 雲雨，舊在黃道外，今在黃道內七度。 雷電，舊在赤道外五度，今在赤道內二度。 霹靂，舊五星並在赤道外四度，今四星在赤道內，一星在外。 土公吏，舊在赤道外，今在赤道內六度。 虛梁，舊在黃道內四度。 外屏，舊在黃道外三度，今當黃道。 八魁，舊九星並在室，今五星在壁，四星在室。 長垣，舊當黃道，今在黃道北五度。 軍井，准《經》，在玉井東南二度半。 天槨，舊在黃道北，今當黃道。 天高，舊在黃道外，今當黃道。 狗國，舊在黃道外，今當黃道。 羅堰，舊當黃道，今在黃道北。
Wings twenty-two stars, eighteen degrees. Old departure from pole ninety-seven degrees; now one hundred three degrees. Chariot four stars, seventeen degrees. Old departure from pole ninety-eight degrees; now one hundred degrees. Literary Refinement—old two stars in Ghost, four stars in Well; now four stars in Willow, one star in Ghost, one star in Well. Northern Dipper—the Dipper's first star old at one degree of Seven Stars; now at thirteen degrees of Extended Net. Second star old at two degrees of Extended Net; now at twelve and a half degrees of Extended Net. Third star old at two degrees of Wings; now at thirteen degrees of Wings. Fourth star old at eight degrees of Wings; now at seventeen degrees strong of Wings. Fifth star old at eight degrees of Chariot; now at ten and a half degrees of Chariot. Sixth star old at seven degrees of Horn; now at four degrees weak of Horn. Seventh star old at four degrees of Neck; now at twelve degrees weak of Horn. Celestial Pass—old south of the ecliptic four degrees; now at the ecliptic. Celestial River—old outside the ecliptic; now at the ecliptic. Celestial Granary—old outside the equator; now at the equator. Three Steps—Upper Step old in Well; now measured in Willow; Middle Step old in Seven Stars; now in Extended Net. Establishment stars—old north of the ecliptic half a degree; now four and a half degrees. Celestial Meadows—old in Hairy Head and Net; now in Stomach and Hairy Head. Wang Liang—old five stars in Wall; now four stars in Strider, one star outside Wall. Screen—old in Beak; now in Net lodge. Cloud and Rain—old outside the ecliptic; now seven degrees within the ecliptic. Thunder and Lightning—old five degrees outside the equator; now two degrees within the equator. Thunderclap—old five stars all four degrees outside the equator; now four stars within the equator, one star outside. Earth Duke Clerk—old outside the equator; now six degrees within the equator. Empty Bridge—old four degrees within the ecliptic. Outer Screen—old three degrees outside the ecliptic; now at the ecliptic. Eight Chieftains—old nine stars all in Encampment; now five stars in Wall, four stars in Encampment. Long Rampart—old at the ecliptic; now five degrees north of the ecliptic. Army Well—according to the Canon, two and a half degrees southeast of Jade Well. Celestial Coffin—old north of the ecliptic; now at the ecliptic. Celestial Height—old outside the ecliptic; now at the ecliptic. Dog Country—old outside the ecliptic; now at the ecliptic. Net Weir—old at the ecliptic; now north of the ecliptic.
23
黃道，春分之日與赤道交於奎五度太； 秋分之日交於軫十四度少； 冬至之日於斗十度，去赤道南二十四度； 夏至之日于井十三度少，去赤道北二十四度。 其赤道帶天之中，用分列宿之度； 黃道斜運，以明日月之行。 其冬至，洛下閎起于牛初，張衡等遷於斗度，由每歲差分不及舊次也。
The ecliptic—on the day of the spring equinox crosses the equator at five degrees strong of Strider; on the day of the autumn equinox crosses at fourteen degrees weak of Chariot; on the day of the winter solstice at ten degrees of Dipper, south from the equator twenty-four degrees; on the day of the summer solstice at thirteen degrees weak of Well, north from the equator twenty-four degrees. The equator girdles heaven's center, used to array the lodges' degrees; the ecliptic runs obliquely, to clarify the Sun and Moon's travel. At the winter solstice, Luoxia Hong began at the beginning of Oxherd; Zhang Heng and others shifted to Dipper degrees—because each year's fractional difference did not reach the old sequence.
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日晷：《周禮》大司徒，常「以土圭之法測土深，正日景，以求地中。 日東則景夕多風，日西則景朝多陰。 日至之景尺五寸，謂之地中，天地之所合也，四時之所交也，風雨之所會也，陰陽之所合也。 然則百物阜安，乃建王國焉。」 鄭氏以為「凡日景於地，千里而差一寸。」 「景尺有五寸者，南戴日下萬五千里，地與星辰四游升降於三萬里之中，是以半之，得地之中焉。」 鄭司農云：「土圭之長尺有五寸，以夏至之日立八尺之表，其景適與土圭等，謂之地中。 今潁川陽城為然。」
Solar shadows: The Rites of Zhou's Grand Minister of Education constantly "by the earth-altar method measured earth's depth, corrected the Sun's shadow, to seek the earth's center. When the Sun is east, then the shadow at evening—much wind; when the Sun is west, then the shadow at morning—much yin. At solstice the shadow one chi five cun—this is called the earth's center, where heaven and earth unite, where the four seasons cross, where wind and rain gather, where yin and yang unite. Then the hundred things flourish in peace, and the royal domain is established."" Master Zheng thought "wherever the Sun's shadow falls on earth, for one thousand li it differs one cun." "Shadow one chi five cun—south bearing the Sun below fifteen thousand li; earth and stars fourfold travel ascending and descending within thirty thousand li—therefore halving it obtains the earth's center." Assistant Minister Zheng of the Ministry of Education said: "The earth-altar's length one chi five cun—at the summer solstice day set an eight-chi gnomon; its shadow exactly equals the earth-altar—this is called the earth's center. Now Yingchuan Yangcheng is so."
25
謹按《南越志》：「宋元嘉中，南征林邑，以五月立表望之，日在表北，影居表南。 交州日影覺北三寸，林邑覺九寸一分，所謂開北戶以向日也。」 交州，大略去洛九千餘里，蓋水陸曲折，非論圭表所度，惟直考實，其五千乎， 詔太史交州測景，夏至影表南長三寸三分，與元嘉中所測大同。 然則距陽城而南，使直路應弦，至於日下，蓋不盈五千里也。 測影使者大相元太云：「交州望極，才出地二十餘度。 以八月自海中南望老人星殊高。 老人星下，環星燦然，其明大者甚眾，圖所不載，莫辨其名。 大率去南極二十度以上，其星皆見。 乃古渾天家以為常沒地中，伏而不見之所也。」 又按貞觀中，史官所載鐵勒、回紇部在薛延陀之北，去京師六千九百里。 又有骨利幹居回紇北方瀚海之北，草多百藥，地出名馬，駿者行數百里。 北又距大海，晝長而夕短，既日沒後，天色正曛，煮一羊胛才熟，而東方已曙。 蓋近日出入之所云。 凡此二事，皆書契所未載也， 太史監南宮說擇河南平地，以水準繩，樹八尺之表而以引度之。 始自滑州白馬縣，北至之晷，尺有五寸七分。 自滑州臺表南行一百九十八里百七十九步，得汴州浚儀古臺表，夏至影長一尺五寸微強。 又自浚儀而南百六十七里二百八十一步，得許州扶溝縣表，夏至影長一尺四寸四分。 又自扶溝而南一百六十里百一十步，至豫州上蔡武津表，夏至影長一尺三寸六分半。 大率五百二十六里二百七十步，影差二寸有餘。 而先儒以為王畿千里，影移一寸，又乖舛而不同矣。
Respectfully according to the Record of Southern Yue: "In the Yuanjia era of the Song Yuan reign, campaigning south against Linyi, in the fifth month a gnomon was set and sighted—the Sun was north of the gnomon, the shadow south of the gnomon. At Jiaozhou the Sun's shadow was felt north three cun; at Linyi felt nine cun one fen—the so-called opening the north door to face the Sun." Jiaozhou, roughly nine thousand-odd li from Luoyang—mostly the land and water winding, not discussing what the altar and gnomon measure; only direct verification—perhaps five thousand! An edict ordered the Astronomy Bureau to measure shadows at Jiaozhou; at the summer solstice the shadow south of the gnomon three cun three fen—greatly the same as measured in the Yuanjia era. Then south from Yangcheng, if a straight path answered the chord to where the Sun stands below, it would not fill five thousand li. The shadow-measuring envoy Grand Chancellor Yuan Tai said: "At Jiaozhou sighting the pole, it barely emerged from earth twenty-odd degrees. In the eighth month from the sea looking south, the Old Man star was exceptionally high. Below the Old Man star, encircling stars glittered; the bright great ones were very many—not recorded in charts, names indistinguishable. Generally south from the south pole twenty degrees and above, those stars are all visible. This is what ancient armillary-celestial-globe masters thought constantly submerged in the earth's center, hidden and not seen." Also according to the Zhenguan era, the astronomy officers recorded the Tiele and Uighur tribes north of Xueyantuo, nine thousand seven hundred li from the capital. Also the Guligan lived north of the Uighur, north of the vast sea—the grass had many hundred medicines; the land produced famous horses; swift ones ran several hundred li. Further north bordering the great sea—day long and evening short; after the Sun had set, the sky was just dusk; boiling one sheep's shoulder just cooked, and the east was already dawn. This is roughly where the Sun recently enters and leaves. These two matters are what written records had not carried. Astronomy Director Nangong Shuo selected level land in Henan, using water-level cord, set an eight-chi gnomon and measured by leading. Beginning from Baima County, Huazhou, north to the solstice shadow one chi five cun seven fen. From Huazhou's gnomon south two hundred ninety-eight li one hundred seventy-nine bu, obtained Bianzhou Junyi's old gnomon; summer solstice shadow one chi five cun slightly strong. Again from Junyi south one hundred sixty-seven li two hundred eighty-one bu, obtained Xuzhou Fugou County's gnomon; summer solstice shadow one chi four cun four fen. Again from Fugou south one hundred sixty li one hundred ten bu, to Yuzhou Shangcai Wujin gnomon; summer solstice shadow one chi three cun six fen half. Generally five hundred twenty-six li two hundred seventy bu, shadow difference two cun odd. But earlier scholars thought the royal domain one thousand li, shadow shifts one cun—again erroneous and not the same.
26
今以句股圖校之，陽城北至之晷，一尺四寸八分弱； 冬至之晷，一丈二尺七寸一分半； 春秋分，其長五尺四寸三分。 以覆矩斜視，北極出地三十四度四分。 凡度分皆以十分為法。 自滑臺表視之，高三十五度三分。 差陽城九分。 自浚儀表視之，高三十四度八分。 差陽城四分。 自武津表視之，高三十三度八分。 差陽城九分。 雖秒分稍有盈縮，雖以目校，然大率五百二十六里二百七十步而北極差一度半，三百五十一里八十步而差一度。 樞極之遠近不同，則黃道之軌景固隨而遷變矣。
Now using the gnomon-triangle diagram to verify, Yangcheng north to solstice shadow one chi four cun eight fen weak; winter solstice shadow one zhang two chi seven cun one fen half; spring and autumn equinoxes, length five chi four cun three fen. By the covering-square oblique sighting, north pole emerging from earth thirty-four degrees four fen. All degrees and fractions use ten parts as method. Viewed from Huatai gnomon, height thirty-five degrees three fen. Difference from Yangcheng nine fen. Viewed from Junyi gnomon, height thirty-four degrees eight fen. Difference from Yangcheng four fen. Viewed from Wujin gnomon, height thirty-three degrees eight fen. Difference from Yangcheng nine fen. Though seconds and fractions slightly have surplus and deficit, though verified by eye, generally five hundred twenty-six li two hundred seventy bu and north pole differs one and a half degrees; three hundred fifty-one li eighty bu and differs one degree. The pivot pole's distance near and far being different, then the ecliptic's track and shadow naturally follow and shift.
27
自此為率，推之比歲朗州測影，夏至長七寸七分，冬至長一丈五寸三分，春秋分四尺三寸七分半。 以圖測之，定氣長四尺四寸七分。 按圖斜視，北極出地二十九度半。 差陽城五度二分。 蔚州橫野軍測影，夏至長二尺二寸九分，冬至長一丈五尺八寸九分，春秋分長六尺四寸四分半。 以圖測之，定氣六尺六寸三分半。 按圖斜視，北極出地四十度。 差陽城五度二分。 凡南北之差十度半，其徑三千六百八十里九十步。 自陽城至朗州，一千八百二十六里百九十六步，自陽城至蔚州橫野軍，一千八百六十一里二百一十四步。 北至之晷，差一尺五寸三分，自陽城至朗州，差七寸二分，自陽城至橫野軍，差八寸。 南至之晷，差五尺三寸六分。 自陽城至朗州，差二尺一寸八分，自陽城至橫野軍，差三尺一寸八分。 率夏至與南方差步，冬至與北方差多。 又以圖校安南，日在天頂北二度四分，北極高二十度四分，冬至影長七尺九寸四分，定春秋分影長二尺九寸三分。 差陽城十四度三分，其徑五千二十三里。 至林邑國，日在天頂北六度六分強，北極之高十七度四分，周圓三十五度，常見不隱。 冬至影長六尺九寸，其徑六千一百一十二里。 假令距陽城而北，至鐵勒之地亦十七度四分，合與林邑與等，則五月日在天頂南二十七度四分，北極之高五十二度，周圓一百四度，常見不隱。 北至之齕四尺一寸三分，南至之齕二丈九就十寸六分。 定春秋分影長九尺八寸七分。 北方其沒地才十五度餘，昏伏於亥之正西，晨見於醜之正東，以里數推之，已在回紇之北，又南距洛陽九千八百一十里，則五月極長之日，其夕常明，然則骨利幹猶在其南矣。 又先儒以南戴日下萬五千里為句股，斜射陽城為弦，考周徑之率以揆天度，當一千四百六里二十四步有餘。 今測日影，距陽城五千餘里，已居戴日之南，則一度之廣，皆宜三分去二，計南北極相去才八萬餘里，其徑五萬餘里，宇宙之廣，豈若是乎？ 然則王蕃所傳，蓋以管窺天，以蠡測海之義也。 古人所以恃句股之術，謂其有征於近事。 顧未知目視不能遠，浸成微分之差，其差不已，遂與術錯。 如人游於大湖，廣不盈百里，而睹日月朝夕出入湖中。 及其浮於巨海，不知幾千萬里，猶睹日月朝出其中，夕入其中。 若於朝夕之際，俱設重差而望之，必將小大同術而不可分矣。
From this as rate, extrapolating compared with Langzhou's shadow measurement—summer solstice length seven cun seven fen, winter solstice one zhang five cun three fen, spring and autumn equinoxes four chi three cun seven fen half. Measured by the diagram, fixed qi length four chi four cun seven fen. According to the diagram oblique sighting, north pole emerging from earth twenty-nine degrees half. Difference from Yangcheng five degrees two fen. Weizhou Hengye Army shadow measurement—summer solstice length two chi two cun nine fen, winter solstice one zhang five chi eight cun nine fen, spring and autumn equinoxes six chi four cun four fen half. Measured by the diagram, fixed qi six chi six cun three fen half. According to the diagram oblique sighting, north pole emerging from earth forty degrees. Difference from Yangcheng five degrees two fen. Generally north-south difference ten degrees half; its diameter three thousand six hundred eighty li ninety bu. From Yangcheng to Langzhou one thousand eight hundred twenty-six li one hundred ninety-six bu; from Yangcheng to Weizhou Hengye Army one thousand eight hundred sixty-one li two hundred fourteen bu. North to solstice shadow difference one chi five cun three fen; from Yangcheng to Langzhou difference seven cun two fen; from Yangcheng to Hengye Army difference eight cun. South to solstice shadow difference five chi three cun six fen. From Yangcheng to Langzhou difference two chi one cun eight fen; from Yangcheng to Hengye Army difference three chi one cun eight fen. Rate: summer solstice and south differ in bu; winter solstice and north differ more. Again using the diagram to verify Annan—the Sun at two degrees four fen north of the celestial apex, north pole height twenty degrees four fen, winter solstice shadow seven chi nine cun four fen, fixed spring and autumn equinox shadow two chi nine cun three fen. Difference from Yangcheng fourteen degrees three fen; its diameter five thousand twenty-three li. To the kingdom of Linyi, the Sun at six degrees six fen strong north of the celestial apex, north pole height seventeen degrees four fen, full circuit thirty-five degrees, constantly visible never hidden. Winter solstice shadow six chi nine cun; its diameter six thousand one hundred twelve li. Supposing distance north from Yangcheng to Tiele land also seventeen degrees four fen, combined equal with Linyi—then in the fifth month the Sun at twenty-seven degrees four fen south of the celestial apex, north pole height fifty-two degrees, full circuit one hundred four degrees, constantly visible never hidden. North to solstice shadow four chi one cun three fen; south to solstice shadow two zhang nine chi ten cun six fen. Fixed spring and autumn equinox shadow nine chi eight cun seven fen. North there its submerged earth barely fifteen degrees odd; at dusk hiding in hai's due west, at dawn seen in chou's due east—by li number extrapolating, already north of the Uighur; again south nine thousand eight hundred ten li from Luoyang—then at the fifth month pole-longest day, its evening constantly bright; then Guligan is still south of it. Also earlier scholars took south bearing the Sun below fifteen thousand li as gnomon-leg, oblique shooting Yangcheng as hypotenuse, examining Zhou-diameter rate to measure heavenly degrees—should be one thousand four hundred six li twenty-four bu odd. Now measuring Sun shadows, distance from Yangcheng five thousand-odd li, already south of bearing the Sun—then one degree's breadth all should three parts remove two; calculating north and south poles mutually distant barely eighty thousand-odd li, its diameter fifty thousand-odd li—is the cosmos's breadth really so? Then what Wang Fan transmitted is probably the meaning of viewing heaven through a tube, measuring the sea with a gourd dipper. Men of old therefore relied on gnomon-triangle technique, saying it had verification in near matters. But not knowing sight cannot reach far, gradually forming fractional differences; the difference not stopping, then diverging from the method. Like a person traveling on a great lake, breadth not filling one hundred li, yet seeing the Sun and Moon morning and evening enter and leave the lake. When floating on the vast sea, not knowing several thousand ten-thousand li, still seeing the Sun and Moon dawn emerging from within, dusk entering within. If at morning and evening's interval both set double-difference and sight, inevitably small and large same method and indivisible.
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夫橫既有之，縱亦宜然。 假令設兩表，南北相距十里，其崇皆數十里，若置火炬于南表之端，而植八尺之木於其下，則當無影。 試從南表之下，仰望北表之端，必將積微分之差，漸與南表參合。 表首參合，則置炬於其上，亦當無影矣。 又置火炬於北表之端，而植八尺之木於其下，則當無影。 試從北表之下，仰望南表之端，又將積微分之差，漸與北表參合。 表首參合，則置炬於其上，亦當無影矣。 復於二表之間，相距各五里，更植八尺之木，仰而望之，則表首環屈而相會。 若置火炬於兩表之端，皆當無影。 夫數十里之高與十里之廣，然則斜射之影與仰望不殊。 今欲求其影差以推遠近高下，猶尚不可知也； 而況稽周天積裏之數於不測之中，又可必乎！ 假令學者因二十里之高以立句股之術，尚不知其所以然，況八尺之木乎！ 原人所以步圭景之意，將欲節宣和氣，輔相物宜，而不在於辰次之周徑； 其所以重歷數之意，將欲敬授人時，欽若乾象，而不在於渾、蓋之是非。 若乃述無稽之談於視聽之所不及，則君子闕疑而不質，仲尼慎言而不論也。 而或者各守所傳之器以述天體，謂渾元可任數而測，大象可運算而窺，終以六家之說，迭為矛盾。 今誠以為蓋天，則南方之度漸狹； 以為渾天，則北方之極浸高。 此二者，又渾、蓋之家未能有以通其說也。 由是而觀，則王仲任、葛稚川之徒，區區于異同之辨，何益人倫之化哉！
Horizontal having it, vertical also should be so. Supposing two gnomons set, north and south mutually distant ten li, both heights several tens of li—if setting a torch at the south gnomon's top, and planting an eight-chi pole below it, then there should be no shadow. Try from below the south gnomon, looking up at the north gnomon's top—invariably accumulated fractional differences will gradually join with the south gnomon. Gnomon tops joining, then setting a torch on it, also should have no shadow. Again setting a torch at the north gnomon's top, and planting an eight-chi pole below it, then there should be no shadow. Try from below the north gnomon, looking up at the south gnomon's top—again accumulated fractional differences will gradually join with the north gnomon. Gnomon tops joining, then setting a torch on it, also should have no shadow. Again between the two gnomons, mutually distant each five li, further planting eight-chi poles, looking up and sighting—then gnomon tops curve and meet. If setting torches at both gnomons' tops, all should have no shadow. Several tens of li height and ten li breadth—then oblique shooting's shadow and looking up are not different. Now wishing to seek its shadow difference to extrapolate distance near and high and low, still not knowable; how much more examining heaven's full-circuit accumulated li within the unmeasurable—again how can it be certain! Supposing scholars because of twenty li height establish gnomon-triangle method, still not knowing its why—how much more an eight-chi pole! Tracing why men step the altar-shadow's intent—they wished to moderate and harmonize qi, assist and complement things' suitability, not in the full circuit's diameter; why they valued calendric numbers—they wished reverently to grant seasons to the people, reverently accord with the qian image, not in whether armillary or canopy is right or wrong. If narrating unverified talk beyond sight and hearing's reach—then the noble man leaves doubt and does not examine; Confucius is careful in speech and does not discuss. But some each guard the transmitted instrument to narrate the celestial body, saying the armillary origin can bear numbers and measure, the great image can be computed and peeped—ultimately the six schools' sayings iteratively contradict. Now truly thinking it canopy heaven, then the south's degrees gradually narrow; thinking it armillary heaven, then the north's pole gradually rises. These two—again armillary and canopy schools have not been able to penetrate their sayings. From this viewing, then Wang Chong, Ge Hong and such, petty in same-different distinction—what benefit to human relations' transformation!
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又凡日晷差，冬夏至不同，南北亦異，而先儒一以里數齊之，喪其事實。 沙門一行因修《大衍圖》，更為《覆矩圖》，自丹穴以暨幽都之地，凡為圖二十四，以考日蝕之分數，知夜漏之短長。 今載諸州測景尺寸如左：
Also generally solar shadow difference—winter and summer solstices not the same, south and north also different; but earlier scholars uniformly used li number to equalize—losing the matter's reality. The monk Yixing because of revising the Great Evolution diagram, further made the Covering-Square diagram—from Cinnabar Cavern to Dark Capital's lands, altogether twenty-four diagrams, to examine solar eclipse's fractions, know night clepsydra's short and long. Now recorded are the various prefectures' measured shadow dimensions as follows:
30
林邑國，北極高十七度四分。 冬至影在表北六尺九寸。 定春秋分影在表北二尺八寸五分，夏至影在表南五寸七分。 安南都護府，北極高二十六度六分。 冬至影在表北七尺九寸四分。 定春秋分影在表北二尺九寸三分，夏至影在表南三寸三分。 朗州武陵縣，北極高二十九度五分。 冬至影在表北一丈五寸三分。 定春秋分影在表北四尺三寸七分半，夏至影在表北七寸七分。 襄州。 恆春分影在表北四尺八寸。 蔡州上蔡縣武津館，北極高三十三度八分。 冬至影在表北一丈二尺三寸八分。 定春秋分影在表北五尺二寸八分，夏至影在表北一尺三寸六分半。 許州扶溝，北極高三十四度三分。 冬至影在表北一丈二尺五寸三分。 定春秋分影在表北五尺三寸七分，夏至影在表北一尺四寸四分。 汴州浚儀太嶽臺，北極高三十四度八分。 冬至影在表北一丈二尺八寸五分。 定春秋分影在表北五尺五寸，夏至影在表北一尺五寸三分。 滑州白馬，北極高三十五度三分。 冬至影在表北一丈三尺。 定春秋分影在表北五尺三寸六分，夏至影在表北一尺五寸七分。 太原府。 恆春分影在表北六尺。 蔚州橫野軍，北極高四十度。 冬至影在表北一丈五尺八寸九分。 定春秋分影在表北六尺六寸三分，夏至影在表北二尺二寸九分。
Kingdom of Linyi, north pole height seventeen degrees four fen. Winter solstice shadow north of gnomon six chi nine cun. Fixed spring and autumn equinox shadow north of gnomon two chi eight cun five fen; summer solstice shadow south of gnomon five cun seven fen. Protectorate General of Annan, north pole height twenty-six degrees six fen. Winter solstice shadow north of gnomon seven chi nine cun four fen. Fixed spring and autumn equinox shadow north of gnomon two chi nine cun three fen; summer solstice shadow south of gnomon three cun three fen. Langzhou Wuling County, north pole height twenty-nine degrees five fen. Winter solstice shadow north of gnomon one zhang five cun three fen. Fixed spring and autumn equinox shadow north of gnomon four chi three cun seven fen half; summer solstice shadow north of gnomon seven cun seven fen. Xiangzhou. Constant spring equinox shadow north of gnomon four chi eight cun. Yuzhou Shangcai County Wujin Lodge, north pole height thirty-three degrees eight fen. Winter solstice shadow north of gnomon one zhang two chi three cun eight fen. Fixed spring and autumn equinox shadow north of gnomon five chi two cun eight fen; summer solstice shadow north of gnomon one chi three cun six fen half. Xuzhou Fugou, north pole height thirty-four degrees three fen. Winter solstice shadow north of gnomon one zhang two chi five cun three fen. Fixed spring and autumn equinox shadow north of gnomon five chi three cun seven fen; summer solstice shadow north of gnomon one chi four cun four fen. Bianzhou Junyi Taiyue Platform, north pole height thirty-four degrees eight fen. Winter solstice shadow north of gnomon one zhang two chi eight cun five fen. Fixed spring and autumn equinox shadow north of gnomon five chi five cun; summer solstice shadow north of gnomon one chi five cun three fen. Huazhou Baima, north pole height thirty-five degrees three fen. Winter solstice shadow north of gnomon one zhang three chi. Fixed spring and autumn equinox shadow north of gnomon five chi three cun six fen; summer solstice shadow north of gnomon one chi five cun seven fen. Taiyuan Prefecture. Constant spring equinox shadow north of gnomon six chi. Weizhou Hengye Army, north pole height forty degrees. Winter solstice shadow north of gnomon one zhang five chi eight cun nine fen. Fixed spring and autumn equinox shadow north of gnomon six chi six cun three fen; summer solstice shadow north of gnomon two chi two cun nine fen.