Moon calendar
When considering the theory of the lunar calendar, the duration of the synodic month with a reasonable degree of accuracy can be taken as 29.53059 days. It is obvious that the corresponding calendar month may contain 29 or 30 days. The calendar lunar year consists of 12 months. The corresponding duration of the astronomical lunar year is
12 * 29.53059 = 354.36706 days.
It can therefore be assumed that the calendar lunar year consists of 354 days: from six "full" months to 30 days and six "empty" for 29 days, since 6 * 30 + 6 * 29 = 354. And that the beginning of the calendar month ascan be more accurately coincided with the new moon, these months should alternate;for example, all odd months can contain 30, and even - for 29 days.
However, the period of 12 synodic months is 0.36706 days longer than the calendar lunar year in 354 days. For three such years this error will already be 3 * 0.36706 = 1,10118 days. Consequently, in the fourth year since the beginning of the count, the new moon will already occur not in the first, but in the second numbers of the months, in eight years, in the fourth, etc. And this means that the calendar should be corrected from time to time: approximately every three yearsto make an insert in one day, i.e., instead of 354 days count in a year 355 days. A year in 354 days is usually called simple, a year in 355 days - continued or leap year( the origin of this name will be described below).
Therefore, the task of constructing the lunar calendar is as follows: to find such an order of alternation of simple and leap-year lunar years, at which the beginning of the calendar months would not be shifted noticeably from the new moon. Its solution begins with the search for such an integer( which makes up the cycle) of the lunar years, during which a whole( almost whole!) Number of insertable days runs in. This is the number of days that have been inserted and is divided between the individual years within the cycle.
Of course, if the length of the astronomical lunar year is 354.36706 days, and the simple calendar year is 354 days, then 36 706 days of insecurity run on 100 000 lunar years. But this is too long a period of time, in which the insertion days are very difficult. Therefore, the fraction
K = 36 706/100 000 = 18 353/50 000
must be represented by another fraction, K = m / n , in which the numerator m and the denominator n will be smaller, but the fraction itself is the largestwill be close to the original. Such fractions are called suitable.
To find the appropriate fractions, the numerator and denominator of the fraction are sequentially divided by a numerator, as a result of which the correct fraction is represented as a continued fraction. Discarding the remainders after division in the first, second, etc. stages, a sequence of suitable fractions is obtained. In this case, the exact value of the continued fraction is always between two suitable fractions, and closer to the subsequent than to the previous one. The decomposition of the fractional part of the lunar year into a continued fraction is written as follows:
As we shall see later, appropriate fractions 3/8 and 11/30 were used in constructing the lunar calendars. In the first case( the "Turkish cycle") for eight years, an insert of three days is made. In the second( "the Arabic cycle") for 30 years there are 11 leap years. The cycle error of 0.0118 days indicates that for every 30 years( one cycle) of the new moon in relation to the first number of calendar months move to 0.0118 days in advance, and this gives a shift in one day in approximately 2500 years.