*THE POPULAR SCIENCE MONTHLY.*

though they would in nineteen years fall upon the same day of the month, they fell about an hour and a half earlier in the day, and this in sixteen cycles, or about three hundred and twelve years, would make a difference of one day. As these tables were published by ecclesiastical and secular authority, and could not be changed without such authority, another method was resorted to to find the times of the moon without the use of these tables. This method was called the *epact,* which we will now proceed to consider.

The lunar year, as we have already seen, differs from the solar year by about eleven days, i. e., if a new moon occur January 1st of any year, on January 1st of the next year the moon will be eleven days old, on the same day of the next year twenty-two days old, the next thirty-three days old, which equals a whole lunation *plus* three days. This cycle corresponds with the lunar cycle, and is constructed as follows:

Lunar Cycle. | Epact. | Paschal Limits. | Lunar Cycle. | Epact. | Paschal Limits. |

1 | 0 | April 13 | 11 | 20 | March 24 |

2 | 11 | "2 | 12 | 1 | April 12 |

3 | 22 | March 22 | 13 | 12 | April 1 |

4 | 3 | April 10 | 14 | 23 | March 21 |

5 | 14 | March 30 | 15 | 4 | April 9 |

6 | 25 | April 18 | 16 | 15 | March 29 |

7 | 6 | "7 | 17 | 26 | April 17 |

8 | 17 | March 27 | 18 | 7 | "6 |

9 | 28 | April 15 | 19 | 18 | March 26 |

10 | 9 | "4 |

From this table the astronomical moons not only for Easter but for the whole year can be found without variation of more than a day for about three hundred and twelve years, at the end of which time the new moon will fall one day earlier, when a new set of epacts must be made, the first of which will be 1 instead of 0, and the succeeding ones will be changed correspondingly. To find the age of the moon for any day of the year, we add to the epact the date of the month, and one for every month from March inclusive, the epact for a year being eleven days, or a day a month nearly. This sum, casting out thirty if required, will give the age of the moon at the given day: e. g., suppose it be required to find the moon's age on Christmas-day of the year 1868. We find, by the method already explained, that 1868 was the seventh year of the lunar cycle, whose epact in the table we found to be 6, to which adding 25 and 10 gives 41; from this deduct one lunation (29 days) 12 days for the moon's age on that day. The epacts are calculated to show the moon's age on March 1st in any year of the cycle.

The rule for finding the Sunday letter of any year, as given in the "Book of Common Prayer," is constructed upon this principle: The dominical letter of the year of Christ, according to N. S., would have