Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/741

.] Most s agreed that  should be celebrated on a. Others followed the example of the s, and adhered to the 14th of the ; but these, as usually happened to the minority, were accounted s, and received the appellation of. In order to terminate dissensions, which produced both scandal and in the, the  of , which was held in  , ordained that the celebration of  should thenceforth always take place on the  which immediately follows the  that happens upon, or next after, the day of the. Should the 14th of the, which is regarded as the day of , happen on a , the celebration of was deferred to the  following, in order to avoid concurrence with the s and s. The observance of this rule renders it necessaiy to reconcile three periods which have no common measure, namely, the week, the lunar month, and the solar year; and as this can only be done approximately, and within certain limits, the determination of  is an affair of considerable nicety and complication. It is to be regretted that the  who formed the  of  were not advised to abandon the  altogether, and appoint  to be celebrated on the first or second  of. The calendar would in that case have possessed all the simplicity and uniformity of the civil calendar, which only requires the adjustment of the civil to the solar year; but they were probably not sufficiently versed in  to be aware of the practical difficulties which their regulation had to encounter.

Dominical .—The first problem which the construction of the calendar presents is to connect the week with the year, or to find the day of the week corresponding to a given day of any year of. As the number of days in the week and the number in the year are prime to one another, two successive years cannot begin with the same day; for if a common year begins, for example, with, the following year will begin with, and if a leap year begins with , the year following will begin with. For the sake of greater generality, the days of the week are denoted by the first seven s of the, , , , , , , , which are placed in the calendar beside the days of the year, so that stands opposite the first day of , opposite the second, and so on to , which stands opposite the seventh; after which  returns to the eighth, and so on through the 365 days of the year. Now, if one of the days of the week, for example, is represented by,  will be represented by ,  by ,  by , and so on; and every  through the year will have the same character , every  , and so with regard to the rest. The which denotes  is called the Dominical, or the  ; and when the dominical  of the year is known, the s which respectively correspond to the other days of the week become known at the same time.

Cycle.—In the Julian calendar the dominical letters are readily found by means of a short cycle, in which they recur in the same order without interruption. The number of years in the period being four, and the days of the week being seven, their product is 4× 7= 28; twenty-eight years is therefore a period which includes all the possible combinations of the days of the week with the commencement of the year. This period is called the Cycle, or the Cycle of the, and restores the first day of the year to the same day of the week. At the end of the cycle the dominical letters return again in the same order on the same days of the month; hence a table of dominical letters, constructed for twenty-eight years, will serve to show the dominical of any given year from the commencement of the era to the reformation. The cycle, though probably not invented before the time of the of, is regarded as having commenced nine years before , so that the year  was the tenth of the solar cycle. To find the year of the cycle, we have therefore the following rule:—Add nine to the date, divide the sum by twenty-eight; the quotient is the number of cycles elapsed, and the remainder is the year of the cycle. Should there be no remainder, the proposed year is the twenty-eighth or last of the cycle. This rule is conveniently expressed by the formula $\left ( \frac{\scriptstyle x+9}{\scriptstyle\text {28}} \right )_r $,|undefined in which x denotes the date, and the symbol r denotes that the remainder, which arises from the division of x+ 9 by 28, is the number required. Thus, for 1840, we have $1840+ 9⁄28$= 66$1⁄28$; therefore $\left ( \frac{\scriptstyle 1840+9}{\scriptstyle\text {28}} \right )_r $ = 1,|undefined and the year 1840 is the first of the cycle. In order to make use of the cycle in finding the dominical letter, it is necessary to know that the first year of the  began with. The dominical letter of that year, which was the tenth of the cycle, was consequently. The following year, or the 11th of the cycle, the was ; then. The fourth year was bissextile, and the dominical letters were, ; the following year , and so on. In this manner it is easy to find the dominical letter belonging to each of the twenty-eight years of the cycle. But at the end of a century the order is interrupted in the Gregorian calendar by the secular suppression of the leap year; hence the cycle can only be employed during a century. In the reformed calendar the period is four hundred years, which number being multiplied by seven, gives two thousand eight hundred years as the interval in which the coincidence is restored between the days of the year and the days of the week. This long period, however, may be reduced to four hundred years; for since the dominical letter goes back five places every four years, its variation in four hundred years, in the Julian calendar, was five hundred places, which is equivalent to only three places (for five hundred divided by seven leaves three); but the Gregorian calendar suppresses exactly three s in four hundred years, so that after four hundred years the dominical letters must again return in the same order.|undefined Hence the following table of dominical letters for four hundred years will serve to show the dominical letter of any year in the Gregorian calendar for ever. It contains four columns of s, each column serving for a century. In order to find the column from which the in any given case is to be taken, strike off the two last figures of the date, divide the preceding figures by four, and the remainder will indicate the column. The symbol, employed in the formula at the top of the column, denotes the number of centuries, that is, the figures remaining after the last two have been struck off. For example, required the dominical letter of the year ? In this case = 18, therefore $\left ( \frac{\scriptstyle\text {X}}{\scriptstyle\text {4}} \right )_r $ = 2;|undefined and in the second column of s, opposite 39, in the table we find, which is the of the proposed year.|undefined It deserves to be remarked, that as the dominical letter of the first year of was, the first column of the following table will give the dominical letter of every year from the commencement of the era to the reformation. For this purpose divide the date by 28, and the opposite the remainder, in the first column of figures, is the dominical letter of the year. For example, supposing the date to be. On dividing by 28, the remainder is 0, or 28; and opposite 28, in the first column of s, we find, , the dominical letters of.