+-----------------------+-----------------------------------------+
| Month. | Dominical Letter. |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| Jan. Oct. | A | B | C | D | E | F | G |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| Feb. Mar. Nov. | D | E | F | G | A | B | C |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| April July | G | A | B | C | D | E | F |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| May | B | C | D | E | F | G | A |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| June | E | F | G | A | B | C | D |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| August | C | D | E | F | G | A | B |
+-----------------------+-----+-----+-----+-----+-----+-----+-----+
| Sept. Dec. | F | G | A | B | C | D | E |
+---+----+----+----+----+-----+-----+-----+-----+-----+-----+-----+
| 1 | 8 | 15 | 22 | 29 |Sun. |Sat |Frid.|Thur.|Wed. |Tues |Mon. |
| 2 | 9 | 16 | 23 | 30 |Mon. |Sun. |Sat. |Frid.|Thur.|Wed. |Tues.|
| 3 | 10 | 17 | 24 | 31 |Tues.|Mon. |Sun. |Sat. |Frid.|Thur.|Wed. |
| 4 | 11 | 18 | 25 | |Wed. |Tues.|Mon. |Sun. |Sat. |Frid.|Thur.|
| 5 | 12 | 19 | 26 | |Thur.|Wed. |Tues.|Mon. |Sun. |Sat. |Frid.|
| 6 | 13 | 20 | 27 | |Frid.|Thur.|Wed. |Tues.|Mon. |Sun. |Sat. |
| 7 | 14 | 21 | 28 | |Sat. |Frid.|Thur.|Wed. |Tues.|Mon. |Sun. |
+---+----+----+----+----+-----+-----+-----+-----+-----+-----+-----+
By means of the lunar cycle the new moons of the calendar were indicated
before the Reformation. As the cycle restores these phenomena to the same
days of the civil month, they will fall on the same days in any two years
which occupy the same place in the cycle; consequently a table of the
moon's phases for 19 years will serve for any year whatever when we know
its number in the cycle. This number is called the _Golden Number_, either
because it was so termed by the Greeks, or because it was usual to mark it
with red letters in the calendar. The Golden Numbers were introduced into
the calendar about the year 530, but disposed as they would have been if
they had been inserted at the time of the council of Nicaea. The cycle is
supposed to commence with the year in which the new moon falls on the 1st
of January, which took place the year preceding the commencement of our
era. Hence, to find the Golden Number N, for any year x, we have N = ((x +
1) / 19)_r, which gives the following rule: _Add 1 to the date, divide the
sum by 19; the quotient is the number of cycles elapsed, and the remainder
is the Golden Number._ When the remainder is 0, the proposed year is of
course the last or 19th of the cycle. It ought to be remarked that the new
moons, determined in this manner, may differ from the astronomical new
moons sometimes as much as two days. The reason is that the sum of the
solar and lunar inequalities, which are compensated in the whole period,
may amount in certain cases to 10°, and thereby cause the new moon to
arrive on the second day before or after its mean time.
_Dionysian Period._--The cycle of the sun brings back the days of the month
to the same day of the week; the lunar cycle restores the new moons to the
same day of the month; therefore 28 × 19 = 532 years, includes all the
variations in respect of the new moons and the dominical letters, and is
consequently a period after which the new moons again occur on the same day
of the month and the same day of the week. This is called the _Dionysian_
or Great _Paschal Period_, from its having been employed by Dionysius
Exiguus, familiarly styled "Denys the Little," in determining Easter
Sunday. It was, however, first proposed by Victorius of Aquitain, who had
been appointed by Pope Hilary to revise and correct the church calendar.
Hence it is also called the _Victorian Period_. It continued in use till
the Gregorian reformation.
_Cycle of Indiction._--Besides the solar and lunar cycles, there is a third
of 15 years, called the cycle of indiction, frequently employed in the
computations of chronologists. This period is not astronomical, like the
two former, but has reference to certain judicial acts which took place at
stated epochs under the Greek emperors. Its commencement is referred to the
1st of January of the year 313 of the common era. By extending it
backwards, it will be found that the first of the era was the fourth of the
cycle of indiction. The number of any year in this cycle will therefore be
given by the formula ((x + 3) / 15)_r, that is to say, _add 3 to the date,
divide the sum by 15, and the remainder is the year of the indiction_. When
the remainder is 0, the proposed year is the fifteenth of the cycle.
_Julian Period._--The Julian period, proposed by the celebrated Joseph
Scaliger as an universal measure of chronology, is formed by taking the
continued product of the three cycles of the sun, of the moon, and of the
indiction, and is consequently 28 × 19 × 15 = 7980 years. In the course of
this long period no two years can be expressed by the same numbers in all
the three cycles. Hence, when the number of any proposed year in each of
the cycles is known, its number in the Julian period can be determined by
the resolution of a very simple problem of the indeterminate analysis. It
is unnecessary, however, in the present case to exhibit the general
solution of the problem, because when the number in the period
corresponding to any one year in the era has been ascertained, it is easy
to establish the correspondence for all other years, without having again
recourse to the direct solution of the problem. We shall therefore find the
number of the Julian period corresponding to the first of our era.
We have already seen that the year 1 of the era had 10 for its number in
the solar cycle, 2 in the lunar cycle, and 4 in the cycle of indiction; the
question is therefore to find a number such, that [v.04 p.0994] when it is
divided by the three numbers 28, 19, and 15 respectively the three
remainders shall be 10, 2, and 4.
Let x, y, and z be the three quotients of the divisions; the number sought
will then be expressed by 28 x + 10, by 19 y + 2, or by 15 z + 4. Hence the
two equations