First draft: August, 2019.

By examining the times of lunar conjunctions listed in several *Dàtǒng* calendars, which were the official state calendars, published by the Ming government in the 16th and 17th century, I am able to determine the exact method used in the conjunction calculation by astronomers in the Imperial Astronomical Bureau at that time. The validated conjunction calculation is used to compute all lunar conjunctions in 1369-1644 and the dates are compared with those listed in the book *3500 Years of Calendars and Astronomical Phenomena*^{}. I find 11 mismatches and these same 11 mismatches also appear in three other calendar data books: *A Sino-Western Calendar For Two Thousand Years (1-2000)*^{}, *Tables of Historical Lunar Conjunctions and Leap Months*^{} and 《歷代長術輯要》(*Compilation of Historical Calendars*) by Wāng Yuēzhēn (汪曰楨). In 7 of the 11 mismatches, I am able to find the *Dàtǒng* calendars for those years and find that all 7 dates match the conjunction calculations. This means that the 7 dates listed in the four calendar books are wrong.

**Contents**

- Introduction
- Conjunction Calculation in the
*Dàtǒng*System - Examining the True Conjunction Times
- Corrections to Calendar Data

Almost all of the ancient Chinese calendar data on this website are based on the book *3500 Years of Calendars and Astronomical Phenomena*^{}. There are other calendar data books used by historians and chronologers. Three of them are *A Sino-Western Calendar For Two Thousand Years (1-2000)*^{}, *Tables of Historical Lunar Conjunctions and Leap Months*^{} and 《歷代長術輯要》(*Compilation of Historical Calendars*) by Wāng Yuēzhēn (汪曰楨). The ancient Chinese calendar data on the Chinese-Western calendar conversion website created by Academia Sinica in Taiwan are based on the book *A Sino-Western Calendar For Two Thousand Years (1-2000)*. However, some of the data in these books don't agree. The most reliable way to resolve the discrepancies is to compare the dates with those listed in the official state calendars for those years.

In imperial China, the production and distribution of official state calendars were regarded as the imperial prerogative. At times the private possession or use of unofficial calendars was outlawed. The imperial calendar was an integral part of the Chinese ideological and political system. Foreign states that were regarded as vassals of China also adopted the imperial calendars to record events. There were periods when China was divided into two or several states. Each state published its own calendar and the start and end date of a month might differ in these calendars. Therefore, the dates listed in the imperial calendars are the ones any Chinese calendar data book should rely on. While most of the ancient imperial calendars were lost, many of the imperial calendars published in and after the 15th century are still preserved to this date, providing valuable information for the study of the calendar calculation in those times. Here I focus on the study of the *Dàtǒng* calendars, the imperial calendars published in the Ming dynasty between 1369 and 1644.

The term *Dàtǒng lì* (大統曆) can refer to both the imperial calendars published by the Ming government and the system of mathematical astronomy used to perform various astronomical calculations in the Ming dynasty. To prevent confusion, here I use *Dàtǒng* calendars to refer to the imperial calendars and *Dàtǒng* system to refer to the astronomical system. *Dàtǒng* calendars can be found in several places today. The National Central Library at Taipei has a collection of a number of *Dàtǒng* calendars. The first pages of these calendars can be found on Digital Taiwan — Culture & Nature. World Digital Library has the *Dàtǒng* calendar in the third Year of Emperor Jia Jing's Reign in the Ming Dynasty (1524) in pdf format. In 2007, the National Library of China Publishing House published six volumes of *Guó jiā tú shū guǎn cáng míng dài dà tǒng lì rì huì biān* ^{}(《國家圖書館藏明代大統曆日彙編》or *Compilation of Dàtǒng calendars collected by the National Library of China*). The six volumes collected 99 imperial calendars from 1446 to 1641. This page lists the calendars collected by the six volumes for those who can read Chinese. Hereafter I use *Compilation* to refer to the 6 volumes of the book and *Compilation (volume number)* to refer to a specific volume of the book (e.g. *Compilation (3)* means volume 3 of the book). The six volumes show images of the printed copies of the imperial calendars. As these calendars were printed hundreds of years ago, the pages suffer from various degrees of damages. Some pages are missing, some pages have tears, some characters are hard to see and so on. There are also pages that are well-preserved.

The structure of all *Dàtǒng* calendars is the same. The cover prints the year the calendar was for. There is a forgery warning (either on the cover or on the front page):

The Imperial Astronomical Bureau is authorized to publish and distribute this

Dàtǒngcalendar.Those who forge copies are subject to decapitation, and those who inform on the forgers will receive a reward of 50 ounces of silver. Copies without the stamps of the Imperial Astronomical Bureau are regarded as private calendars.

The first two pages of a *Dàtǒng* calendar list the dates of lunar conjunctions and dates and times of 24 solar terms in the year. Dates are represented by their sexagenary cycles. Since in the Chinese calendar a lunar conjunction is designated as the first day of a month, each conjunction is labelled by its month number. A month is also labelled as 小 (literally "small") or 大 (literally "big"), depending on whether it contains 29 or 30 days. (On this website, months with 29 days are called short months and those with 30 days are called long months.) The next two pages display a "Diagram of the Position of the Spirits for the Year" (年神方位之圖). It was used for Chinese geomancy, often called *fengshui*. The following dozens of pages list the calendar dates in the entire year. Each day occupies a column, and a column contains what we now call astrology information: certain activities are indicated as auspicious/inauspicious. The last few pages list the era names and year numbers of the previous 60 years. Only the lunar conjunction data on the first two pages are relevant to the studies here.

While all *Dàtǒng* calendars list the dates and times of 24 solar terms in the year, most of them only list the dates of lunar conjunctions. However, there are a few copies of the calendars in which dates and times of lunar conjunctions are listed. When I saw them, I immediately checked the dates and times with my own calculation and confirmed that they matched. In fact, Yong Li^{} already confirmed that the dates and times of the 24 solar terms listed in the *Dàtǒng* calendar for the year 1527 matched his calculation, but he didn't compare the times of lunar conjunctions, which were not available in the *Dàtǒng* calendar for 1527. Seeing the agreement between my calculation and the data in the imperial calendars reminded me of one thing.

About two months ago, I wrote a code to compute the lunar conjunctions and solar terms in the Ming dynasty. I compared the dates in 1369-1644 with those in the book *3500 Years of Calendars and Astronomical Phenomena*. I found complete agreement in solar terms but 11 mismatches in lunar conjunctions. At the time I wasn't concerned with the mismatches since it is well-known that the dates listed in ancient imperial calendars didn't always follow the rules stated in the official historical documents. Sometimes, the imperial astronomers might change the rules slightly, or adopted a new method, or moved a leap month to avoid a solar eclipse occurring on a New Year day, or altered the dates for political reasons and so on (see, e.g., this article^{} by Yi-Long Huang). Not all of the alternations were recorded. Now that I saw my calculations match all the dates and times listed in several imperial calendars, I decided to investigate the mismatches. I checked the dates of the 11 mismatches in three other calendar data books (*A Sino-Western Calendar For Two Thousand Years (1-2000)*, *Tables of Historical Lunar Conjunctions and Leap Months* and *Compilation of Historical Calendars* mentioned above) and found that the data in these three books agree with *3500 Years of Calendars and Astronomical Phenomena*. By searching the imperial calendars for those years, I was able to find the information of 7 of the 11 mismatched lunar conjunctions. To my surprise, all 7 dates listed in the imperial calendars match my calculation, meaning that agreement of data in calendar books is no guarantee for the data's accuracy. The result will be presented in the last section.

Since the article by Li already confirms that the 24 solar terms listed in the imperial calendars match the calculation stated in the *Dàtǒng* system, I will not repeat the same calculation here and will focus on the lunar conjunction calculation. The article by Li mentions two different methods of conjunction calculation, and these two methods don't agree. The actual method used by the *Dàtǒng* system may be determined by comparing the dates and times of lunar conjunctions listed in the *Dàtǒng* calendars. This is the subject of the following two sections. The *Dàtǒng* system was also used to compute the calendars in the Southern Ming and Zheng dynasty in 1645-1683, but the calculations were slightly different from the method used in the Ming dynasty, as revealed from the dates of those calendars. The issue is briefly analyzed in my follow-up article "Calendar Dates in Southern Ming and Zheng Dynasty".

*Dàtǒng* system adopted the same astronomical system as the *Shòushí* system (授時曆) developed in 1280 with only minor modifications. The conjunction calculation in the *Shòushí* system is explained in and and will not be described here. However, the general method used to compute the true conjunctions will be described below in order to explain a possible difference in the conjunction calculation between the *Dàtǒng* system and *Shòushí* system.

Since the middle of the 7th century, the lunar conjunctions in the imperial calendars had been calculated based on *díngshuò* (true conjunction), which took into account the non-uniform motions of the Sun and Moon. True lunar conjunctions are defined as the times when the apparent longitude of the Moon λ_{M} is equal to the apparent longitude of the Sun λ_{s}. The calculation of lunar conjunctions boils down to solving the non-linear equation

f(t) = λ_{M}(t) - λ_{s}(t) = 0. (1)

One way to solve this equation is to use the Newton-Raphson method. This is an iterative method in which the approximate root t_{n+1} in the (n+1)th iteration is obtained from the approximate root t_{n} in the previous iteration by

t_{n+1} = t_{n} - f(t_{n})/f'(t_{n}), (2)

where f' is the time derivative of f. If we choose the initial guess t_{0} as the time of the mean lunar conjunction, i.e. conjunction determined by the mean motions of the Sun and Moon. The first iterative approximation is given by

Approximate true conjunction time = t_{0} - f(t_{0})/f'(t_{0}). (3)

The function -f can be written as

-f(t) = (λ_{M}(t) - λ_{M}(t)) + (λ_{s}(t) - λ_{s}(t)) + λ_{s}(t) - λ_{M}(t),

where λ_{M} and λ_{s} are the mean longitude of the Moon and Sun, respectively. That is, they are computed by taking into account the mean motions of the Moon and Sun. The first term λ_{M}-λ_{M} is called the *lunar correction function* and the second term λ_{s}-λ_{s} is called the *solar correction function*. Denote these two correction functions by S(t) and T(t):

S(t) = λ_{M}(t) - λ_{M}(t),

T(t) = λ_{s}(t) - λ_{s}(t).

The time derivative of f is

f'(t) = λ'_{M}(t) - λ'_{s}(t) = V_{M}(t) - V_{s}(t),

where V_{M} = λ'_{M} is the Moon's apparent angular speed (projected onto the ecliptic) and V_{s} = λ'_{s} is the Sun's apparent angular speed. At mean lunar conjunction t_{0}, λ_{M}=λ_{s} and so the first iteration gives the following approximation to the true lunar conjunction:

Approximate (true) conjunction time t = t_{0} + [S(t_{0}) + T(t_{0})]/[V_{M}(t_{0}) - V_{s}(t_{0})]. (4)

Computation of true lunar conjunctions in ancient China used formulae that resembled (4). The formula adopted by the *Shòushí* system was

t = t_{0} + 0.082×[S(t_{0}) + T(t_{0})]/V(t_{0})

= t_{0} + [S(t_{0}) + T(t_{0})]/V_{M}(t_{0}). (5)*Shòushí* system adopted 27.5546 days as the anomalistic month and divided the anomalistic month into 336 *Xiàn* (限). As a result, one *Xiàn* has 27.5546 days/336 = 0.082 days. The function V represents the increase in the Moon's longitude λ_{M} in one *Xiàn* in units of *dù* (度). Here *dù* is the increase in the mean Sun's longitude in one day. *Shòushí* system adopted 365.2425 days as the tropical year. Hence one *dù* is 360°/365.2425 or about 0.9856°. As a result, V/0.082 is V_{M} in units of *dù*/day. Using V_{M} instead of V_{M} - V_{s} was not very accurate, but it was an improvement over many of the previous astronomical systems in which the mean lunar angular speed V_{M} was used.

Near the end of the 12th century, astronomer Yáng Zhōngfǔ (楊忠輔) discovered that the tropical year was not a constant. *Shòushí* system accepted this idea and stipulated that the tropical year would be shortened by 0.0001 days every 100 years. However, *Shòushí* system was only used for 88 years and was replaced by the *Dàtǒng* system, which abandoned this shortening of tropical year. So the tropical year was always 365.2425 days in the *Dàtǒng* system. The official history for the Ming dynasty also gives a slightly different formula for the calculation of true lunar conjunctions in the *Dàtǒng* system:

t = t_{0} + 0.082×[S(t_{0}) + T(t_{0})]/[V(t_{0}) - 0.082] (6)

= t_{0} + [S(t_{0}) + T(t_{0})]/[V_{M}(t_{0}) - V_{s}],

where V_{s} = 0.082 *dù*/*Xiàn* = 1 *dù*/day is the mean angular speed of the Sun. Replacing V_{M} by V_{M} - V_{s} was an improvement in theory, but it was found by Li in that the overall accuracy of the conjunction times is worse than using equation (5). It is even worse than the average accuracy of astronomical systems used between the 10th and 13th century. This is not surprising since *Shòushí* system probably chose the specific functions T(t) and S(t) to fit the observation data using equation (5). If equation (6) were to be used instead, T(t) and S(t) would have to be modified in order to fit the observation data accurately. By examining the calculation of eclipses by an astronomer in the Ming dynasty, Li concludes that the Ming astronomer used equation (5) instead of (6) to compute true conjunctions. Li suspects that the formula appeared in the historical record is wrong. This issue may be settled by examining the times of true conjunctions listed in the *Dàtǒng* calendars and see which formula can reproduce the times there. The result will be reported in the next section.

For convenience, I use N_{y} to denote the Chinese year whose New Year day is closest to Jan. 1 in Western year y. For example, N_{1524} denotes the Third Year of Emperor Jia Jing's Reign, which began on Feb. 4, 1524 and ended on Jan. 22, 1525.

I found 6 *Dàtǒng* calendars that list the conjunction times. Three of them were found on Digital Taiwan : *Dàtǒng* calendars for N_{1604}, N_{1629} and N_{1639}. The other three were found in *Compilation* Volumes 2 and 5: *Dàtǒng* calendars for N_{1531} (*Compilation (2)* pp. 157-158), N_{1532} (*Compilation (2)* pp. 247-248) and N_{1616} (*Compilation (5)* pp. 281-282). *Compilation (4)* also has the *Dàtǒng* calendar for N_{1604} and *Compilation (6)* also has the *Dàtǒng* calendar for N_{1639}, but the two calendars there don't have the conjunction times. In addition, *Compilation (2)* has two copies of calendars for N_{1532}, one with conjunction times and one without. *Compilation (5)* has two copies of calendars for N_{1616}, again one with conjunction times and one without. This shows that there were at least two different versions of the imperial calendars in N_{1532}, N_{1604}, N_{1616} and N_{1639}. This was probably also the case for other years. The three calendars from Digital Taiwan only have the first 6 or 7 conjunction data, whereas the three calendars from *Compilation* have the conjunction data in the entire year. These six calendars contain data of 56 lunar conjunctions.

The conjunction times in the six imperial calendars are expressed in units of *shíchén* (時辰) and *kè* (刻): 1 day = 12 *shíchén*s = 100 *kè*s. So 1 *shíchén* = 2 hours = 100/12 *kè*s = 8.3333 *kè*s and 1 *kè* = 0.01 days = 14.4 minutes. *Shíchén*s are labelled by the 12 earthly branches zǐ (子), chǒu (丑), yín (寅), mǎo (卯), chén (辰), sì (巳), wǔ (午), wèi (未), shēn (申), yǒu (酉), xū (戌) and hài (亥). In terms of today's 24-hour system, zǐ refers to 23:00 to 1:00 (on the next day); chǒu refers to 1:00-3:00 and so on. Each *shíchén* is subdivided into chū (初) and zhèng (正), each one hour long. So chǒu chū refers to 1:00-2:00, chǒu zhèng refers to 2:00-3:00 and so on. *Kè* is calculated according to the rules that the time interval 0-1 *kè* is labelled as chū *kè*, 1-2 *kè* is labelled as one *kè*, 2-3 *kè* is labelled as two *kè*s and so on. For example, hài chū two kès (亥初二刻) refers to 21:28.8 - 21:43.2, which may be converted to fractional days as 0.895-0.905 or 0.900±0.005. Similarly, "wèi zhèng three kès" (未正三刻) refers to 14:43.2 - 14:57.6, which can be converted to fractional days as 0.613 - 0.623 or 0.618±0.005. Since one hour is only 1/6 kès longer than four kès, "sì zhèng four kès" (巳正四刻) refers to the 2.4-minute time interval between 10:57.6 and 11:00 or 0.4575±0.0008 when expressed in fractional days. Sexagenary dates can be expressed as integers using the convention that 0 = jiǎ zǐ, 1 = yǐ chǒu and so on, which is simply the order number of sexagenary names listed in this table minus one. Using this convention, we can express the times listed in the imperial calendars in terms of numbers between 0 and 60. For example, "xīn sì wèi zhèng three kès" (辛巳未正三刻) is represented by 17.618±0.005, "wù shēn hài chū two kès" (戊申亥初二刻) is represented by 44.900±0.005. The following table uses this convention to express the times listed in the imperial calendars.

Lunar conjunction times are also computed using equations (5) and (6) in the 276 years from N_{1369} to N_{1644}. Following the notation in , I denote D1 the calculation using (5) and D2 the calculation using equation (6). The mean conjunction time t_{0}, functions T(t), S(t) and V(t) are computed using the formulae given in . Note that in the term "degree" (°) is used to mean *dù*. Since the *Dàtǒng* system abandoned the shortening of tropical year, all A' in should be replaced by A. The following table lists the times of the 56 conjunctions calculated using D1, D2 and times listed in the imperial calendars.

Year | Month | Lunar Conjunction Time | ||
---|---|---|---|---|

Dàtǒng Calendar | D1 | D2 | ||

N_{1531} | 1 | 22.932±0.005 | 22.931 | 22.968 |

2 | 52.608±0.005 | 52.605 | 52.653 | |

3 | 22.150±0.005 | 22.154 | 22.201 | |

4 | 51.588±0.005 | 51.590 | 51.628 | |

5 | 20.942±0.005 | 20.937 | 20.960 | |

6 | 50.223±0.005 | 50.220 | 50.224 | |

leap 6 | 19.493±0.005 | 19.491 | 19.476 | |

7 | 48.817±0.005 | 48.814 | 48.783 | |

8 | 18.213±0.005 | 18.211 | 18.169 | |

9 | 47.713±0.005 | 47.714 | 47.668 | |

10 | 17.327±0.005 | 17.330 | 17.289 | |

11 | 47.077±0.005 | 47.076 | 47.052 | |

12 | 16.900±0.005 | 16.902 | 16.904 | |

N_{1532} | 1 | 46.713±0.005 | 46.716 | 46.743 |

2 | 16.432±0.005 | 16.435 | 16.478 | |

3 | 46.035±0.005 | 46.034 | 46.081 | |

4 | 15.515±0.005 | 15.517 | 15.559 | |

5 | 44.900±0.005 | 44.899 | 44.927 | |

6 | 14.213±0.005 | 14.213 | 14.225 | |

7 | 43.515±0.005 | 43.519 | 43.513 | |

8 | 12.838±0.005 | 12.839 | 12.818 | |

9 | 42.223±0.005 | 42.218 | 42.186 | |

10 | 11.682±0.005 | 11.681 | 11.642 | |

11 | 41.255±0.005 | 41.253 | 41.216 | |

12 | 10.932±0.005 | 10.931 | 10.904 | |

N_{1604} | 1 | 48.452±0.005 | 48.447 | 48.431 |

2 | 18.192±0.005 | 18.190 | 18.192 | |

3 | 47.942±0.005 | 47.939 | 47.961 | |

4 | 17.618±0.005 | 17.622 | 17.656 | |

5 | 47.187±0.021 | 47.204 | 47.242 | |

6 | 16.687±0.021 | 16.677 | 16.709 | |

N_{1616} | 1 | 8.702±0.005 | 8.699 | 8.700 |

2 | 38.108±0.005 | 38.111 | 38.102 | |

3 | 7.535±0.005 | 7.533 | 7.516 | |

4 | 36.993±0.005 | 36.988 | 36.964 | |

5 | 6.483±0.005 | 6.481 | 6.452 | |

6 | 36.035±0.005 | 36.035 | 36.007 | |

7 | 5.682±0.005 | 5.680 | 5.662 | |

8 | 35.380±0.005 | 35.384 | 35.381 | |

9 | 5.077±0.005 | 5.078 | 5.090 | |

10 | 34.743±0.005 | 34.740 | 34.762 | |

11 | 4.358±0.005 | 4.354 | 4.382 | |

12 | 33.910±0.005 | 33.911 | 33.940 | |

N_{1629} | 1 | 53.797±0.005 | 53.801 | 53.820 |

2 | 23.567±0.005 | 23.567 | 23.606 | |

3 | 53.213±0.005 | 53.217 | 53.265 | |

4 | 22.743±0.005 | 22.741 | 22.786 | |

leap 4 | 52.160±0.005 | 52.163 | 52.198 | |

5 | 21.493±0.005 | 21.494 | 21.513 | |

6 | 50.785±0.005 | 50.785 | 50.786 | |

N_{1639} | 1 | 55.307±0.005 | 55.310 | 55.303 |

2 | 25.088±0.005 | 25.089 | 25.105 | |

3 | 54.827±0.005 | 54.823 | 54.857 | |

4 | 24.4575±0.0008 | 24.4572 | 24.4988 | |

5 | 53.983±0.005 | 53.979 | 54.019 | |

6 | 23.400±0.005 | 23.400 | 23.429 |

The result is remarkable. All conjunction times calculated using D1 match the times given in the imperial calendars. Even though the times computed by D2 are close, most of them are inconsistent with the data in the imperial calendars. The month 5 conjunction in N_{1639} was close to midnight. The slight time difference in the D2 calculation makes the D2 conjunction occurring on a different day. The times of the month 5 and month 6 conjunction in N_{1604} listed in the imperial calendar cannot be determined very precisely because of tears on the page. The times can only be determined within a one-hour interval. Even in these two cases the D2 conjunction times are still out of the intervals. The imperial calendar in N_{1639} lists the month 4 conjunction time at "sì zhèng four kès" (巳正四刻). As mentioned above, this means that the conjunction occurred within the 2.4-minute interval between 10:57.6 and 11:00. The D1 conjunction time is within this interval. These findings show that D1 was the method used by the *Dàtǒng* calendars in the conjunction calculation.

As mentioned above, if a conjunction occurred close to midnight, the difference in the D2 calculation might make the conjunction occurring on a different day. This happened in the month 5 conjunction in N_{1639} as shown above. The *Dàtǒng* system was used for the imperial calendars from N_{1369} to N_{1644}. Comparing the dates of the D2 conjunctions with those listed in *3500 Years of Calendars and Astronomical Phenomena* in these years, I find 88 mismatches. In 21 of the mismatches I am able to find the *Dàtǒng* calendars in those years and confirm that the 21 dates computed from D2 are all wrong. Comparing the dates of the D1 conjunctions with those listed in *3500 Years of Calendars and Astronomical Phenomena* between N_{1369} and N_{1644}, I find 11 mismatches. Using D2 to do the calculation does not remove any mismatch. That is to say that the 88 D2 mismatches include these 11 cases. Since the above calculation shows that the D1 method was the method used by the *Dàtǒng* calendars in the conjunction calculation at least for the six years investigated, it is worth to examine these 11 mismatches. The result is unexpected and will be presented in the next section.

The numerical result reported in the last section indicates that D1 [using equation (5)] was the method adopted by the *Dàtǒng* calendars in the computation of lunar conjunctions. However, there are 11 mismatches between the D1 conjunction dates and those in *3500 Years of Calendars and Astronomical Phenomena* in the 276 years between N_{1369} and N_{1644}. I checked the three books *A Sino-Western Calendar For Two Thousand Years (1-2000)*, *Tables of Historical Lunar Conjunctions and Leap Months* and *Compilation of Historical Calendars* mentioned above and found that the four books all agreed with one another on those 11 conjunction dates. Then I tried to find the *Dàtǒng* calendars in those years. I was able to find the information of 7 of the 11 conjunctions. To my surprise, these 7 conjunction dates all match the D1 calculation, meaning that the dates of the 7 conjunctions in those four books are wrong. I haven't been able to find the information of the remaining 4 conjunctions and so whether the D1 conjunction dates are correct or not remains unknown. The following table lists the information of the 11 lunar conjunctions.

Year | Month | Lunar Conjunction Date | ||
---|---|---|---|---|

Calendar Books | D1 | Dàtǒng Calendar | ||

N_{1370} | 2 | Gēng Shēn, Feb. 26 | 57.0024 (Xīn Yǒu, Feb. 27) | |

N_{1378} | 8 | Xīn Chǒu, Aug. 24 | 36.9827 (Gēng Zǐ, Aug. 23) | |

N_{1462} | 11 | Rén Chén (Nov. 22) | 27.8143 (Xīn Mǎo, Nov. 21) | Xīn Mǎo, Nov. 21 (Huang ) |

N_{1495} | 7 | Xīn Sì, July 21 | 18.1775 (Rén Wǔ, July 22) | |

N_{1497} | 10 | Jǐ Sì, Oct. 26 | 4.9997 (Wù Chén, Oct. 25) | |

N_{1581} | 10 | Rén Chén (Oct. 28) | 27.9349 (Xīn Mǎo, Oct. 27) | Xīn Mǎo, Oct. 27 ( Compilation (3) p. 606) |

N_{1588} | 3 | Guǐ Wèi (Mar. 26) | 20.4341 (Jiǎ Shēn, Mar. 27) | Jiǎ Shēn, Mar. 27 (Digital Taiwan, Compilation (4) p. 135) |

4 | Guǐ Chǒu, Apr. 25 | 50.0406 (Jiǎ Yín, Apr. 26) | Jiǎ Yín, Apr. 26 (Digital Taiwan, Compilation (4) p. 139) | |

12 | Gēng Chén, Jan. 17, 1589 | 15.9425 (Jǐ Mǎo, Jan. 16, 1589) | Jǐ Mǎo, Jan. 16, 1589 ( Compilation (4) p. 175) | |

N_{1600} | 1 | Yǐ Sì, Feb. 14 | 42.0834 (Bǐng Wǔ, Feb. 15) | Bǐng Wǔ, Feb. 15 ( Compilation (4) p. 445) |

N_{1609} | 1 | Guǐ Wèi, Feb. 4 | 20.0211 (Jiǎ Shēn, Feb. 5) | Jiǎ Shēn, Feb. 5 ( Compilation (5) p. 67) |

It is very surprising that there are three mistakes in the four calendar books in N_{1588}. In *Compilation of Historical Calendars* by Wāng Yuēzhēn, Wāng noted "*Guóquè* records month 12 conjunction on jǐ mǎo, inconsistent (with the date listed here)" in N_{1588}, "*Guóquè* records month 1 conjunction on bǐng wǔ, inconsistent" in N_{1600} and "*Guóquè* records month 1 conjunction on jiǎ shēn, inconsistent" in N_{1609}. *Guóquè* (《國榷》) refers to a chronicle of the Ming dynasty written by Tán Qiān (談遷, 1594-1658). The quoted conjunction dates are actually correct. However, the conjunction dates appeared in *Guóquè* are not always reliable. For example, in *Compilation of Historical Calendars* Wāng noted "*Guóquè* records month 9 conjunction on gēng shēn, inconsistent" in N_{1602} and "*Guóquè* records month 12 conjunction on dīng wèi, inconsistent" in N_{1604}. These two conjunction dates are indeed inconsistent with the *Dàtǒng* calendars (See *Compilation (4)* p. 510 and p. 576). Wāng also noted "*Guóquè* records month 2 conjunction on xīn yǒu, inconsistent" in N_{1370}. This is the first conjunction listed in Table 2 above. Even though the date in *Guóquè* agrees with the calculation by D1, this is only an indirect evidence that the D1 date may be correct.

From Table 2 we see that apart from the month 2 conjunction in N_{1370} and month 10 conjunction in N_{1497}, all other conjunctions were not very close to midnight according to the D1 calculation. Since the D1 data haven't been found to be inconsistent with the *Dàtǒng* calendars so far, I suspect that among the remaining 4 mismatches, at least two of the D1 dates are correct. However, in the absence of reliable sources, the four conjunction dates on this website are still based on the dates in *3500 Years of Calendars and Astronomical Phenomena*.

Zhāng, Péiyú (張培瑜),

*Sānqiān Wǔbǎiniǎn Lìrì Tiānxiàng*(《三千五百年历日天象》 or*3500 Years of Calendars and Astronomical Phenomena*), Elephant Press, July 1997.Hsueh, Chung-San and Ouyang, Yi,

*A Sino-Western Calendar For Two Thousand Years (1-2000)*(Chinese), The Commercial Press Limited, Changsha, China (1940).Chén, Yuán (陳垣),

*Èr shí shǐ shuò rùn biǎo*(《二十史朔閏表》 or*Tables of Historical Lunar Conjunctions and Leap Months*), Zhonghua Book Company (Beijing), July, 1962.National Library of China,

*Guó jiā tú shū guǎn cáng míng dài dà tǒng lì rì huì biān*(國家圖書館藏明代大統曆日彙編》or*Compilation of Dàtǒng calendars collected by the National Library of China*), National Library of China Publishing House (Beijing), September, 2007.李勇 (Li, Yong), "Míng jiā jìng liù nián dà tǒng lì lì shū de qì shuò tuī bù jīng dù" (明嘉靖六年《大统历》历书的气朔推步精度 or "Accuracy of Calculation for the Solar Terms and the Syzygys in the calendar of Datong Lishu 1527"), Progress in Astronomy, 29, 218-227, 2011.

黃一農 (Huang, Yi-Long), "Zhōng guó shǐ lì biǎo shuò rùn dìng zhèng jǔ yú — yǐ táng lín dé lì xíng yòng shí qī wéi lì" (中國史曆表朔閏訂正舉隅 — 以唐《麟德曆》行用時期為例 or "Examples of Corrections to the Chinese History Calendar Data — using the period 665-728 as examples"), Hàn xué yán jiū (漢學研究 or

*Studies in Sinology*), 10, 279-306, 1992.李勇、張培瑜 (Y. Li & P.-Y. Zhang), "Zhōng guó gǔ lì dìng shuò tuī bù zōng shù" (中国古历定朔推步综述 or "A review on the real syzygy calculation in ancient Chinese almanacs"), Progress in Astronomy, 14, 66-76, 1996.

Li, Y., Zhang, C.Z., "Chinese syzygy calculation established in the 13th century", Astro. Astrophys., 332, 1142-1146 (1998).

The copies of the imperial calendars shown on Digital Taiwan are preserved in the National Central Library at Taipei. The catalog numbers of the calendars for N

_{1604}, N_{1629}and N_{1639}are 6322, 6333 and 6337, respectively. The second pages of the calendars should have the times of the conjunctions in the second half of the years. Anyone who can provide me photos of the second page of any of the three calendars (email: yuktungliu@gmail.com) will be greatly appreciated.In the Introduction section in , Huang wrote:

I found a tattered calendar book without year label in the National Central Library at Taipei. Its catalog number was 6294. This tattered calendar book contains calendar data from month 4 to month 12. Upon examining its content, I was able to determine that it was a calendar for N

_{1462}. The sexagenary name of the month 11 conjunction listed there was xīn mǎo. However, every calendar data book mistakenly lists that year's month 11 conjunction on a rén chén day.In a footnote he explained how the year was determined:

This calendar lists the sexagenary month cycle of month 4 as yǐ sì and the sexagenary date of the month 4 conjunction was bǐng yín. The center of the month's "Nine Palace diagram" is "黃" ("yellow"). It can be inferred from the rules of traditional calendar that the heavenly stem of the year was either dīng or rén, and the earthly branch of the year was either zǐ, wǔ or yǒu. In 1000-1911, only N

_{1462}satisfies the above requirements on the sexagenary year and with conjunction dates closely match the dates listed in the tattered calendar.A scan of the calendar page described above can be seen on this page provided by Digital Taiwan.

The first two pages of the copy of the calendar for N

_{1588}collected in*Compilation(4)*is damaged severely. Fortunately, the lunar conjunction dates can still be obatined from the monthly calendar inside the book. The first page of the copy of the calendar shown on Digital Taiwan is clear, which should be scanned from the copy preserved in the National Central Library at Taipei (catalog number is 6319). Interested readers can go the the library to check the book. The second page is likely to be clear too.