| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||||
| 0 | 11 | 22 | 3 | 14 | 25 | 6 | 17 | 28 |
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | |||||||||
| 9 | 20 | 1 | 12 | 23 | 4 | 15 | 26 | 7 | 18 |
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 18 + 11 = 29, whereas it ought to be a cipher to correspond with the moon’s age and the Golden Number 1. The reason for this is, that the intercalary 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. Thus, 18 + 12 = 30; 30 - 30 = 0; the cipher corresponding with the Golden Number 1.
This method of forming the epacts might have been continued indefinitely if the Julian intercalation had been followed without correction and the cycle had 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 errors 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 intercalary day causes the new moons to arrive one day later in all the following months, so that the moon’s age at the end of the month is one day less than it would have been if the intercalation had been made, and the epacts must accordingly be all diminished by unity. Thus, the epacts 11, 22, 3, 14, etc., become 10, 21, 2, 13, etc.
On the other hand, when the time by which the new moons 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, etc., in consequence of the lunar equation, becomes 12, 23, 4, 15, etc. In order to preserve the uniformity of the calendar, the epacts are changed only at the commencement of the 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 a day in 312½ 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 1700, 1800, 1900, 2100, 2200, 2300, 2500, etc. Those in which the lunar equation occurs are 1800, 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 1800, 2100, 2700, etc., they compensate each other, and the epacts are not changed.
CHAPTER VI.
A NEW AND EASY METHOD OF FIXING THE DATE OF EASTER.
In determining the date of Easter, we make use of the numbers called epacts; and, as these numbers have already been explained in the preceding chapter, (q. v.) it will be necessary to give them only a brief notice here. Epact, as has already been defined, is the excess of the solar year beyond the lunar, employed in the calendar to signify the moon’s age at the beginning of the year; that is, if a new moon fall on the first day of January in any year, it will be eleven days old on the first day of the following year, and twenty-two days old on the first day of the third year, and so on.