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The Jewish calendar is primarily lunar, with each month beginning on the new moon, when the first sliver of moon becomes visible after the dark of the moon. In ancient times, the new months used to be determined by observation. When people observed the new moon, they would notify the Sanhedrin. When the Sanhedrin heard testimony from two independent, reliable eyewitnesses that the new moon occurred on a certain date, they would declare the rosh chodesh (first of the month) and send out messengers to tell people when the month began.
The problem with strictly lunar calendars is that there are approximately 12.4 lunar months in every solar year, so a 12-month lunar calendar loses about 11 days every year and a 13-month lunar gains about 19 days every year. The months on such a calendar "drift" relative to the solar year. On a 12 month calendar, the month of Nissan, which is supposed to occur in the Spring, occurs 11 days earlier each year, eventually occurring in the Winter, the Fall, the Summer, and then the Spring again. To compensate for this drift, an extra month was occasionally added: a second month of Adar. The month of Nissan would occur 11 days earlier for two or three years, and then would jump forward 29 or 30 days, balancing out the drift.
In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations. This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years. Adar II is added in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle. The current cycle began in Jewish year 5758 (the year that began October 2, 1997).
In addition, Yom Kippur should not fall adjacent to Shabbat, because this would cause difficulties in coordinating the fast with Shabbat, and Hoshanah Rabba should not fall on Saturday because it would interfere with the holiday's observances. A day is added to the month of Cheshvan or subtracted from the month of Kislev of the previous year to prevent these things from happening.
Numbering of Jewish Years
The year number on the Jewish calendar represents the number of years since creation, calculated by adding up the ages of people in the Bible back to the time of creation. However, this does not necessarily mean that the universe has existed for only 5600 years as we understand years. Many Orthodox Jews will readily acknowledge that the first six "days" of creation are not necessarily 24-hour days (indeed, a 24-hour day would be meaningless until the creation of the sun on the fourth "day"). For a fascinating (albeit somewhat defensive) article by a nuclear physicist showing how Einstein's Theory of Relativity sheds light on the correspondence between the Torah's age of the universe and the age ascertained by science, see The Age of the Universe.
Jews do not generally use the words "A.D." and "B.C." to refer to the years on the Gregorian calendar. "A.D." means "the year of our L-rd," and we do not believe Jesus is the L-rd. Instead, we use the abbreviations C.E. (Common or Christian Era) and B.C.E. (Before the Common Era).
Months of the Jewish Year
The "first month" of the Jewish calendar is the month of Nissan, in the spring, when Passover occurs. However, the Jewish New Year is in Tishri, the seventh month, and that is when the year number is increased. This concept of different starting points for a year is not as strange as it might seem at first glance. The American "new year" starts in January, but the new "school year" starts in September, and many businesses have "fiscal years" that start at various times of the year. Similarly, the Jewish calendar has different starting points for different purposes.
The names of the months of the Jewish calendar were adopted during the time of Ezra, after the return from the Babylonian exile. The names are actually Babylonian month names, brought back to Israel by the returning exiles. Note that most of the Bible refers to months by number, not by name.
The length of Cheshvan and Kislev are determined by complex calculations involving the time of day of the full moon of the following year's Tishri and the day of the week that Tishri would occur in the following year. I won't pretend to understand the mathematics involved, and I don't particularly recommend trying to figure it out. There are plenty of easily accessible computer programs that will calculate the Jewish calendar for more than a millennium to come.
Note that the number of days between Nissan and Tishri is always the same. Because of this, the time from the first major festival (Passover in Nissan) to the last major festival (Sukkot in Tishri) is always the same.
The Jewish calendar has the following months:
In leap years, Adar I has 30 days. In non-leap years, Adar has 29 days.
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