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The new moons indicated by the epacts also differ from the astronomical new moons, and even from the mean new moons, in general by one or two days. In imitation of the Jews, who counted the time of the new moon, not from the moment of the actual phase, but from the time the moon first became visible after the conjunction, the fourteenth day of the moon is regarded as the full moon: but the moon is in opposition generally on the 16th day; therefore, when the new moons of the calendar nearly concur with the true new moons, the full moons are considerably in error. The epacts are also placed so as to indicate the full moons generally one or two days after the true full moons; but this was done purposely, to avoid the chance of concurring with the Jewish passover, which the framers of the calendar seem to have considered a greater evil than that of celebrating Easter a week too late.

Perpetual Table, showing Easter.

Epact. | Dominical Letter. | ||||||

A | B | C | D | E | F | G | |

* | Apr. 16 | Apr. 17 | Apr. 18 | Apr. 19 | Apr. 20 | Apr. 14 | Apr. 15 |

1 | " 16 | " 17 | " 18 | " 19 | " 13 | " 14 | " 15 |

2 | " 16 | " 17 | " 18 | " 12 | " 13 | " 14 | " 15 |

3 | " 16 | " 17 | " 11 | " 12 | " 13 | " 14 | " 15 |

4 | " 16 | " 10 | " 11 | " 12 | " 13 | " 14 | " 15 |

5 | " 9 | " 10 | " 11 | " 12 | " 13 | " 14 | " 15 |

6 | " 9 | " 10 | " 11 | " 12 | " 13 | " 14 | " 8 |

7 | " 9 | " 10 | " 11 | " 12 | " 13 | " 7 | " 8 |

8 | " 9 | " 10 | " 11 | " 12 | " 6 | " 7 | " 8 |

9 | " 9 | " 10 | " 11 | " 5 | " 6 | " 7 | " 8 |

10 | " 9 | " 10 | " 4 | " 5 | " 6 | " 7 | " 8 |

11 | " 9 | " 3 | " 4 | " 5 | " 6 | " 7 | " 8 |

12 | " 2 | " 3 | " 4 | " 5 | " 6 | " 7 | " 8 |

13 | " 2 | " 3 | " 4 | " 5 | " 6 | " 7 | " 1 |

14 | " 2 | " 3 | " 4 | " 5 | " 6 | Mar. 31 | " 1 |

15 | " 2 | " 3 | " 4 | " 5 | Mar. 30 | " 31 | " 1 |

16 | " 2 | " 3 | " 4 | Mar. 29 | " 30 | " 31 | " 1 |

17 | " 2 | " 3 | Mar. 28 | " 29 | " 30 | " 31 | " 1 |

18 | " 2 | Mar. 27 | " 28 | " 29 | " 30 | " 31 | " 1 |

19 | Mar. 26 | " 27 | " 28 | " 29 | " 30 | " 31 | " 1 |

20 | " 26 | " 27 | " 28 | " 29 | " 30 | " 31 | Mar. 25 |

21 | " 26 | " 27 | " 28 | " 29 | " 30 | " 24 | " 25 |

22 | " 26 | " 27 | " 28 | " 29 | " 23 | " 24 | " 25 |

23 | " 26 | " 27 | " 28 | " 22 | " 23 | " 24 | " 25 |

24 | Apr. 23 | Apr. 24 | Apr. 25 | Apr. 19 | Apr. 20 | Apr. 21 | Apr. 22 |

25 | " 23 | " 24 | " 25 | " 19 | " 20 | " 21 | " 22 |

26 | " 23 | " 24 | " 18 | " 19 | " 20 | " 21 | " 22 |

27 | " 23 | " 17 | " 18 | " 19 | " 20 | " 21 | " 22 |

28 | " 16 | " 17 | " 18 | " 19 | " 20 | " 21 | " 22 |

29 | " 16 | " 17 | " 18 | " 19 | " 20 | " 21 | " 15 |

We will now show in what manner this whole apparatus of methods and tables may be dispensed with, and the Gregorian calendar reduced to a few simple formulae of easy computation.

And, first, to find the dominical letter. Let L denote the number of the dominical letter of any given year of the era. Then, since every year which is not a leap year ends with the same day as that with which it began, the dominical letter of the following year must be L - 1, retrograding one letter every common year. After x years, therefore, the number of the letter will be L - x. But as L can never exceed 7, the number x will always exceed L after the first seven years of the era. In order, therefore, to render the subtraction possible, L must be increased by some multiple of 7, as 7m, and the formula then becomes 7m + L - x. In the year preceding the first of the era, the dominical letter was C; for that year, therefore, we have L = 3; consequently for any succeeding year x, L = 7m + 3 - x, the years being all supposed to consist of 365 days. But every fourth year is a leap year, and the effect of the intercalation is to throw the dominical letter one place farther back. The above expression must therefore be diminished by the number of units in x/4, or by (x/4) (this notation being used to denote the quotient, in a whole number, that arises from dividing x by 4). Hence in the Julian calendar the dominical letter is given by the equation

L = 7m + 3 - x - | x 4 | . |

This equation gives the dominical letter of any year from the commencement of the era to the Reformation. In order to adapt it to the Gregorian calendar, we must first add the 10 days that were left out of the year 1582; in the second place we must add one day for every century that has elapsed since 1600, in consequence of the secular suppression of the intercalary day; and lastly we must deduct the units contained in a fourth of the same number, because every fourth centesimal year is still a leap year. Denoting, therefore, the number of the century (or the date after the two right-hand digits have been struck out) by c, the value of L must be increased by 10 + (c - 16) - ((c - 16) / 4) . We have then

L = 7m + 3 - x - | x 4 | + 10 + (c - 16) - | c - 16 4 | ; |

that is, since 3 + 10 = 13 or 6 (the 7 days being rejected, as they do not affect the value of L),

L = 7m + 6 - x - | x 4 | + (c - 16) - | c - 16 4 | . |

This formula is perfectly general, and easily calculated.

As an example, let us take the year 1839. In this case,

x = 1839, | x 4 | = | 1839 4 | = 459, c = 18, c - 16 = 2, and | c - 16 4 | = 0. |

Hence

L = 7m + 6 - 1839 - 459 + 2 - 0

L = 7m - 2290 = 7 × 328 - 2290.

L = 6 = letter F.

The year therefore begins with Tuesday. It will be remembered that in a leap year there are always two dominical letters, one of which is employed till the 29th of February, and the other till the end of the year. In this case, as the formula supposes the intercalation already made, the resulting letter is that which applies after the 29th of February. Before the intercalation the dominical letter had retrograded one place less. Thus for 1840 the formula gives D; during the first two months, therefore, the dominical letter is E.

In order to investigate a formula for the epact, let us make

E = the true epact of the given year;

J = the Julian epact, that is to say, the number the epact would have been if the Julian year had been still in use and the lunar cycle had been exact;

S = the correction depending on the solar year;

M = the correction depending on the lunar cycle;

then the equation of the epact will be

E = J + S + M;

so that E will be known when the numbers J, S, and M are determined.

The epact J depends on the golden number N, and must be determined from the fact that in 1582, the first year of the reformed calendar, N was 6, and J 26. For the following years, then, the golden numbers and epacts are as follows:

1583, N = 7, J = 26 + 11 - 30 = 7;

1584, N = 8, J = 7 + 11 = 18;

1585, N = 9, J = 18 + 11 = 29;

1586, N = 10, J = 29 + 11 - 30 = 10;

and, therefore, in general J = ((26 + 11(N - 6)) / 30). But the numerator of this fraction becomes by reduction 11 N - 40 or 11 N - 10 (the 30 being rejected, as the remainder only is sought) = N + 10(N - 1); therefore, ultimately,

J = | N + 10(N - 1) 30 | . |

On account of the solar equation S, the epact J must be diminished by unity every centesimal year, excepting always the fourth. After x centuries, therefore, it must be diminished by x - (x/4). Now, as 1600 was a leap year, the first correction of the Julian intercalation took place in 1700; hence, taking c to denote the number of the century as before, the correction becomes (c - 16) - ((c - 16) / 4), which must be deducted from J. We have therefore

S = - (c - 16) + | c - 16 4 | . |

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