On Rektor Chr. Zeller's 1886 Paper
"Kalender-Formeln".

1.   Rule.

Let
    J be the number of the century,
    K the number of the year within the same,
    m the number of the month,
    q the number of the day of the month,
    h the number of the day of the week;
remainders of divisions are discarded; January and February are viewed as the 13th and 14th month of the preceding year —:

then h equals the remainder that results

A.   in the Julian Calendar, if the sum

        (m + 1)26           K
  q  +  ---------  +  K  +  -  +  5  -  J
           10               4

B.   in the Gregorian Calendar, if the sum

        (m + 1)26           K     J
  q  +  ---------  +  K  +  -  +  -  -  2J
           10               4     4

is divided by 7.

The calculation is simplified if multiples of 7 are omitted while adding or if only the seven-remainders are added.

2.   Examples.

A.   On which day of the week did Columbus land, Oct. 12, 1492?

  q = 12,   m = 10,   K = 92,   J = 14

         (10 + 1)26
  12  +  ----------  +  92  +  23  +  5  -  14
             10

  =  12 + 28 + 92 + 23 + 5 - 14
  =  5 + 0 + 1 + 2 + 5  =  13  =  1 * 7 + 6

h = 6; that is the sixth day of the week or Friday.

B.   Jan. 24, 1712. Birthday of Frederick II.

  q = 24,   m = 13,   K = 11,   J = 17

         (13 + 1)26            11     17
  24  +  ----------  +  11  +  --  +  --  -  2 * 17
             10                 4      4

  =  24 + 36 + 11 + 2 + 4 - 34  =  43  =  6 * 7 + 1

h = 1; therefore, Frederick II was born on the first day of the week, or Sunday.

3.   Explanation.

Of the values in the sum above, q changes every day, m every month, K every year, K/4 every leap-year, J every century, J/4 every fourth century. From this we can see that the formula remains correct for all cases if it is correct for any one case, which is easily shown by an arbitrary example, provided that it adapts to the changes that the run of the calendar implies, even where these are irregular and volatile. The greatest irregularity, for example, is due to the lengths of months; however, the expression (m + 1)26/10 compensates for these variations; this value increases by 2.6 every month, which will sometimes cause an increase of 2 integers and sometimes an increase of 3, and actually results in three integers for the months with 31 days or 4 weeks, 3 days, and two for the months with 30 days. By the help of this expression it has become possible to unite all possible numbers in one formula, without using auxiliary tables or numbers.

1.   Rule.

A)   For the Julian calendar.
1)  Divide the Year-number (100J+K) or what gives the same remainder (5J+K) by 19, remainder a;
2)  divide (19a+15) by 30, remainder b;
b is the Easter-full-moon-number and gives how many days the Easter full moon is after 21 March;
3)  to b add (K+K/4-J), divide by 7, remainder d;
then Easter is (b+7-d) days after 21 March, (7-d) days after the Easter full moon.

B)   For the Gregorian calendar.
1)  divide (5J+K) by 19, remainder a;
2)  to (19a+15) add the number g = J-J/4-J/3; divide 19a+15+J-J/4-J/3 by 30, remainder b, the Easter-full-moon-number;
3)  to b add K+K/4+J/4+2-2J, divide by 7, remainder d; then Easter is (b+7-d) days after 21 March.

Note 1.   The value of g in 2) often remains the same for successive centuries and amounts to 7, 8, 9 for the years 1583-1700, 1700-1900, 1900-2200. The formula given thus is correct up to the year 4200; for years which lie after that, put instead of J/3 the exact value (8J+13)/25. So other ? the formula then wholly complete for all centuries of the Gregorian calendar.

Note 2.   When in 3) the division by 7 is exact, then put d=0, except for the Gregorian calendar in two cases: 1) when d=0 and b=29; 2) when d=0, b=28, a>10; then put d=7; or what is the same thing: when the calculation for d=0, b=29 gives 26 April as the date of Easter, then instead put 19 April, and 18 instead of 25 April, when b=28, a>10, d=0 is found.

By the way, leaves itself these exceptions also in the formula itself to introduce, as HERM. KINKELIN of Basel shows in the thorough and instructive treatise Berechnung des christlichen Osterfests, Zeitschrift für Mathematik und Physik, Vol.XV, 1870, pp.217-228. One makes namely f = (b+a/11)/29 so as there will be b no greater than 29, a no greater than 18, the value f will always be zero, except in each of the two exceptional cases b=29 or b=28 and a>10, for which one gets f=1. One now joins the expression for the Easter-date in addition by -7f, so splits itself also each of the exceptional cases.

2.   Examples.

A.   Raphael died 2 days before Easter 1520, on which day of the month?

       J = 15,   K = 20,   K + 5J = 95 = 5.19 + 0,   a = 0
                   19a + 15 = 15,  b = 15
         15 + 20 + 20/4 - 15 = 25 = 7.3 + 4,   d = 4

Easter is 15 + 7 - 4 = 18 days after 21 March, on 39 March or 8 April. Raphael died on Good Friday 6 April.

B.   Easter 1886.

                  J = 18,   K = 86,   K/4 = 21
              86 + 5.18 = 176 = 19.9 + 5,   a = 5
     19.5 + 15 + 18 - 18/4 - 18/3 = 118 = 30.3 + 28,   b = 28
  28 + 86 + 86/4 + 18/4 + 2 - 2.18 = 141 - 36 = 105 = 7.15 + 0,
                    d = 0  (because a<10)

Easter is (28 + 7 - 0) = 35 days after 21 March, on 56 March or 25 April.

3.   Explanation.

The number a found in 1) as remainder by the division by 19 is equal to 1 less than the Golden Number, which gives the position of the year in question in the 19-year Moon-period. a grows by 1 each year and can have all values from 0 to 18.

The Easter full moon in the following year is always either earlier by 11 days or later by 30-11 = 19 days; after even itself Law ? also the progress or retrogression by the remainder of the division of (19a+15) by 30, and so one convinces oneself also here, that the Formula, when it once gives a correct result for the Easter-full-moon-number, always must give such, then the value of b proceeds indeed all the Moon-course corresponding forwards or backwards.

By the way, lies? itself the correctness of the procedures also out of the generally-known instructions on the cyclic reckoning of the Moon-phases by means of the Golden Number to prove.

For the Gregorian calendar, in 2) the expression (19a+15) must be added in addition to the number g. In this calendar is naturally because of the residual? secular-change of the date by (J-J/4-2) days advanced, or (J-J/4-2) is the sum of the so-called Sun-equation, æquatio solaris, omitted days; for-equal however will through the so-called Moon-equation, (æquatio lunaris), each Moon-phase in 300 years by 1 day, exactly in 2500 years by 8 days backwards stacked; the sum of these days to be subtracted as Moon-equation amounts to (J/3-2) or exactly ((8J+13)/25-2) which is to count in the opposite sense to the Sun-equation, so that when the total value, after which itself to go the Julian calendar of the full-moon-day postponement amounts to
      (J-J/4-2) - (J/3-2) = (J-J/4-J/3) = g   days.
These must one the Julian Easter-full-moon-date to number, as is shown above.

The Easter-full-moon-day having been found, so in addition its weekday and how many days from there to the next Sunday there are must be determined. It shows fully after the foregoing weekday-calculation, (d+1) is the weekday-number of the Easter-full-moon and from there to the next Sunday there are either 8 weekdays 8-(d+1) or (7-d) days, as above is known.

Finally can now be added, that the above formulae, but with fewer asides and without explanation, were published in Latin in the Bulletin of the Mathematical Society of France 1883, p.59, also recently in the Mathemat.-naturwissenschaftl. Mitteilungen, H.2, p.54, Tübingen 1885, produced by O. BÖKLEN of Reutlingen.

Afterword.

Instead of J and K one can also use the Year-number N itself and have J = N/100, etc. For K+K/4 one can put N+N/4+N/100, for g the expression N/100-N/300-N/400, etc.

One has somewhat larger numbers in the calculation, but more clarity and apart from b and d no auxiliary numbers. - The expressions in the brackets represent in I) and II) the amount of the Sun-equation, in II) the additional with g denoting combining amount of the Sun- and Moon-equations.