Volume 31 Treatises 25: Astronomy 1

昔者，堯命羲、和，出納日月，考星中以正四時。至舜，則曰「在璿璣玉衡，以齊七政」而已。雖二典質略，存其大法，亦由古者天人之際，推候占測，為術猶簡。至於後世，其法漸密者。必積眾人之智，然後能極其精微哉。蓋自三代以來詳矣。詩人所記，婚禮、土功必候天星。而《春秋》書日食、星變，《傳》載諸國所占次舍、伏見、逆順。至於《周禮》測景求中、分星辨國、妖祥察候，皆可推考，而獨無所謂璿璣玉衡者，豈其不用於三代耶？抑其法制遂亡，而不可復得耶？不然，二物者，莫知其為何器也。至漢以後，表測景晷，以正地中，分列境界，上當星次，皆略依古。而又作儀以候天地，而渾天、周髀、宣夜之說，至於星經、曆法，皆出於數術之學。唐興，太史李淳風、浮圖一行，尤稱精博，後世未能過也。故采其要說，以著於篇。至於天象變見所以譴告人君者，皆有司所宜謹記也。

In ancient times, Emperor Yao appointed Xi and He to observe the passage of the sun and moon, determine the stars at their meridian transits, and thereby set the four seasons aright. By the time of Shun, the charge had been reduced to the single injunction: "Take your place at the armillary sphere and jade level, and align the seven luminaries." Though the two canonical books are terse, they preserve the essential method; in early times, when heaven and humanity met in calendrical practice, prognostication and celestial observation remained comparatively simple arts. In later ages, however, the methods grew steadily more exacting. Only by gathering the wisdom of many minds could one approach their full subtlety. From the Three Dynasties onward, the record grows richly detailed. The Book of Poetry shows that wedding ceremonies and earthworks alike had to wait upon the stars. The Spring and Autumn Annals note solar eclipses and irregularities among the stars, while the Commentary records how the various states computed stellar lodges, occultations, and the planets' retrograde and direct courses. The Rites of Zhou describe measuring the gnomon shadow to find the terrestrial center, mapping the stars to distinguish the realms, and scrutinizing prodigies and portents — all of which can be reconstructed in detail — yet nowhere mention the armillary sphere and jade level. Were these instruments not employed in the Three Dynasties after all? Or had their design been lost beyond recovery? Otherwise, no one today knows what those two instruments actually were. From the Han dynasty onward, observers used marked tables and shadow measurements to fix the terrestrial center, parcel out territorial boundaries, and align them with the celestial mansions — all in rough accord with antiquity. They also built observational instruments for heaven and earth; doctrines such as the Armillary Heaven, Zhou Bi, and Clarified Night theories, together with star catalogs and calendrical systems, all arose from the mathematical sciences. When the Tang rose to power, Grand Astrologer Li Chunfeng and the monk Yixing were celebrated above all for their mastery; later generations have never surpassed them. Their principal teachings are therefore excerpted and recorded in this treatise. Celestial anomalies by which Heaven warns rulers are likewise matters that responsible officials should scrupulously document.

貞觀初，淳風上言：「舜在璿璣玉衡，以齊七政，則渾天儀也。《周禮》，土圭正日景以求地中，有以見日行黃道之驗也。暨於週末，此器乃亡。漢落下閎作渾儀，其後賈逵、張衡等亦各有之，而推驗七曜，並循赤道。按冬至極南，夏至極北，而赤道常定于中，國無南北之異。蓋渾儀無黃道久矣。」太宗異其說，因詔為之。至七年儀成。表裏三重，下據准基，狀如十字，末樹鰲足，以張四表。一曰六合儀，有天經雙規、金渾緯規、金常規，相結於四極之內。列二十八宿、十日、十二辰、經緯三百六十五度。二曰三辰儀，圓徑八尺，有璿璣規、月遊規，列宿距度，七曜所行，轉於六合之內。三曰四游儀，玄樞為軸，以連結玉衡遊筩而貫約矩規。又玄極北樹北辰，南矩地軸，傍轉於內。玉衡在玄樞之間，而南北遊，仰以觀天之辰宿，下以識器之晷度。皆用銅。帝稱善，置於凝暉閣，用之測候。閣在禁中，其後遂亡。

Early in the Zhenguan reign, Chunfeng memorialized the throne: "Shun's charge to take his place at the armillary sphere and jade level and align the seven luminaries refers to the armillary celestial instrument. The Rites of Zhou prescribe using the earth gnomon to correct the solar shadow and find the terrestrial center — proof that the ancients could observe the sun's motion along the ecliptic. By the close of the Zhou dynasty, the instrument had been lost. Under the Han, Luoxia Hong built an armillary instrument, and later Jia Kui, Zhang Heng, and others each devised their own; yet in computing the seven luminaries they all worked along the equator alone. At winter solstice the sun stands farthest south and at summer solstice farthest north, while the equator remains fixed at the center — which is why treating all regions alike along the equator cannot distinguish north from south. The armillary instrument had long gone without an ecliptic ring. Emperor Taizong was struck by this argument and ordered the instrument built. In the seventh year the instrument was finished. It had three nested layers; below stood a leveling base in the form of a cross, with tortoise feet at the corners to support the four sighting tables. The first was the Six Harmonies Instrument, comprising celestial meridian double rings, golden armillary latitude rings, and golden constant rings, all linked within the four cardinal pivots. It bore the twenty-eight lunar mansions, the ten-day stems, the twelve double-hours, and all three hundred sixty-five degrees of celestial longitude and latitude. The second was the Three Luminaries Instrument, eight feet in diameter, fitted with armillary rings and lunar excursion rings that marked the intervals between lodges and the paths of the seven luminaries as they revolved within the Six Harmonies. The third was the Four Excursions Instrument, with the dark pivot as its axis, joining the jade transverse sighting tube and threading the graduated rings. At the dark pole the northern asterism was mounted to the north and the terrestrial axis squared to the south, revolving within the assembly. The jade transverse moved between the dark pivots, traveling north and south: raised, it sighted the celestial mansions; lowered, it read the instrument's graduated shadow scale. The whole was cast in bronze. The emperor approved and installed it in the Ninghui Pavilion for astronomical observation. The pavilion stood within the palace precincts; in time the instrument was lost.

開元九年，一行受詔，改治新曆，欲知黃道進退，而太史無黃道儀，率府兵曹參軍梁令瓚以木為游儀，一行是之，乃奏：「黃道游儀，古有其術而無其器，昔人潛思，皆未能得。今令瓚所為，日道月交，皆自然契合，于推步尤要，請更鑄以銅鐵。」十一年儀成。一行又曰：「靈台鐵儀，後魏斛蘭所作，規制樸略，度刻不均，赤道不動，乃如膠柱。以考月行，遲速多差，多或至十七度，少不減十度，不足以稽天象、授人時。李淳風黃道儀，以玉衡旋規，別帶日道，傍列二百四十九交，以攜月遊，法頗難，術遂寢廢。臣更造遊儀，使黃道運行，以追列舍之變，因二分之中，以立黃道，交於奎、軫之間，二至陟降，各二十四度。黃道內施白道月環，用究陰陽朓，朒，動合天運。簡而易從，可以制器垂象，永傳不朽。」於是玄宗嘉之，自為之銘。

In Kaiyuan 9, Yixing received orders to reform the calendar. To track the ecliptic's advance and retreat, the Grand Astrologer's office lacked an ecliptic instrument; Staff Officer Liang Lingzan of the General-in-Chief's headquarters built a traveling instrument of wood. Yixing endorsed it and memorialized: "The ecliptic traveling instrument — the ancients knew the method but never possessed the device; thoughtful men of old strove in vain to achieve it. Lingzan's design aligns the solar path and lunar nodes naturally and is indispensable for calendrical calculation. I ask that it be recast in bronze and iron. In the eleventh year the instrument was finished. Yixing added: "The Spirit Terrace iron instrument made by Hulu of Northern Wei was crude in design, its degree marks uneven, and its equator fixed immovable — like pegs glued to a zither. Used to track the moon, its errors in lunar speed were severe — sometimes as much as seventeen degrees, never less than ten — far too crude to verify celestial phenomena or set the seasons for mankind. Li Chunfeng's ecliptic instrument used a revolving jade-transverse ring with a separate solar track and two hundred forty-nine lunar nodes marked along the side to guide the moon's motion — the method was cumbersome, and the technique fell into disuse. I have devised a new traveling instrument with a movable ecliptic to track shifts among the lunar mansions. Set at the equinoxes, the ecliptic crosses between Kui and Zhen, rising and falling twenty-four degrees at each solstice. Inside the ecliptic I have placed a white-path lunar ring to track the moon's waxing and waning, its motion conforming to the heavens. The design is simple and practical — an instrument that can embody the heavens and endure for all time. Emperor Xuanzong approved and personally composed an inscription for the instrument.

又詔一行與令瓚等更鑄渾天銅儀，圓天之象，具列宿赤道及周天度數。注水激輪，令其自轉，一晝夜而天運周。外絡二輪，綴以日月，令得運行。每天西旋一周，日東行一度，月行十三度十九分度之七，二十九轉有餘而日月會，三百六十五轉而日周天。以木櫃為地平，令儀半在地下，晦明朔望遲速有准。立木人二于地平上：其一前置鼓以候刻，至一刻則自擊之；其一前置鐘以候辰，至一辰亦自撞之。皆于櫃中各施輪軸，鉤鍵關鎖，交錯相持。置於武成殿前，以示百官。無幾而銅鐵漸澀，不能自轉，遂藏於集賢院。

The emperor also ordered Yixing, Lingzan, and their colleagues to cast a new armillary bronze instrument depicting the round heaven, complete with the lodges, the equator, and the full circuit of celestial degrees. Water drove the wheels so that the sphere turned by itself, completing one celestial revolution each day and night. Two outer rings carried the sun and moon so that they could move along their courses. Each day the sky turned westward one full circuit while the sun moved east one degree and the moon thirteen and seven-nineteenth degrees; after twenty-nine-odd revolutions sun and moon converged, and after three hundred sixty-five revolutions the sun completed its annual circuit. A wooden cabinet formed the horizon, with half the instrument below ground level, so that new moon, full moon, and the moon's varying speed could all be shown accurately. Two wooden figures stood on the horizon: one before a drum to mark the quarters of the hour, which it struck automatically when each quarter arrived; the other before a bell to mark the double-hours, which it likewise struck of itself when each double-hour arrived. Inside the cabinet, wheels, axles, hooks, and locks meshed together to drive the mechanism. It was set up before the Wucheng Hall for all the officials to see. Before long the bronze and iron parts seized up and it could no longer turn on its own; it was then stored away in the Academy of Assembled Worthies.

其黃道遊儀，以古尺四分為度。旋樞雙環，其表一丈四尺六寸一分，縱八分，厚三分，直徑四尺五寸九分，古所謂旋儀也。南北科兩極，上下循規各三十四度。表裏畫周天度，其一面加之銀釘。使東西運轉，如渾天遊旋。中旋樞軸，至兩極首內，孔徑大兩度半，長與旋環徑齊。玉衡望筩，長四尺五寸八分，廣一寸二分，厚一寸，孔徑六分。衡旋於軸中，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。陽經雙環，表一丈七尺三寸，裏一丈四尺六寸四分，廣四寸，厚四分，直徑五尺四寸四分，置於子午。左右用八柱，八柱相固。亦表裏畫周天度，其一面加之銀釘。半出地上，半入地下。雙間挾樞軸及玉衡望筩旋環於中也。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，下為地。橫周陽環，謂之陰渾也。平上為兩界，內外為周天百刻。天頂單環，表一丈七尺三寸，縱廣八尺，厚三分，直徑五尺四寸四分。直中國人頂之上，東西當卯酉之中，稍南，使見日出入。令與陽經、陰緯相固，如鳥殼之裹黃。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。赤道單環，表一丈四尺五寸九分，橫八分，厚三分，直徑四尺五寸八分。赤道者，當天之中，二十八宿之位也。雙規運動，度穿一穴。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。隨穴退交，不復差繆。傍在卯酉之南，上去天頂三十六度，而橫置之。黃道單環，表一丈五尺四寸一分，橫八分，厚四分，直徑四尺八寸四分。日之所行，故名橫道。太陽陟降，積歲有差。月及五星，亦隨日度出入。古無其器，規制不知准的，斟酌為率，疏闊尤甚。今設此環，置於赤道環內，仍開合使運轉，出入四十八度，而極畫兩方，東西列周天度數，南北列百刻，可使見日知時。上列三百六十策，與用卦相准。度穿一穴，與赤道相交。白道月環，表一丈五尺一寸五分，橫八分，厚三分，直徑四尺七寸六分。用行有迂曲遲速，與日行緩急相及。古亦無其器，今設于黃道環內，使就黃道為交合，出入六度，以測每夜月離，上畫周天度數，度穿一穴，擬移交會。皆用鋼鐵。游儀，四柱為龍，其崇四尺七寸，水槽及山崇一尺七寸半，槽長六尺九寸，高、廣皆四寸，池深一寸，廣一寸半。龍能興雲雨，故以飾柱。柱在四維。龍下有山、雲，俱在水準槽上。皆用銅。

The ecliptic traveling instrument used four fen of the ancient foot per degree. The revolving-pivot double ring measured one zhang, four chi, six cun, and one fen on its outer face, eight fen wide and three fen thick, with a diameter of four chi, five cun, and nine fen — the instrument the ancients called the revolving sight. The north and south poles were graduated, with thirty-four degrees marked above and below along each ring. The full circuit of degrees was painted on both inner and outer faces, with silver pins set on one side. It could revolve east and west like the turning armillary sphere. A central revolving-pivot axle ran to the inner ends of the two poles, with a bore two and a half degrees wide and a length equal to the ring's diameter. The jade-transverse sighting tube was four chi, five cun, and eight fen long, one cun and two fen wide, one cun thick, with a bore six fen in diameter. The transverse revolved on the axle, turning smoothly to sight the seven luminaries and measure the angular separation of the fixed stars. Square outside and round inside, with a bore one and a half degrees wide — this was the solar-day wheel. The solar meridian double ring had an outer circumference of one zhang, seven chi, and three cun and an inner one of one zhang, four chi, six cun, and four fen; it was four cun wide and four fen thick, with a diameter of five chi, four cun, and four fen, mounted on the meridian. Eight pillars on either side locked the rings firmly in place. The full circuit of degrees was likewise painted on both faces, with silver pins on one side. Half stood above the horizon and half below. The gap between the rings enclosed the pivot axle and the jade-transverse sighting tube at the center. The lunar latitude single ring matched the solar meridian ring in width, thickness, and circumference, interlocking with it half and half so that inner and outer edges aligned perfectly. Its face was level — above was heaven, below was earth. Crossing horizontally around the solar ring, it was called the lunar armillary. The level upper face bore two boundaries marking the hundred quarters of the full celestial circuit inside and out. The celestial summit single ring measured one zhang, seven chi, and three cun around its outer face, eight chi long and wide, three fen thick, with a diameter of five chi, four cun, and four fen. It stood directly above the observer's head, aligned east-west between mao and you, tilted slightly south so that sunrise and sunset could be observed. It was locked to the solar meridian and lunar latitude rings like a shell enclosing its yolk. It stood thirty-six degrees south of the equator, twelve degrees from the ecliptic, fifty-five degrees from the north pole, and ninety-one-odd degrees from each horizontal plane. The equator single ring measured one zhang, four chi, five cun, and nine fen around its outer face, eight fen wide and three fen thick, with a diameter of four chi, five cun, and eight fen. The equator marks the center of heaven — the station of the twenty-eight lunar mansions. The double rings could move, with a hole pierced at each degree. In antiquity the autumn equinox sun stood at five degrees of Jiao; now it stands at thirteen degrees of Zhen; at the winter solstice it stood at the beginning of Qianniu, but now at ten degrees of Dou. By shifting the hole at each crossing, the instrument no longer drifts out of alignment. It lay south of mao and you, thirty-six degrees below the celestial summit, mounted horizontally. The ecliptic single ring measured one zhang, five chi, four cun, and one fen around its outer face, eight fen wide and four fen thick, with a diameter of four chi, eight cun, and four fen. It marks the sun's path and is therefore called the transverse track. The sun's north-south shift accumulates error over the years. The moon and the five planets likewise enter and leave along the sun's degrees. Antiquity had no such instrument; without a standard design, calendrical estimates were especially crude. This ring is now set inside the equator ring, hinged so it can rotate through forty-eight degrees. The poles are marked on both sides — degrees of the full circuit east and west, the hundred time-quarters north and south — so that one can observe the sun and tell the hour. Three hundred sixty counting rods are marked above, aligned with the hexagrams used in calculation. A hole is pierced at each degree where the ring crosses the equator. The white-path lunar ring measured one zhang, five chi, one cun, and five fen around its outer face, eight fen wide and three fen thick, with a diameter of four chi, seven cun, and six fen. The moon's winding course and varying speed are matched against the sun's daily motion. Antiquity had no such instrument either; it is now set inside the ecliptic ring, joining the ecliptic at conjunction and moving through six degrees to measure the moon's nightly separation. The full circuit of degrees is painted above, with a hole at each degree to track the shifting nodes. All were cast in steel and iron. The traveling instrument stood on four dragon pillars four chi and seven cun high. The water trough and its mountain ornament rose one chi, seven cun, and a half; the trough was six chi and nine cun long, four cun high and wide, with a pool one cun deep and one cun and a half wide. Dragons were chosen to adorn the pillars because they summon clouds and rain. The pillars stood at the four diagonal corners. Mountains and clouds were set below the dragons on the leveling trough. These parts were all cast in bronze.

其所測宿度與古異者：舊經，角距星去極九十一度，亢八十九度，氐九十四度，房百八度，心百八度，尾百二十度，箕百一十八度，南斗百一十六度，牽牛百六度，須女百度，虛百四度，危九十七度，營室八十五度，東壁八十六度，奎七十六度，婁八十度，胃、昴七十四度，畢七十八度，觜觿、八十四度，參九十四度，東井七十度，輿鬼六十八度，柳七十七度，七星九十一度，張九十七度，翼九十七度，軫九十八度。今測，角九十三度半，亢九十一度半，氐九十八度，房百一十度半，心百一十度，尾百二十四度，箕百二十度，南斗百一十九度，牽牛百四度，須女百一度，虛百一度，危九十七度，營室八十三度，東壁八十四度，奎七十三度，婁七十七度，胃、昴七十二度，畢七十六度，觜觿八十二度，參九十三度，東井六十八度，輿鬼六十八度，柳八十度半，七星九十三度半，張百度，翼百三度，軫百度。

The measured lodge degrees that differ from the old records are as follows. In the old canon, the Horn pivot star stood ninety-one degrees from the pole; Neck, eighty-nine; Root, ninety-four; Room, one hundred eight; Heart, one hundred eight; Tail, one hundred twenty; Winnowing Basket, one hundred eighteen; Southern Dipper, one hundred sixteen; Ox Leader, one hundred six; Maid, one hundred; Emptiness, one hundred four; Rooftop, ninety-seven; Encampment, eighty-five; Eastern Wall, eighty-six; Strider, seventy-six; Bond, eighty; Stomach and Hairy Head, seventy-four; Net, seventy-eight; Turtle Beak, eighty-four; Three Stars, ninety-four; Well, seventy; Chariot Ghost, sixty-eight; Willow, seventy-seven; Seven Stars, ninety-one; Extended Net, ninety-seven; Wings, ninety-seven; and Axletree, ninety-eight. The new measurements give Horn at ninety-three and a half degrees; Neck, ninety-one and a half; Root, ninety-eight; Room, one hundred ten and a half; Heart, one hundred ten; Tail, one hundred twenty-four; Winnowing Basket, one hundred twenty; Southern Dipper, one hundred nineteen; Ox Leader, one hundred four; Maid, one hundred one; Emptiness, one hundred one; Rooftop, ninety-seven; Encampment, eighty-three; Eastern Wall, eighty-four; Strider, seventy-three; Bond, seventy-seven; Stomach and Hairy Head, seventy-two; Net, seventy-six; Turtle Beak, eighty-two; Three Stars, ninety-three; Well, sixty-eight; Chariot Ghost, sixty-eight; Willow, eighty and a half; Seven Stars, ninety-three and a half; Extended Net, one hundred; Wings, one hundred three; and Axletree, one hundred.

又舊經，角距星正當赤道，黃道在其南；今測，角在赤道南二度半，則黃道復經角中，與天象合。虛北星舊圖入虛，今測在須女九度。危北星舊圖入危，今測在虛六度半。又奎誤距以西大星，故壁損二度，奎增二度；今復距西南大星，即奎、壁各得本度。畢、赤道十六度，黃道亦十六度。觜觿，赤道二度，黃道三度。二宿俱當黃道斜虛，畢尚與赤道度同，觜觿總二度，黃道損加一度，蓋其誤也。今測畢十七度半，觜觿半度。又柳誤距以第四星，今復用第三星。張中央四星為硃鳥嗉，外二星為翼，北距以翼而不距以膺，故張增二度半，七星減二度半；今復以膺為距，則七星、張各得本度。

In the old canon, moreover, the Horn pivot star lay directly on the equator, with the ecliptic to its south; the new measurement places Horn two and a half degrees south of the equator, so that the ecliptic once again runs through the middle of the lodge — in agreement with the sky. The northern star of Emptiness was placed within Emptiness on the old chart; it is now measured at nine degrees of Maid. The northern star of Rooftop was placed within Rooftop on the old chart; it is now measured at six and a half degrees of Emptiness. Strider had also been wrongly measured from the western great star, reducing Eastern Wall by two degrees and inflating Strider by two; measuring again from the southwest great star restores the proper degree allotments for Strider and Eastern Wall. Net was given sixteen degrees on the equator and sixteen on the ecliptic. Turtle Beak was given two degrees on the equator and three on the ecliptic. Both lodges lie where the ecliptic cuts obliquely across the sky. Net still matched its equatorial span, but Turtle Beak totaled only two degrees while the ecliptic figure had been inflated by one — clearly an error. The new measurements give Net seventeen and a half degrees and Turtle Beak half a degree. Willow had also been wrongly measured from the fourth star; the third star is now restored as pivot. The four central stars of Extended Net form the Vermillion Bird's crop, with the two outer stars as its wings. Because the northern pivot was taken from the wings rather than the breast, Extended Net was inflated by two and a half degrees and Seven Stars reduced by two and a half; measuring again from the breast restores the proper degrees for Seven Stars and Extended Net.

其他星：舊經，文昌二星在輿鬼，四星在東井。北斗樞在七星一度，璿在張二度，機在翼二度，權在翼八度，衡在軫八度，開陽在角七度，杓在亢四度。天關在黃道南四度，天尊、天槨在黃道北，天江、天高、狗國、外屏、雲雨、虛梁在黃道外，天囷、土公吏在赤道外，上臺在東井，中台在七星，建星在黃道北半度，天苑在昴、畢，王良在壁，外屏在觜觿，雷電在赤道外五度，霹靂在赤道外四度，八魁在營室，長垣、羅堰當黃道。今測，文昌四星在柳，一星在輿鬼，一星在東井。北斗樞在張十三度，璿在張十二度半，機在翼十三度，權在翼十七度太，衡在軫十度半，開陽在角四度少，杓在角十二度少。天關、天尊、天槨、天江、天高、狗國、外屏，皆當黃道。雲雨在黃道內七度，虛梁在黃道內四度，天囷當赤道，土公吏在赤道內六度，上臺在柳，中台在張，建星在黃道北四度半，天苑在胃、昴，王良四星在奎，一星在壁，外屏在畢，雷電在赤道內二度，霹靂四星在赤道內，一星在外，八魁五星在壁，四星在營室，長垣在黃道北五度，羅堰在黃道北。

Other stars: in the old canon, two stars of Wenchang lay in Chariot Ghost and four in Well. The Northern Dipper's Pivot stood at one degree of Seven Stars; Xuan, at two degrees of Extended Net; Ji, at two degrees of Wings; Quan, at eight degrees of Wings; Heng, at eight degrees of Axletree; Kaiyang, at seven degrees of Horn; and the Ladle, at four degrees of Neck. Heaven's Gate lay four degrees south of the ecliptic; Heaven's Honor and Heaven's Coffin, north of it. Heaven's River, Heaven's Height, Dog Country, Outer Screen, Cloud and Rain, and Empty Bridge stood outside the ecliptic; Heaven's Granary and Earth Duke's Clerk, outside the equator. Upper Terrace was in Well; Middle Terrace, in Seven Stars; Establishment Stars, half a degree north of the ecliptic; Heaven's Park, in Hairy Head and Net; Wangliang, in Wall; Outer Screen, in Turtle Beak; Thunder and Lightning, five degrees outside the equator; Thunderclap, four degrees outside the equator; Eight Chiefs, in Encampment; and Long Rampart and Net Weir, on the ecliptic. The new measurements place four stars of Wenchang in Willow, one in Chariot Ghost, and one in Well. The Northern Dipper's Pivot now stands at thirteen degrees of Extended Net; Xuan, at twelve and a half degrees of Extended Net; Ji, at thirteen degrees of Wings; Quan, at slightly more than seventeen degrees of Wings; Heng, at ten and a half degrees of Axletree; Kaiyang, at slightly less than four degrees of Horn; and the Ladle, at slightly less than twelve degrees of Horn. Heaven's Gate, Heaven's Honor, Heaven's Coffin, Heaven's River, Heaven's Height, Dog Country, and Outer Screen all lie on the ecliptic. Cloud and Rain lies seven degrees inside the ecliptic; Empty Bridge, four degrees inside; Heaven's Granary, on the equator; Earth Duke's Clerk, six degrees inside the equator. Upper Terrace is in Willow; Middle Terrace, in Extended Net; Establishment Stars, four and a half degrees north of the ecliptic; Heaven's Park, in Stomach and Hairy Head. Four stars of Wangliang lie in Strider and one in Wall; Outer Screen, in Net; Thunder and Lightning, two degrees inside the equator. Four stars of Thunderclap lie inside the equator and one outside. Five stars of Eight Chiefs lie in Wall and four in Encampment. Long Rampart stands five degrees north of the ecliptic, and Net Weir, north of the ecliptic.

黃道，春分與赤道交於奎五度太；秋分交於軫十四度少；冬至在斗十度，去赤道南二十四度；夏至在井十三度少，去赤道北二十四度。其赤道帶天之中，以分列宿之度。黃道斜運，以明日月之行。乃立八節、九限，校二道差數，著之歷經。

On the ecliptic, the spring equinox crosses the equator at slightly more than five degrees of Strider; the autumn equinox, at slightly less than fourteen degrees of Axletree; the winter solstice at ten degrees of Dipper, twenty-four degrees south of the equator; and the summer solstice at slightly less than thirteen degrees of Well, twenty-four degrees north of the equator. The equator girdles the center of heaven and serves to divide the degrees among the lodges. The ecliptic runs obliquely to mark the paths of the sun and moon. The eight seasonal nodes and nine limits were then established, the discrepancies between the two paths calculated, and the results entered in the calendrical canon.

蓋天之說，李淳風以為天地中高而四頹，日月相隱蔽，以為晝夜。繞北極常見者謂之上規，南極常隱者謂之下規，赤道橫絡者謂之中規。及一行考月行出入黃道，為圖三十六，究九道之增損，而蓋天之狀見矣。

In the hemispherical-dome theory, Li Chunfeng held that heaven and earth rise high at the center and slope away on all sides, and that the sun and moon alternately conceal and reveal each other to produce day and night. The region that constantly circles the north pole and stays visible is called the upper gnomon; that which circles the south pole and stays hidden, the lower gnomon; and the equator's horizontal belt, the middle gnomon. When Yixing studied the moon's entry into and departure from the ecliptic, he drew thirty-six diagrams tracing the variations of the nine lunar paths, and the shape of the hemispherical dome took form.

削篾為度，徑一分，其厚半之，長與圖等，穴其正中，植針為樞，令可環運。自中樞之外，均刻百四十七度。全度之末，旋為外規。規外太半度，再旋為重規。以均賦周天度分。又距極樞九十一度少半，旋為赤道帶天之紘。距極三十五度旋為內規。

Bamboo splints were cut to serve as degree measures — one fen in diameter, half that in thickness, and as long as the diagram. A hole was bored at the center, a needle set as pivot, and the splint made to revolve freely. Outward from the central pivot, one hundred forty-seven degrees were evenly marked. At the end of the full degree span, the splint was bent to form the outer gnomon. Slightly more than half a degree beyond that, it was bent again to form a second gnomon. The degrees and fractional parts of the full circuit were evenly distributed across it. At ninety-one and a fraction degrees from the polar pivot, it was bent again to form the equator girdling heaven's circumference. At thirty-five degrees from the pole, it was bent to form the inner gnomon.

乃步冬至日躔所在，以正辰次之中，以立宿距。按渾儀所測，甘、石、巫鹹眾星明者，皆以篾，橫考入宿距，縱考去極度，而後圖之。其赤道外眾星疏密之狀，與仰視小殊者，由渾儀去南極漸近，其度益狹；而蓋圖漸遠，其度益廣使然。若考其去極入宿度數，移之於渾天則一也。又赤道內外，其廣狹不均，若就二至出入赤道二十四度，以規度之，則二分所交不得其正；自二分黃赤道交，以規度之，則二至距極度數不得其正；當求赤道分、至之中，均刻為七十二限，據每黃道差數，以篾度量而識之，然後規為黃道，則周天鹹得其正矣。又考黃道二分二至之中，均刻為七十二候，定陰陽曆二交所在，依月去黃道度，率差一候，亦以篾度量而識之，然後規為月道，則周天鹹得其正矣。

The sun's position at the winter solstice was then stepped off to fix the center of the chronogram sequence and establish the lodge pivots. Using the armillary sphere's measurements, all bright stars from the Gan, Shi, and Wu star catalogs were placed with splints — checked horizontally against lodge pivots and vertically for polar distance — and then plotted on the diagram. The spacing of stars outside the equator differs slightly from what one sees looking up because on the armillary sphere, as one approaches the south pole, the degrees grow progressively narrower; whereas on the hemispherical diagram, farther from the center the degrees grow progressively wider — hence the discrepancy. But if one checks their polar distances and lodge positions and transfers them to the armillary sphere, the two agree. Inside and outside the equator, moreover, the widths are uneven. If one measures from the solstices' twenty-four degrees of departure from the equator, the equinox crossings cannot be placed correctly; if one measures from the equinox crossings of the ecliptic and equator, the solstice polar distances cannot be placed correctly; one must find the midpoints between equinox and solstice on the equator, divide them evenly into seventy-two limits, and mark each ecliptic discrepancy with a splint measure. Only then, by bending the ecliptic, will the full circuit be correct throughout. Similarly, the midpoints between the ecliptic's two equinoxes and two solstices should be divided evenly into seventy-two hou, the positions of the two lunar nodes fixed, and for each hou of discrepancy in the moon's distance from the ecliptic, a splint measure applied. Only then, by bending the lunar path, will the full circuit be correct throughout.

中晷之法。初，淳風造曆，定二十四氣中晷，與祖沖之短長頗異，然未知其孰是。及一行作《大衍曆》，詔太史測天下之晷，求其土中，以為定數。其議曰：

The method of the noon gnomon shadow. When Li Chunfeng first compiled his calendar, he fixed the noon gnomon shadows for the twenty-four qi — figures that differed considerably from Zu Chongzhi's — but which was correct remained unknown. When Yixing compiled the Dayan Calendar, an edict ordered the Grand Clerk to measure gnomon shadows across the empire, determine the terrestrial center, and establish fixed constants from the results. The memorial argued as follows:

《周禮·大司徒》：「以土圭之法測土深。日至之景，尺有五寸，謂之地中。」鄭氏以為「日景於地，千里而差一寸。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內，是以半之，得地中，今潁川陽城是也」。宋元嘉中，南征林邑，五月立表望之，日在表北，交州影在表南三寸，林邑九寸一分。交州去洛，水陸之路九千里，蓋山川回折使之然，以表考其弦當五千乎。開元十二年，測交州，夏至，在表南三寸三分，與元嘉所測略同。使者大相元太言：「交州望極，才高二十餘度。八月海中望老人星下列星粲然，明大者甚眾，古所未識，乃渾天家以為常沒地中者也。大率去南極二十度已上之星則見。」又鐵勒、回紇在薛延陀之北，去京師六千九百里，其北又有骨利幹，居澣海之北，北距大海，晝長而夜短，既夜，天如曛不暝，夕胹羊髀才熟而曙，蓋近日出沒之所。太史監南宮說擇河南平地，設水準繩墨植表而以引度之，自滑台始白馬，夏至之晷，尺五寸七分。又南百九十八里百七十九步，得浚儀嶽台，晷尺五寸三分。又南百六十七里二百八十一步，得扶溝，晷尺四寸四分。又南百六十里百一十步，至上蔡武津，晷尺三寸六分半。大率五百二十六里二百七十步，晷差二寸餘。而舊說王畿千里，影差一寸，妄矣。

The Rites of Zhou, Grand Minister of Education: "By the method of the earth square, measure the depth of the earth. Where the solstice shadow measures one chi and five cun, that is called the terrestrial center." Zheng's commentary explains: "The sun's shadow on earth differs by one cun for each thousand li. One chi and five cun marks the point fifteen thousand li south under the sun's bearing; earth and stars oscillate within thirty thousand li in their four excursions — halving this gives the terrestrial center, which is present-day Yangcheng in Yingchuan." During the Yuanjia era of Song, on the southern campaign against Linyi, a gnomon was erected in the fifth month: the sun stood north of the gnomon. At Jiaozhou the shadow fell three cun south of the gnomon; at Linyi, nine cun and one fen. Jiaozhou lies nine thousand li from Luoyang by water and land routes — the winding terrain surely accounts for the distance — but measured by the gnomon, the straight chord should be about five thousand li. In the twelfth year of Kaiyuan, Jiaozhou was measured again: at the summer solstice the shadow fell three cun and three fen south of the gnomon — roughly the same as the Yuanjia figure. The envoy Da Xiangyuan Tai reported: "Viewed from Jiaozhou, the pole stands barely twenty-odd degrees above the horizon. In the eighth month, looking out to sea, the stars below Canopus blazed brilliantly — many large and bright, unknown to antiquity — the very stars that armillary-heaven scholars say lie permanently below the earth. In general, stars more than twenty degrees from the south pole become visible." The Tiele and Uighur peoples lay north of Xueyantuo, six thousand nine hundred li from the capital. Farther north still lived the Guligan, north of the Han Sea with the ocean beyond. Days were long and nights short; even at night the sky glowed like dusk without full darkness, and lamb shanks roasted at evening were barely done before dawn — surely near the place where the sun rises and sets. Grand Clerk Supervisor Nangong Shuo selected level ground in Henan, set leveling cords and ink lines, erected gnomons, and measured in successive stages. Starting from Baima at Huatai, the summer solstice shadow measured one chi, five cun, and seven fen. One hundred ninety-eight li and one hundred seventy-nine paces farther south, at the Yue Terrace in Junyi, the shadow was one chi, five cun, and three fen. One hundred sixty-seven li and two hundred eighty-one paces farther south, at Fugou, the shadow was one chi, four cun, and four fen. One hundred sixty li and one hundred ten paces farther south, at Wujin in Shangcai, the shadow was one chi, three cun, six fen, and a half. On average, the shadow differed by slightly more than two cun for every five hundred twenty-six li and two hundred seventy paces. The old claim that within the royal domain the shadow differed one cun per thousand li was plainly wrong.

今以句股校陽城中晷，夏至尺四寸七分八厘，冬至丈二尺七寸一分半，定春秋分五尺四寸三分，以覆矩斜視，極出地三十四度十分度之四。自滑台表視之，極高三十五度三分，冬至丈三尺，定春秋分五尺五寸六分。自浚儀錶視之，極高三十四度八分，冬至丈二尺八寸五分，定春秋分五尺五寸。知扶溝表視之，極高三十四度三分，冬至丈二尺五寸五分，定春秋分五尺三寸七分。上蔡武津表視之，極高三十三度八分，冬至丈二尺三寸八分，定春秋分五尺二寸八分。其北極去地，雖秒分微有盈縮，難以目校，大率三百五十一里八十步，而極差一度。極之遠近異，則黃道軌景固隨而變矣。自此為率推之，比歲武陵晷，夏至七寸七分，冬至丈五寸三分，春秋分四尺三寸七分半，以圖測之，定氣四尺四寸七分，按圖斜視，極高二十九度半，差陽城五度三分。蔚州橫野軍夏至二尺二寸九分，冬至丈五尺八寸九分，春秋分六尺四寸四分半，以圖測之，定氣六尺六寸二分半。按圖斜視，極高四十度，差陽城五度三分。凡南北之差十度半，其徑三千六百八十里九十步。自陽城至武陵，千八百二十六里七十六步；自陽城至橫野，千八百六十一里二百十四步。夏至晷差尺五寸三分；自陽城至武陵，差七寸三分；自陽城至橫野，差八寸。冬至晷差五尺三寸六分，自陽城至武陵差二尺一寸八分；自陽城至橫野，差三尺一寸八分。率夏至與南方差少，冬至與北方差多。

Triangulating the noon shadows at Yangcheng gives a summer solstice shadow of one chi, four cun, seven fen, and eight li; a winter solstice shadow of one zhang, two chi, seven cun, one fen, and a half; and fixed equinox shadows of five chi, four cun, and three fen. By inverted square and oblique sighting, the pole rises thirty-four and four-tenths degrees above the horizon. From the Huatai gnomon, the pole stands thirty-five and three-tenths degrees high; the winter solstice shadow, one zhang and three chi; the fixed equinox shadow, five chi, five cun, and six fen. From the Junyi gnomon, the pole stands thirty-four and eight-tenths degrees high; the winter solstice shadow, one zhang, two chi, eight cun, and five fen; the fixed equinox shadow, five chi and five cun. From the Fugou gnomon, the pole stands thirty-four and three-tenths degrees high; the winter solstice shadow, one zhang, two chi, five cun, and five fen; the fixed equinox shadow, five chi, three cun, and seven fen. From the Wujin gnomon in Shangcai, the pole stands thirty-three and eight-tenths degrees high; the winter solstice shadow, one zhang, two chi, three cun, and eight fen; the fixed equinox shadow, five chi, two cun, and eight fen. The pole's altitude, though fine seconds and parts vary slightly beyond what the eye can verify, differs by one degree for roughly every three hundred fifty-one li and eighty paces. As the pole's altitude changes, the ecliptic track and gnomon shadows naturally change with it. Extrapolating from this ratio to Wuling's shadows that year: summer solstice, seven cun and seven fen; winter solstice, one zhang, five cun, and three fen; equinox, four chi, three cun, seven fen, and a half. Measured on the diagram, the fixed-qi shadow is four chi, four cun, and seven fen; by oblique sighting, the pole stands twenty-nine and a half degrees — five and three-tenths degrees lower than at Yangcheng. At Hengye Army in Weizhou: summer solstice shadow, two chi, two cun, and nine fen; winter solstice, one zhang, five chi, eight cun, and nine fen; equinox, six chi, four cun, four fen, and a half. Measured on the diagram, the fixed-qi shadow is six chi, six cun, two fen, and a half. By oblique sighting on the diagram, the pole stands forty degrees — five and three-tenths degrees higher than at Yangcheng. The total north-south difference is ten and a half degrees, spanning a diameter of three thousand six hundred eighty li and ninety paces. From Yangcheng to Wuling is one thousand eight hundred twenty-six li and seventy-six paces; from Yangcheng to Hengye, one thousand eight hundred sixty-one li and two hundred fourteen paces. The total summer solstice shadow difference is one chi, five cun, and three fen; from Yangcheng to Wuling, seven cun and three fen; from Yangcheng to Hengye, eight cun. The total winter solstice shadow difference is five chi, three cun, and six fen; from Yangcheng to Wuling, two chi, one cun, and eight fen; from Yangcheng to Hengye, three chi, one cun, and eight fen. On average, the summer solstice shadow varies less toward the south, while the winter solstice shadow varies more toward the north.

又以圖校安南，日在天頂北二度四分，極高二十度四分。冬至晷七尺九寸四分，定春秋分二尺九寸三分，夏至在表南三寸三分，差陽城十四度三分，其徑五千二十三里。至林邑，日在天頂北六度六分強，極高十七度四分，周圓三十五度，常見不隱。冬至晷六尺九寸，定春秋分二尺八寸五分，夏至在表南五寸七分，其徑六千一百一十二里。若令距陽城而北，至鐵勒之地，亦差十七度四分，與林邑正等，則五月日在天頂南二十七度四分，極高五十二度，周圓百四度，常見不隱。北至晷四尺一寸三分，南至晷二丈九尺二寸六分，定春秋分晷五尺八寸七分。其沒地才十五餘度，夕沒亥西，晨出丑東，校其里數，已在回紇之北，又南距洛陽九千八百一十五里，則極長之晝，其夕常明。然則骨利幹猶在其南矣。

The diagram was also used to verify Annam: the sun stands two and four-tenths degrees north of the zenith, and the pole twenty and four-tenths degrees above the horizon. The winter solstice shadow is seven chi, nine cun, and four fen; the fixed equinox shadow, two chi, nine cun, and three fen; the summer solstice shadow falls three cun and three fen south of the gnomon. The pole stands fourteen and three-tenths degrees lower than at Yangcheng, across a diameter of five thousand twenty-three li. At Linyi, the sun stands slightly more than six and six-tenths degrees north of the zenith; the pole, seventeen and four-tenths degrees high. Stars within thirty-five degrees of the pole are constantly visible and never set. The winter solstice shadow is six chi and nine cun; the fixed equinox shadow, two chi, eight cun, and five fen; the summer solstice shadow falls five cun and seven fen south of the gnomon. The diameter spans six thousand one hundred twelve li. If one travels north from Yangcheng to the land of the Tiele, a difference of seventeen and four-tenths degrees — equal to Linyi — then in the fifth month the sun stands twenty-seven and four-tenths degrees south of the zenith; the pole, fifty-two degrees high. Stars within one hundred four degrees of the pole would be constantly visible and never set. The northern shadow would measure four chi, one cun, and three fen; the southern shadow, two zhang, nine chi, two cun, and six fen; the fixed equinox shadow, five chi, eight cun, and seven fen. Stars would set barely fifteen-odd degrees below the horizon; at evening they would set west of hai, at morning rise east of chou. By the reckoned distance, this point lies north of the Uighur and nine thousand eight hundred fifteen li south of Luoyang — where in the longest days, twilight never fully fades. Guligan, then, would lie even farther north.

吳中常侍王蕃考先儒所傳，以戴日下萬五千里為句股，斜射陽城，考周徑之率以揆天度，當千四百六里二十四步有餘。今測日晷，距陽城五千里，已在戴日之南，則一度之廣皆三分減二，南北極相去八萬里，其徑五萬里。宇宙之廣，豈若是乎？然則蕃之術，以蠡測海者也。

Wang Fan, Attendant-in-Ordinary of Wu, reviewed the earlier scholars' transmission: using fifteen thousand li under the sun's bearing as a right triangle and projecting obliquely toward Yangcheng, he calculated the ratio of circumference to diameter and derived a degree-span of one thousand four hundred six li and twenty-four-odd paces. The new gnomon measurements show that five thousand li from Yangcheng already lies south of the sun's bearing — reducing the span of each degree by two-thirds. The distance between the north and south poles is eighty thousand li, and the diameter fifty thousand li. Can the cosmos really be so small? Fan's method, then, is like measuring the sea with a gourd dipper.

古人所以恃句股術，謂其有證於近事。顧未知目視不能及遠，遠則微差，其差不已，遂與術錯。譬游於太湖，廣袤不盈百里，見日月朝夕出入湖中；及其浮於巨海，不知幾千萬里，猶見日月朝夕出入其中矣。若於朝夕之際，俱設重差而望之，必將大小之同術，無以分矣。橫既有之，縱亦宜然。

The ancients trusted the gougu method because they believed it could be verified by nearby observations. They did not recognize that sight cannot penetrate great distances; over distance, minute discrepancies accumulate without cease until the reckoning itself goes astray. Consider boating on Lake Tai, whose expanse does not exceed a hundred li in any direction: morning and evening one sees the sun and moon rise and set from within the lake itself; Yet when one sails the great ocean, whose span runs to countless thousands of myriads of li, one still sees the sun and moon rise and set within its waters morning and evening. If at dawn and dusk one applied the double-difference method to both alike, the same technique for the small and the vast would yield no distinction between them. If this holds in the horizontal dimension, it ought to hold vertically as well.

又若樹兩表，南北相距十里，其崇皆數十里，置大炬于南表之端，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。又置大炬於北表之端，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。復於二表間更植八尺之木，仰而望之，則表首環屈相合。若置火炬於兩表之端，皆當無影矣。夫數十里之高與十里之廣，然猶斜射之影與仰望不殊。今欲憑晷差以指遠近高下，尚不可知，而況稽周天裏步於不測之中，又可必乎？十三年，南至，岱宗禮畢，自上傳呼萬歲，聲聞於下。時山下夜漏未盡，自日觀東望，日已漸高。據曆法，晨初迨日出差二刻半，然則山上所差凡三刻餘。其冬至夜刻同立春之後，春分夜刻同立夏之後。自嶽趾升泰壇僅二十里，而晝夜之差一節。設使因二十里之崇以立句股術，固不知其所以然，況八尺之表乎！

Suppose two gnomons stand ten li apart on a north-south line, each tens of li high: place a great torch at the top of the southern gnomon and an eight-chi stake beneath it — the shadow ought to vanish entirely. Look up from the foot of the southern gnomon toward the crown of the northern one, and minute fractional errors will gradually merge until the northern gnomon aligns with the southern. Once the tops coincide, a torch placed there should likewise cast no shadow. Place a great torch at the top of the northern gnomon and an eight-chi stake beneath it — again, the shadow ought to vanish. Look up from the foot of the northern gnomon toward the crown of the southern one, and minute fractional errors will again gradually merge until the southern gnomon aligns with the northern. Once the tops coincide, a torch placed there should likewise cast no shadow. Plant another eight-chi stake midway between the two gnomons and gaze upward — the gnomon tops will appear to bow and meet in a curve. Place torches atop both gnomons, and each should cast no shadow. At heights of tens of li and spans of ten li, oblique shadows and lines of sight still appear indistinguishable. If gnomon-shadow differences cannot even reliably gauge nearby height and distance, how can one presume to reckon the circuit of heaven in li within the immeasurable depths of space? In the thirteenth year, after the solstice rites at Mount Tai were complete, voices cried "Ten thousand years!" from the summit — audible all the way below. At the mountain's foot the night water-clock had not yet emptied, yet gazing east from Sun-Viewing Peak, the sun had already climbed high. By calendrical reckoning, first light precedes sunrise by two and a half ke; on the summit the total difference came to slightly more than three ke. The night watches at the winter solstice correspond to those after Beginning of Spring; those at the spring equinox correspond to those after Beginning of Summer. From the mountain's base to the altar at the summit is barely twenty li — yet the day-night difference spans a full seasonal node. Even building the gougu method on twenty li of elevation, one could not grasp the underlying cause — to say nothing of an eight-chi gnomon!

原古人所以步圭影之意，將以節宣和氣，轉相物宜，不在於辰次之周徑。其所以重歷數之意，將欲恭授人時，欽若乾象，不在於渾、蓋之是非。若乃述無稽之法於視聽之所不及，則君子當闕疑而不議也。而或者各守所傳之器以術天體，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，誠以為蓋天邪？則南方之度漸狹；果以為渾天邪？則北方之極浸高。此二者，又渾、蓋之家盡智畢議，未能有以通其說也。則王仲任、葛稚川之徒，區區于異同之辨，何益人倫之化哉。凡晷差，冬夏不同，南北亦異，先儒一以里數齊之，遂失其實。今更為《覆矩圖》，南自丹穴，北暨幽都，每極移一度，輒累其差，可以稽日食之多少，定晝夜之長短，而天下之晷，皆協其數矣。

Consider why the ancients measured the earth-square shadows: to harmonize the seasons and match each thing to its proper time — not to calculate the circumference and diameter of the celestial lodges. Their purpose in honoring calendrical reckoning was to reverently grant the seasons to the people and conform to the celestial pattern — not to settle the dispute between armillary and canopy heaven. To expound unverifiable methods beyond the reach of sight and sound — here a gentleman should preserve his doubts and refrain from debate. Yet some cling to inherited instruments to fathom the heavens, claiming the armillary origin can be measured by number and the great pattern glimpsed through calculation. The six schools' doctrines end in perpetual contradiction. If one truly holds to canopy heaven? Then the degrees in the south ought to grow progressively narrower; If one truly holds to armillary heaven? Then the northern pole ought to climb progressively higher. On these two points even the armillary and canopy schools, exhausting all wisdom and debate, could find no way to reconcile their doctrines. As for Wang Chong, Ge Hong, and their ilk, what profit to the moral transformation of humanity in their petty disputations over doctrine? Gnomon-shadow differences vary between winter and summer and between north and south; earlier scholars forced them into a single li-count ratio and thereby lost the truth. A new Inverted Square Diagram has now been drawn, from Dannue in the south to Youdu in the north: for each degree the pole shifts, the difference is accumulated, enabling one to reckon the magnitude of eclipses and fix the length of day and night — so that gnomon shadows throughout the realm accord with a single standard.

昭宗時，太子少詹事邊岡，脩曆術，服其精粹，以為不刊之數也。

In the Zhaozong era, Vice-Director of the Crown Prince's Household Bian Gang revised the calendrical methods, admired their refinement, and declared them an immutable standard.

初，貞觀中，淳風撰《法象志》，因《漢書》十二次度數，始以唐之州縣配焉。而一行以為天下山河之象存乎兩戒。北戒自三危、積石，負終南地絡之陰，東及太華，逾河，並雷首、厎柱、王屋、太行，北抵常山之右，乃東循塞坦，至濊貊、朝鮮，是謂北紀，所以限戎狄也；南戒自岷山、嶓塚，負地絡之陽，東及太華，連益山、熊耳、外方、桐柏，自上洛南逾江、漢，攜武當、荊山，至於衡陽，乃東循嶺徼，達東甌、閩中，是謂南紀，所以限蠻夷也。故《星傳》謂北戒為「胡門」，南戒為「越門」。

Early in the Zhenguan reign, Li Chunfeng compiled the Astronomical Phenomena Treatise, following the twelve lunar-lodge degree counts in the Book of Han and for the first time matching them to Tang prefectures and counties. Yixing, however, held that the pattern of mountains and rivers throughout the realm resided in the Two Prohibitions. The Northern Prohibition begins at Sanwei and Jishi, bearing the yin side of the Zhongnan terrestrial vein, extends east to Mount Taihua, crosses the Yellow River, joins Leishou, Dizhu, Wangwu, and Taihang, reaches the right flank of Mount Chang, then runs east along the border ramparts to Huimo and Korea — this is called the Northern Thread, which bounds the Rong and Di; The Southern Prohibition begins at Mount Min and Mount Bo, bearing the yang side of the terrestrial vein, extends east to Mount Taihua, connects Mount Yi, Mount Xiong'er, Mount Waifang, and Mount Tongbai, from Shangluo south crosses the Yangtze and Han, embraces Wudang and Mount Jing, reaches Hengyang, then runs east along the ridge borders to Dong'ou and Minzhong — this is called the Southern Thread, which bounds the Man and Yi. The Star Tradition therefore calls the Northern Prohibition the "Barbarian Gate" and the Southern Prohibition the "Yue Gate."

河源自北紀之首，循雍州北徼，達華陰，而與地絡相會，並行而東，至太行之曲，分而東流，與涇、謂、濟瀆相為表裏，謂之「北河」。江源自南紀之首，循梁州南徼，達華陽，而與地絡相會，並行而東，及荊山之陽，分而東流，與漢水、淮瀆相為表裏，謂之「南河」。故于天象，則弘農分陝為兩河之會，五服諸侯在焉。自陝而西為秦、涼，北紀山河之曲為晉、代，南紀山河之曲為巴、蜀，皆負險用武之國也。自陝而東，三川、中嶽為成周；西距外方、大伾，北至於濟，南至於淮，東達钜野，為宋、鄭、陳、蔡；河內及濟水之陽為鄁、衛；漢東濱淮水之陰為申、隨。皆四戰用文之國也。北紀之東，至北河之北，為邢、趙。南紀之東，至南河之南，為荊、楚。自北河下流，南距岱山為三齊，夾右碣石為北燕。自南河下流，北距岱山為鄒、魯，南涉江、淮為吳、越。皆負海之國，貸殖之所阜也。自河源循塞垣北，東及海，為戎狄。自江源循嶺徼南，東及海，為蠻越。觀兩河之象。與雲漢之所始終，而分野可知矣。

The Yellow River rises at the head of the Northern Thread, follows Yongzhou's northern border to Huayin, meets the terrestrial vein, runs east in parallel to the bend of Taihang, then divides and flows eastward, paired inside and out with the Jing, Wei, and Ji waterways — this is called the "Northern River." The Yangtze rises at the head of the Southern Thread, follows Liangzhou's southern border to Huayang, meets the terrestrial vein, runs east in parallel to the sunny side of Mount Jing, then divides and flows eastward, paired inside and out with the Han River and Huai waterway — this is called the "Southern River." In the celestial pattern, therefore, Hongnong at the Fen-Shan divide marks the confluence of the two rivers, where the lords of the five domains were seated. West of Shaan lay Qin and Liang; the bends of the Northern Thread's mountains and rivers formed Jin and Dai; those of the Southern Thread formed Ba and Shu — all lands that relied on natural defenses and martial force. East of Shaan, the Three Rivers and Central Peak formed Chengzhou; bounded on the west by Waifang and Great Pei, north to the Ji River, south to the Huai, east to Juye — the lands of Song, Zheng, Chen, and Cai; Henei and the north bank of the Ji River formed Bei and Wei; East of the Han River, along the south bank of the Huai, lay Shen and Sui. All were lands of four-front warfare that relied on culture rather than terrain. East of the Northern Thread, north of the Northern River, lay Xing and Zhao. East of the Southern Thread, south of the Southern River, lay Jing and Chu. From the lower Northern River, south to Mount Dai, lay the Three Qi; flanking Jieshi to the right lay Northern Yan. From the lower Southern River, north to Mount Dai lay Zou and Lu; south across the Yangtze and Huai lay Wu and Yue. All were sea-facing lands where trade and prosperity flourished. From the river source, following the border ramparts north to the sea, lay the Rong and Di. From the river source, following the ridge borders south to the sea, lay the Man and Yue. Observe the pattern of the two rivers together with the Cloud River's beginning and end, and the territorial divisions become knowable.

于《易》，五月一陰生，而雲漢潛萌於天稷之下，進及井、鉞間，得坤維之氣，陰始達於地上，而雲漢上升，始交於列宿，七緯之氣通矣。東井據百川上流，故鶉首為秦、蜀墟，得兩戒山河之首。雲漢達坤維右而漸升，始居列宿上，觜觿、參、伐皆直天關表而在河陰，故實沈下流得大樑，距河稍遠，涉陰亦深。故其分野，自漳濱卻負恒山，居北紀眾山之東南，外接髦頭地，皆河外陰國也。十月陰氣進逾乾維，始上達于天，雲漢至營室、東壁間，升氣悉究，與內規相接。故自南正達於西正，得雲漢升氣，為山河上流；自北正達於東正，得雲漢降氣，為山河下流。陬訾在雲漢升降中，居水行正位，故其分野當中州河、濟間。且王良、閣道由紫垣絕漢抵營室，上帝離宮也，內接成周、河內，皆豕韋分。十一月一陽生，而雲漢漸降，退及艮維，始下接於地，至斗、建間，復與列舍氣通，于《易》，天地始交，泰象也。逾析木津，陰氣益降，進及大辰，升陽之氣究，而雲漢沈潛於東正之中，故《易》，雷出地曰豫，龍出泉為解，皆房、心象也。星紀得雲漢下流，百川歸焉，析木為雲漢末派，山河極焉。故其分野，自南河下流，窮南紀之曲，東南負海，為星紀；自北河末派，窮北紀之曲，東北負海，為析木。負海者，以其雲漢之陰也。唯陬訾內接紫宮，在王畿河、濟間。降婁、玄枵與山河首尾相遠，鄰顓頊之墟，故為中州負海之國也。其地當南河之北、北河之南，界以岱宗，至於東海。自鶉首逾河，戒東曰鶉火，得重離正位，軒轅之祇在焉。其分野，自河、華之交，東接祝融之墟，北負河，南及漢，蓋寒燠之所均也。自析木紀天漢而南，曰大火，得明堂升氣，天市之都在焉。其分野，自钜野岱宗，西至陳留，北負河、濟，南及淮，皆和氣之所布也。陽氣自明堂漸升，達于龍角，曰壽星。龍角謂之天關，于《易》，氣以陽決陰，夬象也。升陽進逾天關。得純乾之位，故鶉尾直建巳之月，內列太微，為天廷。其分野，自南河以負海，亦純陽地也。壽星在天關內，故其分野，在益、亳西南，淮水之陰，北連太室之東，自陽城際之，亦巽維地也。

In the Book of Changes, when in the fifth month the first yin is born, the Cloud River secretly germinates beneath Tianji, advancing to between Well and Axe, receiving the qi of the kun axis; as yin first reaches the earth, the Cloud River ascends and first crosses the fixed stars — and the qi of the seven regulators flows through. Eastern Well occupies the upper reaches of the hundred streams; hence Qu Shou marks the domain of Qin and Shu, receiving the headwaters of the Two Prohibitions' mountains and rivers. The Cloud River reaches right of the kun axis and gradually ascends, first rising above the fixed stars; Turtle Beak, Three Stars, and Orion all stand directly outside Heaven's Gate on the yin side of the river — hence Shi Shen's lower reach obtains Great Bridge, somewhat farther from the river and deeper in yin. Its territorial division therefore runs from the Zhang River's shore back to Mount Heng, southeast of the Northern Thread's mountains, outward connecting the land of Maotou — all yin lands beyond the northern river. In the tenth month yin qi advances past the qian axis and first reaches heaven; the Cloud River arrives between Encampment and Eastern Wall, ascending qi is fully spent, and it connects with the inner rule. From southern culmination to western culmination, receiving Cloud River ascending qi — the upper reaches of mountains and rivers; From northern culmination to eastern culmination, receiving Cloud River descending qi — the lower reaches of mountains and rivers. Zouzi stands in the middle of the Cloud River's ascent and descent, occupying the orthodox position of the Water phase; hence its division lies between the Yellow and Ji rivers in the central provinces. Moreover Wangliang and Celestial Ford run from the Purple Enclosure across the Han to Encampment — the Supreme Lord's detached palace — inward connecting Chengzhou and Henei, all within Shiwei's division. In the eleventh month the first yang is born; the Cloud River gradually descends, retreating to the gen axis and first connecting with the earth; reaching between Dipper and Establishment, it again connects with the qi of the fixed lodges — in the Book of Changes, heaven and earth first meet: the hexagram Tai. Crossing the Timber-Felling Ford, yin qi descends further, advancing to Great Chen; ascending yang qi is spent and the Cloud River sinks hidden within eastern culmination — hence in the Book of Changes, "thunder emerging from earth is called Yu; dragon emerging from spring is called Xie" — all images of Heart and Heart-and-Body. Xingji receives the Cloud River's lower stream, where the hundred streams converge; Ximu is the Cloud River's final branch, where mountains and rivers reach their limit. Its territorial division therefore runs from the lower Southern River to the end of the Southern Thread's bend, southeast facing the sea — Xingji; From the Northern River's final branch to the end of the Northern Thread's bend, northeast facing the sea — Ximu. Sea-facing lands lie in the yin side of the Cloud River. Only Zouzi connects inward to the Purple Palace, between the Yellow and Ji rivers within the royal domain. Jianglou and Xuanxiao stand far from the head and tail of the mountains and rivers, neighboring Zhuanxu's domain — hence the central provinces' sea-facing lands. Their lands lie north of the Southern River and south of the Northern River, bounded by Mount Dai as far as the Eastern Sea. East of Qu Shou, crossing the river, the eastern prohibition is called Qu Huo, occupying the orthodox position of double Li — the altar of Xuanyuan stands there. Its division runs from the junction of the Yellow River and Mount Hua, east connecting Zhurong's domain, north to the river, south to the Han — a region where cold and heat are evenly balanced. South from Ximu, where it marks the Celestial Han, lies Great Fire, receiving Bright Hall ascending qi — the capital of Celestial Market stands there. Its division runs from Juye and Mount Dai west to Chenliu, north to the Yellow and Ji rivers, south to the Huai — all regions where harmonizing qi is spread. Yang qi gradually ascends from Bright Hall, reaching Horn — called Longevity Star. Horn is called Heaven's Gate; in the Book of Changes, qi uses yang to decide yin — the hexagram Guai. Ascending yang advances beyond Heaven's Gate. Receiving the position of pure Qian, Chunwei directly corresponds to the month jian-si, inward arrayed with Supreme Subtlety — the celestial court. Its division runs from the Southern River to the sea — also purely yang territory. Longevity Star lies within Heaven's Gate; hence its division runs southwest of Yi and Bo, on the yin side of the Huai, north connecting east of Mount Taisi, marked from Yangcheng — also xun axis territory.

夫雲漢自坤抵艮為地紀，北斗自乾攜巽為天綱，其分野與帝車相直，皆五帝墟也。究咸池之政而在乾維內者，降婁也，故為少昊之墟。葉北宮之政而在幹維外者，陬訾也，故為顓頊之墟。成攝提之政而在巽維內者，壽星也，故為太昊之墟。布太微之政，而在巽維外者，鶉尾也，故為列山氏之墟。得四海中承太階之政者，軒轅也，故為有熊氏之墟。木、金得天地之微氣，其神治于季月；水、火得天地之章氣，其神治于孟月。故章道存乎至，微道存乎終，皆陰陽變化之際也。若微者沈潛而不及，章者高明而過亢，皆非上帝之居也。

The Cloud River from kun to gen forms the terrestrial thread; the Big Dipper from qian carrying xun forms the celestial net; their territorial divisions align with the Emperor's Carriage — all domains of the Five Emperors. Completing Xianchi's governance within the qian axis is Jianglou — hence Shaohao's domain. Echoing Northern Palace's governance outside the qian axis is Zouzi — hence Zhuanxu's domain. Completing Sheti's governance within the xun axis is Longevity Star — hence Taihao's domain. Spreading Supreme Subtlety's governance outside the xun axis is Chunwei — hence the domain of the Lieshan Clan. Receiving within the four seas the governance that upholds the Great Steps is Xuanyuan — hence the domain of the Youxiong Clan. Wood and Metal receive heaven and earth's subtle qi; their spirits govern the terminal months; Water and Fire receive heaven and earth's manifest qi; their spirits govern the first months. The manifest way resides at the apex; the subtle way at the end — both at the junctures of yin-yang transformation. If the subtle sinks hidden without attaining, or the manifest rises bright and overreaches — neither is the Supreme Lord's abode.

斗杓謂之外廷，陽精之所布也。斗魁謂之會府，陽精之所復也。杓以治外，故鶉尾為南方負海之國。魁以治內，故陬訾為中州四戰之國。其餘列舍，在雲漢之陰者八，為負海之國。在雲漢之陽者四，為四戰之國。降婁、玄枵以負東海，春神主于岱宗，歲星位焉。星紀、鶉尾以負南海，其神主於衡山，熒惑位焉。鶉首、實沈以負西海，其神主于華山，太白位焉。大樑、析木以負北海，其神主於恒山，辰星位焉。鶉火、大火、壽星、豕韋為中州，其神主於嵩丘，鎮星位焉。

The Dipper's Handle is called the Outer Court, where yang essence is spread. The Dipper's Ladle is called the Assembly Hall, where yang essence returns. The Handle governs the outer realm; hence Chunwei is the southern sea-facing land. The Ladle governs the inner realm; hence Zouzi is the central land of four-front warfare. Of the remaining lodges, eight stand on the northern side of the Cloud River and correspond to sea-facing lands. Four stand on the southern side of the Cloud River and correspond to lands of warfare on every front. Jianglou and Xuanxiao face the Eastern Sea; the spirit of spring holds sway at Mount Tai, and Jupiter takes its station there. Xingji and Chunwei face the Southern Sea; their spirit holds sway at Southern Mount Heng, and Mars takes its station there. Chunshou and Shenshen face the Western Sea; their spirit holds sway at Mount Hua, and Venus takes its station there. Daliang and Ximu face the Northern Sea; their spirit holds sway at Northern Mount Heng, and Mercury takes its station there. Chunhuo, Great Fire, Longevity Star, and Shiwei constitute the central domain; their spirit holds sway at Mount Song, and Saturn takes its station there.

近代諸儒言星土者，或以州，或以國。虞、夏、秦、漢，郡國廢置不同。周之興也，王畿千里，及其衰也，僅得河南七縣。今又天下一統，而直以鶉火為周分，則疆場舛矣。七國之初，天下地形雌韓而雄魏，魏地西距高陵，盡河東、河內，北固漳、鄴，東分梁、宋，至於汝南，韓據全鄭之地，南盡潁川、南陽、西達虢略，距函穀，固宜陽，北連上地，皆綿亙數州，相錯如繡。考雲漢山河之象，多者或至十餘宿。其後魏徙大樑，則西河合于東井；秦拔宜陽，而上党入於輿鬼。方戰國未滅時，星家之言，屢有明效。今則同在畿甸之中矣。而或者猶據《漢書地理志》推之，是守甘、石遺術，而不知變通之數也。

In recent times, scholars who discuss star-territory correspondences have sometimes mapped them to prefectures and sometimes to feudal states. Under the Yu, Xia, Qin, and Han dynasties, commanderies and feudal states were abolished and re-established in different arrangements. When Zhou rose to power, its royal domain spanned a thousand li; by the time it declined, it held no more than seven counties in Henan. Now that the realm is unified under one rule, to assign Chunhuo directly as Zhou's territorial allotment would grossly misalign the boundaries. At the outset of the Warring States period, the realm's terrain favored Han in the south and Wei in the north: Wei reached Gaoling in the west and held Hedong and Henei; in the north it secured Zhang and Ye; in the east it divided Liang and Song down to Runan; Han held all of Zheng, south to Yingchuan and Nanyang, west to Guolue, facing Hangu Pass and holding Yiyang, northward connecting Shangdi — territories spanning several prefectures and interwoven like embroidery. When one examines the celestial image of the Cloud River against mountains and rivers, some regions correspond to more than ten lodges. Later, when Wei moved its capital to Daliang, the Western River came under the Well; When Qin seized Yiyang, Shangdang passed into the Chariot Ghost. While the Warring States still stood, the pronouncements of star specialists repeatedly proved strikingly accurate. Now they all lie within the same capital district. Yet some still derive their allotments from the Han Geography Treatise — clinging to the legacy methods of Gan and Shi without grasping the principle of adaptation.

又古之辰次與節氣相系，各據當時歷數，與歲差遷徙不同。今更以七宿之中分四象中位，自上元之首，以度數紀之，而著其分野，其州縣雖改隸不同，但據山河以分爾。

Moreover, the ancient lodge-sequence positions were tied to the solar terms, each reckoned according to the calendar of its day — unlike the shifting caused by precession. We now take the central positions of the four celestial quadrants from among the seven lodges, record them in degrees from the head of the upper origin, and set forth their territorial divisions; though prefectures and counties may change administrative affiliation, the allotments are determined solely by mountains and rivers.

須女、虛、危，玄枵也。初，須女五度，餘二千三百七十四，秒四少。中，虛九度。終，危十二度。其分野，自濟北東逾濟水，涉平陰，至於山莊，循岱嶽眾山之陰，東南及高密，又東盡萊夷之地，得漢北海、千乘、淄川，濟南、濟郡及平原、渤海、九河故道之南，濱于碣石。古齊、紀、祝、淳於、萊、譚、寒及斟尋、有過、有鬲、蒲姑氏之國，其地得陬訾之下流，自濟東達於河外，故其象著為天津，絕雲漢之陽。凡司人之星與群臣之錄，皆主虛、危，故岱宗為十二諸侯受命府。又下流得婺女，當九河末派，比于星紀，與吳、越同占。

Maid, Emptiness, and Rooftop constitute Xuanxiao. At the beginning: five degrees into Maid, with a remainder of 2374 and four-odd seconds. At the middle: nine degrees into Emptiness. At the end: twelve degrees into Rooftop. Its territorial division runs from Jibei eastward across the Ji River, through Pingyin to Shanzhaung, along the northern slopes of Mount Tai and its surrounding peaks, southeast to Gaomi, then east to the lands of the Lai barbarians — comprising Han Beihai, Qiancheng, Zichuan, Jinan, Jijun, and the country south of the old courses of Pingyuan, Bohai, and the Nine Rivers, reaching the shore at Jieshi. The ancient states of Qi, Ji, Zhu, Chunyu, Lai, Tan, Han, Zhenxun, Youguo, Youge, and the Pugu clan occupied the lower reaches of Zouzi, from east of Ji to beyond the river; hence their celestial image appears as the Celestial Ford, standing on the southern side of the Cloud River. The stars governing the director of men and the register of ministers all preside over Emptiness and Rooftop; hence Mount Tai is the hall where the twelve feudal lords receive their mandates. Further downstream lies Maid, at the final branch of the Nine Rivers — comparable to Xingji and sharing the same omen with Wu and Yue.

營室、東壁，陬訾也。初，危十三度，餘二千九百二十六，秒一太。中，營室十二度。終，奎一度。自王屋、太行而東，得漢河內，至北紀之東隅，北負漳、鄴，東及館陶、聊城。又自河、濟之交，涉滎波，濱濟水而東，得東郡之地，古邶、庸阝、衛、凡、胙、邗、雍、共、微、觀、南燕、昆吾、豕韋之國。自閣道、王良至東壁，在豕韋，為上流。當河內及漳、鄴之南，得山河之會，為離宮。又循河、濟而東接玄枵為營室之分。

Encampment and Eastern Wall constitute Zouzi. At the beginning: thirteen degrees into Rooftop, with a remainder of 2926 and one-odd second. At the middle: twelve degrees into Encampment. At the end: one degree into Kui. From Wangwu and the Taihang eastward through Han Henei to the eastern corner of the Northern Thread, north facing Zhang and Ye, east to Guantao and Liaocheng. Further, from the confluence of the Yellow and Ji Rivers, crossing Yingbo and following the Ji River eastward, one reaches the lands of Dongjun — the ancient states of Bei, Yong, Wei, Fan, Zuo, Han, Yong, Gong, Wei, Guan, Nanyan, Kunwu, and Shiwei. From the Celestial Passage and Wangliang to Eastern Wall, within Shiwei, marks the upper reach. Where Henei and the region south of Zhang and Ye meet at the confluence of mountains and rivers lies the detached palace. Further, following the Yellow and Ji Rivers eastward to join Xuanxiao constitutes the allotment of Encampment.

奎、數，降婁也。初，奎二度，餘千二百一十七，秒十七少。中，婁一度。終，胃三度。自蛇丘、肥成，南屆钜野，東達梁父，循岱嶽眾山之陽，以負東海。又濱泗水，經方與、沛、留、彭城，東至於呂梁，乃東南抵淮，並淮水而東，盡徐夷之地，得漢東平、魯國、琅邪、東海、泗水、城陽，古魯、薛、邾、莒、小邾、徐、郯、鄫、鄅、邳、邿、任、宿、須句、顓臾、牟、遂、鑄夷、介、根牟及大庭氏之國。奎為大澤，在陬訾下流，當钜野之東陽，至於淮、泗。婁、胃之墟，東北負山，蓋中國膏腴地，百穀之所阜也。胃得馬牧之氣，與冀之北土同占。

Kui and Lou constitute Jianglou. At the beginning: two degrees into Kui, with a remainder of 1217 and seventeen-odd seconds. At the middle: one degree into Lou. At the end: three degrees into Stomach. From Sheqiu and Feicheng south to Juye and east to Liangfu, along the southern slopes of Mount Tai and its surrounding peaks, facing the Eastern Sea. Further, following the Si River through Fangyu, Pei, Liu, and Pengcheng east to Lüliang, then southeast to the Huai and along the Huai eastward through the lands of the Xu barbarians — comprising Han Dongping, Lu, Langye, Donghai, Sishui, and Chengyang, and the ancient states of Lu, Xue, Zhu, Ju, Xiao Zhu, Xu, Tan, Zeng, Yu, Pi, Shi, Ren, Su, Xuju, Zhuanyu, Mou, Sui, Zhuyi, Jie, Genmou, and the Datin clan. Kui is the Great Marsh, in the lower reaches of Zouzi, at the eastern yang of Juye, reaching the Huai and Si. The domain of Lou and Stomach, backed by mountains to the northeast, is the fertile heartland of the central domain, where the hundred grains flourish. Stomach receives the qi of horse pasture and shares the same omen with the northern lands of Ji.

胃、昴、畢，大樑也。初，胃四度，餘二千五百四十九，秒八太。中，昴六度。終，畢九度。自魏郡濁漳之北，得漢趙國、廣平、钜鹿、常山，東及清河、信都，北據中山、真定，全趙之分。又北逾眾山，盡代郡、雁門、雲中、定襄之地與北方群狄之國。北紀之東陽，表裏山河，以蕃屏中國，為畢分。循北河之表，西盡塞垣，皆髦頭故地，為昴分。冀之北土，馬牧之所蕃庶，故天苑之象存焉。

Stomach, Hairy Head, and Net constitute Daliang. At the beginning: four degrees into Stomach, with a remainder of 2549 and eight-odd seconds. At the middle: six degrees into Hairy Head. At the end: nine degrees into Net. From north of the Zhuo River in Wei Commandery through Han Zhao, Guangping, Julu, and Changshan, east to Qinghe and Xindu, north holding Zhongshan and Zhending — the full allotment of Zhao. Further north, crossing the mountain ranges, one reaches the lands of Dai, Yanmen, Yunzhong, and Dingxiang and the barbarian states of the northern frontier. The eastern yang of the Northern Thread, with mountains and rivers forming inner and outer barriers, screens the central domain — this is the allotment of Net. Following the outer side of the Northern River west to the frontier ramparts — all the former lands of Maotou — constitutes the allotment of Hairy Head. The northern lands of Ji, where horse pasture flourishes, preserve the image of the Celestial Park.

觜觿、參、伐，實沈也。初，畢十度，餘八百四十一，秒四之一。中，參七度。終，東井十一度。自漢之河東及上黨、太原，盡西河之地，古晉、魏、虞、唐、耿、楊、霍、冀、黎、郇與西河戎狄之國。西河之濱，所以設險限秦、晉，故其地上應天闕。其南曲之陰，在晉地，眾山之陽；南曲之陽，在秦地，眾山之陰。陰陽之氣並，故與東井通。河東永樂、芮城、河北縣及河曲豐、勝、夏州，皆東井之分。參、伐為戎索，為武政，當河東，盡大夏之墟。上党次居下流，與趙、魏接，為觜觿之分。

Beak, Three Stars, and Attack constitute Shenshen. At the beginning: ten degrees into Net, with a remainder of 841 and four seconds minus one quarter. At the middle: seven degrees into Three Stars. At the end: eleven degrees into Well. From Han Hedong through Shangdang and Taiyuan to the lands of the Western River — the ancient states of Jin, Wei, Yu, Tang, Geng, Yang, Huo, Ji, Li, and Xun, together with the Rong and Di of the Western River. The banks of the Western River, where natural barriers were set to divide Qin and Jin, correspond on high to the Celestial Gate. The yin of its southern bend lies in Jin territory, on the sunny side of the massed peaks; The yang of its southern bend lies in Qin territory, on the shaded side of the massed peaks. Yin and yang qi converge there, hence it connects with the Well. Yongle, Ruicheng, and Hebei County in Hedong, together with Feng, Sheng, and Xiazhou in the River Bend, all fall within the allotment of the Well. Three Stars and Attack form the Rong cord and govern martial affairs; facing Hedong, they encompass the domain of Great Xia. Shangdang next lies in the lower reach, bordering Zhao and Wei — the allotment of Beak.

東井、輿鬼，鶉首也。初，東井十二度，餘二千一百七十二，秒十五太。中，東井二十七度。終，柳六度。自漢三輔及北地、上郡、安定，西自隴坻至河右，西南盡巴、蜀、漢中之地，及西南夷犍為、越雋、益州郡，極南河之表，東至牂柯，古秦、梁、豳、芮、豐、畢、駘杠、有扈、密須、庸、蜀、羌、髳之國。東井居兩河之陰，自山河上流，當地絡之西北。輿鬼居兩河之陽，自漢中東盡華陽，與鶉火相接，當地絡之東南。鶉首之外，雲漢潛流而未達，故狼星在江、河上源之西，弧矢、犬、雞皆徼外之備也。西羌、吐蕃、吐谷渾及西南徼外夷人，皆占狼星。

Well and Chariot Ghost constitute Chunshou. At the beginning: twelve degrees into Well, with a remainder of 2172 and fifteen-odd seconds. At the middle: twenty-seven degrees into Well. At the end: six degrees into Willow. From Han's Three Adjuncts through Beidi, Shangjun, and Anding, west from Longqi to the right bank of the Yellow River, southwest through Ba, Shu, and Hanzhong, and the southwestern Yi of Jianwei, Yuexi, and Yizhou Commandery, to the far south beyond the Southern River and east to Zangke — the ancient states of Qin, Liang, Bin, Rui, Feng, Bi, Taigang, Youhu, Mixu, Yong, Shu, Qiang, and Mao. The Well lies on the northern side of the Two Rivers, at the upper reaches of mountains and rivers, corresponding to the northwest of the terrestrial thread. The Chariot Ghost lies on the southern side of the Two Rivers, from Hanzhong east to Huayang, connecting with Chunhuo and corresponding to the southeast of the terrestrial thread. Beyond Chunshou, the Cloud River flows hidden and has not yet reached its terminus; hence the Wolf Star lies west of the upper sources of the Yangtze and Yellow Rivers, while Bow and Arrow, Dog, and Cock all serve as frontier defenses beyond the border. The Western Qiang, Tubo, Tuyuhun, and the southwestern barbarians beyond the frontier all take their omen from the Wolf Star.

柳、七星、張，鶉火也。初，柳七度，餘四百六十四，秒七少。中，七星七度。終，張十四度。北自滎澤、滎陽，並京、索，暨山南，得新鄭、密縣，至外方東隅，斜至方城，抵桐柏，北自宛、葉，南暨漢東，盡漢南陽之地。又自雒邑負北河之南，西及函谷，逾南紀，達武當、漢水之陰，盡弘農郡，以淮源、桐柏、東陽為限，而申州屬壽星，古成周、虢、鄭、管、鄶、東虢、密、滑、焦、唐、隨、申、鄧及祝融氏之都。新鄭為軒轅、祝融之墟，其東鄙則入壽星。柳。在輿鬼東，又接漢源，當益、洛之陽，接南河上流。七星系軒轅，得土行正位，中嶽象也，河南之分。張，直南陽，漢東，與鶉尾同占。

Willow, Seven Stars, and Extended Net constitute Chunhuo. At the beginning: seven degrees into Willow, with a remainder of 464 and seven-odd seconds. At the middle: seven degrees into Seven Stars. At the end: fourteen degrees into Extended Net. North from Xingze and Xingyang, along Jing and Suo to the south of the mountains, through Xinzheng and Mi County to the eastern corner of Waifang, slanting to Fangcheng and Tongbai, north from Wan and Ye and south to the east of Han — encompassing all of Han Nanyang. Further, from Luoyi facing the south of the Northern River, west to Hangu Pass, crossing the Southern Thread to the yin side of Wudang and the Han River, encompassing Hongnong Commandery and bounded by Huaiyuan, Tongbai, and Dongyang — while Shenzhou belongs to Longevity Star — lie the ancient Chengzhou, Guo, Zheng, Guan, Kuai, Eastern Guo, Mi, Hua, Jiao, Tang, Sui, Shen, Deng, and the capital of the Zhurong clan. Xinzheng is the domain of Xuanyuan and Zhurong; its eastern border falls within Longevity Star. Willow. East of the Chariot Ghost, connecting with the Han source, facing the sunny side of Yi and Luo and joining the upper reach of the Southern River. Seven Stars is tied to Xuanyuan and holds the orthodox position of the Earth phase; it is the image of the Central Peak — the allotment of Henan. Extended Net faces Nanyang and the east of Han, sharing the same omen with Chunwei.

翼、軫，鶉尾也。初，張十五度，餘千七百九十五，秒二十二太。中，翼十二度。終，軫九度。自房陵、白帝而東，盡漢之南郡、江夏，東達廬江南部，濱彭蠡之西，得長沙、武陵，又逾南紀，盡郁林、合浦之地，自沅、湘上流，西達黔安之左，皆全楚之分。自富、昭、象、龔、繡、容、白、廉州已西，亦鶉尾之墟。古荊楚、鄖、鄀、羅、權、巴、夔與南方蠻貊之國。翼與咮張同象，當南河之北，軫在天關之外，當南河之南，其中一星主長沙，逾嶺徼而南，為東甌、青丘之分。安南諸州在雲漢上源之東陽，宜屬鶉火。而柳、七星、張皆當中州，不得連負海之地，故麗於鶉尾。

Wings and Chariot Crossboard constitute Chunwei. At the beginning: fifteen degrees into Extended Net, with a remainder of 1795 and twenty-two-odd seconds. At the middle: twelve degrees into Wings. At the end: nine degrees into Chariot Crossboard. From Fangling and Baidi eastward through Han Nanjun and Jiangxia to the southern part of Lujiang, along the west of Poyang, comprising Changsha and Wuling, further crossing the Southern Thread to the lands of Yulin and Hepu, from the upper reaches of Yuan and Xiang west to the left of Qian'an — the full allotment of Chu. From Fu, Zhao, Xiang, Gong, Xiu, Rong, Bai, and Lian prefectures westward also lies the domain of Chunwei. The ancient Jing-Chu, Yun, Ruo, Luo, Quan, Ba, and Kui, together with the southern barbarian and Mo states. Wings shares the same omen with Beak and Extended Net, facing the north of the Southern River; Chariot Crossboard lies beyond the Celestial Pass, facing the south of the Southern River; one star among them governs Changsha, and south beyond the mountain frontier lies the allotment of Dong'ou and Qingqiu. The prefectures of Annan lie in the eastern yang of the upper source of the Cloud River and ought to belong to Chunhuo. Yet Willow, Seven Stars, and Extended Net all face the central domain and cannot connect to sea-facing lands; hence they are attached to Chunwei.

角、亢，壽星也。初，軫十度，餘八十七，秒十四少。中，角八度。終，氐一度。自原武、管城，濱河、濟之南，東至封丘、陳留，盡陳、蔡、汝南之地，逾淮源，至於弋陽，西涉南陽郡至於桐柏，又東北抵嵩之東陽，中國地絡在南北河之間，首自西傾，極於陪尾，故隨、申，光皆豫州之分，宜屬鶉火，古陳、蔡、許、息、江、黃、道、柏、沈、賴、蓼、須頓、胡、防、弦、厲之國。氐涉壽星，當洛邑眾山之東，與亳土相接，次南直潁水之間，曰太昊之墟，為亢分。又南涉淮氣連鶉尾，在成周之東陽，為角分。

Horn and Gullet constitute Longevity Star. At the beginning: ten degrees into Chariot Crossboard, with a remainder of 87 and fourteen-odd seconds. At the middle: eight degrees into Horn. At the end: one degree into Root. From Yuanwu and Guancheng along the south bank of the Yellow River and Ji, east to Fengqiu and Chenliu, through Chen, Cai, and Runan, past the Huai's source to Yiyang, west into Nanyang as far as Tongbai, then northeast to the eastern flank of Mount Song — China's terrestrial network runs between the Northern and Southern Rivers, from Western Slope to Companion Tail. Sui, Shen, and Guang all belong to Yuzhou and should be assigned to Vermillion Fire: the ancient lands of Chen, Cai, Xu, Xi, Jiang, Huang, Dao, Bai, Shen, Lai, Liao, Xudun, Hu, Fang, Xian, and Li. Root overlaps Longevity Star, east of the Luoyang mountains and adjoining Bo territory; farther south, along the Ying River, lies Taihao's domain — the allotment of Gullet. Still farther south it crosses the Huai's influence, linking with Vermillion Tail at the eastern flank of Chengzhou — the allotment of Horn.

氐、房、心，大火也。初，氐二度，餘千四百一十九，秒五太。中，房二度。終，尾六度。自雍丘、襄邑、小黃而東，循濟陰，界于齊、魯，右泗水，達于呂梁，乃東南接太昊之墟，盡漢濟陰、山陽、楚國、豐、沛之地，古宋、曹、郕、滕、茅、郜、蕭、葛、向城、逼陽、申父之國。商、亳負北河，陽氣之所升也，為心分、豐、沛負南河，陽氣之所布也，為房分。其下流與尾同占，西接陳、鄭為氐分。

Root, Chamber, and Heart constitute Great Fire. At the beginning: two degrees into Root, with a remainder of 1419 and five-odd seconds. At the middle: two degrees into Chamber. At the end: six degrees into Tail. From Yongqiu, Xiangyi, and Xiaohuang east along Jiyin, between Qi and Lu, with the Si to the right as far as Liang, then southeast to Taihao's domain through Han Jiyin, Shanyang, the Chu kingdom, Feng, and Pei — ancient Song, Cao, Cheng, Teng, Mao, Gao, Xiao, Ge, Xiangcheng, Biyang, and Shenfu. Shang and Bo lie along the Northern River where yang qi rises — Heart's allotment; Feng and Pei along the Southern River where yang qi spreads — Chamber's allotment. Its lower course shares omens with Tail; to the west it borders Chen and Zheng — Root's allotment.

尾、箕，析木津也。初，尾七度，餘二千七百五十，秒二十一少，中，箕五度，終，南斗八度。自渤海、九河之北，得漢河間、涿郡、廣陽及上谷、漁陽、右北平、遼西、遼東、樂浪、玄菟，古北燕、孤竹、無終、九夷之國。尾得雲漢之末派，龜、魚麗焉，當九河之下流，濱於渤碣，皆北紀之所窮也。箕與南斗相近，為遼水之陽，盡朝鮮三韓之地，在吳、越東。

Tail and Winnowing Basket form the Ford of Split Wood. At the beginning: seven degrees into Tail, remainder 2750 and twenty-one-odd seconds; at the middle: five degrees into Winnowing Basket; at the end: eight degrees into Southern Dipper. North of Bohai and the Nine Rivers lie Han Hejian, Zhuo, Guangyang, Shanggu, Yuyang, Youbeiping, Liaoxi, Liaodong, Lelang, and Xuantu — the lands of ancient Northern Yan, Guzhu, Wuzhong, and the Nine Yi. Tail marks the Cloud River's last branch, rich in turtles and fish, at the lower Nine Rivers along Bohai and Jieshi — the farthest reach of the Northern Thread. Winnowing Basket lies near Southern Dipper, on the north bank of the Liao, through Korea and the Three Han, east of Wu and Yue.

南斗、牽牛，星紀也。初，南斗九度，餘千四十二，秒十二太。中，南斗二十四度。終，女四度。自廬江、九江，負淮水，南盡臨淮、廣陵，至於東海，又逾南河，得漢丹楊、會稽、豫章，西濱彭蠡，南涉越門，迄蒼梧、南海，逾嶺表，自韶、廣以西，珠崖以東，為星紀之分也。古吳、越、群舒、廬、桐、六、蓼及東南百越之國。南斗在雲漢下流，當淮、海間，為吳分。牽牛去南河浸遠，自豫章迄會稽，南逾嶺徼，為越分。島夷蠻貊之人，聲教所不暨，皆系于狗國云。

Southern Dipper and Lead Ox form Star Chronicle. At the beginning: nine degrees into Southern Dipper, remainder 1042 and twelve-odd seconds. At the middle: twenty-four degrees into Southern Dipper. At the end: four degrees into Maid. From Lujiang and Jiujiang along the Huai, south through Linhuai and Guangling to the Eastern Sea, then across the Southern River through Danyang, Kuaiji, and Yuzhang, west along Poyang, south through Yuemen to Cangwu and Nanhai, beyond the mountain ranges — from Shao and Guang west to Zhuya east — all belong to Star Chronicle. Ancient Wu, Yue, the Shu states, Lu, Tong, Liu, Liao, and the Hundred Yue of the southeast. Southern Dipper stands in the Cloud River's lower reach between the Huai and the sea — Wu's allotment. Lead Ox lies far from the Southern River; from Yuzhang to Kuaiji, south beyond the frontier passes — Yue's allotment. Island peoples and the Man and Mo tribes beyond the reach of civilization are all assigned to the Dog Kingdom — or so the tradition runs.

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