With regard to the lunar equation M, we have already stated that in the Gregorian calendar the epacts are increased by unity at the end of every period of 300 years seven times successively, and then the increase takes place once at the end of 400 years. This gives eight to be added in a period of twenty-five centuries, and x/25 in x centuries. But 8x/25 = 1/3 (x - x/25). Now, from the manner in which the intercalation is directed to be made (namely, seven times successively at the end of 300 years, and once at the end of 400), it is evident that the fraction x/25 must amount to unity when the number of centuries amounts to twenty-four. In like manner, when the number of centuries is 24 + 25 = 49, we must have x/25 = 2; when the number of centuries is 24 + 2 × 25 = 74, then x/25 = 3; and, generally, when the number of centuries is 24 + n × 25, then x/25 = n + 1. Now this is a condition which will evidently be expressed in general by the formula n - ((n + 1) / 25). Hence the correction of the epact, or the number of days to be intercalated after x centuries reckoned from the commencement of one of the periods of twenty-five centuries, is {(x - ((x+1) / 25)) / 3}. The last period of twenty-five centuries terminated with 1800; therefore, in any succeeding year, if c be the number of the century, we shall have x = c - 18 and x + 1 = c - 17. Let ((c - 17) / 25) = a, then for all years after 1800 the value of M will be given by the formula ((c - 18 - a) / 3); therefore, counting from the beginning of the calendar in 1582,

M =right bracec - 15 - a
line
3
left brace.

By the substitution of these values of J, S and M, the equation of the epact becomes

E =left bracketN + 10(N - 1)
line
30
right bracket - (c - 16) +left bracketc - 16
line
4
right bracket +left bracketc - 15 - a
line
3
right bracket.

It may be remarked, that as a = ((c - 17) / 25), the value of a will be 0 till c - 17 = 25 or c = 42; therefore, till the year 4200, a may be neglected in the computation. Had the anticipation of the new moons been taken, as it ought to have been, at one day in 308 years instead of 312½, the lunar equation would have occurred only twelve times in 3700 years, or eleven times successively at the end of 300 years, and then at the end of 400. In strict accuracy, therefore, a ought to have no value till c - 17 = 37, or c = 54, that is to say, till the year 5400. The above formula for the epact is given by Delambre (Hist. de l'astronomie moderne, t. i. p. 9); it may be exhibited under a variety of forms, but the above is perhaps the best adapted for calculation. Another had previously been given by Gauss, but inaccurately, inasmuch as the correction depending on ''a'' was omitted.

Having determined the epact of the year, it only remains to find Easter Sunday from the conditions already laid down. Let

P = the number of days from the 21st of March to the 15th of the paschal moon, which is the first day on which Easter Sunday can fall;

p = the number of days from the 21st of March to Easter Sunday;

L = the number of the dominical letter of the year;

l = letter belonging to the day on which the 15th of the moon falls:

then, since Easter is the Sunday following the 14th of the moon, we have

p = P + (L - l),

which is commonly called the number of direction.

The value of L is always given by the formula for the dominical letter, and P and l are easily deduced from the epact, as will appear from the following considerations.

When P = 1 the full moon is on the 21st of March, and the new moon on the 8th (21 - 13 = 8), therefore the moon's age on the 1st of March (which is the same as on the 1st of January) is twenty-three days; the epact of the year is consequently twenty-three. When P = 2 the new moon falls on the ninth, and the epact is consequently twenty-two; and, in general, when P becomes 1 + x, E becomes 23 - x, therefore P + E = 1 + x + 23 - x = 24, and P = 24 - E. In like manner, when P = 1, l = D = 4; for D is the dominical letter of the calendar belonging to the 22nd of March. But it is evident that when l is increased by unity, that is to say, when the full moon falls a day later, the epact of the year is diminished by unity; therefore, in general, when l = 4 + x, E = 23 - x, whence, l + E = 27 and l = 27 - E. But P can never be less than 1 nor l less than 4, and in both cases E = 23. When, therefore, E is greater than 23, we must add 30 in order that P and l may have positive values in the formula P = 24 - E and l = 27 - E. Hence there are two cases.

When E < 24,left braceP = 24 - E
l = 27 - E, orleft bracket27 - E
line
7
right bracket,
When E > 23,left braceP = 54 - E
l = 57 - E, orleft bracket57 - E
line
7
right bracket.

By substituting one or other of these values of P and l, according as the case may be, in the formula p = P + (L - l), we shall have p, or the number of days from the 21st of March to Easter Sunday. It will be remarked, that as L - l cannot either be 0 or negative, we must add 7 to L as often as may be necessary, in order that L - l may be a positive whole number.

By means of the formulae which we have now given for the dominical letter, the golden number and the epact, Easter Sunday may be computed for any year after the Reformation, without the assistance of any tables whatever. As an example, suppose it were required to compute Easter for the year 1840. By substituting this number in the formula for the dominical letter, we have x = 1840, c - 16 = 2, ((c - 16) / 4) = 0, therefore

L = 7m + 6 - 1840 - 460 + 2

= 7m - 2292

= 7 × 328 - 2292 = 2296 - 2292 = 4

L = 4 = letter D . . . (1).

For the golden number we have N = ((1840 + 1) / 19); therefore N = 17 . . . (2).

For the epact we have

left bracketN + 10(N - 1)
line
30
right bracket = left bracket17 + 160
line
30
right bracket = left bracket177
line
30
right bracket = 27;
likewise c - 16 = 18 - 16 = 2, c - 15
line
3
= 1, a = 0; therefore

E = 27 - 2 + 1 = 26 . . . (3).

Now since E > 23, we have for P and l,

P = 54 - E = 54 - 26 = 28,

l =left bracket57 - E
line
7
right bracket = left bracket57 - 26
line
7
right bracket = left bracket31
line
7
right bracket = 3;

consequently, since p = P + (L - l),

p = 28 + (4 - 3) = 29;

that is to say, Easter happens twenty-nine days after the 21st of March, or on the 19th April, the same result as was before found from the tables.