The Jewish calendar is based on both the lunar and solar cycles. Torah months are lunar and holidays are dated by the month. The holiday of Pesah is always to be in the spring month, that is to say the month of the Spring Equinox. Seasons are a function of the solar cycle. As lunar months are 29 days 12 hours 44 minutes and several seconds each, twelve lunar months comprise 354.37 days. A solar year is slightly more than 365.24 days. In just a few years of twelve lunar months each, Pesah would have drifted backwards relative to the seasons and occur in winter. To assure Pesah occurring in the spring, lunar months are intercalated (merged) with the solar year. It is the High Court's responsibility to intercalate as it sees fit. As it was halachically established that a year be completely composed of whole months, the basic principle of intercalation was that an extra month is periodically added to the year to compensate for the difference between cycles.
Originally, intercalation was based on astronomical calculation and direct observation of the signs of spring. Early Talmudic sages checked the state of the crops and relied on the weather to determine if an adjustment (an extra month) had to be made in a particular year. As the primary purpose of adding the `leap' month is to assure Pesah remaining in the Spring month, it is added just before Pesah's month (Nissan), and is called Adar II. By later Talmudic times intercalation was accomplished strictly by calculation.
It is also a halachic imperative that months be composed of whole days only. Thus, the calendar is so structured that the 29 1/2 days of each lunar cycle are resolved into months of either twenty nine or thirty days each. In a `regular' year (as explained below) six months have thirty days and six months have twenty nine days. These `full' and `short' months rotate; Nissan is always thirty days, Iyar always twenty nine, Sivan always thirty, etc. Whenever a month has thirty days, two days are celebrated as Rosh Hodesh of the following month - the thirtieth of the outgoing month and the first of the new month. Whenever a month has twenty nine days, only one day is celebrated as Rosh Hodesh of the following month - the first of the month.
When a set calendar was established, intercalation was achieved by adding an extra thirty day month to the Jewish year seven times every nineteen years, as the number of days in 19 solar years is extremely close to the number of days in 235 lunar months, both comprising about 6939.6 days. Rounding out slightly:
365.2422 days per yr x 19 yrs = 6939.60 days
29.5305 days per mo x 235 months = 6939.67 days
An event's Hebrew date (luni-solar) and civil date (solar) will bevery close to each other every nineteen years but not necessarily identical, as nineteen year cycles have varying internal patterns based on considerations which will be explained shortly.
In the calendar eventually settled upon, the seven extra months in each nineteen year cycle are added during years 3, 6, 8, 11, 14, 17 and 19. The set calendar was established by Hillel II, president of the Sanhedrin, in 359 C.E., based for the most part on the calculations of the third century sage Rab Adda. Basically, it is the calendar we have been using the past 16 centuries.
Additional calendrical considerations are that Yom Kippur not fall on Friday or Sunday (it would entail great difficulty having two consecutive days on which cooking, carrying, etc. would be prohibited), and that Hoshannah Rabbah not fall on Shabbat (which would interfere with habatat araba). This means that Rosh Hashanah, the first day of the year, cannot be set on Sunday, Wednesday or Friday. To accomplish this, one day is periodically added to or subtracted from the year. If added, it is always added to Heshvan (the month immediately following the New Year's month, Tishre); if subtracted, it is always subtracted from Kislev (the very next month). In `regular' years, Heshvan has twenty nine and Kislev thirty days. In `short' years, both have twenty nine days. In `complete' years, both have thirty days.
As a result of the above, non-leap years have either 353, 354 or 355 days; leap years have either 383, 384 or 385 days. That the total days of a year be one of these six amounts became an imperative of the calendar. Altogether, encompassing exactly which days of the week the holidays of a year will occur and the number of days of that year, there are fourteen formats - seven for non-leap years and seven for leap years.
A problem is slowly developing. Rab Adda's solar year comprises 365 days 5 hours 55 minutes and 25 seconds. This measurement is about 6 minutes 39 seconds greater than the actual value of 365 days, 5 hours, 48 minutes and 46 seconds, which was not determined until centuries later. This 6 2/3 minutes annual `lengthening' of the solar year adds up to about one day every 216 years, or about seven days since the third Century. This means that our average calendar dates have `moved forward' seven days relative to astronomical reality as seasons have `moved backward' by that amount relative to our calendar. The problem developing is that Pesah is slowly drifting away from the month in which the Spring Equinox occurs toward the second month of Spring.
The discrepancies involved in Tal Umatar and Bircat Hahamah are greater, as their settings are based on the less accurate calculations of Shemuel. This was explained in the study on Tal Umatar.
The originators of the calendar were undoubtedly aware that a slight discrepancy might exist. Just as their predecessors had corrected the solar-lunar calibration by direct observation, theyexpected the same would be done when the system would revert back to observation by witnesses or whenever necessary. It was always taken for granted that halakha was in harmony with reality. In the future, when a national Bet Din will be established, it will make an adjustment based on astronomical observation.
It is interesting to note that the Talmudic measurement for an average lunar month is strikingly close to our present-day measurement. It is claimed that the average Halachic lunar month is greater than the true mean by less that half a second.