Mathematics of the Jewish Calendar/Slonimsky's Formula for the Year Type

Slonimsky's Formula for the Year Type

In 1844, Rabbi Chaim Zelig Slonimsky (1810-1904) published a modification of Gauss' formula that can be used to calculate the year type of any given year. The version below is his modified formula of 1852.

Let the Jewish year be A. Calculate the remainder r of the division


 * (7A-6)/19

If r < 12, the year is ordinary, otherwise it is leap.

Calculate


 * K = 0.178117457A + 0.777965458r + 0.2533747

and take the decimal part, discarding the integer part. The year type may then be read from the following table:


 * r < 5
 * K
 * ≥ 0.000000 ; 1
 * ≥ 0.090410 ; 2
 * ≥ 0.271103 ; 3
 * ≥ 0.376121 ; 4
 * ≥ 0.661835 ; 5
 * ≥ 0.714282 ; 6
 * ≥ 0.752248 ; 7


 * r = 5, 6
 * K
 * ≥ 0.000000 ; 1
 * ≥ 0.090410 ; 2
 * ≥ 0.271103 ; 3
 * ≥ 0.376121 ; 4
 * ≥ 0.661835 ; 5
 * ≥ 0.714282 ; 6
 * ≥ 0.804693 ; 7


 * r = 7 to 11
 * K
 * ≥ 0.000000 ; 1
 * ≥ 0.090410 ; 2
 * ≥ 0.285711 ; 3
 * ≥ 0.376121 ; 4
 * ≥ 0.661835 ; 5
 * ≥ 0.714282 ; 6
 * ≥ 0.804693 ; 7


 * r > 11
 * K
 * ≥ 0.000000 ; 8
 * ≥ 0.157466 ; 9
 * ≥ 0.285711 ; 10
 * ≥ 0.428570 ; 11
 * ≥ 0.533590 ; 12
 * ≥ 0.714282 ; 13
 * ≥ 0.871750 ; 14

The formula calculates a function of the Molad. The lower limits are calculated as the same function of the lower bounds for the Molads of each year type. The function is as follows:


 * 1) Take the lower Molad limits for each year type.
 * 2) Add 0.25; if the result exceeds 8, subtract 7 from the result.
 * 3) Subtract 1 and divide by 7 to get a number in the range 0 to 1.

Reference

 * Ch. Z. Slonimsky (1844), "Eine allgemeineformel fur die gesammte judische Kalenderberechnung", Crelle's Journal, 28, p.179
 * Ch. Z. Slonimsky (1852), Yesodei ha'ibbur, p. 26