"Kalender-Formeln" von Rektor Chr. Zeller in Markgröningen, Mathematisch-naturwissenschaftliche Mitteilungen des mathematisch-naturwissenschaftlichen Vereins in Württemberg, ser. 1, 1 (1885), pp.54-58 - in German.
The translation is as yet low-grade.
Scanned images of (my copies of) the original printed pages are in zel-85px.htm.
VI. Calendar Formulae
from Rector Chr. Zeller of Markgröningen.
I. Day of the Week Calculation.
J be the century-number,
K the remaining part of the year-number,
e the residue, which remains when J is divided by 4,
m the number of the month,
q the day of the month,
h the number of the day of the week.
For fraction- or division-expressions, only the whole number therein contained comes into consideration.
Then h equals the remainder that results when
a) in the new calendar the sum :-
(m + 1) 26 K q + ---------- + K + - - 2 e (or + 5 e) 10 4
b) in the old calendar the sum :-
(m + 1) 26 K q + ---------- + K + - - (J + 2) or + (6 J + 5) 10 4
is divided by 7.
January and February are to be seen as the 13th and 14th month of the preceding year and to be so taken in the calculation.
Example 1. Birthday of Frederick the Great. 24 Jan 1712.
J + 17 e = 1 K = 11 m = 13 q = 24 (13 + 1) 26 11 24 + ----------- + 11 + -- - 2. 1 = 24 + 36 + 11 10 4 + 2 - 2 = 71 = 7. 10 + 1
h = 1. therefore, Frederick was born on the first day of the week, or Sunday.
Example 2. Coronation of Charlemagne 25 Dec 800.
J = 8 K = 0 m = 12 g = 25 13. 26 25 + ------ - (8 + 2) = 25 + 33 - 10 = 48 = 7. 6 + 6 10
h = 6 . . on the sixth day of the week or Friday.
The calculation is simplified if multiples of 7 are omitted while adding or if only the seven-remainders are added.
The first item in the formula changes every day, the second every month, the third every year, the fourth every leap-year, the fifth every century.
From this it is apparent also that the formula remains correct for all cases if it is correct for any one case, provided that it adapts to the changes that the calendar itself implies. These are certainly irregular and happen step-wise, every 4 years, every hundred years etc. and therefore it is striking that for these a formula is correct, which in other cases indeed only takes place where a successive and regular change of values is called for. In this case, however, it is achieved with the help of those expressions by which not the exact value is coming into the calculation but only the integers contained therein." These [expressions] lend themselves to the sequence of leaps that happen irregularly, as e.g. the month-formula (m+1)26/10 shows, whose exact value grows by 2.6 with each month, [and] which sometimes causes an increase of 2 units, sometimes of 3, and which, in fact, amounts to three days precisely for the months that have 31 days or 4 weeks and 3 days, and to two, for the months of 4 weeks and two days.
These expressions replace what formerly would have been compiled in a table with entries for each month, With their help it is possible to collect all numbers needing consideration within a single formula, which, naturally, can still be simplified further for particular cases, or for special values of a single argument, e.g. for fixed days-of-the-month in past years or for fixed centuries etc.
II. Easter Calculation.
A) In the old calendar.
1) Divide the Year-number K + 100 J or what is the same thing K + 5 J by 19. Remainder a.
2) divide 19 a + 15 by 30, remainder b; b is the Easter-full-moon-number and gives how many days after 21 March the Easter full moon is;
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.
Example. Raphael died 2 days before Easter 1520, when was Easter in that year?
J = 15 K = 20 1) K + 5J = 20+75 = 95 = 19.5 + 0, a = 0 2) 19a + 15 = 15 b = 15 3) 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.
B) In the new calendar.
1) divide K + 5 J by 19, remainder a;
2) to 19 a + 15 add the number g = J - J/4 - (8J+13)/25 (which number often remains the same for successive centuries and amounts to 7, 8, 9 for year between 1583 and 1700, 1700 and 1900, 1900 and 2200) divide by 30, remainder b, the Easter-full-moon-number;
3) to b add K + K/4 - 2 - 2 e, divide by 7, remainder d.
Then Easter is b + 7 - d days after 21 March, 7 - d days after the Easter-full-moon.
When in 3) the division by 7 is exact, then put d=0, except for the new calendar in two cases, 1) when also b=29 2) when b=28 or a is greater than 10; then put d=7 and what is the same thing, when the calculation for b=29 and d=0 gives the 26 April as the date of Easter, then instead put the 19 April, and put the 18 April when for a greater than 10 b=28 or d=0 is found instead of Easter-date the 25 April. This case will first occur in 1954.
Example. Easter 1886.
J = 18 = 4. 4 + 2, e = 2 K = 86 K/4 = 21 86 + 5. 18 = 176 = 19.9 + 5, a = 5 19. 5 + 15 + 8 = 118 = 30. 3 + 28, b = 28 28 + 86 + 21 + 2 - 2. 2 = 133 = 7. 19 + 0, d = 0 (because a equals smaller than 10)
Easter 28 + 7 - 0 = 35 days after 21 March, on 56 March or 25 April.
The author has derived these formulae, which were published both in the Württemb. Vierteljahresheften für Landesgeschichte 1882 and in Latin in the Bulletin de la Société Mathém. de France 1883 p.59, partly from Gauss's Easter Rules, partly from a calculation-guide which Brinckmeier, Praktisches Handbuch der histor. Chronologie Leipzig 1843 from Delambre, Histoire de l'Astronomie moderne l. p. 25 communicated. By the way, the example in the latter place contains several errors. Also the "Easter-tables in the new Style" given by Brinckm. p.84 are only accurate for the years 1700-1900. Even so have in the very useful and compendious Universal-Calendar of A. v. Eck 3. Aufl. Berlin 1865 themselves errors generated, namely in the beginning-way given instruction to the continuation of time and feast-calculation past 1900 out and in the following by the statements for the years 2049, 2076, 2106, 2133. These examples show, that also for the owner of such reference-books formulae are not wholly superfluous, which allows for the control of handbooks through recalculation without much difficulty.
Whoever desires more exact proof for Easter-formulae, will find the material also in the painstaking and fundamental discussion on the calculation of the Christian Easter-feasts, which Prof. D. Herm. Kinkelin of Basel has provided in the Zeitschr. für Mathem. und Phys. Vol.XV Leipzig 1870 p.217-228. This has not only confirmed the correctness and completeness of the foregoing expressions, but also has brought to my attention [?] that the exception indicated for b=29 & 28 is no longer necessary. As Kinkelin shows for the said formula, one can indeed put f = (b + a/11)/29, where b must be below 30, hence f will always be 0 except for b = 29 or b = 28 or a > 10. If now 7f is also subtracted in the expression for the Easter date then the formula holds for all cases without exception.
See Likely Errors in Implementations and Common Notes.
Probable typos in the printed maths have been preserved above, but underlined.
Read and adapt the 'Day of the Week Calculation' notes for the 1886 paper.
Typo: Day-of-Week, Example 2 : g should be q ?
The dividend of the division by 7 is sometimes negative.
Typo : Easter Date, B), 3) : term -2 should be +2 ?
Also, I suspect a muddling of "and" and "or" in the Easter text, when referring to a>10.
To avoid "mod of negative" error for the years 300, 702, 1101, 1503, 2600, 3401, ..., the term -2e in 3) can be changed to +5e.