© J R Stockton, ≥ 2008-11-13

"Die Grundaufgaben der Kalenderrechnung auf neue und vereinfachte Weise gelöst".

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**This Page**:-- The Original 1882 Publication - Images (85kB)
- Other Papers published by Zeller :-
- "Problema duplex Calendarii fundamentale" (1883)
- "Kalender-Formeln" (1885)
- "Kalender-Formeln" (1886)

- The Date of Easter Sunday
- For more date/time pages, and links, see Date and Time Index and Links
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Rektor Zeller, "Die Grundaufgaben der Kalenderrechnung auf neue und
vereinfachte Weise gelöst", *Württembergische
Vierteljahrshefte für Landesgeschichte*, Jahrgang V (1882), pp.
313-314. - in German.

Scanned images of (my copies of) the original printed pages are in zel-82px.htm.

The places named above are all in the south-west of Germany. Franconia and Swabia were ancient Duchies. The Sülchgau area is the site of modern Rottenburg.

*** from the German by JRS, late autumn
2002; edited by JRS with help by PN ***

The Basic Problems of the Calendar-Reckoning solved in a new and simplified Manner

I. Notation.

Let
*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.

Fraction- or division- expressions like *k/4* etc. shall here
not mean fractional numbers, but that whole number, which one by the
carrying out of the specified division as quotient obtains.

II. Day of the Week Determination.

Problem 1. From the Date to find its Weekday.

Rule: One forms the total

a. for the new calendar :-

(m + 1) 26 k q + ---------- + k + - - 2 e (or + 5 e) 10 4

b. for the old calendar :-

(m + 1) 26 A q + ---------- + A + - - J (or + 6 J) 10 4whereA = k + 4

to be divided by *7*, the remainder gives the desired
Weekday-number.

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.

Now it is only necessary to add four easily obtainable items to the
date *q* and to divide by *7*; this can be done without
written reckoning, especially if one omits multiples of seven during the
addition of the sums or if one only adds the seven- remainders.

The first term changes every day, the second every month, the third every year, the fourth every leap-year, the fifth every century.

Now, if one uses instead of the last three the seven-remainder of
*k + k/4 -2e*, which we will call = *+ t*, then *q +
(m+1)×2,6 + t* is the weekday-formula for the entire calendar
year *k*; one must only add the number *t + (m+1)×2,6*,
without fraction, to the date *q* in order to find the number,
whose seven-remainder gives the day of the week.

Example 1. 1882, 11th Sept.
*J = 18 = 4.4 + 2 k = 82 m = 9*

e = 2 k/4 = 20 q = 11 (9 + 1) 26 11 + ---------- + 82 + 20 - 2.2 = 135 = 7.19 + 2 10

thus the 11th of September 1882 was on the second day of the week or Monday.

Example 2. 1492, 12th Oct. Discovery of the New World.
*A = 92 + 4 = 96*

(10 + 1) 26 12 + ----------- + 96 + 24 - 14 = 146 = 7.20 + 6 10 = 28

thus the 12th of October 1492 was on the sixth day of the week or Friday.

Example 3. 1712, 24th Jan. Birthday of Frederick II.
*k = 11 m = 13 e = 1*

(13 + 1) 26 24 + ----------- + 11 + 2 - 2.1 = 24 + 36,4 + 11 + 2.2 = 71 = 7.10 + 1 10

thus Frederick II was born on the 1st day of the week or Sunday.

III. Easter Calculation.

Problem 2. To find the date of Easter

a. in the old calendar.

1) *k + 5 J* to be divided by *19*, remainder *a*;

2) *19 a + 15* to be divided 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*, to be divided by *7*,
remainder *d*;

then Easter is *b + 7 - d* days after 21 March,
or *7 - d* days after the Easter full moon.

b. in the new calendar.

1) *k + 5 J* to be divided by *19*, remainder *a*;

2) to *19 a + 15* add the number
*h = J - J/4 - (8J+13)/25*,
which number often remains the same for
successive centuries and amounts to *7, 8, 9* for years between
*1583* and *1700, 1700-1900, 1900-2200*;
to be divided by *30*, remainder *b*,
the Easter-full-moon-number;

3) to *b* add *k + k/4 + 2 - 2 e*, to be divided by
*7*, remainder *d*;

then Easter is *b + 7 - d* days after 21 March,
or *7 - d* days after the Easter-full-moon.

When in 3) the division by *7* is exact, then put
*d=o*,
except
in the case, when 1) also *b=29* or 2)
*b=28* and *a* is greater than *10*, then put
*d=7*.
Or what is the same thing, when the calculation for *b=29* and
*d=o*
gives the 26th of April as the date of Easter, then
instead put the 19th, and the 18th of April when one finds for
*a* greater than *10* *b=28* and
*d=o* as
Easter-date the 25th of April.
This case will occur for the first time in 1954. --

Example 1. Easter 1355. Coronation of Charles IV in Rome.

55 + 5.13 = 120 = 19.6 + 6 a = 6 | 19.6 + 15 = 129 = 4.30 + 9 b = 9 9 + 55 + 55/4 - 13 = 64 = 7.9 + 1 d = 1

Easter is *9 + 7 - 1 = 15* days after the 21st of March = 36th
of March = 5th of April.

Example 2. Easter 1886. * J = 18 = 4.4 + 2
e = 2 *

86 + 5.18 = 176 = 19.9 + 5 19.5 + 15 + 8 = 118 = 30.3 + 28 b = 28 28 + 86 + 86/4 + 2 - 2.2 = 133 = 7.19 + 0 d = 0(becauseaequals smaller than10)

Easter is *28 + 7 - 0 = 35* days after the 21st of March = 56th
of March = 25th of April.

Markgröningen.

Rektor Zeller.

See Likely Errors in Implementations and Common Notes.

Note the Continental usage of comma as a decimal separator, allowing the use of a dot to indicate multiplication.

Three mathematical 'typos' -
*d=o*
- in the last paragraph
have been preserved above, but underlined.

Read and adapt the 'Day of the Week Calculation' notes for the 1886 paper.

The dividend of the division by 7 is sometimes negative.

To avoid "mod of negative" error for the years 300, 702, 1101, 1503,
2600, 3401, 5500, 6301, the term *-2e* in 3) should be changed to
*+5e*.