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In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations. It is unique in its incorporation of accurate lunar and solar cycles.
Since the lunar cycle is about 29 ½ days, months alternate between 29 and 30 days long. Two months, Cheshvan and Kislev, vary between 29 and 30 days; thus, a standard year can have either 353, 354 or 355 days depending on whether the combined length is 29+29, 29+30, 30+29 or 30+30.
In order to harmonize the lunar and solar cycles, an extra month is added on leap years in a 19-year cycle. These leap years occur in the 3rd, 6th, 8th, 11th, 14th, 17th and 19th years of the cycle. The month is added to Adar, the last of the twelve months. On leap years, there are two Adars, Adar I and Adar II.
Thus, depending on the other cycle of Cheshvan and Kislev mentioned above, a leap year can have either 383, 384 or 385 days.
Remarkably, this intricately woven system maintains the festivals and their proper seasons in an accurate and precise manner.