# Lunar-solar calendar

** Theory. ** Two astronomical values are based on the theory of lunar-solar calendars:

1 tropical year = 365.242 20 days,

1 synodic month = 29,530 59 days.

Hence we get:

1 tropical year = 12,368 26 synodic months.

In other words, in the solar year there are 12 full lunar months and another about one-third. Consequently, the year in the lunisolar calendar can consist of 12 or 13 lunar months. In the latter case, the year is called embolism( from the Greek "embolismos" - insert).

Note that in ancient Rome and medieval Europe, the insertion of an additional day or month was called intercalation( from Latin inter-calatio - an insert), and the added month is an intercalary.

In the lunisolar calendar, the beginning of each calendar month should be as close as possible to the new moon, and the average during the cycle the duration of the calendar year should be close to the duration of the tropical year. The insertion of the 13th month is made from time to time so that the beginning of the calendar year is maintained as close as possible to some point in the astronomical solar year, for example, to the equinox.

We decompose the fractional part of the ratio of the duration of the tropical year to the duration of the synodic month, that is, the value K = 0.36828 in the continued fraction:

K = M / N = 1/2;1/3;3/8;4/11;7/19;123/334;. ..

The lunar-solar calendars created in antiquity correspond to the third( 3/8) and fifth( 7/19) suitable fractions.

** Trietherid. ** The simplest case of the lunisolar calendar is a period of two years, during which one lunar month is inserted. In chronology, this system received the conventional name tri-teride, as many peoples, in particular the Romans, inclusive inclusive, inclusive, that is, including the second year of the previous biennium.

Obviously, the first of two years could consist of 12 lunar months, the second - out of 13, so there was only 25 months in trieteride. But as the 25 synodic months are

29,53059 * 25 = 738,26475( days), then in the indicated period of time there could be 13 full( 30 days) and 12 empty( for 29 days) months, since 13 * 30+ 12 * 29 = 738( days).

Meanwhile, the duration of the two tropical years is 730.4844 days. Therefore, a calendar built on a tri-teride, for every eight years, ahead of the moon for one day, but lagged behind the Sun for two years at 7d, 78, and in eight years - for a whole month.

But the ancient people did not know the true duration of the tropical year for a long time. That is why there is every reason to believe that it was such a time account that was originally used by many peoples. For an approximate agreement with the Sun, it was enough to take the second year of every fourth triethide at 12 months, and with the Moon - to shorten the full month every 24 hours every eight years. Of course, from time to time the system needed a more rigorous adjustment.

Much more accurate was the true three-year cycle. In this case, 37 synodic months = 1092.6318 days, 3 tropical years = 1095.7266 days.

Thus, the three-year cycle( 19 * 30 + 18 * 29 = 1092) is only three days ahead of the solar year;for 10 such cycles( over 30 years), this error increases to 30.95 days. The insertion of a lunar month once in 30 years made it possible to coordinate the beginning of the calendar year with the solar year with a sufficient degree of accuracy.

** Octaethide **.The eight-year cycle, octaetheride, was used in ancient Babylon and, apparently, independently of the Babylonians, was discovered by the ancient Greeks. It was described by the Greek astronomer Cleostratus about 540 g, BC.e, in a special composition. In this case, 8 tropical years = 2921.9376 = 2922 days, 93 synodic months = 2923.5284 days.

Therefore, the 8-year calendar cycle will consist of 99 months: 53 full and 46 empty, since 53 * 30 + 46 * 29 = 2924( days).

The period error with respect to the Moon is 0d, 47, i.e. after two such cycles, the specific phase of the moon appears one day earlier than at the beginning of the cycle, therefore, the calendar cycles must contain alternately 2924 and 2923 days. But in relation to the Sun, the error is 1.53 days for 8 years or about three days for 16 years. And if at the beginning of the cycle the new moon took place at the moment of the equinox, then in 16 years it will happen only three days later.

The internal structure of the period, i.e., the distribution of the days by months, becomes clear if this time interval is described as follows:

2924 = [(8 * 354) + 2] +( 3 * 30) or 2924 = 8 [6 * 30+ 6 * 29] +( 3 * 30).

As you can see, in the 8-year period, except for the correct alternation of complete and empty months, an insert of two days( in the second cycle - one) and three full months should be carried out. These latter were most often inserted in the 3rd, 6th and 8th calendar years of the cycle. Thus, it turns out that the 8-year cycle is actually a combination of two three-year and one two-year cycles.

** Generalizations of the eight-year cycle. ** In ancient Greece for a while longer cycles were used, resulting from the eight-year period. A natural generalization of octaetereid is the 16-year cycle - ekkadeketerid. Here the period consists of 105 full and 93 empty months, which provides quite good agreement of the calendar with the phases of the Moon:

105 * 30 + 93 * 29 = 5847,

29.53059 * 198 = 5847.0568.

The specific phase of the Moon in this case moves forward one day only for 281.69 years. But 365.2422 * 16 = 5843.875≈ 5844.

Therefore, for every 16 years, the beginning of the count( the 1st day of the spring month of the lunisolar calendar) goes forward in relation to the spring equinox for the same three days ahead. After ten such cycles, to reconcile the ** calendar with the solar ** year, it is necessary to discard exactly one full month in 30 days from the account.

By way of such reasoning a 160-year cycle was opened. It has 1979 months, and for the last 8 years there are three months and two months. At the same time 1979 synodic months = 58,441,037 days, 160 tropical years = 58,438.752 days;

the divergence from the Sun for 160 years is only a little over two days. It can therefore be said that in the 160-year cycle, octaetheride was brought to a high degree of perfection and could survive in this form for quite a long time, without giving noticeable deviations from the solar year. The invention of the 160-year cycle is attributed to the outstanding Alexandrian scientist Eratosthenes( about 276 - about 196 BC.).

And, finally, in Western Europe in the III-VI centuries., And in Britain and before the beginning of the IX.n.e. In determining the date of the spring full moon, an 84-year cycle was used( l0 * 8 + 1/2 * 8).In this period, there are 84 tropical years = 30,680.365 days, 1,039 synodic months = 30,682.283 days.

It was assumed that the cycle consists of 1,039 months, of which 551 are full( including 31 months of insertions) and 488 empty ones. Consequently, at the end of the cycle, the full moon is shifted one day forward, since there are only 30 682 days in the calendar cycle. The 84-year cycle was convenient for calculations because after its expiry the days of the week in the Julian calendar fell on the same calendar number of months( since 84 = 3 * 28).

** Methane cycle. ** More accurate is still the 19-year cycle used in ancient China, Babylon, independently discovered by the Greek astronomer Meton in 432 BC.e. In this cycle, the

ratio of 19 tropical years = 235 synodic months is fulfilled.

In fact,

19 X365,242 20 = 6939,602 days

and

235X 29,530 59 = 6939,689 days.

The error of the metonic cycle is 0,087 days, ie 2,1 hours - for as much phase of the Moon shift forward for every 19 years. This is one day for 219 years( Fig.).

Fig. The shift of a specific phase of the moon( for example, the full moon) to the cts: / - Gregorian, 2 - Julian calendars because of the inaccuracy of the methonic cycle

. The Methonov cycle served as the basis for constructing many lunar-solar calendars. And so.as in the calendar year and month there should be an integer number of days, it was actually accepted that 235 lunar months = 6940 days.

Thus, the cycle should have 110 empty( for 29 days) and 125 full( for 30 days) months: 110 * 29 + 125 * 30 = 6940. The numerator of a suitable fraction shows that the insertion of the 13th month should be done 7 timesin every 19 years.