© J R Stockton, ≥ 2008-11-13

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

The 1886 Paper : "Kalender-Formeln"

"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.

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.

Notes on the 1886 Paper

Notes on 'Day of the Week Calculation'

• 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.

• 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.

Notes on 'Easter Calculation'

A) For the Julian Calendar

The dividend of the division by 7 is sometimes negative.

B) For the Gregorian Calendar

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.

Afterword

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.

Tests for the 1886 Paper

Sub-Sections "2. Examples"

JavaScript Day-of-Week MOD 7 Test - 1886
to test for MOD 7 errors in   -
and wait for a while for the results. The dates shown
are those for which the expression in I.1.B gives a
different result after the modulo 7 operation
to one with +5J instead of -2J.

Sub-Section "Afterword"

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".

One day per month : years : Julian 28, Gregorian 400. OK.
532 years. OK.
Note 1       Note 2
8000 years. Both Notes are needed, else the test fails for some years.
Zeller is continued after zeller-c.htm#Opus.
1882, 1883, 1885, Card.