Hebrew calendar arithmetic requires that the time calculated for the molad does not exceed the first day of any month. In the calendar's present format, the molad zaqen postponement rule is used to guard against any such excess.
The correction made by the molad zaqen rule applies only to Rosh Chodesh of Kislev or Shevat of the previous year. Consequently, this rule is in no way related to the visibility of the new moon on Rosh Hashannah.
A brief summary of the following analysis may be seen at Overpost Overview.
The excess of time of the molad over the first day of any month will be referred to as the overpost.
The overpost problem arises when the leap month of 30 days is placed anywhere after the month of Heshvan during a deficient or regular leap year (ie, years of either 383 or 384 days).
Because it was decided at some point in the calendar's history to place the leap month of 30 days as the sixth month of any leap year, corrective action also had to be made to overcome the overpost.
Calendar arithmetic shows that postponing Rosh Hashannah to the next allowable day whenever the molad of Tishrei exceeds 18h;656p on one of the permissible days for the holiday eliminates the overpost for the immediately preceding year.
The molad zaqen rule sets that postponement time at or after exactly 18h (noon) of the day.
For some Hebrew year H
Let R = the number of days that have elapsed up to the first of Tishrei H since day 0 of year 1H Let t = the time of the molad on day R Let R' = the number of days that have elapsed up to the first of Tishrei H+1 since day 0 of year 1H Let t' = the time of the molad on day R' Let A = its annual lunation period Let L = its length in days Let d' = 1d  1p = 23h 1079pAs an example of the above, let H = 5758H. Then
R = 2,102,728d t = 4h 129p R' = 2,103,082d t' = 12h 1005p A = 354d 8h 876p L = 354d
In general,
R + t + A = R' + t' R + L = R'
Let u = the molad period = 29d 12h 793p Let i = the number of months that have elapsed in year H Let m(i) = the i^{th} month in year H Let s(i) = the number of days that have elapsed in year H up to the start of month m(i+1) Let D(i) = i*u  s(i) for any month m(i+1) Let E(i) = the excess time of the molad on the first day of any month m(i+1) in year H
Then R + t + i*u = time of the molad for month m(i+1) R + s(i)= number of days that have elapsed up to the start of month m(i+1)
E(i), the excess time of the molad on the first day of any month in year H, is the difference between those two values.
Hence, E(i) = t + i*u  s(i) = t + D(i)
The following table shows the value of D(i) for any given month m(i+1) in any year of length L.
The table follows the traditional placement of the leap month of 30 days as the sixth month of the leap year.
A minus sign following the time of D(i) indicates a negative value for D(i).
YEAR LENGTH L IN DAYS  

MONTH  353 days  354 days  355 days  383 days  384 days  385 days 
Heshvan  11h 287p  11h 287p  11h 287p  11h 287p  11h 287p  11h 287p 
Kislev  1h 506p  1h 506p  22h 574p  1h 506p  1h 506p  22h 574p 
Tevet  14h 219p  9h 861p  1d 9h 861p  14h 219p  9h 861p  1d 9h 861p 
Shevat  1d 2h 1012p  2h 1012p  21h 68p  1d 2h 1012p  2h 1012p  21h 68p 
Adar  15h 725p  8h 355p  1d 8h 355p  15h 725p  8h 355p  1d 8h 355p 
v'Adar  4h 438p  19h 642p  1d 19h 642p  
Nisan  1d 4h 438p  4h 438p  19h 642p  17h 151p  6h 929p  1d 6h 929p 
Iyar  17h 151p  6h 929p  1d 6h 929p  5h 944p  18h 136p  1d 18h 136p 
Sivan  1d 5h 944p  5h 944p  18h 136p  18h 657p  5h 423p  1d 5h 423p 
Tammuz  18h 657p  5h 423p  1d 5h 423p  7h 370p  16h 710p  1d 16h 710p 
Av  1d 7h 370p  7h 370p  16h 710p  20h 83p  3h 997p  1d 3h 997p 
Elul  20h 83p  3h 997p  1d 3h 997p  8h 876p  15h 204p  1d 15h 204p 
Using the following two conditions, values of D(i) which do not lead to overposts can be found. 1. When D(i) < 0 there can be no overpost for any value of t. When D(i) < 0 , t + D(i) < d' since t <= d' (by def'n). 2. When d' + A  D(i) > L + d' there can be no overpost for any value of t. By definition t + D(i) = d' + O(i) so that t  O(i) = d'  D(i) t  O(i) + A = d'  D(i) + A When d' + A  D(i) > L + d' then t + A  O(i) > L + d' since t  O(i) = d'  D(i) hence,  O(i) > L + d'  (t + A) now L + d' => t + A L + d'  (t + A) => 0  O(i) > L + d'  (t + A) => 0 O(i) < 0 (the overpost value O(i) is negative!) Therefore, when d' + A  D(i) > L + d' there can be no overpost since the overpost value O(i) is less than zero. From the above, it is necessary only to look for positive values of D(i) for which d' + A  D(i) < L + d' To avoid a potential overpost, it is necessary that the maximum value of t be decreased by O(i). That leads to the following relationships R + (t  O(i)) + A = R + (d'  D(i)) + A = R' + t' d'  D(i) + A + (R  R') = d'  D(i) + (A  L) = t' If more than one value for t' is found, then the smallest value of t' would become the limiting value for the maximum permissible time of the molad on Rosh Hashannah. Otherwise an overpost will occur. The table for the values of D(i) may now be scanned.
For L = 353 Days
The largest excess for the 353 day year is 1d;7h;370p. Applying the formula
in condition 2,
d'  D(i) + A cannot fall below 354d; 1h;505p.
Since this value is larger than 354d  1p, the overpost problem cannot arise
for the largest possible excess time value. Hence, the overpost cannot take
place in a 353 day year.
For L = 354 Days
The largest excess for the 354 day year is 7h;370p. Applying the formula in
condition 2,
d'  D(i) + A cannot fall below 355d; 1h;505p. Since this
value is larger than 355d  1p, the overpost problem cannot arise for
the largest possible excess time value. Hence, the overpost cannot take place
in a 354 day year.
For L = 355 Days
All of the excesses in the 355 day year are negative. So according to condition 1, there is no possibility of an overpost problem in a 355 day year.
For L = 383 Days
Applying the largest excess value of 1d;2h;1012p to the formula in condition 2, the result is 383d;18h;656p. Consequently, corrective action must be taken to prevent an overpost from occurring after the first day of Shevat in such a year.
Using the formula t' = d'  D(i) + (A  L) (developed above) the result is t' = 1d  1p  (1d;2h;1012p) + (383d;21h;589p  383d) = (2;1013p) + 21h;589p = 18h;656p
Hence, 18h;656p becomes one of the values of t' to be considered for the maximum allowable time of the molad on Rosh Hashannah.
In the 383 day year, the second largest molad excess is 20h;83p. Applying the
formula of
condition 2, the result of d'  D(i) + A cannot fall below
384d;1h;505p. Since this value is larger than 384d  1p, the overpost problem
cannot arise for any other month in the 383 day year.
For L = 384 Days
In the 384 day year only two values are positive, and both of these values when used in the formula of condition 2 yield results which indicate a molad overpost problem taking place in both the months of Kislev and Shevat. In this case, the smaller of the two times possible for t' is 18h;656p.
For L = 385 Days
All of the molad excesses in the 385 day year are negative and so cannot lead to a molad overpost.
Final Value of t'
The above shows that 18h;656p must be the maximum allowable time for the
molad on
Rosh Hashannah of the subsequent year if an overpost is to be avoided.
The scholars, possibly to be on the safe side, chose an even lesser time of 18h for the molad zaqen postponement rule. However, it may also have been chosen for the reason that with the 18h limit exactly 1 out every seven years will be postponed due to the molad zaqen rule, thus bringing together two numbers of considerable significance in Jewish numerological traditions.
The time for molad of Tishrei 76H is 27,377d;1h;915p
and corresponds to
Saturday 21 August 3685g. The time of this molad
bypasses all of the postponement rules.
The year 76H is a leap year. So the next molad of Tishrei (77H) is
27,377d;1h;915p + 383d;21h;589p = 27,760d;23h;424p ==> 5d;23h;424p
If the molad zaqen rule is not applied to the molad of Tishrei 77H then the year 76H is 2 days shorter and must be 383 days long. On that basis it may be seen that Rosh Chodesh Shevat will occur on day 27,494d and end at 27,494d;23h;1079p. Now, the molad of Shevat will be at
27,377d;1h;915p + 4 * (29d;12h;793p) = 27,377d;1h;915p + 118d;2h;1012p = 27,495d;4h;847p
Subtracting the maximum time of Rosh Chodesh Shevat from the time of the molad of Shevat, the result is
From that simple calculation, the molad of Shevat of 76H can be seen to occur 4h;848p after the end of Rosh Chodesh Shevat.27,495d;4h;847p  27,494d;23h;1079p = 4h;848p
Because of its molad timing of Saturday;1h;915p, Rosh Hashannah 76H cannot be postponed. Therefore the correction for the overpost must come from a postponement of the year 77h. The molad zaqen rule can be applied to the year 77H because its molad of Tishrei is on Thursday past 18h. That causes two extra days to be inserted into the year 76H, and so, no Rosh Chodesh for 76H will end prior to the time of its corresponding molad.
A similar set of arithmetic can be found to take place for the molad of Tishrei 5874H, corresponding to Monday 11 September 2113g. The molad of Tishrei will arrive at 2d;1h;51p.
The molad of Shevat will occur 3h;1068p past Rosh Chodesh unless some corrective action is taken. Since there can be no postponement for Rosh Hashannah 5874H, the corrective action must come from a postponement of Rosh Hashannah 5875H.
The molad of Tishrei 5875H is 0d;22h;640p. Since the time is on Saturday past 18h, the molad zaqen rule can be applied causing 5875H to be postponed by 2 days. These two days will be added to the months of Heshvan and Kislev, in 5874H, thereby correcting the overpost found for the month of Shevat.
From these examples it is clear that the molad zaqen rule is an arithmetical device that applies a necessary correction only to the previous year. Therefore the rule has nothing whatever to do with the visibility of the moon on Rosh Hashannah.
The molad zaqen rule is the corrective action needed to overcome the overpost problem when the leap month is placed as the sixth month of a leap year.
A simple scan of the 353 and 354 day years in the above table shows that if the leap month is placed prior to the month of Heshvan, then absolutely NO corrective action is required. Hence, the molad zaqen rule can be eliminated.
Similarly, a simple scan of the 353 and 354 day years shows that if the leap month is placed past the month of Adar, then the corrective action will have to be the postponement of the subsequent Rosh Hashannah for times that are earlier than currently specified in the molad zaqen rule.
The need for the molad zaqen rule could also have been eliminated if the month of Heshvan had been made a 30 day month, and a day removed from some month past Adar I, such as the mournful month of Av, to be returned to the month in abundant years.
To maintain the existing keviyyot when the molad zaqen is eliminated, it is necessary to add 6 hours to the time specified in both Dehiyyah GaTaRad and Dehiyyah BeTU'TeKaPoT.
Hence, the limiting times of these two rules become 15h;204p and 21h;589p respectively.
Once the adjustments are made, it is possible to develop the following distribution table of the keviyyot over the full calendar cycle of 689,472 years.
YEAR LENGTH IN DAYS  

DAY  353  354  355  383  384  385  TOTALS 
Mon  39369  0  81335  40000  0  32576  193280 
Tue  0  43081  0  0  36288  0  79369 
Thu  0  124416  22839  26677  0  45899  219831 
Sat  29853  0  94563  40000  0  32576  196992 
TOTALS  69222  167497  198737  106677  36288  111051  689472 
The only change to the calculated results is that the number of Rosh Hashanot that are not postponed is increased by 98,496 from 268,937 to 367,433. The increase comes from the loss of the 98,496 molad zaqen postponements over the full calendar cycle of 689,472 years.
The Molad Overpost Distribution
The above distribution table does not show that in some of the years the time of the molad can exceed either the first day of the month of Kislev, or the first day of the month of Shevat by as much as 5 hours and 422 parts. And, in some years the overpost will take place in both of these two months.
The overpost occurs in no other month.
Over the full calendar cycle the overpost is found to occur in 16,304 deficient leap years (383 day years) and in 4,440 regular leap years (384 day years).
Over this same cycle, the overpost will occur in both the months of Kislev and Shevat 2,220 times, and only 2,220 times for the month of Kislev.
Assuming calendar calculations with all the rules in place except for the molad zaqen, then for the range of years from 3760H (0g) to 4180H (420g), the overpost can be seen to occur in the following years:
YEAR  OVERPOST  MONTH  YEAR LENGTH 


3773H  13g  3h 565p  Shevat  383 days 
3844H  84g  219p  Shevat  383 days 
3855H  95g  4h 67p  Shevat  383 days 
3860H  99g  127p  Kislev  384 days 
3860H  100g  1h 633p  Shevat  384 days 
3922H  162g  824p  Shevat  383 days 
3933H  173g  4h 672p  Shevat  383 days 
4004H  244g  1h 326p  Shevat  383 days 
4020H  260g  2h 740p  Shevat  383 days 
4102H  342g  3h 242p  Shevat  383 days 
4107H  347g  808p  Shevat  384 days 
4118H  358g  4h 656p  Shevat  383 days 
4180H  420g  3h 847p  Shevat  383 days 
The range was selected because it covers a period during which the Talmud was being developed.
First Paged 20 Jul 1997 Next Revised 7 Nov 2006