Discrepancies Between Modern Calculation and Historical Data (1645 – 1911)

First draft: February 2019

Update on 2020-12-12: I created a web-based platform to calculate the solar and lunar positions as well as the times of solar terms and moon phases according to the Shixian system. It may be useful for people interested in the study of the history of Chinese astronomy in the Qing dynasty.

Following the calendar reform, the rules for the Chinese calendar from 1645–1911 were the same as the current rules, except that time was calculated for the Beijing local time instead of 120°E. The discrepancies between the old calculation and modern calculation are caused by the time difference as well as the inaccuracy of the old methods. Here I list the discrepancies and comment on them.

Lunar Conjunctions

The following table lists the discrepancies in the dates of lunar conjunction calculated using the method at the time (third column) and by the modern method (last column) described in the calendar rules page.

Click to show/hide table

It appears that most of the discrepancies are caused by the time difference between UT1+8 and Beijing local time. One might think that if the Beijing local mean solar time is used in the modern calculation, the discrepancies will be much reduced. This does not turn out to be the case. In fact, there will be several more discrepancies using the Beijing local mean solar time instead of UT1+8. This is because there are many cases in which the first days from the old calculation were one day later than they should be. They did not show up in the table because they fell on the same days as the modern calculation when expressed in UT1+8. For example, the first day of the seventh month in N₁₇₂₈ was listed on Aug. 6, 1728. Modern calculation has the lunar conjunction time at 00:05 UT1+8 on Aug. 6, 1728. That is why it is not in the table. If the Beijing local time is used, the lunar conjunction date and time is Aug. 5, 1728 at 23:51 and so there will be a discrepancy. When UT1+8 is used, most of the discrepancies arising from the inaccuracy of the old calculation are hidden by the time difference offset. However, the discrepancies will be reduced from 22 to 14 if the the difference between the apparent solar time and mean solar time (i.e. the equation of time) is taken into account. So about half of the discrepancies are caused by the equation of time.

24 Solar Terms

The following table lists the discrepancies in the dates of 24 solar terms calculated using the method at the time (third column) and by the modern method (last column) described in the calendar rules page.

Click to show/hide table

We see that there are much more discrepancies compared to the lunar conjunction dates. Accurate computation of the 24 solar terms (defined using dìngqì) is harder than the that of the lunar conjunction because the apparent motion of the Sun is much slower. The table clearly shows that the discrepancies are much reduced after 1733. We know the reason from historical records: the computation of the Sun's position was based on Tycho's model before 1730s, and was switched to using the more accurate Kepler laws around 1730s. There are 53 discrepancies after 1733. If the time difference between the apparent and mean solar time is taken into account, the number of discrepancies will be reduced to 19.

Leap Months

The discrepancies in the dates of lunar conjunctions and 24 solar terms lead to differences in some of the leap months. The following table shows the discrepancies in the leaps months listed in the calendar at the time (second column) and those based on the modern calculation (last column).

Chinese Year	Actual Leap Month*	Retrodicted Leap Month*
N1645	6	5
N1651	2	1
N1661	7	8
N1727	3	2
N1789	5	6

^* Actual leap months refer to the leap months listed in the calendar. They were based on the calculation using the metod at the time. Retrodicted leap months refer to leap months that would have been based on the modern calculation.

All discrepancies after N₁₆₄₅ are caused by the discrepancies of the lunar conjunction dates or 24 solar term dates between the old and modern methods. The case of N₁₆₄₅ is special. Both the old and modern methods result in the same dates for the lunar conjunctions and solar terms, and leap month 6 in N₁₆₄₅ contained the major solar term Z6. This broke the rule that a leap month must not contain any major solar term. Wāng Yuēzhēn (汪曰楨), a Chinese mathematician in the 19th century, explained that even though the solar term Z6 and the lunar conjunction associated with the month occurred on the same day, Z6 occurred earlier in the day than the lunar conjunction and was counted as a major solar term of the previous month. As a result, leap month 6 did not contain any major solar term. This "rule" was first proposed by Lǐ Tiānjīng (李天經), an astronomer in the Imperial Astronomical Bureau, in 1630s. This rule was only used in N₁₆₄₅. It was never used again after this year.

N₁₆₄₅ was the first year the calendar was issued in which the 24 solar terms were calculated using dìngqì. As pointed out in the solar terms page, the "no zhōngqì rule" used previously to insert leap months is incompatible with dìngqì, where two major solar terms can occur in a month and there can have two months without a major solar term in a suì. It is apparent that the rule for leap months was not ironed out in N₁₆₄₅. Historical document indicates that the rule that was used later was to place a leap month in the first month that did not contain a major solar term after the winter solstice in a leap suì. This is the origin of rule 5 stated in the rules page.