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|
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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)_w (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)_w.

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 le