CALENDAR [ECCLESIASTICAL. following year must be L - 1, retrograding one letter every com mon year. After x years, therefore, the number of the letter Avill 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 (this notation being used by the number of units iu , or by 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= 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) - (- T 16 ) (x - X - I 7 ) /w We have then L = 7m + 3 + 10 + that is, since 3 + 10 = 13 or 6 (the 7 days being rejected, as they do not affect the value of L), This formula is perfectly general, and easily calculated. As an example, let us take the year 1839. In this case, and C -1T^X= - 4 Hence L = 7m + 6 - 1839 -459 + 2-0 L = 7m - 2290 = 7 x 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 inter calation 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, ami M are deter mined. The epact J depends on the golden number N, and must be de termined 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 = 2<, + 11 - 30 - 10; and, therefore, in general J = ( ^ ) . Cut numerator of this fraction becomes by reduction UN - 40 or 11 N - 10 (the 30 being rejected, as the remainder only is sought) = N + 10 (N - 1); therefore, ultimately, T /N -f 10 (N - l) " I 7T^ the On account of the solar equation S, the cpacfc J must be dimin ished by unity every centesimal year, excepting always the fourth. After x centuries, therefore, it must be diminished by x - ( . V,. Now, as 1600 was a leap year, the first correction of the Julian in tercalation took place in 1700 ; hence, taking c to denote the number of the century as before, the correction becomes (c - l(j) _ ( I , which must be deducted from J. We have thcre- V 4 )w fore S- -(c - 10) + (^ 4 -J 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 8.c eight to be added in a period of twenty-five centuries, and --- in x centuries. But rr^ = TT ( * KK ) Now, from the manner in ZO O lj J 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 ^ 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 ^r = 2; when the number of centuries is 24 -f- 2 x 25 74, then ~ - 3 ; and, generally, when the number of centuries is 24 + n x 25, then = n + 1. Now this is a condition which will 25 evidently be expressed in general by the formula n - ( ^~ ) w Hence the correction of the epact, or the number of days to be in tercalated after x centuries reckoned from the commencement of OIK of the periods of twenty-five centuries, is 25 The last period of twenty-five centuries terminated with 1800 ; there fore, in any succeeding year, if c be the number of the century, v. e shall have x c - 18 and x + 1 = c - 17. Let f - () , 1 = a, then for all years after 1800 the value of M will be given by the formula ( 5 "I > therefore, counting from the begin-
" J w
ning of the calendar in 1582, ( f IP; n M - i 3 * By the substitution of these values of J, S, and M, the equation of the epact becomes It may be remarked, that as a = { -- 1 , the value of a will x^ ^.D j iv be 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 312i, 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, dc V Astronomic Modcrnc, torn. 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 ; I 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 - I),
which is commonly called the number of direction.