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Overview of Calendars
Julian calendar
Julias Ceasar created the calendar in 46 B.C. as a modified form of the old Roman republican calendar which was based on lunar cycles. The new Julian calendar set fixed lengths for the months, abandoning the lunar cycle. It also specified that there would be exactly 12 months per year and 365.25 days per year with every 4th year being a leap year.
Note that the current accepted value for the tropical year is 365.242199 days, not 365.25. This lead to an 11 day shift in the calendar with respect to the seasons by the 16th century when the Gregorian calendar was created to replace the Julian calendar.
The difference between the Julian and today's Gregorian calendar is that the Gregorian does not make centennial years leap years unless they are a multiple of 400, which leads to a year of 365.2425 days. In other words, in the Gregorian calendar, 1700, 1800 and 1900 are not leap years, but 2000 is. All centennial years are leap years in the Julian calendar.
The details are unknown, but the lengths of the months were adjusted until they finally stablized in 8 A.D. with their current lengths:
 January 31
 February 28/29
 March 31
 April 30
 May 31
 June 30
 Quintilis/July 31
 Sextilis/August 31
 September 30
 October 31
 November 30
 December 31
In the early days of the calendar, the days of the month were not numbered as we do today. The numbers ran backwards (decreasing) and were counted from the Ides (15th of the month  which in the old Roman republican lunar calendar would have been the full moon) or from the Nonae (9th day before the Ides) or from the beginning of the next month.
In the early years, the beginning of the year varied, sometimes based on the ascension of rulers. It was not always the first of January.
Also, today's epoch, 1 A.D. or the birth of Jesus Christ, did not come into use until several centuries later when Christianity became a dominant religion.
Gregorian calendar
The Gregorian calendar is a modified version of the Julian calendar. The only difference being the specification of leap years. The Julian calendar specifies that every year that is a multiple of 4 will be a leap year. This leads to a year that is 365.25 days long, but the current accepted value for the tropical year is 365.242199 days.
To correct this error in the length of the year and to bring the vernal equinox back to March 21, Pope Gregory XIII issued a papal bull declaring that Thursday October 4, 1582 would be followed by Friday October 15, 1582 and that centennial years would only be a leap year if they were a multiple of 400. This shortened the year by 3 days per 400 years, giving a year of 365.2425 days.
Another recently proposed change in the leap year rule is to make years that are multiples of 4000 not a leap year, but this has never been officially accepted and this rule is not implemented in these algorithms.
Hebrew calendar
The Hebrew calendar is based on lunar as well as solar cycles. A month always starts on or near a new moon and has either 29 or 30 days (a lunar cycle is about 29 1/2 days). Twelve of these alternating 2930 day months gives a year of 354 days, which is about 11 1/4 days short of a solar year.
Since a month is defined to be a lunar cycle (new moon to new moon), this 11 1/4 day difference cannot be overcome by adding days to a month as with the Gregorian calendar, so an entire month is periodically added to the year, making some years 13 months long.
For astronomical as well as ceremonial reasons, the start of a new year may be delayed until a day or two after the new moon causing years to vary in length. Leap years can be from 383 to 385 days and common years can be from 353 to 355 days.
Note that the month names and other words that appear in this file have multiple possible spellings in the Roman character set. I have chosen to use the spellings found in the Encyclopedia Judaica.
Adar II, the month added for leap years, is sometimes referred to as the 13th month, but I have chosen to assign it the number 7 to keep the months in chronological order. This may not be consistent with other numbering schemes.
Leap years occur in a fixed pattern of 19 years called the metonic cycle. The 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of this cycle are leap years. The first metonic cycle starts with Hebrew year 1, or 3761/60 B.C. This is believed to be the year of creation.
To construct the calendar for a year, you must first find the length of the year by determining the first day of the year (Tishri 1, or Rosh HaShanah) and the first day of the following year. This selects one of the six possible month length configurations listed above.
Finding the first day of the year is the most difficult part. Finding the date and time of the new moon (or molad) is the first step. For this purpose, the lunar cycle is assumed to be 29 days 12 hours and 793 halakim. A heleq (singular form of halakim) is 1/1080th of an hour or 3 1/3 seconds. (This assumed value is only about 1/2 second less than the value used by modern astronomers  not bad for a number that was determined so long ago.) The first molad of year 1 occurred on Sunday at 11:11:20 P.M. This would actually be Monday, because the Hebrew day is considered to begin at sunset.
Since sunset varies, the day is assumed to begin at 6:00 P.M. for calendar calculation purposes. So, the first molad was 5 hours 204 halakim after the start of Tishri 1, 0001 (which was Monday September 7, 3761 B.C. by the Gregorian calendar). All subsequent molads can be calculated from this starting point by adding the length of a lunar cycle.
Once the molad that starts a year is determined the actual start of the year (Tishri 1) can be determined. Tishri 1 will be the day of the molad unless it is delayed by one of the following four rules (called dehiyyot). Each rule can delay the start of the year by one day, and since rule #1 can combine with one of the other rules, it can be delayed as much as two days.
 Tishri 1 must never be Sunday, Wednesday or Friday. (This is largely to prevent certain holidays from occurring on the day before or after the Sabbath.)
 If the molad occurs on or after noon, Tishri 1 must be delayed.
 If it is a common (not leap) year and the molad occurs on Tuesday at or after 3:11:20 A.M., Tishri 1 must be delayed.
 If it is the year following a leap year and the molad occurs on Monday at or after 9:32:43 and 1/3 sec, Tishri 1 must be delayed.
Glossary
dehiyyot
The set of 4 rules that determine when the new year starts relative to the molad.
heleq/halakim
1/1080th of an hour or 3 1/3 seconds. The singular form is heleq, the plural is halakim.
lunar cycle
The period of time between mean conjunctions of the sun and moon (new moon to new moon). This is assumed to be 29 days 12 hours and 793 halakim for calendar purposes.
metonic cycle
A 19 year cycle which determines which years are leap years and which are common years. The 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of this cycle are leap years.
molad
The date and time of the mean conjunction of the sun and moon (new moon). This is the approximate beginning of a month.
Rosh HaShanah
The first day of the Hebrew year (Tishri 1).
Tishr
The first month of the Hebrew year.
