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, P = 24 - E; l = 27 - E, or ((27 - E) / 7)_r,
When E > 23, P = 54 - E; l = 57 - E, or ((57 - E) / 7)_r.
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)_w = 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)_r; therefore N = 17 . .
. (2).