x = 1839, (x/4)_w = (1839/4)_w = 459, c = 18, c - 16 = 2,
and ((c - 16) / 4)_w = 0.
((26 + 11(N - 6)) / 30)_r. 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)_r.
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)_w. 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)_w, which [v.04 p.0999] must be
deducted from J. We have therefore
S = - (c - 16) + ((c - 16) / 4)_w.
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)_w. 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)_w) / 3}_w.
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)_w = a, then for all years
after 1800 the value of M will be given by the formula ((c - 18 - a) /
3)_w; therefore, counting from the beginning of the calendar in 1582,
M={(c - 15 - a) / 3}_w.
By the substitution of these values of J, S and M, the equation of the
epact becomes
E = ((N + 10(N - 1)) / 30)_r - (c - 16) + ((c - 16) / 4)_w + ((c - 15 -
a) / 3)_w.
It may be remarked, that as a = ((c - 17) / 25)_w, 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;