Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/742

670 —Dominical Letters.

The Day of the Week.

Cycle and en .—In connecting the lunar month with the solar year, the framers of the ecclesiastical calendar adopted the period of Meton, or cycle, which they supposed to be exact. A different arrangement has, however, been followed with respect to the distribution of the months. The s are supposed to consist of twenty-nine and thirty days alternately, or the year of 354 days; and in order to make up nineteen solar years, six  or  months, of thirty days each, are introduced in the course of the cycle, and one of twenty-nine days is added at the end. This gives 19x 354+ 6x 30+ 29= 6935 days, to be distributed among 235 lunar months. But every leap year one day must be added to the lunar month in which the 29th of is included. Now if leap year happens on the first, second, or third year of the period, there will be five leap years in the period, but only four when the first leap year falls on the fourth. In the former case the number of days in the period becomes 6940 and in the latter 6939. The mean length of the cycle is therefore 6939$3⁄4$ days, agreeing exactly with nineteen Julian years. By means of the cycle the s 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 '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, or because it was usual to mark it with red s in the calendar. The Golden Numbers were introduced into the calendar about, but disposed as they would have been if they had been inserted at the time of the. The cycle is supposed to commence with the year in which the new falls on the 1st of, which took place the year preceding the commencement of our era. Hence, to find the Golden Number N, for any year x, we have $\scriptstyle\text {N = } \left ( \frac{\scriptstyle x \scriptstyle\text { + 1}}{\scriptstyle\text {19}} \right )_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 s, determined in this manner, may differ from the s sometimes as much as two days. The reason is, that the sum of the and  inequalities, which are compensated in the whole period, may amount in certain cases to 10°, and thereby cause the  to arrive on the second day before or after its mean .|undefined

' Period.—The cycle of the brings back the days of the month to the same day of the week; the cycle restores the new s to the same day of the month; therefore 28x 19= 532 years, includes all the variations in respect of the new s and the dominical letters, and is consequently a period after which the s again occur on the same day of the month and the same day of the week. This is called the ' or Great Period, from its having been employed by, familiarly styled &ldquo;Denys the Little,&rdquo; in determining. It was, however, first proposed by of, who had been appointed by  to revise and correct the  calendar. Hence it is also called the Period. It continued in use till the Gregorian reformation.

Besides the and cycles, there is a third of 15 years, called the cycle of, frequently employed in the computations of. This period is not, like the two former, but has reference to certain s which took place at stated epochs under the. Its commencement is referred to the 1st of of the year  of the. By extending it backwards, it will be found that the was the fourth of the cycle of. The number of any year in this cycle will therefore be given by the $\left ( \frac{\scriptstyle x \scriptstyle\text { + 3}}{\scriptstyle\text {15}} \right )_r $, that is to say, add 3 to the date, divide the sum by 15, and the remainder is the year of the. When the remainder is 0, the proposed year is the fifteenth of the cycle.|undefined

 Period.—The period, proposed by the celebrated  as an universal measure of, is formed by taking the continued product of the three cycles of the , of the , and of the indiction, and is consequently 28x 19x 15= 7980