1
淩犯視差新法上道光中，欽天監秋官正司廷棟所撰，較舊法加密，附著卷末，以備參考。
The upper section of the new occultation-parallax method was written in the Daoguang era by Si Tingdong, Autumnal Official of the Directorate of Astronomy. It refines the older procedure and is appended at the end of this fascicle for reference.
2
求用時
Finding apparent time
3
推諸曜之行度，皆以太陽為本； 而太陽之實行，又以平行為根。 其推步之法，總以每日子正為始，此言子正者，乃為平子正，即太陽平行之點臨于子正初刻之位也。 今之推步時刻，雖以兩子正之實行為比例，而所得者亦皆平行所臨之點，則實行所臨之點，自有進退之殊。 設太陽在最卑後實行大於平行，則太陽所臨之點必在平行之東，以時刻而言，乃為未及。 若太陽過最高後實行小於平行，則太陽所臨之點必在平行之西，以時刻而言，乃為已過。 故以應加之均數變時為應減之時差，應減之均數變時為應加之時差，此因太陽有平行實行之別，以生均數時差也。 然太陽所行者黃道，時刻所據者赤道，因黃道與赤道斜交，則同升必有差度。 如二分後赤道小於黃道，其差應減，在時刻為未及。 二至後赤道大於黃道，其差應加，在時刻為已過。 故以正弧三角形法求得黃赤升度差，變為時分，二分後為加，二至後為減，此因經度有黃道赤道之分，以生升度時差也。 按本時之日行自行所生之二差，各加減于平時而得用時，由用時方可以推算他數，故交食亦必以推用時為首務，即日月食之第一求也。 其法理圖說已載于考成前編，講解最詳，其圖分而為二，且均數時差圖系用小輪。 至考成後編求均數改為橢圓法，其法理亦備悉於求均數篇內，然未言及時差。 今依太陽實行所臨黃道之點，以均數之分取得黃道上平行點，即以平實二點依過二極、二至經圈作距等圈法，引於赤道，可使二差合為一圖。 其太陽之經度所臨之時刻及二時差之加減，皆可按圖而稽矣。
When computing the celestial motions of the planets, all procedures take the Sun as the reference; while the Sun's true motion is itself derived from mean motion. Stepwise calculation always begins at midnight each day. Here "midnight" means mean midnight: the instant when the Sun's mean longitude reaches the first quarter-mark of the midnight hour. Modern time-stepping may scale between two midnights using true motion, yet the result still marks where mean motion stands; the point where true motion stands therefore advances or lags on its own. If, after perigee, true motion exceeds mean motion, the Sun's apparent position lies east of the mean; in clock time this reads as behind schedule. If, after apogee, true motion falls short of mean motion, the Sun's apparent position lies west of the mean; in clock time this reads as ahead of schedule. Thus an equation due to be added becomes a time correction to subtract, and one due to be subtracted becomes a correction to add. This equation-of-time correction comes from the difference between mean and true solar motion. The Sun moves along the ecliptic, but civil time is measured on the equator. Because the two circles meet obliquely, the same rising instant yields different degrees on each. After the equinoxes the equatorial arc is shorter than the ecliptic arc, so the correction is subtracted and the clock reads as behind. After the solstices the equatorial arc is longer, so the correction is added and the clock reads as ahead. Use a right spherical triangle to find the ecliptic–equator ascension difference, convert it to time: add after equinoxes, subtract after solstices. This ascension time correction comes from measuring longitude on the ecliptic while time runs on the equator. The two corrections from the Sun's daily and anomalistic motion are applied to mean time to get apparent time, and only then can other quantities follow. Eclipse work therefore begins with apparent time — the first requirement in solar and lunar eclipse calculation. The theory and diagrams appear in the front fascicle of the Kaocheng with full explanation. The figures are split in two, and the equation-of-time diagram uses an epicycle. The rear fascicle replaces epicycles with ellipses for the equation, and the theory is complete in that chapter, but it does not treat the time correction. From the ecliptic point of true solar longitude, use the equation to find the mean point. Project both mean and true points onto the equator with parallel circles along the polar and solstitial meridians, so both corrections appear in a single figure. The instant of solar longitude and how to add or subtract both time corrections can then be read straight from the diagram.
4
如道光十二年壬辰三月初六日癸丑戌正二刻十一分，月與司怪第四星同黃道經度，是為淩犯時刻。 本日太陽引數三宮三度五十五分，太陽黃道經度三宮十五度五十三分，求用時。 如圖甲為北極，乙丙丁戊為赤道，乙甲丁為子午圈，乙為子正，丁為午正，己庚辛壬為黃道，丙甲戊為過二極二至經圈，己為冬至，辛為夏至，庚為春分，壬為秋分。 子為太陽實行之點，當赤道於醜，則醜點即太陽實臨之用時。 卯為太陽平行之點，而當赤道於辰。 其卯子之分，即應加之均數一度五十五分四十五秒，試自卯子二點與丙甲戊過極至經圈平行作卯午、子未二線，即如距等圈，將太陽平行、實行之度皆引於赤道，則庚午必與庚卯等，庚未必與庚子等，其赤道之午未亦必與卯子均數等。 變時得七分四十三秒，為赤道午未之分，即均數時差也。 次用庚丑子正弧三角形求庚丑弧，此形有醜直角，有庚角黃赤交角二十三度二十九分，有庚子弧太陽距春分後黃道度十五度五十三分。 乃以半徑為一率，庚角之餘弦為二率，庚子弧之正切為三率，求得四率為庚丑弧之正切，檢表得庚丑弧十四度三十七分三十六秒，為太陽距春分後赤道度。 乃與庚子黃道弧相等之庚未弧相減，得醜未弧一度十五分二十四秒，為應減之黃赤升度差。 變時得五分二秒，即升度時差也。 蓋太陽平行卯點，距春分之庚卯弧與庚午弧等，則午點乃為平時，即今之淩犯時刻。 而太陽實行子點，距春分之庚子與庚未弧等，則午未為平行與實行之差。 如以太陽右旋而言之，為實行已過平行，然以隨天左旋而計之，為實行未及平行，是未點轉早於午點，故必減午未均數時差，乃得未點時刻，此太陽在黃道虛映於赤道之時刻也。 然子點太陽實當赤道之醜，則醜未為黃道與赤道之差。 若以經度東行而言之，為赤道未及黃道，茲以時刻西行而計之，為赤道已過黃道，是醜點復遲於未點，故必加醜未升度時差，方得醜點時刻，即太陽在黃道實當於赤道之時刻也。 其兩時差既為一加一減，而所減者又大於應加之分，故先以兩時差相減，得醜午時分二分四十一秒，而為時差此因兩時差加減異號故相減，若同號則相加，所謂兩數通為一數也。 又因減數大於加數，故仍從減，若加數大者則從加矣。 乃減于午點淩犯時刻戌正二刻十一分，即得醜點戌正二刻八分十九秒，為淩犯用時也。
For example, on the sixth day of the third month, Daoguang year 12 (renchen), at 8:11 p.m., the Moon and the fourth star of Simai share the same ecliptic longitude — the mean occultation time. That day the Sun's argument is 3°55′ in mansion 3; ecliptic longitude 15°53′ in mansion 3. Find apparent time. In the figure: A is the north celestial pole; B, C, D, E mark the equator; B–A–D is the meridian, with B at midnight and D at noon; F, G, H, I are the ecliptic, with F at winter solstice, H at summer solstice, G at spring equinox, and I at autumn equinox; C–A–E is the polar–solstitial meridian. Point Z marks true solar motion; projected to the equator at Chou, Chou is the apparent time when the Sun is actually there. Point Mao marks mean solar motion, projected to the equator at Chen. The arc Mao–Zi is the +1°55′45″ equation. Through Mao and Zi, draw parallels to the polar–solstitial meridian (like small circles) to project mean and true longitude onto the equator: arc G–Wu equals G–Mao, arc G–Wei equals G–Zi, and on the equator Wu–Wei equals the equation Mao–Zi. In time this is 7m43s on the equator between Wu and Wei — the equation-of-time correction. Next solve the right spherical triangle G–Chou–Zi for arc G–Chou: right angle at Chou, angle G = obliquity 23°29′, side G–Zi = 15°53′ ecliptic longitude after the spring equinox. Use the proportion: radius : cos(obliquity) :: tan(ecliptic arc G–Zi) : tan(equatorial arc G–Chou). The table gives G–Chou = 14°37′36″ — equatorial longitude after the spring equinox. Subtract the ecliptic arc G–Zi from the equal arc G–Wei to get Chou–Wei = 1°15′24″ — the ascension correction to subtract. In time this is 5m2s — the ascension time correction. At mean point Mao, arc G–Mao from the equinox equals G–Wu, so Wu marks mean time — the stated occultation moment. At true point Zi, G–Zi equals G–Wei from the equinox, so Wu–Wei is the mean–true difference. In heliocentric terms true motion has passed mean; in diurnal terms it lags. Wei therefore comes before Wu, so subtract the equation correction Wu–Wei to get Wei — the time of the Sun's ecliptic position projected onto the equator. At Zi the Sun actually stands on the equator at Chou, so Chou–Wei is the ecliptic–equator difference. In longitude the equator lags the ecliptic; in clock time it leads. Chou therefore comes after Wei, so add the ascension correction Chou–Wei to reach Chou — when the Sun's true ecliptic position meets the equator. One correction adds, one subtracts, and the subtracted term is larger. Subtract them to get Chou–Wu = 2m41s as the net correction. Opposite signs subtract; like signs add — combining the two into one. Because subtraction dominates, apply the net as a subtraction; if addition were larger, apply it as an addition instead. Subtract from the mean occultation at 8:11 p.m. to get 8:08:19 p.m. at Chou — the apparent occultation time.
5
一率半徑
First ratio: radius
6
二率庚角餘弦
Second ratio: cosine of angle G
7
三率庚子弧正切
Third ratio: tangent of arc G–Zi
8
四率庚丑弧正切
Fourth ratio: tangent of arc G–Chou
9
圖略
Diagram omitted
10
又設淩犯時刻醜正一刻，太陽引數三宮十三度二十九分，黃道實行三宮二十五度三十四分，求用時。 如子為太陽實行之點，當赤道於醜，其醜點即所臨之用時。 卯為太陽平行之點，當赤道於辰，其子卯為應加之均數一度五十二分二十五秒，亦自卯子二點與過極至經圈平行作卯醜、子未二距等圈，其平行卯點映於赤道，恰與實行當赤道之醜點合，是由平行所得之時刻，已合實行實臨赤道之用時，遇此可無庸求其時差也。 然何以知之，蓋兩時差之數相等，必減盡無餘，即無時差之總數矣。 今試按法求之，既作卯醜、子未二線，其庚丑與庚卯等，庚未與庚子等，則醜未必與卯子均數等，變時得七分三十秒，即赤道上應減之均數時差。 次用庚丑子正弧三角形，求得庚丑弧赤道度，與庚子弧黃道度相等之庚未弧相減，得醜未弧，黃赤升度差恰與均數等。 變時亦得七分三十秒，即赤道上應加之升度時差。 其時差一為加、一為減，而兩數相等，乃減盡無餘，既無時差之總數，則其淩犯時刻即為用時可知矣。 此法以醜點淩犯時刻減去均數時差，得未點實行虛映之時刻，而復加相等之升度時差，所得用時，固仍在醜點之位，蓋因太陽平行距春分後黃道度等於太陽實行距春分後赤道度故也。 又如太陽正當本天之最卑或最高，乃無平行實行之差，自無均數時差，止加減升度時差一數。 設太陽當本天最卑，又當子正，如太陽在黃道之子點，則庚乙與庚子等，以庚丑子正弧形求得醜乙黃赤升度差。 變時減於乙點時刻，即得醜點用時，乃在乙點子正之前也。 若太陽當本天最高，又當午正，如太陽在黃道之午點，則壬丁與壬午等，以壬寅午正弧形求得寅丁黃赤升度差，變時減于丁點時刻，即得寅點用時，乃在丁點午正之前也。
Suppose another occultation at 1:00 a.m., with argument 13°29′ in mansion 3 and true ecliptic longitude 25°34′ in mansion 3. Find apparent time. With Zi as true solar motion projected to Chou on the equator, Chou is the apparent time. Mean point Mao projects to Chen, with equation Zi–Mao = +1°52′25″. Parallel circles Mao–Chou and Zi–Wei bring mean motion onto the equator exactly at Chou, where true motion stands. Mean time then equals apparent time, so no net correction is needed. This is known because both corrections are equal and cancel, leaving no net time difference. Following the method with lines Mao–Chou and Zi–Wei: G–Chou = G–Mao and G–Wei = G–Zi, so Chou–Wei need not equal the equation Mao–Zi. In time this is 7m30s to subtract on the equator. Solve triangle G–Chou–Zi for equatorial arc G–Chou; subtract the equal ecliptic arc from G–Wei to get Chou–Wei — the ascension difference exactly equals the equation. In time this is also 7m30s — the ascension correction to add on the equator. One correction adds, one subtracts, but both are 7m30s and cancel. With no net correction, the mean occultation time is the apparent time. Subtract the equation correction from the Chou occultation to get Wei's projected time, then add the equal ascension correction — apparent time remains at Chou because mean ecliptic longitude after the equinox equals true equatorial longitude after the equinox. When the Sun is at perigee or apogee there is no mean–true difference and no equation correction — only the ascension correction applies. At perigee and midnight, with the Sun at ecliptic Zi, G–Yi = G–Zi. Triangle G–Chou–Zi gives ascension difference Chou–Yi. Convert to time, subtract from the Yi moment, and Chou apparent time falls before midnight at Yi. At apogee and noon, with the Sun at ecliptic Wu, R–D = R–Wu. Triangle R–Yin–Wu gives ascension difference Yin–Ding; subtract from the Ding moment to get Yin apparent time before noon.
11
圖略
Diagram omitted
12
又如太陽實行正當冬、夏至或正當春、秋分，此四點皆無黃道赤道之差，自無升度時差，止加減均數時差一數。 設太陽實行六宮初度為正當夏至，在黃道之辛點，當赤道於戊，而平行卯點，當赤道於辰，自卯點與丙甲戊過極至經圈平行作卯午距等圈，則午點為淩犯時刻，其戊午與辛卯均數等，變時得均數時差。 減於午點而得戊點，即用時也。
At the solstices or equinoxes there is no ecliptic–equator difference and no ascension correction — only the equation correction applies. At summer solstice (mansion 6°0), true Sun at Xin projects to Wu on the equator; mean Mao to Chen. Parallel circle Mao–Wu gives occultation at Wu; Wu–Wu equals equation Xin–Mao, which converts to the equation time correction. Subtract from the noon occultation point to reach Wu on the equator — the apparent time.
13
圖略
Diagram omitted
14
求春分距午時分、黃平象限宮度及限距地高
Finding equinox offset from noon, yellow ecliptic quadrant longitude, and limiting altitude
15
推算太陰淩犯視差，固依後編求日食三差之法，而其為用不同。 蓋日食之東西差為求視距弧，而南北差為求視緯，其視距弧、視緯則為求視相距及視行之用。 緣太陰行于白道，是必以白平象限為准焉。 若五星之距恆星、五星之互相距，皆以黃道同經度之時為相距時刻，而較黃緯南北相距之數為其上下之分也。 至月距五星、月距恆星，亦皆以黃道經度相同之時為淩犯時刻，不更問白道經度，其于白平象限又何與焉？ 然其以東西差定視時之進退，以南北差判視緯之大小，以定視距之遠近者，其差皆黃道經緯之差，故必以黃平象限之宮度為准。 黃平象限者，地平上黃道半周適中之點也。 顧黃道與赤道斜交，地平上赤道半周適中之點，恆當子午圈，而地平上黃道半周適中之點，則時有更易。 蓋黃極由負黃極圈每日隨天左旋，繞赤極一周，如黃極在赤極之南，則冬至當午正，其黃道斜升斜降； 若黃極在赤極之北，則夏至當午正，其黃道正升正降，而黃平象限亦皆恰當子午圈； 設黃極在赤極之西，則春分當午正，其黃道之勢斜倚，出自東北而入西南，黃平象限乃在午正之東； 設黃極在赤極之東，則秋分當午正，其黃道出自東南而入西北，黃平象限乃在午正之西。 是則黃道之向，隨時不同，故以黃道之逐度，推求黃平象限及限距地高以立表。
Lunar occultation parallax uses the rear fascicle's solar-eclipse three-difference method, but applies it differently. In solar eclipse, east–west difference yields apparent distance arc and north–south difference yields apparent latitude; both serve to find apparent separation and apparent motion. Because the Moon follows the lunar path, calculations must use the white ecliptic quadrant. Planet–star and planet–planet separations use the moment of equal ecliptic longitude, comparing ecliptic latitudes for north–south offset. Moon–planet and Moon–star occultations likewise use equal ecliptic longitude and ignore lunar-path longitude — so the white ecliptic quadrant does not apply. Yet east–west difference sets apparent-time advance or lag, north–south difference sets apparent latitude, and both fix apparent distance — all are ecliptic longitude–latitude differences, so the yellow ecliptic quadrant longitude is required. The yellow ecliptic quadrant is the midpoint of the ecliptic arc above the horizon. Because the ecliptic and equator meet obliquely, the equatorial midpoint above the horizon always lies on the meridian, but the ecliptic midpoint shifts with the season. The ecliptic pole, carried on the small circle of the negative ecliptic pole, rotates westward with the sky once per day around the celestial pole. When the ecliptic pole lies south of the celestial pole, winter solstice comes at noon and the ecliptic rises and sets obliquely; if the ecliptic pole lies north of the celestial pole, summer solstice is at noon, the ecliptic rises and sets vertically, and the yellow ecliptic quadrant always lies on the meridian; if the ecliptic pole lies west of the celestial pole, spring equinox is at noon, the ecliptic tilts from northeast to southwest, and the yellow ecliptic quadrant lies east of the meridian; if the ecliptic pole lies east of the celestial pole, autumn equinox is at noon, the ecliptic runs from southeast to northwest, and the yellow ecliptic quadrant lies west of the meridian. The ecliptic's orientation changes continually, so tables are built by computing the yellow ecliptic quadrant and limiting altitude for each ecliptic degree.
16
先設太陽正當春分點，黃道實行為三宮初度，求午正初刻黃平象限宮度及限距地高度分。 如圖甲乙丙丁為子午圈，甲為天頂，丙丁為地平，乙為北極，乙丙為京師北極出地，高三十九度五十五分，戊己庚為赤道，交于地平之己點，其戊點當午正，為地平上赤道半周適中之點，戊丁為赤道距地高五十度五分，當戊己丁角，辛子壬為負黃極圈，子為黃極，乙子己丑為過極至經圈，戊丑庚為黃道，而交地平於寅點，庚為秋分，醜為冬至，戊為春分，即太陽之所在，臨於午正，乃無春分距午之時分。 試自黃極子點出弧線過天頂作子甲卯黃道經圈，為本時黃平象限，其辰點為地平上黃道半周適中之點，而在正午之東，即黃平象限宮度也。 辰寅卯角為黃道與地平相交之角，而當辰卯弧，即本時限距地高之度也。 法用戊辰甲正弧三角形求戊辰、甲辰二弧，此形有辰直角，有戊甲弧赤道距天頂，與乙丙北極高度等。 以赤道交子午圈之戊直角九十度內減己戊丑角黃赤交角二十三度二十九分，得寅戊丁角六十六度三十一，為黃道交子午圈角； 亦名黃道赤經交角。 與辰戊甲角為對角，其度等。 乃以半徑為一率，戊角黃道赤經交角之餘弦為二率，戊甲弧赤道距天頂，亦即太陽距天頂其正切為三率，求得四率，為黃平象限距午之正切，檢表得十八度二十六分十四秒，為戊辰弧黃平象限距午正之黃道度。 與戊點春分三宮相加，因黃平象限在午東，故加。 得辰點三宮十八度二十六分十四秒，即本時黃平象限之經度也。 又以半徑為一率，戊角黃道赤經交角之正弦為二率，戊甲弧太陽距天頂之正弦為三率，求得四率，為黃平象限距天頂之正弦，檢表得三十六度三分九秒，為甲辰弧黃平象限距天頂。 與甲卯象限九十度相減，得辰卯弧五十三度五十六分五十一秒，即本時限距地高，而當辰寅卯角之度也。
First take the Sun at the spring equinox (mansion 3°0) and find, at the first quarter after noon, the yellow ecliptic quadrant longitude and limiting altitude. In the figure: A–B–C–D is the meridian (A zenith, C–D horizon, B north pole). B–C is Beijing's latitude, 39°55′. F–G–H is the equator (H on the horizon, F at noon as the equatorial midpoint; altitude F–D = 50°05′). Circle Xin–Zi–Ren carries ecliptic pole Zi; B–Zi–Ji–Chou is the polar–solstitial meridian; F–Chou–H is the ecliptic (Yin on the horizon; H autumn equinox, Chou winter solstice, F spring equinox where the Sun stands at noon, so equinox offset from noon is zero). From ecliptic pole Zi through the zenith draw meridian Zi–A–Mao — the yellow ecliptic quadrant. Chen marks the ecliptic midpoint above the horizon, east of noon — the quadrant longitude. Angle Chen–Yin–Mao is the ecliptic–horizon angle; arc Chen–Mao is the limiting altitude. Use right spherical triangle F–Chen–A for arcs F–Chen and A–Chen: right angle at Chen, side F–A = co-latitude (equal to Beijing's 39°55′). From 90° at F where the equator crosses the meridian, subtract obliquity 23°29′ (angle Ji–F–Chou) to get angle Yin–F–D = 66°31′ — the ecliptic–meridian angle; also called the angle between ecliptic and equatorial right ascension. It equals the opposite angle Chen–F–A. Proportion: radius : cos(F angle) :: tan(co-latitude F–A) : tan(quadrant offset). Table gives 18°26′14″ for arc F–Chen — ecliptic degrees of quadrant east of noon. Add to spring equinox at F (mansion 3) because the quadrant lies east of noon. Chen = mansion 3, 18°26′14″ — the yellow ecliptic quadrant longitude. Second proportion: radius : sin(F angle) :: sin(co-latitude) : sin(zenith distance). Table gives A–Chen = 36°03′09″ — quadrant zenith distance. Subtract from 90° quadrant A–Mao to get Chen–Mao = 53°56′51″ — the limiting altitude opposite angle Chen–Yin–Mao.
17
一率半徑
First ratio: radius
18
二率戊角餘弦
Second ratio: cosine of angle F
19
三率戊甲弧正切
Third ratio: tangent of arc F–A
20
四率戊辰弧正切
Fourth ratio: tangent of arc F–Chen
21
一率半徑
First ratio: radius
22
二率戊角正弦
Second ratio: sine of angle F
23
三率戊甲弧正弦
Third ratio: sine of arc F–A
24
四率甲辰弧正弦
Fourth ratio: sine of arc A–Chen
25
圖略
Diagram omitted
26
又設太陽正當秋分點，黃道實行為九宮初度，求午正初刻春分距午時分並黃平象限及限距地高，即以秋分當于正午之戊，則庚未戊為黃道，交地平於寅，庚為春分，未為夏至，子乙未己為過極至經圈，亦自黃極子點出弧★過天頂，作子甲卯弧黃平象限，而地平上黃道適中之辰點，在正午之西。 先以春分距午西之庚戊赤道半周變十二時為春分距午之時分，次仍用戊辰甲正弧三角形求戊辰、甲弧二弧，此形有辰直角，有戊甲赤道距天頂。 以戊直角內減己戊未角黃赤交角，得辰戊甲角黃道赤經交角，亦六十六度三十一分，求得戊辰弧黃平象限距午正之黃道度，亦十八度二十六分十四秒。 與戊點秋分九宮相減，因黃平象限在午西，故減。 得辰點八宮十一度三十三分四十六秒，即本時黃平象限之經度。 又求得甲辰弧8888與甲卯象限相減，得辰卯弧，亦為五十三度五十六分五十一秒，即本時限距地高，而當辰寅卯角之度也。
Next take the Sun at the autumn equinox (mansion 9°0) and find equinox offset from noon, quadrant longitude, and limiting altitude at the first quarter after noon. With autumn equinox at noon (Wu), ecliptic G–Wei–F meets the horizon at Yin (G spring equinox, Wei summer solstice); polar meridian Zi–Yi–Wei–Ji; quadrant meridian Zi–A–Mao places Chen west of noon. First convert arc G–F (equator semicircle west of equinox) into hours as equinox offset from noon; then solve right triangle F–Chen–A (right angle at Chen, side F–A = co-latitude) for arcs F–Chen and A–Chen. From 90° at F subtract obliquity (angle Ji–F–Wei) to get angle Chen–F–A = 66°31′; arc F–Chen = 18°26′14″ east of noon in ecliptic degrees. Subtract from autumn equinox at Wu (mansion 9) because the quadrant lies west of noon. Chen = mansion 8, 11°33′46″ — the yellow ecliptic quadrant longitude. Subtract arc A–Chen (36°03′09″) from the 90° quadrant A–Mao to get Chen–Mao = 53°56′51″ — the limiting altitude opposite angle Chen–Yin–Mao.
27
又設太陽距春分後三十度，黃道實行為四宮初度，求午正初刻黃平象限諸數。 乃以黃道經度四宮初度當午正如辛點，即太陽之所在，辛壬癸為黃道，交地平於寅。 醜為冬至，壬為春分，乙子醜為過極至經圈。 仍自黃極子點過天頂甲點作子甲卯弧黃平象限，其黃道適中之辰點，在午正之東。 求法先用辛戊壬正弧三角形求壬戊、辛戊二弧及壬辛戊角，此形有戊直角，有壬角黃赤交角，有壬辛太陽距春分後黃道弧三十度。 乃以半徑為一率，黃赤交角之餘弦為二率，黃道弧之正切為三率，求得四率，為赤道弧之正切，檢表得二十七度五十四分一十秒，為壬戊弧赤道同升度，亦即本時春分距午後赤道度。 變時得一時五十一分三十七秒，即本時春分距午時分。 又以半徑為一率，黃赤交角之正弦為二率，黃道弧之正弦為三率，求得四率，為黃赤距度之正弦，檢表得十一度二十九分三十三秒，為辛戊弧太陽距赤道北緯度。 又以黃道弧之餘弦為一率，黃赤交角之餘切為二率，半徑為三率，求得四率，為黃道交子午圈角之正切，檢表得六十九度二十二分五十一秒，為壬辛戊角黃道交子午圈角，即黃道赤經交角。 次用辛辰甲正弧三角形求辛辰、甲辰二弧，此形有辰直角，有辛角，與壬辛戊角為對角，其度等。 以甲戊弧赤道距天頂內減辛戊黃赤距度，得甲辛弧二十八度二十五分二十七秒，為本時太陽距天頂。 乃以半徑為一率，辛角黃道赤經交角之餘弦為二率，甲辛弧太陽距天頂之正切為三率，求得四率，為黃平象限距午之正切，檢表得十度四十七分二十八秒，為辛辰弧黃平象限距午正之黃道度。 與辛點四宮初度相加，因黃平象限在午東，故加。 得辰點四宮十度四十七分二十八秒，即本時黃平象限之經度也。 又以半徑為一率，辛角黃道赤經交角之正弦為二率，甲辛弧太陽距天頂之正弦為三率，求得四率，為黃平象限距天頂之正弦，檢表得二十六度二十七分二十秒，為甲辰弧黃平象限距天頂。 與甲卯象限九十度相減，得辰卯弧六十三度三十二分四十秒，為本時限距地高，即當辰寅卯角之度也。
Next take the Sun 30° after the spring equinox (mansion 4°0) and find the yellow ecliptic quadrant quantities at the first quarter after noon. Place mansion 4°0 at noon at Xin (the Sun); ecliptic Xin–Ren–G meets the horizon at Yin. Chou winter solstice, Ren spring equinox; polar meridian B–Zi–Chou. Again from ecliptic pole Zi through zenith A draw quadrant Zi–A–Mao; Chen, the ecliptic midpoint, lies east of noon. First solve right triangle Xin–F–Ren for arcs Ren–F, Xin–F and angle Ren–Xin–F: right angle at F, angle Ren = obliquity, side Ren–Xin = 30° ecliptic arc after equinox. Proportion: radius : cos(obliquity) :: tan(ecliptic arc) : tan(equator arc). Table gives Ren–F = 27°54′10″ — equatorial co-ascension, i.e. spring equinox offset west of noon. Converted to time: 1h51′37″ — spring equinox offset from noon. Second proportion: radius : sin(obliquity) :: sin(ecliptic arc) : sin(declination). Table gives Xin–F = 11°29′33″ — Sun's north declination. Third proportion gives tan(ecliptic–meridian angle) = 69°22′51″ for angle Ren–Xin–F — the ecliptic–equator right-ascension angle. Next solve right triangle Xin–Chen–A (right angle at Chen; angle Xin equals opposite angle Ren–Xin–F) for arcs Xin–Chen and A–Chen. From co-latitude A–F subtract declination Xin–F to get A–Xin = 28°25′27″ — the Sun's zenith distance. Proportion: radius : cos(Xin angle) :: tan(zenith distance A–Xin) : tan(quadrant offset). Table gives 10°47′28″ for arc Xin–Chen — ecliptic degrees of quadrant east of noon. Add to Xin (mansion 4°0) because the quadrant lies east of noon. Chen = mansion 4, 10°47′28″ — the yellow ecliptic quadrant longitude. Second proportion: radius : sin(Xin angle) :: sin(zenith distance) : sin(quadrant zenith distance). Table gives A–Chen = 26°27′20″ — quadrant zenith distance. Subtract from 90° quadrant A–Mao to get Chen–Mao = 63°32′40″ — the limiting altitude opposite angle Chen–Yin–Mao.
28
一率半徑
First ratio: radius
29
二率壬角餘弦
Second ratio: cosine of angle Ren
30
三率壬角弧正切
Third ratio: tangent of arc Ren–Xin
31
四率壬戊弧正切
Fourth ratio: tangent of arc Ren–F
32
一率半徑
First ratio: radius
33
二率壬角正弦
Second ratio: sine of angle Ren
34
三率壬辛弧正弦
Third ratio: sine of arc Ren–Xin
35
四率辛戊弧正弦
Fourth ratio: sine of arc Xin–F
36
一率壬辛弧餘弦
First ratio: cosine of arc Ren–Xin
37
二率壬角餘切
Second ratio: cotangent of angle Ren
38
三率半徑
Third ratio: radius
39
四率辛角正切
Fourth ratio: tangent of angle Xin
40
一率半徑
First ratio: radius
41
二率辛角餘弦
Second ratio: cosine of angle Xin
42
三率甲辛弧正切
Third ratio: tangent of arc A–Xin
43
四率辛辰弧正切
Fourth ratio: tangent of arc Xin–Chen
44
一率半徑
First ratio: radius
45
二率辛角正弦
Second ratio: sine of angle Xin
46
三率甲辛弧正弦
Third ratio: sine of arc A–Xin
47
四率甲辰弧正弦
Fourth ratio: sine of arc A–Chen
48
又設太陽距秋分前三十度，黃道實行為八宮初度，求午正初刻黃平象限諸數。 乃以辛點太陽實行當正午，其申點為秋分，而在午東，壬為春分，未為夏至，子乙未為過極至經圈，亦自黃極子點過天頂，作子甲卯弧本時黃平象限，而在午西。 法用辛戊申正弧三角形，此形戊為直角，申角為黃赤交角，申辛黃道弧亦為三十度，求得申戊赤道同升度，亦為二十七度五十四分一十秒。 乃與壬申赤道之半周相減，得壬戊弧五宮二度五分五十秒，為本時春分距午後赤道度。 變時得十時八分二十三秒，即本時春分距午時分也。 次用辛辰甲正弧三角形，辰為直角，其辛角黃道赤經交角及甲辛弧太陽距天頂，皆與前圖之度等。 求得辛辰弧黃平象限距午正黃道度，亦為十度四十七分二十八秒。 與辛點八宮初度相減，因黃平象限在午西，故減。 得辰點七宮十九度十二分三十二秒，即本時黃平象限之經度。 又求得甲辰弧與甲卯象限相減，得辰卯弧，亦為六十三度三十二分四十秒，即本時限距地高，亦當辰寅卯角之度也。
Next take the Sun 30° before the autumn equinox (mansion 8°0) and find the yellow ecliptic quadrant quantities at the first quarter after noon. Place Xin (the Sun's true longitude) at noon; Shen autumn equinox lies east of noon; Ren spring equinox, Wei summer solstice; polar meridian Zi–B–Wei. From ecliptic pole Zi through the zenith draw quadrant Zi–A–Mao west of noon. Use right triangle Xin–F–Shen: right angle at F, angle Shen = obliquity, ecliptic arc Shen–Xin = 30°; equatorial co-ascension Shen–F = 27°54′10″. Subtract from semicircle Ren–Shen to get Ren–F = mansion 5, 2°05′50″ — spring equinox offset west of noon. Converted to time: 10h08′23″ — spring equinox offset from noon. Next use right triangle Xin–Chen–A (right at Chen): angle Xin and zenith distance A–Xin match the previous example. Arc Xin–Chen = 10°47′28″ — quadrant offset from noon in ecliptic degrees. Subtract from Xin (mansion 8°0) because the quadrant lies west of noon. Chen = mansion 7, 19°12′32″ — the yellow ecliptic quadrant longitude. Subtract A–Chen from the 90° quadrant A–Mao to get Chen–Mao = 63°32′40″ — the limiting altitude opposite angle Chen–Yin–Mao.
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又設太陽當正午實行距春分前三十度為二宮初度，乃以辛點太陽當午正，則春分壬點在午正之東，申為秋分，醜為冬至，乙子醜為過極至經圈，其子甲卯本時黃平象限亦在午正之東。 法用辛戊壬正弧三角形，有戊直角，有壬角黃赤交角，有壬辛黃道弧三十度。 求得壬戊赤道弧，亦為二十七度五十四分一十秒。 乃與赤道全周相減，得十一宮二度五分五十秒，為本時春分距午後赤道度。 變時得二十二時八分二十三秒，即本時春分距午時分也。 又求得辛戊弧亦為十一度二十九分三十三秒，為太陽距赤道南緯度，並求得壬辛戊角亦為六十九度二十二分五十一秒，為本時黃道赤經交角。 次用辛辰甲正弧三角形，此形有辰直角，有辛角，以甲戊赤道距天頂與辛戊黃赤距度相加，得甲辛弧太陽距天頂五十一度二十四分三十三秒。 乃以半徑為一率，辛角之餘弦為二率，甲辛弧之正切為三率，求得四率，為黃平象限距午之正切，檢表得二十三度四十八分四十秒，即辛辰弧黃平象限距午正之黃道度。 與辛點二宮初度相加，得辰點二宮二十三度四十八分四十秒，即本時黃平象限之經度也。 又以半徑為一率，辛角之正弦為二率，甲辛弧之正弦為三率，求得四率，為甲辰弧黃平象限距天頂之正弦，檢餘弦表得四十二度五十九分一秒，即卯辰弧本時限距地高之度也。
Next take the Sun at noon 30° before equinox (mansion 2°0): Xin at noon places Ren spring equinox east of the meridian; Shen autumn equinox, Chou winter solstice; polar meridian B–Zi–Chou; quadrant Zi–A–Mao also east of noon. Use right triangle Xin–F–Ren: right angle at F, angle Ren = obliquity, ecliptic arc Ren–Xin = 30°. Arc Ren–Wu (equator) = 27°54′10″. Subtract from the full equator to get mansion 11, 2°05′50″ — equatorial degrees of equinox offset east of noon. In time that is 22h08′23″ — the equinox offset from noon. Also Xin–Wu = 11°29′33″ (Sun south of the equator), and angle Ren–Xin–Wu = 69°22′51″ — the ecliptic–equatorial right-ascension angle. Next solve right triangle Xin–Chen–A (right angle at Chen, angle at Xin): co-latitude A–Wu plus yellow-red distance Xin–Wu gives A–Xin = 51°24′33″ — the Sun's zenith distance. Proportion: radius : cos(Xin angle) :: tan(zenith distance A–Xin) : tan(quadrant offset). Table gives Xin–Chen = 23°48′40″ — ecliptic degrees of quadrant east of noon. Add to Xin at mansion 2°0 to get Chen = mansion 2, 23°48′40″ — the yellow ecliptic quadrant longitude. Second proportion: radius : sin(Xin angle) :: sin(zenith distance) : sin(quadrant zenith distance). Versed-sine table gives Mao–Chen = 42°59′01″ — the limiting altitude.
50
一率半徑
First ratio: radius
51
二率辛角餘弦
Second ratio: cosine of angle Xin
52
三率甲辛弧正切
Third ratio: tangent of arc A–Xin
53
四率辛辰弧正切
Fourth ratio: tangent of arc Xin–Chen
54
一率半徑
First ratio: radius
55
二率辛角正弦
Second ratio: sine of angle Xin
56
三率甲辛弧正弦
Third ratio: sine of arc A–Xin
57
四率甲辰弧正弧
Fourth ratio: sine of arc A–Chen
58
又設太陽當午正實行距秋分後三十度為十宮初度，乃以辛點太陽當午正，則申點秋分在午正後，而春分必在午正前，未為夏至，子乙未為過極至經圈，其子甲卯本時黃平象限在午正之西。 求法仍用辛戊申正弧三角形，此形邊角之度與前圖之辛戊壬形同，惟申戊弧所變之一時五十一分三十七秒，乃秋分距午後之時分，是以加赤道半周之十二時，得十三時五十一分三十七秒，始為本時春分距午時分也。 次用辛辰甲正弧三角形，此形邊與角之度亦與前圖之辛辰甲形同，惟因辰點在辛點之西，是以十宮初度內減辛辰弧二十三度四十八分四十秒，得九宮六度十一分二十秒，即本時黃平象限之經度。 其辰卯弧限距地高四十二度五十九分一秒，亦與前數相同也。 由此則逐度皆以距春、秋分前後各相對之度推之，其求午正太陽距天頂之加減，則以緯南、緯北而分。 求黃平象限宮度之加減，則以冬至、夏至為斷。 蓋冬至過午西，黃平象限恆在午正之東，夏至過午西，黃平象限恆在午正之西，此加減所由定也。
Next take the Sun at noon 30° past autumn equinox (mansion 10°0). With Xin as the Sun at noon, Shen (autumn equinox) is after the meridian and spring equinox before it; Wei is summer solstice; polar meridian Zi–Yi–Wei; quadrant Zi–A–Mao lies west of noon. Use the same right triangle Xin–Wu–Shen as Xin–Wu–Ren above; arc Shen–Wu converts to 1h51′37″ as autumn-equinox offset east of noon. Add 12 hours (half the equator) to get 13h51′37″ — the spring-equinox offset from noon. Again solve Xin–Chen–A with the same values as before; because Chen lies west of Xin, subtract Xin–Chen = 23°48′40″ from mansion 10°0 to get Chen = mansion 9, 6°11′20″ — the quadrant longitude. Chen–Mao limiting altitude = 42°59′01″ — unchanged from the earlier case. Degree-by-degree values follow the symmetric offsets before and after spring and autumn equinox; noon zenith-distance addition or subtraction depends on south versus north latitude. Quadrant-longitude addition or subtraction is determined by winter versus summer solstice. When winter solstice is west of the meridian the quadrant always lies east of noon; when summer solstice is west of the meridian it always lies west — hence the sign of the correction.
59
今設太陽黃道經度三宮十六度四十四分，用時為戌正二刻八分十九秒，求春分距午時分及黃平象限宮度、限距地平高度。 如申辛壬癸為黃道，交地平於寅，壬為春分，醜為夏至，申為秋分，子乙丑亥為過二極二至經圈。 乃自黃極子點過天頂甲點作子甲卯黃道經圈，其黃道適中之辰點，乃在午正之西。 今太陽在春分後之未點，當赤道之午點，自子正計之，即用時之時刻。 先用未午壬正弧三角形求壬午弧，此形午為直角，有壬角黃赤交角二十三度二十九分，有壬未弧太陽距春分後黃道度十六度四十四分，求得壬午弧十五度二十四分五十八秒，為太陽距春分後赤道度。 變時得一小時一分四十秒，與午點用時相加，得二十一小時三十九分五十九秒，為壬點春分距子正後之時分。 內減十二時，得九小時三十九分五十九秒，即壬戊弧本時春分距午時分。 次用甲戊辛正弧三角形，因壬戊春分距午後之度已過象限，故用申戊辛正弧形。 求辛角及辛戊、辛申二弧。 此形戊為直角，有申角黃赤交角，有申戊弧秋分距午前時分所變之赤道度三十五度零十五秒，求得戊辛弧十三度五十九分四十秒，為本時正午之黃赤距度。 求得申辛戊角七十度五十六分五十八秒，為黃道交子午圈角，即黃道赤經交角。 與甲辛辰角為對角，其度等。 求得申辛弧三十七度二十一分五十秒，為秋分距午正前黃道度。 與申點秋分九宮相減，得七宮二十二度三十八分一十秒，即辛點正午黃道經度。 次用甲辰辛正弧三角形求辛辰、甲辰二弧，此形辰為直角，有辛角黃道赤經交角。 以甲戊弧京師赤道距天頂三十九度五十五分，內減辛戊正午黃赤距度，得甲辛弧二十五度五十五分二十秒，為本時正午黃道距天頂度，求得辛辰弧九度零五十三秒，為黃平象限距午西之黃道度。 與辛點正午黃道經度相減，得辰點七宮十三度三十七分十七秒，即本時黃平象限之經度，並求得甲辰弧二十四度二十四分二十四秒，為黃平象限距天頂之度。 與甲卯象限相減，得辰卯弧六十五度三十五分三十六秒，為本時黃平象限距地平之高度，即當辰寅卯角之度也。
Now take the Sun at ecliptic longitude mansion 3, 16°44′ with applied time 8:08:19 at the second quarter of Xu; find equinox offset from noon, quadrant longitude, and limiting altitude. Ecliptic Shen–Xin–Ren–Gui meets the horizon at Yin (Ren spring equinox, Chou summer solstice, Shen autumn equinox); polar–solstitial meridian Zi–Yi–Chou–Hai. From ecliptic pole Zi through zenith A draw meridian Zi–A–Mao; Chen, the ecliptic midpoint, lies west of noon. The Sun stands at Wei after spring equinox, on the equator at Wu — counting from midnight Zi, this is the applied time. First solve right triangle Wei–Wu–Ren (right angle at Wu): obliquity 23°29′ at Ren, arc Ren–Wei = 16°44′ ecliptic after equinox; Ren–Wu = 15°24′58″ equatorial degrees after equinox. In time that is 1h01′40″; add to the time at Wu to get 21h39′59″ — spring equinox at Ren measured from midnight Zi. Subtract 12 hours to get 9h39′59″ — arc Ren–Wu, the equinox offset from noon. Next one would use triangle A–Wu–Xin, but because the equinox offset east of noon has passed the quadrant, use the Shen–Wu–Xin form instead. Find angle Xin and arcs Xin–Wu and Xin–Shen. Right angle at Wu, obliquity at Shen; arc Shen–Wu = 35°00′15″ equatorial degrees from autumn-equinox offset west of noon; Wu–Xin = 13°59′40″ — noon yellow-red distance. Angle Shen–Xin–Wu = 70°56′58″ — the ecliptic–meridian angle, i.e. the ecliptic–equatorial right-ascension angle. It equals the opposite angle A–Xin–Chen. Shen–Xin = 37°21′50″ — ecliptic degrees from autumn equinox west of noon. Subtract from autumn equinox at Shen (mansion 9) to get Xin at noon = mansion 7, 22°38′10″. Next solve right triangle A–Chen–Xin (right angle at Chen, angle Xin = ecliptic–equatorial right-ascension angle) for arcs Xin–Chen and A–Chen. Beijing co-latitude A–Wu = 39°55′ minus noon yellow-red distance Xin–Wu gives A–Xin = 25°55′20″ (ecliptic zenith distance); Xin–Chen = 9°00′53″ — quadrant longitude west of noon in ecliptic degrees. Subtract from Xin's noon longitude to get Chen = mansion 7, 13°37′17″ — quadrant longitude; A–Chen = 24°24′24″ — quadrant zenith distance. Subtract from quadrant A–Mao to get Chen–Mao = 65°35′36″ — limiting altitude above the horizon, opposite angle Chen–Yin–Mao.
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求距限差
Finding quadrant distance difference
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距限差者，乃月距黃平象限之差度也。 蓋舊法月距限以九十度為率，因黃道麗天，其向隨時不同，而出於地平之上者，恆為半周，其適中之點，距地平東西皆九十度。 故以九十度之限，以察月在地平之上下，若月距限逾九十度者，為在地平下，遂不入算，然此以黃道為立算之端也。 顧白道與黃道斜交，月行白道，不無距黃道南北之緯度。 緯南者早入遲出，月當地平時，其距黃平象限不及九十度； 緯北者早出遲入，月當地平時，其距黃平象限已過九十度； 是則九十度之率未足為據也。 於是立法以求其差，猶五星伏見距日限度有距日加減差之義也。 其法以限距地平之高及月距黃道之緯，依正弧三角形法求之。 蓋黃道之勢，隨天左旋，其升降正斜，時時不同。 正升正降者，京師限距地高至七十三度餘，高度大，則月緯所當之距限差轉小； 斜升斜降者，京師限距地高只二十六度餘，高度小，則月緯所當之距限差轉大。 若值月緯最大，其差可至十度有奇，此距限差之不可不立也。 故依京師黃平象限距地平高度，逐度求其太陰黃道實緯度所當距限差以立表。
Quadrant distance difference is the correction to the Moon's distance from the yellow ecliptic quadrant. The old method took ninety degrees as the Moon's horizon limit. The ecliptic rides the sky with a changing orientation, but the arc above the horizon is always a semicircle whose midpoint stands ninety degrees from the horizon on either side. Ninety degrees served to decide whether the Moon was above or below the horizon: beyond ninety degrees meant below and out of the reckoning — but all of this assumes the ecliptic as the frame of reference. The lunar path and ecliptic meet obliquely, however, and the Moon on the white path always carries some latitude north or south of the ecliptic. With south latitude the Moon sets early and rises late; at horizon crossing its distance from the yellow ecliptic quadrant falls short of ninety degrees; With north latitude the Moon rises early and sets late; at horizon crossing its distance from the yellow ecliptic quadrant already exceeds ninety degrees; The ninety-degree standard is therefore not a reliable basis. A correction was therefore introduced — on the same principle as the distance-from-Sun additive correction used in the five planets' visibility limits. The correction is derived from limiting altitude and lunar ecliptic latitude by right spherical triangle. The ecliptic rides the sky as it turns westward; its rise and set are now steep, now slanting, and change from hour to hour. With steep rise and set, Beijing's limiting altitude exceeds 73°; the higher the arc, the smaller the quadrant distance difference for a given lunar latitude; With oblique rise and set, Beijing's limiting altitude is only about 26°; the lower the arc, the larger the quadrant distance difference. At maximum lunar latitude the correction can exceed 10° — which is why quadrant distance difference must be tabulated. A table is therefore built for Beijing: at each yellow ecliptic quadrant altitude, the quadrant distance difference for each true lunar ecliptic latitude.
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設京師限距地平高度三十四度，太陰距黃道實緯度南北各五度，求距限差。 如圖甲為天頂，乙丙為地平，丁為黃極，甲丁乙丙為黃道經圈，戊己庚為黃道，交地平於己點，其戊點即黃平象限。 戊丙為限距地高三十四度，與甲丁黃極距天頂之度等，而當戊己丙角與乙己庚角為對角，其度亦等。 如月恰在正交或中交，合于黃道之己點，正當地平，則戊己為月距限九十度，若過九十度，自必在地平之下。 今設月在黃道南五度，則辛壬癸為黃道距等圈，月在地平時為壬點，當于黃道之卯，其戊卯月距限乃不及九十度。 又設月距黃道北五度，則子醜寅為黃道距等圈，月在地平時為醜點，當于黃道之辰，其戊辰月距限乃已過九十度，故必求其差數以加減之。 法用己卯壬正弧三角形求己卯弧，此形有卯直角，有己角，當限距地高，有卯壬弧月距黃道緯度。 乃以己角之正切為一率，半徑為二率，卯壬弧之正切為三率，求得四率，為距限差度之正弦，檢表得七度四十二分，即己卯弧為所求之距限差，而與己辰弧之度分等，蓋己辰醜正弧三角形與己卯壬形同用己角，而辰醜弧月距黃道緯度，亦與卯壬等是兩正弧形為相等形，故所得之己卯弧必與己辰弧相等無疑矣。 既得己卯距限差，與戊己九十度相減，得八十二度十八分，即戊卯距限，而與距等圈辛壬之度相應，為月在緯南之地平限度。 以己辰距限差與戊己九十度相加，得九十七度四十二分，即戊辰距限，而與距等圈子醜之度相應，為月在緯北之地平限度也。
Given Beijing limiting altitude 34° and true lunar ecliptic latitude ±5°, find the quadrant distance difference. In the figure: A is the zenith; B–C the horizon; D the ecliptic pole; A–D–B–C the ecliptic meridian; F–G–H the ecliptic (meeting the horizon at G); F is the yellow ecliptic quadrant. Arc F–C = limiting altitude 34°, equal to ecliptic-pole zenith distance A–D; angles F–G–C and B–G–H are opposite and equal. If the Moon is at a node on the ecliptic at G on the horizon, F–G is the lunar horizon limit at 90°; beyond 90° it would lie below the horizon. Suppose the Moon lies 5° south of the ecliptic: circle Xin–Ren–Gui is a latitude parallel; at horizon crossing the Moon is at Ren (ecliptic Mao); F–Mao, the lunar horizon limit, falls short of 90°. If the Moon is 5° north of the ecliptic: circle Zi–Chou–Yin is a latitude parallel; at the horizon the Moon is at Chou (ecliptic Chen); F–Chen already exceeds 90°, so an additive correction is required. Use right spherical triangle Ji–Mao–Ren for arc Ji–Mao: right angle at Mao, angle Ji = limiting altitude, side Mao–Ren = lunar ecliptic latitude. Proportion: tan(Ji angle) : radius :: tan(Mao–Ren) : sin(quadrant distance difference). Table gives 7°42′ for arc Ji–Mao. It equals Ji–Chen because triangles Ji–Chen–Chou and Ji–Mao–Ren share angle Ji with equal latitude sides — congruent right triangles, so the two arcs must match. Quadrant distance difference Ji–Mao subtracted from 90° (F–G) gives 82°18′ for F–Mao — matching parallel Xin–Ren: the horizon limit with south latitude. Adding Ji–Chen to 90° gives 97°42′ for F–Chen — matching parallel Zi–Chou: the horizon limit with north latitude.
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一率己角正切
First ratio: tangent of angle Ji
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二率半徑
Second ratio: radius
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三率卯壬弧正切
Third ratio: tangent of arc Mao–Ren
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四率己卯弧正弦
Fourth ratio: sine of arc Ji–Mao
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圖略
Diagram omitted
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求黃經高弧交角及月距天頂
Finding ecliptic altitude arc intersection angle and Moon's zenith distance
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舊法推日食三差，原以黃平象限為本。 自考成前編謂三差並生於太陰，而太陰之經緯度為白道經緯度，用白道較之用黃道為密，故求三差則按月距白平象限之度，以白道高弧交角及太陰高弧為據。 後編變通其法，乃以白經高弧交角及日距天頂以求三差，而求白經高弧交角，系赤經高弧交角加減赤白二經交角而得，並不求月距白平象限之度，是法較前頗為省算。 今推視差者，乃求其星月黃道同經之視距視時，故三差應由黃平象限而定也。 是則其法原可仿於後編不求黃平象限而竟求黃經高弧交角之術，即黃道高弧交角之餘度。 然非月距黃平象限度與地平限度相較，其月在地平之上下無由可知。 故今求交角，乃先求得月距黃平象限之東西、黃平象限去地之高下、太陰距黃極之遠近，然後按後編用斜弧形求赤經高弧交角日距天頂之法，則黃經高弧交角及月距天頂之度可得矣。
The old solar-eclipse three-difference method took the yellow ecliptic quadrant as its frame. The Comprehensive Calendar front fascicle derives all three differences from the Moon, whose coordinates are white-path coordinates; the white path is tighter than the ecliptic, so the three differences were taken from lunar distance off the white ecliptic quadrant, the white-path altitude arc intersection angle, and the lunar altitude arc. The rear fascicle streamlines this: it derives the three differences from the white-path ecliptic altitude arc intersection angle and solar zenith distance, forming that angle by adding or subtracting the equatorial–white-path longitudinal intersection angle from the equatorial altitude arc intersection angle, and never computes lunar distance off the white ecliptic quadrant — saving work over the earlier method. Occultation parallax seeks apparent separation and time when star and Moon share ecliptic longitude; the three differences must therefore be fixed by the yellow ecliptic quadrant. One may follow the rear fascicle and skip the yellow ecliptic quadrant, computing the ecliptic altitude arc intersection angle directly — the supplement of the ecliptic altitude arc angle. Without comparing lunar distance off the yellow ecliptic quadrant against the horizon limit, one cannot tell whether the Moon is above or below the horizon. The intersection angle is found by first computing lunar distance west of the yellow ecliptic quadrant, quadrant altitude, and lunar distance from the ecliptic pole, then applying the rear fascicle's oblique-spherical method (originally for equatorial altitude arc intersection angle and solar zenith distance) to obtain ecliptic altitude arc intersection angle and lunar zenith distance.
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設星、月黃道經度同為申宮二十六度二十二分十一秒，月距正交前四十三度四十八分五十六秒，黃白交角五度四分一十秒，黃平象限七宮十三度三十七分十七秒，限距地高六十五度三十五分三十六秒，求太陰實緯黃經高弧交角月距天頂。 如圖甲為天頂，甲乙丙丁為子午圈，丙丁為地平，乙為北極，戊己庚為赤道，戊為午正，己為酉正，庚為子正，卯為黃極，辛壬癸子為黃道，壬為春分，癸為夏至，午為黃道交地平之點。 午未弧為九十度，其未點即黃平象限，宮度為七宮十三度三十七分十七秒。 未辰弧當午角為六十五度三十五分三十六秒，即限距地高度，而與甲卯黃極距天頂之度等。 巳寅醜為白道，寅為正交，寅角為黃白交角五度四分一十秒，申為太陰當黃道于酉，申寅為月距正交前白道度四十三度四十八分五十六秒，申酉為月距黃道緯度，其酉點為星月所當之黃道經度五宮二十六度二十二分十一秒，與未點黃平象限宮度相減，得未酉弧四十七度十五分六秒，為月距黃平象限西之度。 乃當未卯酉角，甲申戌為高弧，卯申甲角為黃經高弧交角，甲申為月距天頂。 求法，先用寅酉申正弧三角形，此形酉為直角，有寅角黃白交角，有寅申弧月距正交前白道度，求得申酉弧三度三十分二十七秒，即太陰距黃道南實緯度。 與卯酉象限相加，得卯申弧九十三度三十分二十七秒，為月距黃極。 次用甲卯申斜弧三角形，此形有甲卯邊黃極距天頂，有申卯邊月距黃極，有申卯甲角當酉未弧月距限度為所夾之角，求申角及甲申邊。 乃自天頂作甲亥垂弧，分為甲亥卯、甲亥申兩正弧三角形。 先用甲亥卯正弧三角形，此形亥為直角，有卯角，有甲卯邊，求得卯亥弧五十六度十四分十五秒，為距極分邊。 與申卯弧月距黃極相減，得申亥弧三十七度十六分十二秒，為距月分邊。 次用甲亥申正弧三角形，此形亥為直角，有申亥邊，兼甲亥卯正弧三角形之亥卯邊及卯角。 用合率比例法，求得申角五十六度二分五十一秒，即黃經高弧交角。 仍以甲卯申斜弧形，用對邊對角法，求得甲申弧五十三度四十三分二十四秒，即月距天頂之度也。
Given shared ecliptic longitude 26°22′11″ in mansion Shen, Moon 43°48′56″ before the ascending node on the white path, yellow–white intersection 5°04′10″, yellow ecliptic quadrant at mansion 7, 13°37′17″, limiting altitude 65°35′36″: find true lunar latitude, ecliptic altitude arc intersection angle, and lunar zenith distance. In the figure: A is the zenith; A–B–C–D the meridian (C–D horizon, B north pole); F–G–H the equator (F noon, G west, H midnight); Mao the ecliptic pole; Xin–Ren–Gui–Zi the ecliptic (Ren spring equinox, Gui summer solstice); Wu where the ecliptic crosses the horizon. Arc Wu–Wei = 90°; Wei is the yellow ecliptic quadrant at mansion 7, 13°37′17″. Arc Wei–Chen (angle Wu) = limiting altitude 65°35′36″, equal to ecliptic-pole zenith distance A–Mao. Si–Yin–Chou is the white path (Yin = ascending node, angle Yin = yellow–white intersection 5°04′10″). Shen marks the Moon on the ecliptic at You; Shen–Yin = 43°48′56″ on the white path before the node; Shen–You = lunar ecliptic latitude. You = shared ecliptic longitude 26°22′11″ in mansion 5. Wei–You = 47°15′06″ — lunar distance west of the yellow ecliptic quadrant. Angle Wei–Mao–You: A–Shen–Xu is the altitude arc; angle Mao–Shen–A is the ecliptic altitude arc intersection angle; A–Shen is lunar zenith distance. First solve right triangle Yin–You–Shen (right angle at You, angle Yin = yellow–white intersection, Yin–Shen = lunar arc before ascending node) for Shen–You = 3°30′27″ — true latitude south of the ecliptic. Add to quadrant Mao–You to get Mao–Shen = 93°30′27″ — lunar distance from the ecliptic pole. Next use oblique triangle A–Mao–Shen: sides A–Mao = ecliptic-pole zenith distance, Shen–Mao = lunar pole distance; angle Shen–Mao–A = arc You–Wei (lunar quadrant distance). Solve for angle Shen and side A–Shen. Drop perpendicular arc A–Hai from the zenith to split the figure into right triangles A–Hai–Mao and A–Hai–Shen. First triangle A–Hai–Mao (right angle at Hai, angle Mao, side A–Mao): Mao–Hai = 56°14′15″ — polar component leg. Subtract from Shen–Mao to get Shen–Hai = 37°16′12″ — lunar component leg. Next triangle A–Hai–Shen: right angle at Hai, side Shen–Hai, plus Hai–Mao and angle Mao from the first triangle. Compound proportion gives angle Shen = 56°02′51″ — the ecliptic altitude arc intersection angle. On oblique triangle A–Mao–Shen, opposite sides and angles give A–Shen = 53°43′24″ — lunar zenith distance.
71
圖略
Diagram omitted
72
求太陰距星及淩犯視時
Finding lunar distance from a star and occultation apparent time
73
太陰距地平上之高弧，自地心立算者為實高，在地面所見者為視高，其相差之分，即地半徑差也。 月當地平時，距天頂為九十度，其相差之數最大，而角之正弦即當地之半徑。 迨月上升，則距地漸高，距地愈高，則差數愈小，其所差之分，皆與本時月距天頂之正弦相應，故用比例法而得本時高下差也。 夫高下既差，則有視經、視緯之別。 其視經、實經之差者，東西差也； 視緯、實緯之差者，南北差也。 今求三差，乃依後編日食求三差法用直線三角形算之。 然後編三差圖乃寫渾于平，今則用以渾測渾之圖，求其三差，其所得之南北差，與本時太陰實緯之度相較，而得視緯。 得以視緯與星緯相較，觀其緯之南北而定相距之上下也。 其所得之東西差，與一小時之太陰實行為比例，而得用時距視時之距分。 辨其月距限之東西加減淩犯用時，而得淩犯之視時也。
Lunar altitude above the horizon: geocentric value is true altitude, surface value is apparent altitude; the minute difference is the terrestrial-radius correction (parallax). At the horizon the Moon's zenith distance is 90° and parallax is maximal; the sine of that angle equals the local Earth radius (in the scale used). As the Moon climbs, zenith distance shrinks and parallax diminishes in proportion to the sine of instantaneous lunar zenith distance; the current altitude correction follows by proportion. Altitude displacement then splits true coordinates from apparent longitude and latitude. The difference between apparent and true longitude is the east–west correction; The difference between apparent and true latitude is the north–south correction. The three corrections are found by the rear fascicle's solar-eclipse three-difference method, using rectilinear triangles. The rear fascicle projects the sphere onto a plane in its three-difference diagram; here the sphere-on-sphere diagram is used instead. The north–south correction, applied to the Moon's true latitude, yields apparent latitude. Comparing apparent latitude with the star's latitude shows whether the Moon passes north or south of it. The east–west correction, scaled to the Moon's hourly true motion, gives the time offset between applied and observed times. Apply the east–west correction, with the proper sign for the Moon's position relative to the limit, to the occultation applied time to get the observed occultation time.
74
前求得道光十二年壬辰三月初六日癸丑，月距司怪第四星淩犯用時戌正二刻八分十九秒，黃經高弧交角五十六度二分五十一秒，月距天頂五十三度四十三分二十四秒，本日太陰最大地半徑差六十分七秒，太陰黃道實緯度南三度三十分二十七秒，司怪第四星黃道緯度南三度十一分四十四秒，一小時太陰實行三十六分三十三秒，求星月相距分秒淩犯視時。 如圖甲為天頂，甲未辰巳為黃道經圈，辰午巳為地平，卯為黃極，未午辛為黃道，未點即黃平象限宮度，未辰弧即限距地高，與卯甲黃極距天頂之度等。 申點為太陰，子點為司怪第四星，同當黃道於酉。 其酉點即月與星之黃道經度，酉未弧即月距限西之度，子酉為星距黃道南緯度三度十一分四十四秒，申酉為太陰距黃道南實緯度三度三十分二十七秒，申卯弧即月距黃極，甲申戌為高弧，申甲為月距天頂度五十三度四十三分二十四秒，卯申甲角為黃經高弧交角五十六度二分五十一秒，而與戌申亥角為對角，其度等。 此皆自地心立算之實度也。 然人居地面高於地心，故視高常低於實高，而月當地平時，其地半徑差為最大，今乃六十分七秒。 於是依後編求本時高下差之法，以半徑與甲申弧正弦之比同於最大地半徑差與本時高下差之比，得本時高下差四十八分二十八秒。 如申火之分，其火點即太陰之視高，自火點與黃道平行，作火木線，遂成申木火直角三角形。 因弧度甚小，乃作直線算，與後編求日食三差之理同。 此形木為直角，有申角黃經高弧交角，有申火邊本時高下差，求得木火邊四十分十二秒為東西差，求得申木邊二十七分四秒為南北差，加于申酉太陰實緯，得木酉太陰視緯三度五十七分三十一秒。 內減子酉星緯，得子木弧四十五分四十七秒，為人目仰視太陰距司怪第四星月在星下之分也。 夫星、月同當酉點之經度，固為相距。 今太陰視高在火，其視緯雖差至木，而距星之子點尚在一度內，其火點當黃道之視經度則差至土，是用時時星經度雖在酉，而太陰視經度之土點乃在其西，是為未及。 然土酉之分與火木等，故以一小時太陰實行與火木東西差為比例，得距分一時六分，為月行火木之時分。 加于月視高臨火點之用時，得亥初二刻十四分十九秒，即人目視太陰臨於木點與星，同當酉點經度之視時也。
Using values already found for the third day of the third month, Daoguang 12 (day Guichou): occultation applied time 8:08:19 at the second quarter of Xu; ecliptic longitude altitude-arc angle 56°02′51″; lunar zenith distance 53°43′24″; maximum geocentric parallax 6′07″; Moon true latitude 3°30′27″ S; Fourth Star of Si Guai latitude 3°11′44″ S; hourly true lunar motion 36′33″. Find star–Moon separation and observed occultation time. In the figure: A = zenith; A–Wei–Chen–Si = ecliptic meridian; Chen–Wu–Si = horizon; Mao = ecliptic pole; Wei–Wu–Xin = ecliptic. Wei marks the yellow ecliptic quadrant longitude; arc Wei–Chen is the limiting altitude, equal to co-latitude A–Mao. Shen is the Moon, Zi the Fourth Star of Si Guai; both lie on the ecliptic at You. You is star and Moon ecliptic longitude; arc You–Wei = Moon's westward distance from the limit. Zi–You = 3°11′44″ S (star); Shen–You = 3°30′27″ S (Moon true latitude); Shen–Mao = lunar polar distance; A–Shen–Xu = altitude arc; Shen–A = 53°43′24″ zenith distance; angle Mao–Shen–A = 56°02′51″ (ecliptic longitude altitude-arc angle), equal to opposite angle Xu–Shen–Hai. All of these are geocentric true values. Observers stand above the geocenter, so apparent altitude is always lower than true altitude. At the horizon geocentric parallax is greatest — here 6′07″. By the rear fascicle's altitude-difference method: radius : sin(Shen–A) :: maximum geocentric parallax : present-time altitude difference. The present-time correction is 48′28″. At Shen–Huo, Huo is the Moon's apparent altitude. From Huo draw Mu parallel to the ecliptic, forming right triangle Shen–Mu–Huo. The arc is small enough to treat as rectilinear — the same principle as the rear fascicle's solar-eclipse three-difference method. Right angle at Mu; angle Shen = ecliptic longitude altitude-arc angle; side Shen–Huo = present-time altitude difference. Mu–Huo = 40′12″ (east–west correction); Shen–Mu = 27′04″ (north–south correction). Add to true latitude Shen–You to get apparent latitude Mu–You = 3°57′31″. Subtract star latitude Zi–You to get arc Zi–Mu = 45′47″ — the observer sees the Moon 45′47″ below the Fourth Star of Si Guai. Star and Moon at the same ecliptic longitude You are in conjunction by longitude. The Moon's apparent altitude is at Huo and apparent latitude at Mu; the star at Zi remains within one degree. But apparent longitude shifts from Huo to Tu on the ecliptic — at applied time the star is at You, while the Moon's apparent longitude at Tu still lies west: conjunction has not yet occurred. Tu–You equals Huo–Mu, so scale hourly true lunar motion to the east–west correction Huo–Mu to get 1h06m — the time for the Moon to cover Huo–Mu. Add to the applied time at Huo to get 10:14:19 at the first quarter of Hai — the observed time when the Moon at Mu and the star share longitude You.
75
圖略
Diagram omitted
76
求視時月距限
Finding the Moon's distance from the limit at observed time
77
視時月距限，必大於用時月距限，因其視經差所當之距分既有加減，則太陰與星隨天西移自有進退也。 蓋太陰以地半徑差由高而變下，則視經之差於實經、視緯之差於實緯必矣。 茲據黃平象限在天頂南之地面而言之，視緯恆差而南，如實緯北者，視緯常小於實緯，其差為減； 實緯南者，視緯常大於實緯，其差為加。 故緯南之星、月實距雖在一度內，而視距轉在一度外者有之； 緯北之星、月實距雖在一度外，而視距轉在一度內者有之。 南北相距一度外者不入淩犯之限，故不取用。 至若視經之差，所當月行距分之最大者或至二小時，而二小時之際，諸曜隨天左旋，幾至一宮，故視經之差，關於月行之進退矣。 如月在黃平象限西者，視經度差之而西，視時必遲于用時； 月在黃平象限東者，視經度差之而東，視時必早于用時。 以致用時星、月未入地平，而視時星、月已入地平者有之，或用時星、月已出地平，而視時星、月未出地平者有之。 是故于求用時之後，即以月距黃平象限與地平限度相較，可知斯時月在地平之上下。 月距限小於地平限度者，為月在地平上； 大於地平限度者，為月在地平下。 如遇月距限微小于地平限度者，用時星、月必在地平上，視時星、月或在地平下，其所差者，即視經之差當月行距分之諸曜左旋度。 今取最小實經、視經之差所當左旋之度為視經差，法見下卷求地平限度節下。 減于地平限度，所得視地平限度，而與月距限度考之。 如月距限小於地平限度而大於視地平限度者，則為用時月雖在地平上，視時月必在地平下矣； 既知月必在地平下，故遇此者去之。 如月距限小於視地平限度者，則為視時月在地平之上。 夫猶有不然者，以視經差所取皆最小之數也。 若知月行實跡非由視時，再推月距限度，則其時月果在地平之上下，未可得其確准。 故今于既得視時之後，必詳察太陰實緯及用時月距限度。 如實緯南月距限過六十度，或實緯北月距限過七十度者，用時月距限在此限度內者，視時月必在地平之上。 皆以視時復求月距黃平象限之度。 如其度大於地平限度者，乃視時月在地平之下，仍不取用。 必其度小於地平限度，始為視時月必在地平之上，而可證諸實測。 此視差之所以必逐細詳推，然後可得而取用也。
Observed-time distance from the limit exceeds applied-time distance because the apparent-longitude correction shifts the Moon and star westward with the sky, advancing or delaying conjunction. Geocentric parallax lowers the Moon's apparent altitude, so apparent longitude and latitude must differ from true longitude and latitude. From the surface south of the zenith (where the yellow ecliptic quadrant lies), apparent latitude always shifts south. If true latitude is north, apparent latitude is smaller — a subtractive correction; if true latitude is south, apparent latitude is greater — an additive correction. Star and Moon south of the ecliptic may be within one degree geocentrically yet over one degree apart to the observer; those north of the ecliptic may be over one degree apart geocentrically yet within one degree apparently. North–south separation beyond one degree lies outside the occultation limit and is excluded. The apparent-longitude correction can correspond to as much as two hours of lunar motion; in two hours the sky rotates nearly one mansion westward, so apparent longitude governs whether the Moon leads or lags. If the Moon lies west of the yellow ecliptic quadrant, apparent longitude shifts west and observed time lags applied time; if east of the quadrant, apparent longitude shifts east and observed time precedes applied time. Applied-time values may show star and Moon above the horizon when observed-time values show them below, or applied above horizon when observed still below. After finding applied time, compare the Moon's distance from the yellow ecliptic quadrant with the horizon limit to determine whether it is above or below the horizon. If distance from the limit is less than the horizon limit, the Moon is above the horizon; if greater, it is below the horizon. When distance from the limit is only slightly less than the horizon limit, applied-time star and Moon are above the horizon but observed-time values may place them below — the gap equals the sky's westward rotation over the lunar motion corresponding to the apparent-longitude correction. Use the minimum westward rotation corresponding to the true–apparent longitude difference as the apparent-longitude correction; see the lower fascicle, section on finding the horizon limit. Subtract it from the horizon limit to get the apparent horizon limit and compare with the Moon's distance from the limit. If distance from the limit is less than the horizon limit but greater than the apparent horizon limit, applied-time Moon is above the horizon but observed-time Moon is below; Knowing the observed Moon must be below the horizon, such cases are discarded. If distance from the limit is less than the apparent horizon limit, the observed Moon is above the horizon. Exceptions remain because the apparent-longitude correction uses minimum values. If one recomputes distance from the limit from the Moon's true path rather than observed time, whether it is truly above or below the horizon cannot be determined with certainty. After obtaining observed time, one must therefore examine the Moon's true latitude and applied-time distance from the limit in detail. If true latitude is south and distance from the limit exceeds 60°, or north and exceeds 70°, an applied-time distance within these bounds guarantees the observed Moon is above the horizon. In every case recompute the Moon's distance from the yellow ecliptic quadrant from observed time. If that value exceeds the horizon limit, the observed Moon is below the horizon and the case is still excluded. Only when the corrected distance falls below the horizon limit can one say that at apparent time the Moon must stand above the horizon — a claim that actual observation can confirm. That is why parallax must be derived in careful detail before it can be applied in practice.