© J R Stockton, ≥ 2008-11-13

"Kalender-Formeln".

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"Kalender-Formeln" von Chr. Zeller,
*Acta Mathematica*, vol.9 (1886-7), pp.131-6. (Nov 1886) -
in German.

Sebastian Koppehel of the University of Hamburg kindly provided me,
by E-mail in April 2002, with a report on Zeller's
*Acta Mathematica* paper.

The box below contains his translation of the first part
of the article, and mine of the second part. He wrote :
*I also made a PDF version which looks much nicer and
which can be found, together with the German version, the LaTeX
sources, and the Pascal implementation
(84,6 kB)*.

Scanned images of (my copies of) the original printed pages are in zel-86px.htm.

*** Section I from the German by Sebastian Koppehel,
April 2002; edited by JRS ***

Calendar Formulae

by

Christian Zeller

of Markgröningen.

I. Day of the Week Calculation.

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.

*** Section II translated by JRS!
To be yet further tidied (English & HTML) ***

II. Easter Calculation.

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.

Now one has only the three Rules | |||

I) for the weekday :- | |||

Julian: | q + 2m + (3m+3)/5 + N + N/4 | | | divide by 7, remainder h,the weekday-number; |

Gregorian: | q + 2m + (3m+3)/5 + N + N/4 - (N/100 - N/400 - 2) |
| | |

II) for the full moon :- | |||

Julian: | 19N - N/19 + 15 | | | divide by 30, remainder b,the Easter-full-moon-number; |

Gregorian: | 19N - N/19 + 15 + (N/100 - N/300 - N/400) |
| | |

by which for the calculation only the Year-number
N will be used; | |||

III) for the Easter-day :- | |||

Julian: | b + N + N/4 | | | divide by 7, remainder d; |

Gregorian: | b + N + N/4 - (N/100 - N/400 - 2) |
| | |

then Easter is (b + 7 - d) days after the 21 March. |

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.

Acta mathematica. 9. Imprimé le 20 Novembre
1886.

See Likely Errors in Implementations and Common Notes.

SK added :-

- Zeller's use of the "equals" sign is a little unusual, to say the least. It's clear, of course, what he means: The remainder doesn't change.
- Zeller writes that
*h*is equal to the remainder of the division by seven. He treats Sunday as the first day of the week, which is - and especially was in former times - a common perception in Germany. It means, however, that*h = 0*has to be interpreted as the seventh day,*i.e.*Saturday. - The sum sometimes turns out to be negative. For example, for 2002-04-19, it is -1. A multiple of seven should be added so that the number becomes positive and can be used as regular output.

I add :-

- We know that
*q≥1, m≥3, K≥0*and therefore the Julian expression in I.1 A is at least*16-J*which cannot go negative before 1700 (but does so on 1700-03-01). By a similar argument, the Gregorian expression in I.1 B can readily go negative. Adding*7*J*into the last term of each expression seems to be a suitable fix. - We know that
*q≤31, m≤14, K≤99*and therefore the Julian expression in I.1 A is at most 198, which fits in a byte. By a similar argument, the Gregorian expression in I.1 B is at most 193. - With
*7*J*added, however, both readily exceed a byte, but not a 16-bit unsigned word.

The dividend of the division by 7 is sometimes negative.

To avoid "mod of negative" error for some years, including 2002, the
term *-2J* in 3) can be changed to *+5J*.

Zeller II.B.2 uses an approximate Gregorian form, valid only up to the year 4200. Note 1 gives the change needed to cover all years.

Considering Note 2, the Easter-full-moon-number (*die
Ostervollmondszahl*) of expression II.B.2
cannot always be the date of the Paschal Full Moon, since Easter is the
Sunday *after* that.

The Afterword is rather like Zeller's card
"Das Ganze der Kalender-Rechnung", but it
apparently lacks anything handling the special Gregorian cases with
*b≥28*.

One must remember to use the modified year and month.

The Afterword omits clear mention of Notes 1 & 2 of Section II.1. To get correct results for all years, those Notes must be applied here.

As printed, the test for Old-and-New Day-of-Week passes, and the test for Old Easter passes.

The test for New Easter passes when Notes 1 & 2 of paper
Section II.1. are heeded.

• Without Note 1, the test shows, after 4200, in some
centuries, about one error per four years.

• Without Note 2, the test shows about one error per
century.

The Paschal Full Moon is not tested; see in On Chr. Zeller's Card "Das Ganze der Kalender-Rechnung".