Analysis of interval constants of Shoushili-affiliated calendars
aa r X i v : . [ a s t r o - ph . E P ] N ov Research in Astron. Astrophys.
Vol. X No. XX , 000–000 R esearchin A stronomyand A strophysics Analysis of interval constants of Shoushili-affiliated calendars
Byeong-Hee Mihn , , Ki-Won Lee , Young Sook Ahn Advanced Astronomy and Space Science Division, Korea Astronomy and Space Science Institute,Daejeon 305-348, Korea Department of Astronomy and Space Science, Chungbuk National University, Cheongju 361-763,Korea Institute of Liberal Education, Catholic University of Daegu, Gyeongsan 712-702, Korea; [email protected]
Received 2013 June 12; accepted 2013 July 27
Abstract
We study the interval constants that are related to the motions of the Sunand the Moon, i.e., the Qi, Intercalation, Revolution, and Crossing interval constants,in calendars affiliated with the Shoushi calendar (Shoushili), such as Datongli andChiljeongsannaepyeon. It is known that those interval constants were newly introducedin Shoushili and revised afterward, except for the Qi interval constant, and the revisedvalues were adopted in the Shoushili-affiliated calendars. In this paper, we investigatefirst the accuracy of the interval constants and then the accuracy of the Shoushili-affilatedcalendars in terms of the interval constants by comparing the times of the new moonand the solar eclipse maximum calculated by each calendar with modern calculations.During our study, we found that the Qi and Intercalation interval constants used in theearly Shoushili were well determined, whereas the Revolution and Crossing interval con-stants were relatively poorly measured . We also found that the interval constants usedby the early Shoushili were better than those of the later one, and hence than those ofDatongli and Chiljeongsannaepyeon. On the other hand, we found that the early Shoushiliis, in general, a worse calendar than Datongli for use in China but a better one thanChiljeongsannaepyeon for use in Korea in terms of the new moon and the solar eclipsetimes, at least for the period 1281 – 1644. Finally, we verified that the sunrise and sun-set times recorded in Shoushili-Li-Cheng and Mingshi are those at Beijing and Nanjing,respectively.
Key words: history and philosophy of astronomy: general — celestial mechanics —ephemerides
The Shoushi calendar (Shoushili, hereinafter) was developed by Shoujing Guo and his colleaguesfrom the Yuan dynasty (Bo 1997). It is known as one of most the famous calendars in Chinesehistory (Needham 1959). Its affiliated calendars, the Datongli of the Ming dynasty and theChiljeongsannaepyeon (Naepyeon, hereinafter) of the Joseon dynasty, are extremely similar. Comparedto previous Chinese calendars, the most distinguishing characteristic of the Shoushili is the abolition ofthe use of the Super Epoch, an ancient epoch where the starting points of all interval constants are thesame (Sivin 2009). Instead, Shoushili adopted the winter solstice of 1280 (i.e., 1280 December 14.06
B.-H. Mihn et al. in Julian date) as the epoch and introduced seven interval constants from observations in order to re-duce the large number of accumulated days in the calculations arising from the use of the Super Epoch.Of the seven interval constants, four are related to the motions of the Sun and the Moon: the Qi, Ren(Intercalation), Zhuan (Revolution), and Jiao (Crossing) interval constants. According to the records ofMingshi (History of the Ming Dynasty), the early values of these interval constants, except for the Qiinterval constant, were revised later. Currently, it is known that Datongli and Naepyeon adopted the re-vised values of the early Shoushili; hence, both calendars are essentially identical to the later Shoushili.The main purpose of this study is to evaluate the effect of the values of these four interval constants onthe accuracy of calendar calculations. In this paper, therefore, “Shoushili” refers to the early one unlessstated ohterwise.In this study, we first estimate the accuracy of the values of the Qi, Intercalation, Revolution, andCrossing interval constants of the Shoushili-affiliated calendars. We then investigate the accuracy ofeach calendar in terms of the interval constants by comparing the timings of the new moon and thesolar eclipse maximum calculated by each calendar with modern calculations. Because sunrise andsunset times are needed to calculate the timings of the solar eclipse maximum in the Shoushili-affiliatedcalendars, we verify those times presented in calendar books as well.This paper is composed as follows. In section 2, we briefly introduce the Shoushili-affiliated calen-dars and modern astronomical calculations. In section 3, we present the results on the analysis of the fourinterval constants, the new moon times, the sunrise and sunset times, and the solar eclipse maximumtimes. Finally, we summarize our findings in section 4.
In the chapter related to calendars in the Yuanshi (History of the Yuan Dynasty), Shoushili is described ascovering two parts: Shoushili-Yi (Discussion) and Shoushili-Jing (Method). It is known that the calendarwas used in the Goryeo dynasty of Korea since the reign of King Chungseon (1308 – 1313) (Jeon 1974).For this reason, Shoushili is also recorded in the chapter related to calendars in the Goryeosa (History ofthe Goryeo Dynasty). However, the Goryeosa does not include the Shoushili-Yi. Conversely, the Yuanshicontains no Li-Cheng (Ready-reference Astronomical Tables), which is presented in the Goryeosa. Inaddition, both historical books have no tables on the sunrise and sunset times.A book entitled Shoushili-Li-Cheng (Ready-reference Astronomical Tables for Shoushili) is pre-served in the Kyuganggak Institute for Korean Study (Kyuganggak, hereinafter) in Korea. This bookcontains various tables for the use of calendar calculations by Shoushili along with the times of sun-rise and sunset times (for further details, see Lee & Jing 1998). We have to refer to three books (i.e.,Yuanshi, Goryeosa, and Shoushili-Li-Cheng) to completely understand Shoushili. Kyuganggak also pos-sesses a book called Shoushili-Jie-Fa-Li-Cheng (Expeditious Ready-reference Astronomical Tables forShoushili). According to the preface of the Kyuganggak edition, this book was brought from China byBo Gang (an astronomer in the Goryeo court). It was printed in 1346, and reprinted in 1444, around thepublication year of Naepyeon.
There are two versions of Datongli in the Ming dynasty; Wushen-Datongli (Datongli of the WushenYear) made by Ji Liu in 1368, and Datong-Lifa-Tonggui (Comprehensive Guide to the CalendricalMethod by Datongli) made by Tong Yuan in 1384. Although the epochs of the former and the lattercalendars are the winter solstices of 1280 and 1383, respectively, both are based on the Shoushili (Lee1996). In the chapter related to calendars in the Mingshi, Datongli is described as having three parts:the first is on the origin of the techniques, the second on Li-Cheng including the tables on the timingsof sunrise and sunset, and the third on calendar methods. nterval constants of Shoushili-affiliated calendars 3
Table 1
Summary of the four interval constants adopted in the Shoushili-affiliated calendars.
Interval Shoushili Datongli NaepyeonConstants Yuanshi Goryeosa Mingshi Tonggui SeriesQi 550,600 550,600 550,600 550,600 550,600Intercalation 201,850 202,050 202,050 202,050 202,050Revolution 131,904 131,904 130,205 130,205 130,205Crossing 260,187.86 260,388 260,388 260,388 260,388
Notes: Values when the epoch is the winter solstice of 1280.
It is widely known that Datongli was used in the Goryeo dynasty since 1370, although this fact isarguable (refer to Lee et al. 2010). Unlike Shoushili, Datongli is not included in the chapter related tocalendars in the Goryeosa. Instead, a series with a name similar to Datong-Lifa-Tonggui of Tong Yuan,for example, Datong-Liri-Tonggui (Comprehensive Guide to the Calendar Day by Datongli), is pre-served in the Kyuganggak. According to the work of Lee (1988), the series (Tonggui series, hereinafter)is related to the Datongli and was published around 1444 and its main purpose was being a referenceduring the compilation of the Naepyeon.
King Sejong of the Joseon dynasty ordered In-Ji Jeong et al. to compile the Naepyeon in 1433. Althoughthe data on when the compilation was completed is not clear, the oldest extant version is the one pub-lished in 1444 by Sun-Ji Yi and Dam Kim. In terms of the contents, each chapter of the Naepyeoncorresponds to the Tonggui series and the timings of sunrise and sunset are contained in the Taeum(Moon) chapter. In addition, these times are different in the Shoushili-affiliated calendars. Because thecompilation of the Naepyeon was considered to be one of the greatest works of King Sejong, this bookis also appended in his Veritable Record, unlike Veritable Records of other king’s (Lee et al. 2008).A book entitled Jeongmyoyeon-Gyeosik-Garyeong (Example Supplement for the Calculations ofthe Solar and Lunar Eclipses Occurred in 1447; shortly Garyeong), which is related to the Naepyeon,also remains in the Kyuganggak. This book contains modified interval constants for the epoch of 1442and is valuable for step-by-step calculations of not only the solar eclipse but also the new moon by theNaepyeon, and hence by the Shoushili or Datongli. In this study, we refer to Garyeong to calculate thenew moon and solar eclipse maximum times by the Shoushili-affiliated calendars, and we refer to thework of Lee (1988), which introduced the differences among the calendars.In Table 1, we summarize the values of the four interval constants in each calendar. All dates aregiven in the Julian calendar and all values of interval constants are in units of Part; one day is 10,000Parts. The Qi interval constant is the interval between the epoch and the midnight of the first day in asexagenary cycle on counting backwards from the epoch. This value is same in all Shoushili-affiliatedcalendars. Intercalation interval constant is the interval from the epoch to the ‘mean’ new moon ofthe month belonging to the epoch. The time of a new moon is determined by correcting the slowness orfastness of the solar and lunar motions to the ‘mean’ new moon time, which is obtained by accumulatingthe Intercalation interval constant. Revolution and Crossing interval constants are the lengths betweenthe epoch, and the times of lunar perigee and descending node passage, respectively (see also Sivin2009).As can be seen in Table 1, the values of interval constants in Datongli and Naepyeon are identicalto each other. Interestingly, the numbers for the Shoushili of the Goryeosa are the same as those for theNaepyeon and the Datongli, except for the Revolution interval constant.
In modern calculations, we use the astronomical algorithms of Meeus (1989, 1998) and the DE406ephemeris of Standish et al. (1997). In addition, we use Besselian elements to calculate the solar eclipse
B.-H. Mihn et al.
Table 2
Summary of the dates in the Shoushili-affiliated calendars and modern calculationsrelating to the four interval constants.
Item Shoushili-affiliated calendars (A) Modern Calculation (B) B − A Calendar RemarkJulian Date JD − − Dec. 14.060000 0.000000 0.011638 16.8 S, D, N EpochMFDSC Oct. 20.000000 − − Nov. 23.875000 − − Nov. 24.211966 − − − − Nov. 30.869600 − − − − Nov. 18.041214 − − − − − − Notes: Winter solstice of 1280, Midnight of the first day in a sexagenary cycle on counting backwardsfrom the epoch, Mean new moon, new moon, Lunar perigee passage, Lunar descending node passage, S:Shoushili, D: Datongli, N: Naepyeon time in a local circumstance. Although Muckes & Meeus (1983) tabulated Besselian elements for thesolar eclipses ranging from − ∆ T ,difference between the universal time (UT) and the dynamical time (TD). To estimate ∆ T for a givenyear, we employ the cubic spline interpolation method (see Press et al. 1992) using the data obtainedrecently by Morrison & Stephenson (2004). To directly compare the results of modern calculations withthose by the Shoushili-affiliated calendars, we convert the universal time into the local apparent solartime by correcting the equation of time. Lastly, we assume that the locations of Beijing, Nanjing, andSeoul are at 39 ◦ ′ N and 116 ◦ ′ E, 32 ◦ ′ N and 118 ◦ ′ E, and 37 ◦ ′ N and 126 ◦ ′ E,respectively.
A winter solstice was used as the epoch in ancient Chinese calendars. In the Shoushili, the values ofthe interval constants were based on the winter solstice of 1280, as mentioned earlier. It is known thatShoujing Guo determined the date of the winter solstice from gnomon shadow measurements (Chen1983; Li 2005). He used a tall gnomon and estimated the moment when the length of the shadow,caused by the Sun, is the longest, based on several days’ observations. In modern times, the wintersolstice is calculated as the time when the Sun is passing the ecliptic longitude ( λ ) of 270 ◦ , using anastronomical ephemeris such as DE406. For the details on how Shoujing Guo determined the times ofthe winter solstice and of the lunar perigee and descending node passages, refer to Yuanshi.In Table 2, we present the dates related to four interval constants in the Shoushili-affiliated calen-dars along with the results from modern calculations. All dates are in the Julian calendar, in units ofthe apparent solar time at Beijing unless otherwise mentioned. The difference in the equation of timeaccording to the regions (i.e., Beijing, Nanjing, and Seoul), is negligible. Hence, we can easily convertthe times at Beijing into times at other regions by correcting only the longitudinal difference. Therefore,the time at Seoul, for example, is obtained by adding the time at Beijing with +42.26 min (i.e., 10.566 ◦ of longitudinal difference between Beijing and Seoul).In the table, the first column contains items related to the interval constants; WS1280 is the wintersolstice of 1280, i.e., the epoch of the Shoushili-affiliated calendars. MFDSC is the midnight of thefirst day in the sexagenary cycle (i.e., Jiazi day) before the epoch. MNM and MN are mean new moon nterval constants of Shoushili-affiliated calendars 5 and new moon, respectively, and LPP and LDNP are lunar perigee and descending node passages,respectively. The second column contains the Julian dates derived from the epoch and the values ofinterval constants in the Shoushili-affiliated calendars, except for WS1280, the epoch itself, and NM.The third column is the day number obtained by subtracting 2188925.56 d (i.e., the epoch) from theJulian day number (JD) corresponding to the date given in the second column. The fourth and fifthcolumns are the results of modern calculations and the difference between the modern calculations andthe values derived from the Shoushili-affiliated calendars, respectively. The sixth column represent thecalendar; S is Shoushili, D Datongli, and N Naepyeon. The last column contains the interval constantsrelated with the items in the first column.According to modern calculations, ∆ T in 1280 is 532.6 s and JD of the winter solstice of the yearat Beijing is 2188925.571638d (i.e., 1280 December 14.071638), which is obtained by correcting theequation of time by − ∆ T = 65.9 s (U.S. Nautical Almanac Office 2009), com-pare the result with the data of the Purple Moumtain Observatory Chinese Academy of Science (2009,shortly CAS2009), and find a good agreement in the values, with the JD being 2455552.485037d (i.e.2010 December 21.985037) in UT (see als Korea Astronomy and Space Science Institute 2009).Although it is known that Shoujing Guo also determined the Intercalation interval constant basedon observations, there is not much detail on the calculations he used (Li & Zhang 1998a). Hence, wecalculated the date of the new moon in 1280 November as an indirect method to verify the Intercalationinterval constant. The dates are calculated to be JD 2188905.711966 and JD 2188905.719181 d (or1280 November 21.344808 in UT) by Shoushili and modern calculations, respectively, which gives justa difference of only +10.4 min. To check our calculation, we also compare the time of the new moonwith the data provided by NASA and find that the difference is less than 1 min. The exact times atwhich the astronomers of the Yuan dynasty modified the values of the interval constants of the Shoushiliare not known. On using Datongli’s value of the Intercalation interval constant, the difference increases,becoming ∼ ∼ ∼
216 min; Chen 2006)and ∼ ∼ ∼ In the Shoushili-affiliated calendars, the time of any phase of the Moon, for example, new moon, isdetermined in the following manners. First calculate the mean new moon time using the Intercalationinterval constants and the length of the synodic month (i.e., 295305.93 Parts in the Shoushili-affiliatedcalendars). The new moon time is then determined by using the Revolution interval constant and byconsidering the motions of the Sun and the Moon. Because there is no Li-Cheng in Yuanshi, we use thevalues of Shoushili-Li-Cheng preserved in Kyuganggak for the the motions of the Sun and the Moon. http://eclipse.gsfc.nasa.gov/ phase/phasecat.html B.-H. Mihn et al.
Fig. 1
The number distribution showing the differences in the new moon times between theShoushili-affiliated calendars and modern calculations at (a) Beijing and (b) Seoul. The red-solid and blue-dotted lines represent the results for Shoushili and Datongli (Naepyeon in (b)),respectively.For the sake of completeness, we check the values of Shoushili-Li-Cheng against Mingshi and Naepyeonand find that all values are identical to each other, except for a few typos in each book.Comparing the length of the synodic month in the Shoushili with that obtained by modern calcula-tions (i.e., 29.530587 d in 1280; refer to CAS2009), the difference is less than 1 s. Hence, we ignore theerror in the length of the synodic month in the Shoushili. Instead, it is worth noting that the motion of theSun speeds up or slow down before and after the perihelion or aphelion and not the winter or summersolstice as mentioned in the Shoushili. However, it is well known that the time of the winter solsticewas very close to the perihelion passage time of the Earth around 1280. According to our investigations,the Earth passed the perihelion on the JD 2188925.078000d resulting in the difference of 0.48200 dcompared to the time of the winter solstice in 1280.To evaluate the effect of the Intercalation and the Revolution interval constants in determining thenew moon times, we compute those times for the Shoushili-affiliated calendars for the years rangingfrom 1280 to 1644, compare them with the results of modern calculations, and present the numberdistribution of the differences in Figure 1 along with the root-mean-square (RMS) values. In the figure,the panels (a) and (b) show the results at Beijing and Seoul, respectively. The red-solid and blue-dottedlines represent the results for Shoushili and Datongli (Naepyeon in Fig. 1(b)), respectively. In eachpanel, the horizontal and vertical axes represent the difference in units of minutes with the intervals of15 min and the number, respectively.As shown in Figure 1, the new moon times given by the Datongli are on an average more accuratethan those given by Shoushili at Beijing, i.e., RMSs are ∼ ∼
21 min; Li & Zhang (1998a,b))and ∼ ∼ ∼ ∼ ∼ nterval constants of Shoushili-affiliated calendars 7 Fig. 2
Nighttime lengths (a) from Shoushili-Li-Cheng at Beijing and (b) from Mingshi(Taiyin-Tonggui) at Nanjing. The blue-dotted and red-solid lines indicate results from theliterature and the modern calculations, respectively, at each region. In each panel, the upperand bottom are nighttime lengths after the winter and summer solstices, respectively. For moredetails, refer to text.
One way to assess the overall accuracy of the four interval constants is to check the solar eclipse time.Before that, we verify the timings of sunrise (SR) and sunset (SS) presented in the calendar booksbecause these times are used for calculating the times of the solar eclipses. In Figure 2, we depict thenighttime lengths (i.e., period from SS to SR) from the Shoushili-Li-Cheng and the Mingshi along withthe results of modern calculations. In the figure, the blue-dotted lines represent the results (a) fromShoushili-Li-Cheng at Beijing and (b) from Mingshi at Nanjing, while the red-solid lines are the resultsfrom modern calculations at each region. In each panel, the upper and bottom panels show the nighttimelengths after winter and summer solstices, respectively. The horizontal and vertical axes represent theday number and the nighttime length in units of hours, respectively. According the Shoushili-affiliatedcalendars, the summer solstice of 1281 is June 14, i.e., JD = 2189107.5 d (Lee et al. 2010). In this study,we define the SR/SS time as the zenith distance ( z ) of 90 ◦ , which is different from the modern definitionof z = 90 ◦ ′ . The result for Naepyeon at Seoul is given in the work of Lee et al. (2011), and shows adifference of less than 1 min on average.A table on the sunrise and sunset times is also contained in the Taiyin-Tonggui (ComprehensiveGuide on the Sun), which is one of Tonggui series. According to our observations, there were somedisagreements between Mingshi and Taiyin-Tonggui in the SR and SS times. We can easily figure outwhich document is incorrect by checking Daybreak (period from midnight to sunrise or from sunset tomidnight) and Dusk (period from midnight to sunset) Parts because the sum of both parts should be10,000 Parts, i.e., one day. In spite of this check, there were two discrepancies. However, these can beignored because the differences were less than 1 Part. Hence, we can consider that the values of the SRand SS times are identical in both documents. Although there is no statement in the Taiyin-Tonggui,the Mingshi explicitly states that the SR and SS times are those at Nanjing. As shown in Figure 2, theRMSs are less than ∼ B.-H. Mihn et al.
Fig. 3
Differences in the solar eclipse maximum time between the Shoushili-affiliated calen-dars and modern calculations according to the region (a) at Beiing and (b) at Seoul. The circlesand crosses represent the results for Shoushili and Datongli (Naepyeon in (b)), respectively.
Based on the four interval constants and the SR/SS times discussed above, we calculate the timings ofthe solar eclipse maximum according to the calendars and compare these with the results of moderncomputations. Prior to the comparison, we validate our calculations by the Shoushili-affiliated calen-dars using the records in the historical literature. In the Garyeong, the procedure for calculating solarand lunar eclipses by Naepyeon is described in great detail. A calendar book entitled Jiaosi-Tonggui(Comprehensive Guide on the Eclipse), which is preserved in Kyuganggak, also lists step-by-step val-ues in the process of calculating several eclipses by Datongli. We find that the results of our calculationsof Naepyeon and Datongli show exact agreement with those of the documents. In particular, we findthat the solar eclipse times of Jiaosi-Tonggui come from the result that used the SR/SS times at Nanjingand not Beijing. We compare the calculations in the Shoushili with the records in the Mingshi. In thehistory book, the times for a total of 32 solar eclipses, calculated by Shoushili, are recorded (see alsoYabuuchi & Nakayama 2006). We find that all times match well except for four records: 707 June 1,1059 January 1, 1061 June 1, and 1162 January 1 in the luni-solar calendar. According to our computa-tions based on the Shoushili, there is no solar eclipse in 707, whereas the others show a difference of 1Mark ( ∼
15 min). However, we think that the actual differences would be smaller than 1 Mark, which isthe significant digit in the records. For the hour systems used in ancient China and Korea, (refer to Saito1995; Lee et al. 2011).In this study, we restrict ourselves to the cases where the eclipse maximum occurred during daytime,i.e., the Sun’s altitude is greater than zero at the eclipse maximum. Figure 3 shows the difference betweenthe times given by the Shoushili-affiliated calendars (Ts) and by modern calculations (Tm), i.e., Ts − Tm, between 1281 and 1644, along with RMS. We use SR/SS times (a) at Beijing and (b) at Seoul. Ineach panel, the circles and crosses represent the results for Shoushili and Datongli (Naepyeon in (b)),respectively.According to this study, the RMSs for Shoushili and Datongli at Beijing are about ∼ ∼ ∼
24 min. From the figure, we can easily see that the interval constants of the nterval constants of Shoushili-affiliated calendars 9
Datongli are better than those of Shoushili at Beijing, which is similar to the case of the new moontimes. The situation is reversed at Seoul (RMSs for Shoushili and Naepyeon are ∼ ∼ It is known that Shoushili, of the Yuan dynasty, is one of the most accurate calendars in the history ofChina. The court of the next dynasty, i.e., the Ming dynasty, revised the calendar and titled it as Datongli.An example of the revisions is that the annual precession was abandoned. In the Joseon dynasty of Korea,both calendars were referred to for the compilation of Naepyeon by the Joseon royal astronomers. Withregard to the interval constants, the Joseon court adopted the values of Datongli (i.e., the later Shoushili)in Naepyeon. Although there are some differences, particularly in the values of the interval constants,the Datongli and the Naepyeon are basically identical to the (early) Shoushili.In this paper, we study the four interval constants given in the Shoushili-affiliated calendars, whichare related to the motions of the Sun and the Moon: the Qi, Intercalation, Revolution, and Crossinginterval constants. We first compared the values of those interval constants with the results of moderncalculations, and then investigate on the accuracy of the timings of the new moon and the solar eclipsemaximum given by the Shoushili-affiliated calendars using the interval constants values adopted in eachcalendar, along with timings of sunrise and sunset. The following is the summary of our findings.(1) In the Shoushili, the Qi and Intercalation interval constants are well determined (errors of ∼ ∼ ∼ ∼ ∼ ∼ Acknowledgements
Ki-Won Lee is supported by Basic Science Research Program throughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education(2013R1A1A2013747).
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