1
志第八曆二
Treatise Eight: Calendrics, Part Two.
2
▲大統曆法一上法原
▲ Grand Concordance Calendar Method, Part One (Upper): Origins of the Method.
3
造曆者各有本原，史宜備錄，使後世有以考。 如《太初》之起數鐘律，《大衍》之造端蓍策，皆詳本志。 《授時曆》以測算術爲宗，惟求合天，不牽合律呂、卦爻。 然其法所以立，數之所從出，以及晷影、星度，皆有全書。 郭守敬、齊履謙傳中，有書名可考。 《元史》漫無採摭，僅存李謙之《議祿》、《曆經》之初稿。 其後改三應率及立成之數，與夫割圓弧矢之法，平立定三差之原，盡削不載。 使作者精意湮沒，識者憾焉。 今據《大統因通軌》及《歷草》諸書，稍爲編次，首法原，次立成，次推步。 而法原之目七：曰句股測望，曰弧矢割圓，曰黃赤道內外度，曰白道交周，曰日月五星平立定三差，曰裏差刻漏。
Every calendar rests on its own foundations, and the dynastic history ought record them in full so later generations can investigate. The Taichu calendar's grounding in pitch-pipes and musical regulation and the Dayan's in yarrow-casting are both set forth at length in their respective treatises. The Season Granting Calendar takes observational computation as its foundation, seeking only agreement with the heavens and not forcing alignment with pitch-pipes or hexagram lines. Yet the rationale of its methods, the derivation of its constants, and the full treatment of gnomon shadows and stellar degrees all survive in complete texts. Titles of those works can still be traced in the biographies of Guo Shoujing and Qi Lüqian. The Yuan History excerpts almost nothing; only Li Qian's draft Discourse on Official Emoluments and Calendar Classic remains. Later revisions to the three response rates and tabular constants, the arc-sagitta circle-division methods, and the foundations of the mean, upright, and fixed three differences were all excised and omitted. The compilers' careful thought was thereby lost—a source of regret for those who understand the matter. The present account draws on the Grand Concordance Comprehensive Track, the Draft Calendar, and related works, arranged as follows: first the method's origins, then the tabular constants, then stepwise computation. The origins of the method comprise seven topics: gnomonic surveying; arc-sagitta circle division; ecliptic and equatorial inner and outer degrees; the white path's crossing circuit; the mean, upright, and fixed three differences for the sun, moon, and five planets; and terrestrial parallax with clepsydra graduations.
4
▲句股測望
▲ Gnomonic Surveying.
5
北京立四丈表，冬至日午正，測得景辰七丈九尺八寸五分。 隨以簡儀測到太陽南至地平二十六度四十六分半，爲半弧背。 求得矢度，五度九十一分半。 置周天半徑，截矢餘五十四度九十六分爲股，乃本地支戴日下之度。 以弦股別句術，求得句二十六度一下七分六十六秒，爲日出地半弧弦。
At Beijing a four-zhang gnomon was set up; at true noon on the winter solstice its shadow measured seven zhang, nine chi, eight cun, and five fen. The simplified armillary then gave the Sun's south declination above the horizon as 26°46′30″, taken as the half-arc back. The sagitta came to 5°91′30″. Take the celestial half-diameter and subtract the sagitta; the remainder, 54°96′, is the leg—the local colatitude under the subsolar point. Applying the chord–leg separation to obtain the other leg yielded 26°07′66″ as the half-arc chord of the Sun's altitude above the horizon.
6
北京立四丈表，夏至日午正，測得景長一丈一尺七寸一分。 隨以簡儀測到太陽南至地平七十四度二十六分半，爲半弧背。 求得矢度，四十三度七十四分少。 置周天半徑，截矢餘一十七度一十三分二十五秒爲句，乃本地去戴日下之度。 以句弦別股術，求得股五十八度四十五分半，爲日出地半弧弦。
At Beijing a four-zhang gnomon was set up; at true noon on the summer solstice its shadow measured one zhang, one chi, seven cun, and one fen. The simplified armillary then gave the Sun's south declination above the horizon as 74°26′30″, taken as the half-arc back. The sagitta came to 43°74′, slightly less. Take the celestial half-diameter and subtract the sagitta; the remainder, 17°13′25″, is the leg—the local distance from the subsolar point. Applying the leg–chord separation to obtain the base leg yielded 58°45′30″ as the half-arc chord of the Sun's altitude above the horizon.
7
以二至日度相併，得一百度七十三分，折半得五十度三十六分半，爲北京赤道出地度。 以赤道出地度轉減周天四之一，餘四十度九十四分九十三秒七十五微，爲北京北極出地度。
Adding the two solstice altitudes gives 100°73′; halving yields 50°36′30″ as Beijing's equatorial altitude above the horizon. Subtracting the equatorial altitude from a quadrant of the celestial circuit leaves 40°94′93″75 as Beijing's polar altitude above the horizon.
8
▲弧矢割圓
▲ Arc-Sagitta Circle Division.
9
周天經一百二十一度七十五分少。 少不用。 半徑六十零度八十七分半。 又爲黃赤道大弦。 二至黃赤道內外半弧背二十四度。 所測就整。 二至黃赤道弧矢四度八十四分十二秒。 黃赤道大句二十三度八十分七十秒。 黃赤道大股五十六度零二分六十八秒。 半徑內減去矢度之數。
The celestial diameter is fixed at 121°75′, slightly less. The 'slightly less' fraction is dropped in use. The half-diameter is 60°87′30″. This value also serves as the great chord linking ecliptic and equator. At the solstices the inner and outer half-arc backs on the ecliptic and equator are 24°. Observed values are used as rounded integers. At the solstices the ecliptic–equatorial arc sagitta is 4°84′12″. The great altitude leg of the ecliptic–equatorial triangle is 23°80′70″. The great base leg of the ecliptic–equatorial triangle is 56°02′68″. This is the half-diameter minus the sagitta.
10
割圓求矢術置半弧度自之，爲半弧背幕，周天徑自之，爲上廉。 上廉乘半弧背幕，爲正實。 上廉乘徑，爲益從方。 半弧背倍之，乘徑，爲下廉。 以初商乘上廉，得數以減益從方，餘爲從方。 置初商自之以下廉，餘以初商乘之，爲從廉。 從方、從廉相併，爲下法。 下法乘初商，以減正實，實不足減，改初商。 實有不盡，次第商除之。 倍初商數，與次商相併以乘上廉，得數以減益從方，餘爲從方。 並初商次商而自之，又以初商自之，並二數以減下廉，餘以初商倍數並次商乘之，爲從廉。 從方、從廉相併，爲下法。 下法乘次商，以減餘實，而定次商。 有不盡者，如法商之，皆以商得數爲矢度之數。 黃赤道同用。
Method for finding the sagitta by circle division: square the half-arc back to obtain the half-arc-back power; square the celestial diameter to obtain the upper excess. Multiply the upper excess by the half-arc-back power to form the positive product. Multiply the upper excess by the diameter to form the augmenting companion square. Double the half-arc back and multiply by the diameter to form the lower excess. Multiply the first trial quotient by the upper excess, subtract from the augmenting companion square, and take the remainder as the companion square. Square the first trial quotient, subtract from the lower excess, and multiply the remainder by the first trial quotient to obtain the companion excess. Add the companion square and companion excess to form the lower divisor. Multiply the lower divisor by the first trial quotient and subtract from the positive product; if the product is insufficient, adjust the first quotient. If a remainder remains, continue successive quotient divisions. Double the first quotient, add the second quotient, multiply by the upper excess, subtract from the augmenting companion square, and take the remainder as the companion square. Square the sum of the first and second quotients and also square the first quotient alone; subtract both from the lower excess; multiply the remainder by twice the first quotient plus the second quotient to obtain the companion excess. Add the companion square and companion excess to form the lower divisor. Multiply the lower divisor by the second quotient, subtract from the remaining product, and thereby fix the second quotient. If a remainder still remains, continue dividing by the same procedure; each quotient obtained contributes to the sagitta in degrees. The ecliptic and equator employ the same procedure.
11
如以半弧背一度求矢。 術曰：置半弧背一度自之，得一度，爲半弧幕。 置周天徑一百二十一度太自之，得一萬四千八百二十三度零六分二十五秒，爲上廉。 上廉乘半弧背幕，得一萬四千八百二十三度零六分二五，爲正實。 上廉又乘徑，得一百八十零萬四千七百零七度八十五分九十三秒七五，爲益從方。 半弧背一度倍之，得二度，以乘徑得二百四十三度五十分，爲下廉。 初商八十秒。 置初商八十秒乘上廉一萬四千八百二十三度零六二五，得一百一十八度五八四五，以減益從方一百八十零萬四千七百零七度八五九三七五，餘一百八十零萬四千五百八十九度二七四八七五，爲從方。 又置初商八十秒自之，得六十四微，以減下廉餘二百四十三度四九九三六。 仍以八十秒乘之，得一度九四七九九九四八八，爲從廉。 以從廉、從方並之，共得一百八十零萬四千五百九十一度二二二八七四四八八，爲下法。 下法乘初商，得一萬四千四百三十六度七十二分九七八二九九五九零四，以減正實，餘實三百八十六度三十三分二七一七零零四零九六。 次商二秒。 置初商八十秒倍之，得一分六十秒。 加次商二委六十二秒，乘上廉一萬四千八百二十三度零六二五，得二百四十零度一三三六一二五，以減益從方，餘一百八十零萬四千四百六十七二五七六二五，爲從方。 又置初次商八十二秒自之，得六十七微。 加初商八十秒自之之數，得一秒三十一微，以減下廉，餘二百四十三度四九九八六九。 以前所得一分六十二秒乘之，得三度九十四分四六九七八七七八，爲從廉。 以從廉、從方並，得一百八十零萬四千四百七十一度六十七分零四六零三七八，爲下法。 下法乘次商，得三百六十零度八九四三三四零九二零七五五六，以減餘實，仍餘二十五度四三八三八二九一二零二零四四。 不足一秒葉不用，下同。
Example: to find the sagitta for a half-arc back of one degree. The procedure runs: square a half-arc back of one degree to obtain 1° as the half-arc power. Square the celestial diameter 121° slightly more to obtain 14,823°06′25″ as the upper excess. Multiply the upper excess by the half-arc power to obtain 14,823°06′25″ as the positive product. Multiply the upper excess again by the diameter to obtain 1,804,707°85′93″75 as the augmenting companion square. Double the one-degree half-arc back to obtain 2°; multiply by the diameter to obtain 243°50′ as the lower excess. First trial quotient: 80″. Multiply the first quotient 80″ by the upper excess 14,823°0625 to obtain 118°5845; subtract from the augmenting companion square 1,804,707°859375; the remainder 1,804,589°274875 is the companion square. Square the first quotient 80″ to obtain 64 micro-units; subtract from the lower-excess remainder 243°49936. Multiply again by 80″ to obtain 1°947999488 as the companion excess. Add companion excess and companion square to obtain 1,804,591°222874488 as the lower divisor. Multiply the lower divisor by the first quotient to obtain 14,436°729782995904; subtract from the positive product; the remainder is 386°332717004096. Second trial quotient: 2″. Double the first quotient 80″ to obtain 1′60″. Add the second quotient 2″ to obtain 82″; multiply by the upper excess 14,823°0625 to obtain 240°1336125; subtract from the augmenting companion square; the remainder 1,804,467°257625 is the companion square. Square the combined first and second quotient 82″ to obtain 67 micro-units. Add the square of the first quotient alone (80″) to obtain 1″31μ; subtract from the lower excess; the remainder is 243°499869. Multiply by the 1′62″ obtained above to get 3°9446978778 as the companion excess. Add companion excess and companion square to obtain 1,804,471°670460378 as the lower divisor. Multiply the lower divisor by the second quotient to obtain 360°89433409207556; subtract from the remaining product; the remainder is 25°43838291202044. Fractions below one second are discarded; the same rule applies below.
12
凡求得矢度八十二秒，餘度各如上法，求到矢度，以爲黃赤相求及其內外度之根。 數詳後。
In all, the sagitta obtained is 82″; for other arc lengths apply the same procedure to obtain the sagitta—the foundation for converting between ecliptic and equator and for their inner and outer degrees. The full tables appear below.
13
▲黃赤道差
▲ Ecliptic–Equatorial Difference.
14
求黃赤道各度下赤道積度術。 置周天半徑內減去黃道矢度，餘爲黃赤道小弦。 置黃赤道小弦，以黃赤道大股乘之大股見割圓爲實。 黃赤道大弦半徑爲法。 實如法而一，爲黃赤道小股。 直黃道矢自乘爲實，以周天全徑爲法，實如法而一，爲黃道半背弦差。 以差去減黃赤道積度，即黃道半弧背。 餘爲黃道半弧弦。 置黃赤道半弧弦自之爲股幕，黃赤道小股自之爲句幕，二幕並之，以開平方法除之，爲赤道小弦。 置黃赤道半弧弦，以周天半徑亦爲赤道大弦乘之爲實，以赤道小弦爲法而一，爲赤道半弧弦。 置黃赤道小股，亦爲赤道橫小句以赤道大弦即半徑乘之爲實，以赤道小弦爲法而一，爲赤道橫大句，以減半徑，餘爲赤道磺弧矢。 橫弧矢自之爲實，以全徑爲法而一，爲赤道半背弦差。 以差加赤道半弧，爲赤道積度。
Method for finding the accumulated equatorial degree corresponding to each ecliptic degree. Take the celestial half-diameter and subtract the ecliptic sagitta; the remainder is the small ecliptic–equatorial chord. Set the small chord and multiply by the great base leg of the ecliptic–equatorial triangle (given under circle division) to form the product. Use the great chord—that is, the half-diameter—as divisor. Divide the product by the divisor to obtain the small base leg of the ecliptic–equatorial triangle. Square the ecliptic sagitta for the product, divide by the full celestial diameter, and obtain the ecliptic half-back chord correction. Subtract this correction from the accumulated ecliptic degree to obtain the ecliptic half-arc back. The remainder is the ecliptic half-arc chord. Square the ecliptic half-arc chord for the leg power and the small base leg for the altitude power; add the two powers and take the square root to obtain the small equatorial chord. Multiply the ecliptic half-arc chord by the half-diameter (the great equatorial chord) for the product and divide by the small equatorial chord to obtain the equatorial half-arc chord. Take the small base leg as the equatorial transverse small altitude, multiply by the half-diameter for the product, divide by the small equatorial chord to obtain the transverse great altitude; subtract from the half-diameter; the remainder is the equatorial transverse arc sagitta. Square the transverse arc sagitta, divide by the full diameter, and obtain the equatorial half-back chord correction. Add this correction to the equatorial half-arc to obtain the accumulated equatorial degree.
15
如黃道半弧背一度，求赤道積度。 術曰：「置半徑六十零度八十七分五十秒，即黃赤道大弦。 內減黃道矢八十二秒餘六十零度八六六八，爲黃赤道小弦。 置黃赤道小弦，以黃赤道大股五十六度零二六八乘之，得三千四百一十零度一七二零三零二四爲實，以黃赤道大弦六十零度八七五爲法，實如法而一，得五十六度零一分九十二秒，爲黃赤道小股。 又爲赤道小句。 置矢度八十二秒自之，得六十七微，以全徑一百二十一度七五爲法，除之得五十五纖，爲黃道平半背弦差。 置黃道半弧弦一度，內減黃道半背弦差，餘爲半弧弦，因因差在微以下不減，即用一度爲半弧弦。 置黃道半弧弦一度自之，得一度爲股幕。 黃赤道小股五十六度零一矣二自之，得三千一百三十八度一五零七六八六四爲句幕。 二幕並得三千一百三十九度一五零七六八六四爲弦實，平方開之，得五十六度零二八一，爲赤道小弦。 置黃道半弧弦一度，以半徑即赤道大弦乘之，得六十零度八七五爲實，以赤道小股五十六度零二八一爲法除之，得一度零八分六十五秒，爲赤道半弧弦。 置黃赤道小股五十六度零一九二，又爲赤道小句。 以赤道大弦半徑六十零度八七五乘之，得三千四百一十零度一六八八爲實，以赤道小弦爲法除之，得六十零度八十六分五十三秒，爲赤道橫大句。 置半徑六十零度八十七分五十秒，內減赤道大句六十零度八十六分五十三秒，餘九十七秒，爲赤道橫弧矢。 置赤道橫弧矢九十七秒自之，得九十四微零九，以全徑爲法除之，得七十纖，爲赤道背弦差。 置赤道半弧弦一度零八分六十五秒，加赤道背弦差，爲赤道積度，今差在微已下不加，即用半弧弦爲積度。
Example: for an ecliptic half-arc back of one degree, find the accumulated equatorial degree. The procedure runs: "Set the half-diameter at 60°87′50″—this is the great ecliptic–equatorial chord. Subtract the ecliptic sagitta 82″; the remainder 60°8668 is the small ecliptic–equatorial chord. Multiply the small chord by the great base leg 56°0268 to obtain 3410°17203024 as product; divide by the great chord 60°875; the quotient is 56°0192 as the small base leg. This value also serves as the small equatorial altitude leg. Square the sagitta 82″ to obtain 67 micro-units; divide by the full diameter 121°75 to obtain 55 fine units as the ecliptic half-back chord correction. Set the ecliptic half-arc chord at 1°; subtract the half-back chord correction; because the correction lies below the micro-unit it is not subtracted—take 1° directly as the half-arc chord. Square the ecliptic half-arc chord 1° to obtain 1° as the leg power. Square the small base leg 56°0192 to obtain 3138°15076864 as the altitude power. Add the two powers to obtain 3139°15076864 as the chord product; take the square root to obtain 56°0281 as the small equatorial chord. Multiply the ecliptic half-arc chord 1° by the half-diameter (the great equatorial chord) to obtain 60°875 as product; divide by the small equatorial chord 56°0281 to obtain 1°0865 as the equatorial half-arc chord. Set the small base leg at 56°0192; it also serves as the small equatorial altitude leg. Multiplying by the equatorial great-chord semidiameter of 60°87′50″ yields 3410.1688 as the dividend; dividing by the equatorial small chord gives 60°86′53″ as the equatorial transverse great leg. Take the semidiameter of 60°87′50″ and subtract the equatorial great leg of 60°86′53″; the remainder of 97″ is the equatorial transverse arc-versine. Square the equatorial transverse arc-versine of 97″ to get 94 wei 09; dividing by the full diameter yields 70 xian as the equatorial back-chord correction. Add the equatorial back-chord correction to the equatorial half-arc chord of 1°08′65″ to obtain the equatorial accumulated degree; since the correction falls below the wei, it is not added—the half-arc chord itself serves as the accumulated degree.
16
凡求得赤道積度一度零八分六十五秒。 餘度各如上法，求到各黃道度下赤道積，兩數相減，即得黃赤道差，乃至後之率。 其分後，以赤道度求黃道，反此求之，其數並同。
The equatorial accumulated degree thus obtained is 1°08′65″. For each remaining degree, repeat the procedure to find the equatorial accumulation under each ecliptic degree; the difference between the two values is the ecliptic–equator correction, carried through to the later rate tables. For the fractional parts, to derive ecliptic from equatorial degrees, invert the same procedure; the quantities are identical.
17
▲黃赤道相求弧矢諸率立成上
▲ Ecliptic–equator mutual arc-versine rates: ready-made tables (upper section)
18
表格略
[Table omitted.]
19
▲黃赤道相求弧矢諸率立成下
▲ Ecliptic–equator mutual arc-versine rates: ready-made tables (lower section)
20
表格略
[Table omitted.]
21
按郭敬創法五端，內一曰黃道差，此其根率也。 舊法以一百一度相減乘。 《授時》立術，以句股、弧矢、方圓、斜直所容，求其數差，合於渾象之理，視古爲密。 顧《至元曆經》所載略，又誤以黃道矢度爲積差，黃道矢差爲率，今正之。
Commentary: Guo Shoujing established five methods; ecliptic difference is one of them, and these are its fundamental rates. The older method relied on subtracting and multiplying by 101 degrees. The Season Granting calendar established its procedure through right-triangle, arc-versine, circle–square, and oblique–rectilinear relations to derive the numerical differences—conforming to the armillary model and finer than antiquity. Yet the Calendar Classic of the Zhiyuan reign records only an abridgment and also mistakenly treated ecliptic versine degrees as accumulated difference and ecliptic versine difference as rate—here corrected.
22
▲割圓弧矢圖
▲ Diagram: circle-cutting and arc-versine
23
凡渾圓中剖，則成平圓。 任割平圓之一分，成弧矢形，皆有弧背，有弧弦，有矢。 剖弧矢形而半之，則有半弧背，有半弧弦，有矢。 因弦矢句股形，以半弧弦爲句，矢減半徑之餘爲股，半徑爲弦。 句股內成小句股，則有小句、小股、小弦、而大小可互求，平側可互用，渾圓之理，斯爲密近。
Whenever a sphere is bisected through its center, a plane circle results. Cut any segment of the plane circle and an arc-versine figure is formed, each with arc-back, arc-chord, and versine. Bisect the arc-versine figure and one has half arc-back, half arc-chord, and versine. From the chord–versine right triangle: the half arc-chord is the leg, the semidiameter minus the versine is the base, and the semidiameter is the hypotenuse. Within the legs a smaller right triangle arises, with small leg, small base, and small hypotenuse, so large and small quantities and plane and lateral relations can be interconverted—the spherical principle is thereby rendered precise.
24
平者爲赤道，斜者爲黃道。 因二至黃道赤之距，生大句股。 因各度黃赤之距，生小句股。
The plane circle is the equator; the oblique circle is the ecliptic. From the ecliptic–equator separation at the two solstices arises the great right triangle. From the ecliptic–equator separation at each degree arises the small right triangle.
25
外大圓爲赤道。 從北極平視，則黃道在赤道內，有赤道各度，即各有其半弧弦，以生大名股。 又各有其相當之黃道半弧弦，以生小句股。 此二者皆可互求。
The outer great circle is the equator. Viewed from the north pole in the plane, the ecliptic lies inside the equator; each equatorial degree has its half arc-chord, yielding the great base. Each also has its corresponding ecliptic half arc-chord, yielding the small right triangle. The two can be derived from each other.
26
按舊史無圖，然表亦圖之屬也。 今句股割弧矢之法，實爲歷家測算之本。 非圖不明，因存其要者數端。
Commentary: Earlier histories carried no diagrams, yet ready-made tables are themselves a kind of diagram. The leg–hypotenuse method for cutting arcs and versines is in fact the foundation of calendrical computation. Without a diagram it cannot be made clear; therefore a few essentials are preserved here.
27
▲黃赤道內外度
▲ Ecliptic and equator: degrees within and without
28
推黃道各度，距赤道內外及去極遠近術。 置半徑內減去赤道小弦，餘爲赤道二絃差。 又爲黃赤道小弧矢，又爲內外矢，又爲股弦差。 置半徑內外減去黃道矢度，餘爲黃赤道小弦，以二至黃赤道內外半弧弦乘之爲實，以黃赤道大弦爲法，即半徑。 除之爲黃赤道小弧弦。 即黃赤道內外半弧弦，又爲黃赤道小句。 置黃赤道小弧矢自之，即赤道二絃差。 以全徑除之，爲半背弦差。 以差加黃赤道小弧弦爲黃赤道小弧半背，即黃赤道內外度。 置黃赤道內外度，視在盈初縮末限以加，在縮初盈天限以減，皆加減象限度，即各得太陽去北極度分。
Procedure for deriving, at each ecliptic degree, the distance within or without the equator and the distance from the pole. Take the semidiameter and subtract the equatorial small chord; the remainder is the equatorial double-chord difference. This value also serves as the ecliptic–equator small arc-versine, as the inner-and-outer versine, and as the base–hypotenuse difference. Subtract the ecliptic versine degree from the semidiameter (within or without as the case requires); the remainder is the ecliptic–equator small chord. Multiply it by the solstitial inner-and-outer half arc-chord as dividend and divide by the ecliptic–equator great chord (the semidiameter). The quotient is the ecliptic–equator small arc-chord. This is the ecliptic–equator inner-and-outer half arc-chord, and also the ecliptic–equator small leg. Square the ecliptic–equator small arc-versine to obtain the equatorial double-chord difference. Divide by the full diameter to obtain the half back-chord correction. Add the correction to the ecliptic–equator small arc-chord to obtain the small half arc-back—that is, the ecliptic–equator inner-and-outer degree. Take the ecliptic–equator inner-and-outer degree: add the quadrant limit where it falls within the waxing-initial and waning-final limit, subtract where within the waning-initial and waxing-final limit—in each case yielding the Sun's distance from the north pole in degrees and minutes.
29
如冬至後四十四度，求太陽去赤道內外及去極度。 術曰：「置半徑六十零度八十七分半，內減黃道四十四度下赤道小弦五十八度三十五分六十九秒，餘二度五十一分八十一秒，爲黃赤道小弧矢。 即內外矢。 置半徑六十零度八七五，內減黃道四十四度，矢一十六度五十六分八十二秒，餘四十四三十零分六十八秒，爲黃赤道小弦。 置黃赤道小弦，以二至黃赤道內外半弧弦二十三度七十一分乘之，得一千零五十零度五十一分四二三八爲實，以黃赤道大弦六十零度八七五爲法除之，得一十七度二十五分十九秒爲黃赤道小弧弦。 即內外半弧弦。 置黃赤道小弧矢二度五十一分八十一秒自之爲實，以全徑地百二十一度七十五分除之，得五分二十一秒爲背弦差，以差加黃赤道小弧弦一十七度二十五分六十九秒，得一十七度三十零分八十九秒，爲二至前後四十四度，太陽去赤道內外度。 置象限九十一度三十一分四十三秒七五，以內外度一十七度三零八九加之，得一百零八度六十二分三十二秒七五，爲冬至後四十四度太陽去北極度。
Example: after the winter solstice at 44°, find the Sun's distance within or without the equator and from the pole. The procedure says: "Set the semidiameter at 60°87′30″. Subtract the equatorial small chord at ecliptic 44° (58°35′69″); the remainder of 2°51′81″ is the ecliptic–equator small arc-versine. That is, the inner-and-outer versine. Set the semidiameter at 60°87′50″. Subtract the ecliptic versine at 44° (16°56′82″); the remainder of 44°30′68″ is the ecliptic–equator small chord. Multiply the ecliptic–equator small chord by the solstitial inner-and-outer half arc-chord of 23°71′ to obtain 1050°51′4238 as dividend; dividing by the ecliptic–equator great chord of 60°87′50″ yields 17°25′19″ as the ecliptic–equator small arc-chord. That is, the inner-and-outer half arc-chord. Square the ecliptic–equator small arc-versine of 2°51′81″ and divide by the full diameter of 121°75′ to obtain a back-chord correction of 5′21″; adding this to the small arc-chord of 17°25′69″ gives 17°30′89″ as the Sun's equatorial inner-and-outer distance at 44° from the solstice. Take the quadrant limit of 91°31′43″75 wei and add the inner-and-outer degree of 17°30′89″, obtaining 108°62′32″75 wei as the Sun's distance from the north pole at 44° after the winter solstice.
30
▲黃道每度去赤道內外及去北極立成
▲ Ready-made tables: for each ecliptic degree, distance within or without the equator and from the pole
31
表格略
[Table omitted.]
32
▲白道交周
▲ White path: nodal period
33
推白赤道正交，距黃赤道正交北極數。 術曰：「置實測白道出入黃道內外六度爲半徑弧弦，又爲大圖弧矢，又爲股弦差。 置半徑六十零度七五自之，得三千七百零五度七六五六二五，以矢六度而一，得六百一十七度六十三分爲股弦和，加矢六度，共六百二十三度六十三分爲大圓徑。 依法求得容闊五度七十分，又爲小句。 又以二至出入半弧弦二十三度七十一分爲大句。 以大句爲法，除大股五十六度零六分五十秒，得二度三十七分就整爲度差。 以度差乘小句，得小股一十三度四十七分八十二秒，爲容半長。 置半徑六十零度八七五爲大弦，以乘小句五度七十分爲實，以大句二十三度七十一分爲法除之，得一十四度六十三分爲小弦，又爲白赤道正交，距黃赤道正交半弧弦。 依法求行半弧背一十四度六十六分，爲白赤道正交距黃赤道正交極婁數。
Procedure for deriving the white-path equatorial node and its polar distance from the ecliptic–equator node. The procedure says: "Take the empirically measured white path's separation from the ecliptic at 6° as semidiameter arc-chord, also as great-circle arc-versine, and also as base–hypotenuse difference. Square the semidiameter of 60°75″ to obtain 3705°765625; dividing by the 6° versine yields 617°63′ as the base–hypotenuse sum; adding the 6° versine gives 623°63′ as the great-circle diameter. By the prescribed method the contained breadth of 5°70′ is obtained, which also serves as the small leg. Take the solstitial entry-and-exit half arc-chord of 23°71′ as the great leg. Divide the great base of 56°06′50″ by the great leg as divisor, obtaining 2°37′ rounded to the nearest degree as the degree difference. Multiply the degree difference by the small leg to obtain the small base of 13°47′82″ as the contained half-length. Take the semidiameter of 60°87′50″ as great hypotenuse and multiply by the small leg of 5°70′; dividing by the great leg of 23°71′ yields 14°63′ as small chord, and also as the half arc-chord between the white-path equatorial node and the ecliptic–equator node. By the prescribed method the half arc-back of 14°66′ is obtained as the polar distance of the white-path equatorial node from the ecliptic–equator node.