The Mahommedan era, or era of the Hegira, used in Turkey, Persia, Arabia, etc., is dated from the first day of the month preceding the flight of Mahomet from Mecca to Medina, i.e. Thursday the 15th of July A.D. 622, and it commenced on the day following. The years of the Hegira are purely lunar, and always consist of twelve lunar months, commencing with the approximate new moon, without any intercalation to keep them to the same season with respect to the sun, so that they retrograde through all the seasons in about 32½ years. They are also partitioned into cycles of 30 years, 19 of which are common years of 354 days each, and the other 11 are intercalary years having an additional day appended to the last month. The mean length of the year is therefore 354-11/30 days, or 354 days 8 hours 48 min., which divided by 12 gives 29-191/360 days, or 29 days 12 hours 44 min., as the time of a mean lunation, and this differs from the astronomical mean lunation by only 2.8 seconds. This small error will only amount to a day in about 2400 years.

To find if a year is intercalary or common, divide it by 30; the quotient will be the number of completed cycles and the remainder will be the year of the current cycle; if this last be one of the numbers 2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29, the year is intercalary and consists of 355 days; if it be any other number, the year is ordinary.

Or if Y denote the number of the Mahommedan year, and

R =left bracket11 Y + 14
line
30
right bracket,

the year is intercalary when R < 11.

Also the number of intercalary years from the year 1 up to the year Y inclusive = ((11 Y + 14) / 30); and the same up to the year Y - 1 = (11 Y + 3 / 30).

To find the day of the week on which any year of the Hegira begins, we observe that the year 1 began on a Friday, and that after every common year of 354 days, or 50 weeks and 4 days, the day of the week must necessarily become postponed 4 days, besides the additional day of each intercalary year.

Hence if w = 1
indicate Sun.

2
Mon.

3
Tue.

4
Wed.

5
Thur.

6
Frid.

7
Sat.

the day of the week on which the year Y commences will be

w = 2 + 4left bracketY
line
7
right bracket +left bracket11 Y + 3
line
30
right bracket (rejecting sevens).
But, 30left bracket11 Y + 3
line
30
right bracket +left bracket11 Y + 3
line
30
right bracket = 11 Y + 3
gives 120left bracket11 Y + 3
line
30
right bracket = 12 + 44 Y - 4left bracket11 Y + 3
line
30
right bracket,
orleft bracket11 Y + 3
line
30
right bracket = 5 + 2 Y + 3left bracket11 Y + 3
line
30
right bracket (rejecting sevens).

So that

w = 6left bracketY
line
7
right bracket + 3left bracket11 Y + 3
line
30
right bracket (rejecting sevens),

the values of which obviously circulate in a period of 7 times 30 or 210 years.

Let C denote the number of completed cycles, and y the year of the cycle; then Y = 30 C + y, and

w = 5left bracketC
line
7
right bracket + 6left brackety
line
7
right bracket + 3left bracket11 y +3
line
30
right bracket (rejecting sevens).

From this formula the following table has been constructed: -

Table VIII.

Year of the
Current Cycle (y)

Number of the Period of Seven Cycles = (C/7)

0

1

2

3

4

5

6

0

8

Mon.

Sat.

Thur.

Tues.

Sun.

Frid.

Wed.

1

9

17

25

Frid.

Wed.

Mon.

Sat.

Thur.

Tues.

Sun.

*2

*10

*18

*26

Tues.

Sun.

Frid.

Wed.

Mon.

Sat.

Thur.

3

11

19

27

Sun.

Frid.

Wed.

Mon.

Sat.

Thur.

Tues.

4

12

20

28

Thur.

Tues.

Sun.

Frid.

Wed.

Mon.

Sat.

*5

*13

*21

*29

Mon.

Sat.

Thur.

Tues.

Sun.

Frid.

Wed.

6

14

22

30

Sat.

Thur.

Tues.

Sun.

Frid.

Wed.

Mon.

*7

15

23

Wed.

Mon.

Sat.

Thur.

Tues.

Sun.

Frid.

*16

*24

Sun.

Frid.

Wed.

Mon.

Sat.

Thur.

Tues.

To find from this table the day of the week on which any year of the Hegira commences, the rule to be observed will be as follows: -

Rule

Divide the year of the Hegira by 30; the quotient is the number of cycles, and the remainder is the year of the current cycle. Next divide the number of cycles by 7, and the second remainder will be the Number of the Period, which being found at the top of the table, and the year of the cycle on the left hand, the required day of the week is immediately shown.

The intercalary years of the cycle are distinguished by an asterisk.

For the computation of the Christian date, the ratio of a mean year of the Hegira to a solar year is

Year of Hegira
line
Mean solar year
=354-11/30
line
365.2422
= 0.970224.

The year 1 began 16 July 622, Old Style, or 19 July 622, according to the New or Gregorian Style. Now the day of the year answering to the 19th of July is 200, which, in parts of the solar year, is 0.5476, and the number of years elapsed = Y - 1. Therefore, as the intercalary days are distributed with considerable regularity in both calendars, the date of commencement of the year Y expressed in Gregorian years is

0.970224 (Y - 1) + 622.5476,

or 0.970224 Y + 621.5774.

This formula gives the following rule for calculating the date of the commencement of any year of the Hegira, according to the Gregorian or New Style.

Rule

Multiply 970224 by the year of the Hegira, cut off six decimals from the product, and add 621.5774. The sum will be the year of the Christian era, and the day of the year will be found by multiplying the decimal figures by 365.

The result may sometimes differ a day from the truth, as the intercalary days do not occur simultaneously; but as the day of the week can always be accurately obtained from the foregoing table, the result can be readily adjusted.

Example

Required the date on which the year 1362 of the Hegira begins.

970224
1362
- - - -
1940448
5821344
2910672
970224
- - - - -
1321.445088
621.5774
- - - - -
1943.0225
365
- -
1125
1350
675
- - -
8.2125

Thus the date is the 8th day, or the 8th of January, of the year 1943.

To find, as a test, the accurate day of the week, the proposed year of the Hegira, divided by 30, gives 45 cycles, and remainder 12, the year of the current cycle.

Also 45, divided by 7, leaves a remainder 3 for the number of the period.

Therefore, referring to 3 at the top of the table, and 12 on the left, the required day is Friday.

The tables, page 571, show that 8th January 1943 is a Friday, therefore the date is exact.