Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/744

672 contained 6939 days 18 hours, whereas the exact of 235 s, as we have already seen, is 235× 29·530588= 6939 days 16 hours 31 s. The difference, which is 1 hour 29 s, amounts to a day in 308 years, so that at the end of this  the s occur one day earlier than they are indicated by the golden numbers. During the 1257 years that elapsed between the and the reformation, the error had accumulated to four days, so that the  which were marked in the calendar as happening, for example, on the 5th of the month, actually fell on the 1st. It would have been easy to correct this error by placing the golden numbers four lines higher in the new calendar; and the suppression of the ten days had already rendered it necessary to place them ten lines lower, and to carry those which belonged, for example, to the 5th and 6th of the month, to the 15th and 16th. But, supposing this correction to have been made, it would have again become necessary, at the end of 308 years, to advance them one line higher, in consequence of the accumulation of the error of the cycle to a whole day. On the other hand, as the golden numbers were only adapted to the Julian calendar, every omission of the centenary would require them to be placed one line lower, opposite the 6th, for example, instead of the 5th of the month; so that, generally speaking, the places of the golden numbers would have to be changed every century. On this account thought fit to reject the golden numbers from the calendar, and supply their place by another set of numbers called Epacts, the use of which we shall now proceed to explain.

Epacts.—Epact is a of, employed in the calendar to signify the 's age at the beginning of the year. The common solar year containing 365 days, and the lunar year only 354 days, the difference is eleven; whence, if a fall on the 1st of  in any year, the moon will be eleven days old on the first day of the following year, and twenty-two days on the first of the third year. The numbers eleven and twenty-two are therefore the epacts of those years respectively. Another addition of eleven gives thirty-three for the epact of the fourth year; but in consequence of the insertion of the month in each third year of the lunar cycle, this epact is reduced to three. In like manner the epacts of all the following years of the cycle are obtained by successively adding eleven to the epact of the former year, and rejecting thirty as often as the sum exceeds that number. They are therefore connected with the golden numbers by the formula $\left ( \frac{\scriptstyle\text { 11 } \scriptstyle n}{\scriptstyle\text {30}} \right )_r $, in which n is any whole number; and for a whole lunar cycle (supposing the first epact to be 11), they are as follows: 11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29. But the order is interrupted at the end of the cycle; for the epact of the following year, found in the same manner, would be 29+ 11= 40 or 10, whereas it ought again to be 11 to correspond with the 's age and the golden number 1. The reason of this is, that the month, inserted at the end of the cycle, contains only twenty-nine days instead of thirty; whence, after 11 has been added to the epact of the year corresponding to the golden number 19, we must reject twenty-nine instead of thirty, in order to have the epact of the succeeding year; or, which comes to the same thing, we must add twelve to the epact of the last year of the cycle, and then reject thirty as before.|undefined

This method of forming the epacts might have been continued indefinitely if the Julian had been followed without correction, and the cycle been perfectly exact; but as neither of these suppositions is true, two equations or corrections must be applied, one depending on the error of the Julian year, which is called the solar equation; the other on the error of the lunar cycle, which is called the lunar equation. The solar equation occurs three times in 400 years, namely, in every secular year which is not a leap year; for in this case the omission of the day causes the  to arrive one day later in all the following months, so that the 's age at the end of the month is one day less than it would have been if the  had been made, and the epacts must accordingly be all diminished by unity. Thus the epacts 11, 22, 3, 14, &c., become 10, 21, 2, 13, &c. On the other hand, when the by which the  anticipate the lunar cycle amounts to a whole day, which, as we have seen, it does in 308 years, the new moons will arrive one day earlier, and the epacts must consequently be increased by unity. Thus the epacts 11, 22, 3, 14, &c., in consequence of the lunar equation, become 12, 23, 4, 15, &c. In order to preserve the uniformity of the calendar, the epacts are changed only at the commencement of a century; the correction of the error of the lunar cycle is therefore made at the end of 300 years. In the Gregorian calendar this error is assumed to amount to one day in 312$1⁄2$ years, or eight days in 2500 years, an assumption which requires the line of epacts to be changed seven times successively at the end of each period of 300 years, and once at the end of 400 years; and, from the manner in which the epacts were disposed at the reformation, it was found most correct to suppose one of the periods of 2500 years to terminate with the year 1800. The years in which the solar equation occurs, counting from the reformation, are,, 1900, 2100, 2200, 2300, 2500, &c. Those in which the lunar equation occurs are, 2100, 2400, 2700, 3000, 3300, 3600, 3900, after which, 4300, 4600, and so on. When the solar equation occurs, the epacts are diminished by unity; when the lunar equation occurs, the epacts are augmented by unity; and when both equations occur together, as in, 2100, 2700, &c., they compensate each other, and the epacts are not changed. In consequence of the solar and lunar equations, it is evident that the epact, or 's age at the beginning of the year, must, in the course of centuries, have all different values from one to thirty inclusive, corresponding to the days in a full lunar month. Hence, for the construction of a perpetual calendar, there must be thirty different sets or lines of epacts. These are exhibited in the subjoined table (TableIII.) called the Extended Table of Epacts, which is constructed in the following manner. The series of golden numbers is written in a line at the top of the table, and under each golden number is a column of thirty epacts, arranged in the order of the, beginning at the bottom and proceeding to the top of the column. The first column, under the golden number 1, contains the epacts, 1, 2, 3, 4, &c., to 30 or 0. The second column, corresponding to the following year in the lunar cycle, must have all its epacts augmented by 11; the lowest, therefore, in the column is 12, then 13, 14, 15, and so on. The third column, corresponding to the golden number 3, has for its first epact 12+ 11= 23; and in the same manner all the nineteen columns of the table are formed. Each of the thirty lines of epacts is designated by a of the, which serves as its index or argument. The order of the s, like that of the s, is from the bottom of the column upwards. In the tables of the calendar the epacts are usually  in, excepting the last, which is designated by an  ( * ), used as an indefinite  to denote 30 or 0, and 25, which in the last eight columns is expressed in , for a reason that will immediately be explained. In the table 