Gaussian Passover formula

The Passover formula by Carl Friedrich Gauß from 1802 makes it possible to calculate the date of the Passover festival in the Julian calendar for a Jewish year or a year of the Christian era . The formula, which is less well known than Gaussian Easter formula , gives the first day of the Jewish festival of Passover, which corresponds to the 15th Nisan in the Jewish calendar . It should be noted, however, that in the Jewish calendar the day begins on the evening before with sunset.

The calculated day in the Julian calendar can then easily be converted into the Gregorian calendar .

calculation

Let A be the year in the Jewish calendar (A> 0). Let B be the year in the Christian era (B ≥ 0). The following applies: A - 3760 = B.

First divide (12 · A + 17) by 19 or (12 · B + 12) by 19 and denote the remainder by a. Then divide A or B by 4 and call the remainder b.

Calculate: 32.0440932 + 1.5542418 * a + 0.25 * b - 0.003177794 * A, or: 20.0955877 + 1.5542418 * a + 0.25 * b - 0.003177794 * B and set the result is M + m. M means the resulting whole number and m the real decimal fraction.

Formulated with fractions:

${\ displaystyle M + m = 32 {\ frac {4343} {98496}} + 1 {\ frac {272953} {492480}} a + {\ frac {1} {4}} b - {\ frac {313} { 98496}} A}$ ,

respectively

${\ displaystyle M + m = 20 {\ frac {9415} {98496}} + 1 {\ frac {272953} {492480}} a + {\ frac {1} {4}} b - {\ frac {313} { 98496}} B}$ .

Then one divides (M + 3 · A + 5 · b + 5) or (M + 3 · B + 5 · b + 1) by 7 and calls the remainder c.

Finally, a distinction must be made between the following cases:

I: Is c = 2, 4 or 6: The Passover festival is on (M + 1) March in the Julian calendar.

II: If c = 1, a> 6 and m ≥ 1367/2160, then the Passover festival is on (M + 2) -th March in the Julian calendar.

III: If c = 0, a> 11 and m ≥ 23269/25920, then the Passover festival is on (M + 1) March in the Julian calendar.

In all other cases, Passover is on M-th March according to the Julian calendar. If the results are greater than 31, the day numbers of the previous months must be subtracted.

Cases I, II and III correspond to the postponements (Dechijoth) Adu, Gatrat and Betutakpat ( Jewish calendar #calendaric exception rules ) on the Jewish New Year's Day, which follows the Passover festival. The Jach shift is already incorporated into the formula.

The Jewish New Year's Day always follows the calculated first day of the Passover festival by 163 days, so that New Year's Day in the Jewish calendar ( Rosh Hashanah ) can also be accessed using the Gaussian Passover formula .

Conversion Julian - Gregorian calendar

From 1583 to 1699 is 10 days, from 1700 to 1799 is 11 days, from 1800 to 1899 is 12 days, from 1900 to 2099 is 13 days, and from 2100 to 2199 is 14 days to add to the date in the Julian calendar to get the date in the Gregorian calendar. A generally applicable conversion algorithm can be found under Conversion between the Julian and Gregorian calendar . Bach gives a list with the dates of the Passover festival from 500 to 2999 AD.

Proof of the formula

Hamburger provided proof of this formula and extensive explanations for it in 1896, as did Bach.

Calculation example

The first day of the Passover festival is to be calculated for the year 2017 AD.

So B = 2017 and A = 5777. One gets a = 10 and b = 1. Furthermore, M + m = 29.47839526. So M = 29 and m = 0.47839526. Finally we get c = 3 from this.

Cases I, II and III do not occur here, so the Passover festival is on March Mth and thus on March 29, 2017 in the Julian calendar. To get the corresponding date in the Gregorian calendar, 13 days have to be added, resulting in March 42 or April 11, 2017. The Passover festival thus begins on the evening of April 10, 2017 with sunset.

The following New Year's Day of the Jewish year 5778 (A + 1) is 163 days later and therefore on September 21, 2017 (beginning on September 20 with sunset).

Individual evidence

^ Gauss: Calculation of the Jewish Easter festival. In: Monthly correspondence for the promotion of geography and celestial science. Edited by Franz Xaver von Zach . Volume 5. Becker, Gotha 1802, pp. 435–437, urn : nbn: de: bvb: 12-bsb10538597-4 ( scan from the Bayerische Staatsbibliothek ). -
Gauss: works. Volume VI, p. 80 f. ( Text archive - Internet Archive ).
↑ ^a ^b ^c M. Hamburger: Derivation of the Gaussian formula for determining the Jewish Easter festival . In: L. Fuchs (Ed.): Journal for pure and applied mathematics . tape 116 . de Gruyter, 1896, ISSN 0075-4102 , p. 90-96 , doi : 10.1515 / crll.1896.116.90 .
↑ ^a ^b ^c Joseph Bach: The time and fixed calculation of the Jews with special consideration of the Gaussian Easter formula together with a perpetual calendar (= yearbook of the Bischöfliche Gymnasium in Strasbourg 1908. Supplement). Herder, Freiburg im Breisgau 1908, OCLC 6612522 .

[1] Gauss: Calculation of the Jewish Easter festival. In: Monthly correspondence for the promotion of geography and celestial science. Edited by Franz Xaver von Zach . Volume 5. Becker, Gotha 1802, pp. 435–437, urn : nbn: de: bvb: 12-bsb10538597-4 ( scan from the Bayerische Staatsbibliothek ). -
Gauss: works. Volume VI, p. 80 f. ( Text archive - Internet Archive ).

[Hamburger-2] M. Hamburger: Derivation of the Gaussian formula for determining the Jewish Easter festival . In: L. Fuchs (Ed.): Journal for pure and applied mathematics . tape 116 . de Gruyter, 1896, ISSN 0075-4102 , p. 90-96 , doi : 10.1515 / crll.1896.116.90 .

[Bach-3] Joseph Bach: The time and fixed calculation of the Jews with special consideration of the Gaussian Easter formula together with a perpetual calendar (= yearbook of the Bischöfliche Gymnasium in Strasbourg 1908. Supplement). Herder, Freiburg im Breisgau 1908, OCLC 6612522 .