a 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 | 2 | 3 | 4 | 5 | 6 | 7
indicate Sun. | Mon. | Tue. | Wed. | Thur. | Frid. | Sat.

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

w = 2 + 4(Y / 7)_r + ((11 Y + 3) / 30)_w (rejecting sevens).

But, 30 ((11 Y + 3) / 30)_w + ((11 Y + 3) / 30)_r = 11 Y + 3

gives 120((11 Y + 3) / 30)_w = 12 + 44 Y - 4((11 Y + 3) / 30)_r,

or ((11 Y + 3) / 30)_w = 5 + 2 Y + 3((11 Y + 3) / 30)_r (rejecting
sevens).

So that

w = 6(Y / 7)_r + 3((11 Y + 3) / 30)_r (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 = 5(C / 7)_r + 6(y / 7)_r + 3((11 y +3) / 30)_r (rejecting sevens).

From this formula the following table has been constructed:--

TABLE VIII.

Year of the Number of the Period of Seven Cycles = (C/7)_r
Current Cycle (y) 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 c