Thus it is approximately eleven days shorter than the solar year, with the result that in each cycle of 32.5 years, the individual months pass through all the solar seasons.
If, for example, the Hajj (which takes place in the last month of the Muslim calendar) occurs at the height of summer, it will occur in the coolest season 16.25 years later, and the same time in summer again after another 16.25 years. But the mean length of a lunar year is 354 days, 8 hours, 48 minutes, 36 seconds, and the period of mean lunation is 29 days, 12 hours, 44 minutes, 3 seconds.
The resulting cumulative error became about ten days by the close of the sixteenth century, which was rectified in 1582 by Pope Gregory X111 (1502-85), who ordained that 5 October in that year be called 15 October.
The intercalary day it self is always added to the twelfth month, i.e., the month of Dhu-al-Hijja which has twenty-nine days in the common year - in a kabisha year it has 30 days.
Thus, to determine whether a Muslim year is common or kabisha, divide it by 30.
Also, Caesar decreed the year began with the first of January, not with the vernal equinox in late March.
The new calendar officially began on 1 January 45 BCE.
This difference of 8 hours, 48 minutes, 36 seconds (which is almost equal to 11/30th of a day), by which the astronomical lunar year is the longer, is compensated by the intercalation of eleven days in every cycle of thirty years at the stated intervals.
The most commonly used method of intercalation is to make years 2, 5, 7, 10, 13, 16, 18, 21, 24, 26 and 29 in the cycle into leap years, called kabisha.
Therefore, the Romans invented an extra month called Mercedonius of 22 or 23 days. In the Christian or Julian calendar, named after Julius Caesar (100 or 102-44 BCE), who introduced it at the suggestion of the Egyptian astronomer Sosigenes as a reform of the old Roman calendar, the years are reckoned from the birth of Christ, and so are known as years AD (Anno Domini: 'in the year of the Lord'), or simply CE (Common Era) as non-Christian usage in place of AD.
It fixed the average length of the year as 365.25 days, each fourth year being a Leap year, making up the omitted quarters by having 366 days.
Hence their months were 29 or 31 days long with the exception of February, which had 28 days.
However, 4 months of 31 days, 7 months of 29 days, and 1 month of 28 days added up to only 355 days.
Thus, 1600 was a Leap year but not 1700, 1800 or 1900, while 2000 CE will be a Leap year.