The origins of both these calendars are lost in antiquity but essentially the Muslim calendar is based on the lunar calendar, which probably originated in Sumeria, and the Christian calendar traces its origins to the solar calendar of the ancient Egyptians.
Though the Sumerian year was of 12 lunar months, that is about 355 days, it was necessary to reconcile this with the solar year of 365 days for agricultural purposes and this was roughly effected by the addition of a 13th lunar month every three years.
The Egyptian year also had 12 months of 30 days each with a five or six day addition each year, the adjustment being made by the priests through close observation of the winter solstice.
In the time of the Prophet Mohammed, a version of the Sumerian calendar was in use in Mecca but this was, over the years, slipping back against the solar calendar; that is, the first day of the new lunar year was occurring ever earlier in the solar year and the insertion of an extra month every third year made it very difficult to regulate affairs in the new Islamic state. So, in AD 638 the Caliph Omar ordered that a new calendar be introduced of 12 lunar months only, starting with the day of the flight of the Prophet from Mecca to Medina on the 16th July 622 AD, this being denoted as 1st Muharram AH (AH is short for Anno Hyjrae, the year of the flight).
This decision has given Islam a calendar that is approximately 10 days shorter than the Christian year, with the result that over a 30-year period the Hyjrae new year and more noticeably the religious festivals and the holy month of Ramadan change throughout the Christian year.
In detail, the Caliph's decree required that there should be alternate months of 30 and 29 days. The holy month of Ramadan must start and finish on the sighting of the new moon. No month may have more than 30 days, but to provide a further element of adjustment for the periodicity of the moon, eleven years of each 30-year cycle had an additional day added at the end of the year, i.e. 30 Dhu al Hijja. These years are known as "khabisha" years (c.f. leap years) and are the 2, 5, 7, 10, 13, 16, 18, 21, 24, 26 and 29th years of the 30-year cycle.
However, in spite of all these various adjustments or perhaps because of them, the Muslim calendar is very regular and calculations of the date of the 1st Muharram or the Eids can be made to an accuracy of +/- (plus or minus) one day for any year of the Muslim era.
The Christian calendar contains its own aberrations due to the solar year being 365 days plus a little less than a quarter of a day. In AD 324 it was found necessary to adjust the calendar and the Council of Nicea ruled that every fourth year should have an additional day, i.e. 366 days instead of 365. This, however, was not good enough and by 1582 the Christian year was 10 days in advance of the solar year so Pope Gregory XIII decreed that 10 days were to be omitted from the month of October that year, the fourth of that month being followed by the fifteen. In addition, every centurial year would not be a leap year unless the first two digits were divisible by four, i.e. 1600 AD was a leap year but 1700 AD was not.
The old Christian calendar is known as the Julian or Old Style (OS) calendar and the reformed calendar as the Gregorian or New Style (NS) calendar. England, Sweden and Russia did not conform to the New Style calendar but by 1752 England realized the need for this and in that year 11 days were omitted in September, the second being followed by the 14th.
With the advent of the pocket calculator, it is now possible to calculate the correlation between Muslim and Christian dates within +/- (plus or minus) one day. Previously one had to rely on tables of dates in books not very easy of access, or make a crude approximation.
The best ratio between the Muslim lunar year and the Christian (OS) year is 0.970203. Since the Muslim era started on the 16th of July 622 as AH (zero had not been invented then) we must assume that a datum of AH 0 of 27th July 621 or 621.57.
Thus we key in the Muslim year, multiply by 0.970203, add 621.57 and we obtain the Christian year with a decimal figure which represents the number of days in the year. Multiply the decimal figure by 365,or by 366 if the Christian year is a leap year and the number of days will appear.
Example 1. Find 1st Muharram 1400 AH1400 x .970203 = 1358.2842
Multiply decimals by 3650.8542 x 365 = 311.783
This is the Old Style date, so we must add 13 days to bring it to New Style.311 + 13 = 324
This corresponds to 20 November 1979.
Example 2. For those interested only in current dates we can use a different factor, i.e. 621.6055 which takes into account the difference between Old and New Style dates.1400 x .970203 = 1358.2842
which is the same as Example 1.
The accuracy of this calculation is +/- (plus or minus) one day.