Hanukkah is an eight-day festival, celebrated starting on the 25th of Kislev, which commemorates the retaking of the Temple after it was profaned by Antiochus.
The title of this page, though, is chosen simply to fit with that of the previous page, which discussed the complexities in the computation of Easter. The topic here is the Jewish calendar in general.
The determination of which years have 12 lunar months in them, and which have 13 lunar months, is based strictly on the 19-year Metonic cycle, and so no additional complications are introduced in the calendar by any attempt to approximate the solar year of the seasons more closely. Instead, the complications of this calendar largely come from the close approximation to the lunar month that it uses.
As noted on a previous page, September 30, 2000 was Tishri 1, 5761 by the Jewish calendar.
The possible form of the two basic types of years, with 12 months and with 13 months, in this calendar are shown below:
SR NR LR SL NL LL
Tishri 30 30
Marheshvan 29 29 30 29 29 30
Kislev 29 30 30 29 30 30
Tevet 29 29
Shevat 30 30
Adar 29 30
ve-Adar 29
Nisan 30 30
Iyyar 29 29
Sivan 30 30
Tammuz 29 29
Ab 30 30
Elul 29 29
--- --- --- --- --- ---
353 354 355 383 384 385
The six columns show the lengths of the months for the short, normal, and long forms of the regular 12-month year, and for the short, normal, and long forms of the leap 13-month year.
The alternation of 30 day and 29 day months reflects the fact that a lunar month is approximately 29 1/2 days in length. The current value for the synodic month is 29.530588853 days. The approximation used in calculating the Jewish calendar is very close to that; it is exactly 29 and 13753/25920 days, or about 29.5305941358... days. 1/25920 of a day is 1/1080 of an hour or 1/18 of a minute, a unit of time known as a part, and so this amounts to 29 days, 12 hours, 44 minutes, and 1 part.
With this approximation to the lunar month, it follows that the nominal length of a 12-lunar-month year should be 354 and 9516/25920 days and the nominal length of a 13-lunar-month year should be 383 and 23269/25920 days.
Thus, to keep the months of each type of year as closely synchronized as possible with the lunar month, 12-lunar-month years will occasionally have to be lengthened from 354 days to 355 days, and 13-lunar-month years will, rarely, have to be shortened from 384 days to 383 days.
Over the course of a 19-year cycle, the displacements in units of 1/25920 of a day will be:
1) 9516 8)* 13707 15) 4145 2) 19032 9) 23223 16) 13661 3)* 16381 10) 6819 17)* 11010 4) 25897 11)* 4168 18) 20526 5) 9493 12) 13684 19)* 17875 6)* 6842 13) 23200 7) 16358 14)* 20549
These are the displacements after the end of a year, so the starting displacement of the first year in the cycle is 0, that for any other year is given in the entry for the year above, and the net displacement of a full cycle is 17875/25920 of a day.
The Metonic cycle allots 235 lunations to 19 years, so this cycle is 6,939 and 17875/25920 days. (As both numbers are divisible by 5, this shortens the cycle of this calendar somewhat.)
Knowing that the year 1 of the Hebrew calendar was also the year 1 of this cycle, and that the New Moon which indicated its beginning is calculated by this to have taken place at 5604/25920 of the day after 12 noon, modular multiplication lets us work out the nominal starting point for any year in this calendar.
The year starts on the actual day of the New Moon only if it takes place before noon, and on the next day when it takes place after noon, so 5604 is the useful value of the odd fraction of a day for that year.
The information I have read about the Jewish calendar seems to claim that there is one additional complication: the time of noon as affected by the Equation of Time is apparently what is used to set the threshhold for starting a year, but this appears to be contradicted later.
The calendar is made somewhat more complicated by certain rules, known as qeviyyot, which ensure that certain religious holidays do not start on the wrong day of the week. (Actually, the rule that a month's nominal start is one day later than the day of the New Moon when the New Moon takes place after 12 noon is also included among the qeviyyot, but it makes matters simpler to incorporate it into the basic, uniform calculations here.)
If the start of a year, as calculated from its nominal length, would be on a Sunday, a Wednesday, or a Friday, then the start of the year is always delayed by one day to avoid this.
As this always lengthens the preceding year, and shortens the following year, it can shorten a 12-month year instead of lengthening it, and it can lengthen a 13-month year instead of shortening it. This is why both types of year come in all three forms.
But this could also have the effect of lengthening a 12-month year twice, or shortening a 13-month year twice, which is not provided for. When this would happen, the start of another year is moved.
This happens in the following two cases:
If the nominal first day of a regular year that has a nominal length of 355 days falls on a Tuesday, the first day of the next year would nominally fall on Saturday, and be delayed until Sunday. This would result in lengthening this regular year twice, so instead its own start is delayed by one day, to Wednesday.
If the first day of a leap year that has a nominal length of 383 days, due to the odd fractions of a day in the accumulating lunar months, and its start was also delayed from Wednesday to Thursday, then the start of the following year must be delayed from Monday to Tuesday to keep this year from being shortened twice.
This recounts the facts of the calendar, sufficient to allow one to write a computer program to deal with it. But can anything be done to allow it to be calculated from the simple use of a limited number of tables?
The first thing we might do is construct a table that shows, from the starting odd fraction of a day for the New Moon, measured from noon on the day previous to the nominal 1 Tishri of the first year in the cycle, the nominal lengths of each of the years in the cycle. A short BASIC program can do this work:
In this section, a cycle occupies 6,939 days.
0 9516 19032 16381 25897 9493 6842 16358 13707 23223 6819 4168 13684 23200 20549 4145 13661 11010 20526 17875
354 0 354 4 384 1 354 0 355 4 384 2 354 1 384 5 354 4 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 384 3
23 9539 19055 16404 0 9516 6865 16381 13730 23246 6842 4191 13707 23223 20572 4168 13684 11033 20549 17898
354 0 354 4 384 1 355 0 354 5 384 2 354 1 384 5 354 4 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 384 3
2697 12213 21729 19078 2674 12190 9539 19055 16404 0 9516 6865 16381 25897 23246 6842 16358 13707 23223 20572
354 0 354 4 384 1 355 0 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 384 3
2720 12236 21752 19101 2697 12213 9562 19078 16427 23 9539 6888 16404 0 23269 6865 16381 13730 23246 20595
354 0 354 4 384 1 355 0 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 355 2 383 0 355 5 354 3 384 0 354 6 384 3
5371 14887 24403 21752 5348 14864 12213 21729 19078 2674 12190 9539 19055 2651 0 9516 19032 16381 25897 23246
354 0 354 4 384 1 355 0 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 354 6 384 3
5394 14910 24426 21775 5371 14887 12236 21752 19101 2697 12213 9562 19078 2674 23 9539 19055 16404 0 23269
354 0 354 4 384 1 355 0 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 383 4
6888 16404 0 23269 6865 16381 13730 23246 20595 4191 13707 11056 20572 4168 1517 11033 20549 17898 1494 24763
354 0 355 4 383 2 355 0 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 383 4
In this section, a cycle occupies 6,940 days
8045 17561 1157 24426 8022 17538 14887 24403 21752 5348 14864 12213 21729 5325 2674 12190 21706 19055 2651 0
354 0 355 4 383 2 355 0 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 384 4
9539 19055 2651 0 9516 19032 16381 25897 23246 6842 16358 13707 23223 6819 4168 13684 23200 20549 4145 1494
354 0 355 4 384 2 354 1 354 5 384 2 354 1 384 5 355 4 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 384 4
9562 19078 2674 23 9539 19055 16404 0 23269 6865 16381 13730 23246 6842 4191 13707 23223 20572 4168 1517
354 0 355 4 384 2 354 1 354 5 384 2 355 1 383 6 355 4 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 384 4
12213 21729 5325 2674 12190 21706 19055 2651 0 9516 19032 16381 25897 9493 6842 16358 25874 23223 6819 4168
354 0 355 4 384 2 354 1 354 5 384 2 355 1 384 6 354 5 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 384 4
12236 21752 5348 2697 12213 21729 19078 2674 23 9539 19055 16404 0 9516 6865 16381 25897 23246 6842 4191
354 0 355 4 384 2 354 1 354 5 384 2 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 354 3 384 0 355 6 384 4
12259 21775 5371 2720 12236 21752 19101 2697 46 9562 19078 16427 23 9539 6888 16404 0 23269 6865 4214
354 0 355 4 384 2 354 1 354 5 384 2 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 355 3 383 1 355 6 384 4
14910 24426 8022 5371 14887 24403 21752 5348 2697 12213 21729 19078 2674 12190 9539 19055 2651 0 9516 6865
354 0 355 4 384 2 354 1 354 5 384 2 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 355 3 384 1 354 0 384 4
16404 0 9516 6865 16381 25897 23246 6842 4191 13707 23223 20572 4168 13684 11033 20549 4145 1494 11010 8359
355 0 354 5 384 2 354 1 354 5 384 2 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 355 3 384 1 354 0 384 4
16427 23 9539 6888 16404 0 23269 6865 4214 13730 23246 20595 4191 13707 11056 20572 4168 1517 11033 8382
355 0 354 5 384 2 354 1 355 5 383 3 355 1 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 355 3 384 1 354 0 384 4
19078 2674 12190 9539 19055 2651 0 9516 6865 16381 25897 23246 6842 16358 13707 23223 6819 4168 13684 11033
355 0 354 5 384 2 354 1 355 5 384 3 354 2 384 6 354 5 354 2 384 6 355 5 354 3 384 0 354 6 355 3 384 1 354 0 384 4
19101 2697 12213 9562 19078 2674 23 9539 6888 16404 0 23269 6865 16381 13730 23246 6842 4191 13707 11056
355 0 354 5 384 2 354 1 355 5 384 3 354 2 384 6 354 5 355 2 383 0 355 5 354 3 384 0 354 6 355 3 384 1 354 0 384 4
21752 5348 14864 12213 21729 5325 2674 12190 9539 19055 2651 0 9516 19032 16381 25897 9493 6842 16358 13707
355 0 354 5 384 2 354 1 355 5 384 3 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 354 6 355 3 384 1 354 0 384 4
21775 5371 14887 12236 21752 5348 2697 12213 9562 19078 2674 23 9539 19055 16404 0 9516 6865 16381 13730
355 0 354 5 384 2 354 1 355 5 384 3 354 2 384 6 354 5 355 2 384 0 354 6 354 3 384 0 355 6 354 4 384 1 354 0 384 4
A 6,939 day cycle advances the day of the week by two days, one of 6,940 days advances the day of the week by three days for its successor.
First, there is a row giving the number of parts from noon at which the New Moons are calculated to take place; this row shows the earliest part (or molad) at which the New Moon might take place for the pattern of nominal year lengths shown below. Below it, therefore, is a row giving the starting molad just after the end of the range for the given pattern of lengths, the one beginning the next pattern of lengths.
The next row gives the nominal lengths of the years, as derived from the calculated times of the New Moon only, followed by their relative starting days of the week.
It is useful to be able to tell at a glance how far the possible year lengths advance the day of the week:
353 3 383 5 354 4 384 6 355 5 385 0
and, in fact, a table more than seven times as large as the large one above, giving the adjustments to be made for a cycle starting on each day of the week already made would be the most convenient. The table would have to be broken up into more rows, since a change in the starting point might change the row for the cycle preceding or following.
Can a small table give the nominal day of the week on which each 19-year cycle starts, and the number of odd parts in the computed time of the New Moon?
If one divided the number of the year of interest by 19 before beginning, then the number of the cycle could be divided into hundreds, tens, and units, with three numbers to be added modulo 25,920.
Some simplifications are available. As it happens, years of each length only start on some of the four possible days of the week. As well, there are approximate repetitions of the calendar that can allow tables to be simplified.
Also, a cycle of 6,939 and 17875/25920 days is 179,876,755 parts long. Thirteen such cycles, making 247 years, are less than 90,216 days by 905 parts, and 90,216 is a multiple of 7. However, many of the rows in the table above are separated by less than 905 parts, so exact repetitions will be limited in their frequency.
But this at least serves as a starting point for a first table:
Year Molad W JD Year Molad W JD Year Molad W JD Year Molad W JD 1 5604 2 347990 2224 23379 1 1159933 4447 15234 1 1971877 6670 7089 1 2783821 248 4699 2 438206 2471 22474 1 1250149 4694 14329 1 2062093 6917 6184 1 2874037 495 3794 2 528422 2718 21569 1 1340365 4941 13424 1 2152309 7164 5279 1 2964253 742 2889 2 618638 2965 20664 1 1430581 5188 12519 1 2242525 7411 4374 1 3054469 989 1984 2 708854 3212 19759 1 1520797 5435 11614 1 2332741 7658 3469 1 3144685 1236 1079 2 799070 3459 18854 1 1611013 5682 10709 1 2422957 7905 2564 1 3234901 1483 174 2 889286 3706 17949 1 1701229 5929 9804 1 2513173 8152 1659 1 3325117 1730 25189 1 979501 3953 17044 1 1791445 6176 8899 1 2603389 8399 754 1 3415333 1977 24284 1 1069717 4200 16139 1 1881661 6423 7994 1 2693605 8646 25769 7 3505548
Remember, in this table, the date given for the start of the cycle is the day of the New Moon, not the actual beginning of the year, as no adjustments have yet been applied.
The year is, of course, the year in this calendar's own epoch. The Molad is the number of odd parts left over in the calculated time of the New Moon. Under the W is the day of the week on which the New Moon fell before noon, or following the day of the New Moon if it came in the afternoon. And, to complete matters, the Julian Day is also given, so that this calendar may be related to other calendars.
Although the table is shown in full, as calculated from the year 1 of its era, the method of calculating this calendar shown here is due to Hillel II, and thus was introduced around the year 432 A.D., and thus the table only represents the actual Hebrew calendar starting around the year 4192 or so.
A larger table derived from this table, showing the starting molad for each of the 19-year cycles in each 247-year period will fit, and using the large table above, it is possible to note where that starting molad will change enough from one period to the next to cause possible variances in the repetition of the calendar. The limits to the days on which years may start often mean that changes can take place only in steps of two days, and so not all possible changes will end up resulting in actual change, but these are the cases where checking is required. (A change in a 19-year cycle, of course, might affect the first or last year in an adjacent cycle as well.) Thus, while the repetition is not exact, it might still be used to produce a somewhat more condensed table of the calendar.
After noting where possible changes may exist for the first two opportunities, I see that the cases are frequent enough that I only additionally noted them for the cycles before and after the current cycle. An asterisk shows a possible change, or a row of five dashes, where the change also affects the nominal length in days of the 19-year cycle. Since 8540+17875=25920, these changes always occur in pairs.
1 5604 23479 15434 7389 25264 17219 9174 1129 19004 10959 2914 20789 12744
* * * * * *
248 4699 22574 14529 6484 24359 16314 8269 224 18099 10054 2009 19884 11839
* ----- ----- * *
495 3794 21669 13624 5579 23454 15409 7364 25239 17194 9149 1104 18979 10934
742 2889 20764 12719 4674 22549 14504 6459 24334 16289 8244 199 18074 10029
989 1984 19859 11814 3769 21644 13599 5554 23429 15384 7339 25214 17169 9124
1236 1079 18954 10909 2864 20739 12694 4649 22524 14479 6434 24309 16264 8219
1483 174 18049 10004 1959 19834 11789 3744 21619 13574 5529 23404 15359 7314
1730 25189 17144 9099 1054 18929 10884 2839 20714 12669 4624 22499 14454 6409
1977 24284 16239 8194 149 18024 9979 1934 19809 11764 3719 21594 13549 5504
2224 23379 15334 7289 25164 17119 9074 1029 18904 10859 2814 20689 12644 4599
2471 22474 14429 6384 24259 16214 8169 124 17999 9954 1909 19784 11739 3694
2718 21569 13524 5479 23354 15309 7264 25139 17094 9049 1004 18879 10834 2789
2965 20664 12619 4574 22449 14404 6359 24234 16189 8144 99 17974 9929 1884
3212 19759 11714 3669 21544 13499 5454 23329 15284 7239 25114 17069 9024 979
3459 18854 10809 2764 20639 12594 4549 22424 14379 6334 24209 16164 8119 74
3706 17949 9904 1859 19734 11689 3644 21519 13474 5429 23304 15259 7214 25089
3953 17044 8999 954 18829 10784 2739 20614 12569 4524 22399 14354 6309 24184
4200 16139 8094 49 17924 9879 1834 19709 11664 3619 21494 13449 5404 23279
4447 15234 7189 25064 17019 8974 929 18804 10759 2714 20589 12544 4499 22374
4694 14329 6284 24159 16114 8069 24 17899 9854 1809 19684 11639 3594 21469
4941 13424 5379 23254 15209 7164 25039 16994 8949 904 18779 10734 2689 20564
5188 12519 4474 22349 14304 6259 24134 16089 8044 25919 17874 9829 1784 19659
5435 11614 3569 21444 13399 5354 23229 15184 7139 25014 16969 8924 879 18754
* * * * ----- -----
5682 10709 2664 20539 12494 4449 22324 14279 6234 24109 16064 8019 25894 17849
* * *
5929 9804 1759 19634 11589 3544 21419 13374 5329 23204 15159 7114 24989 16944
6176 8899 854 18729 10684 2639 20514 12469 4424 22299 14254 6209 24084 16039
6423 7994 25869 17824 9779 1734 19609 11564 3519 21394 13349 5304 23179 15134
6670 7089 24964 16919 8874 829 18704 10659 2614 20489 12444 4399 22274 14229
6917 6184 24059 16014 7969 25844 17799 9754 1709 19584 11539 3494 21369 13324
7164 5279 23154 15109 7064 24939 16894 8849 804 18679 10634 2589 20464 12419
7411 4374 22249 14204 6159 24034 15989 7944 25819 17774 9729 1684 19559 11514
7658 3469 21344 13299 5254 23129 15084 7039 24914 16869 8824 779 18654 10609
7905 2564 20439 12394 4349 22224 14179 6134 24009 15964 7919 25794 17749 9704
8152 1659 19534 11489 3444 21319 13274 5229 23104 15059 7014 24889 16844 8799
8399 754 18629 10584 2539 20414 12369 4324 22199 14154 6109 23984 15939 7894
8646 25769 17724 9679 1634 19509 11464 3419 21294 13249 5204 23079 15034 6989
Another convenient table for working out the calendar is one giving the nominal starting day of the week for each 19-year cycle:
1 2 4 7 3 5 1 4 7 2 5 1 3 6 248 2 4 7 3 5 1 4 7 2 5 1 3 6 495 2 4 7 3 5 1 4 6 2 5 1 3 6 742 2 4 7 3 5 1 4 6 2 5 1 3 6 989 2 4 7 3 5 1 4 6 2 5 7 3 6 1236 2 4 7 3 5 1 4 6 2 5 7 3 6 1483 2 4 7 3 5 1 4 6 2 5 7 3 6 1730 1 4 7 3 5 1 4 6 2 5 7 3 6 1977 1 4 7 3 5 1 4 6 2 5 7 3 6 2224 1 4 7 2 5 1 4 6 2 5 7 3 6 2471 1 4 7 2 5 1 4 6 2 5 7 3 6 2718 1 4 7 2 5 1 3 6 2 5 7 3 6 2965 1 4 7 2 5 1 3 6 2 5 7 3 6 3212 1 4 7 2 5 1 3 6 2 4 7 3 6 3459 1 4 7 2 5 1 3 6 2 4 7 3 6 3706 1 4 7 2 5 1 3 6 2 4 7 3 5 3953 1 4 7 2 5 1 3 6 2 4 7 3 5 4200 1 4 7 2 5 1 3 6 2 4 7 3 5 4447 1 4 6 2 5 1 3 6 2 4 7 3 5 4694 1 4 6 2 5 1 3 6 2 4 7 3 5 4941 1 4 6 2 5 7 3 6 2 4 7 3 5 5188 1 4 6 2 5 7 3 6 1 4 7 3 5 5435 1 4 6 2 5 7 3 6 1 4 7 3 5 5682 1 4 6 2 5 7 3 6 1 4 7 2 5 5929 1 4 6 2 5 7 3 6 1 4 7 2 5 6176 1 4 6 2 5 7 3 6 1 4 7 2 5 6423 1 3 6 2 5 7 3 6 1 4 7 2 5 6670 1 3 6 2 5 7 3 6 1 4 7 2 5 6917 1 3 6 2 4 7 3 6 1 4 7 2 5 7164 1 3 6 2 4 7 3 6 1 4 7 2 5 7411 1 3 6 2 4 7 3 5 1 4 7 2 5 7658 1 3 6 2 4 7 3 5 1 4 7 2 5 7905 1 3 6 2 4 7 3 5 1 4 6 2 5 8152 1 3 6 2 4 7 3 5 1 4 6 2 5 8399 1 3 6 2 4 7 3 5 1 4 6 2 5 8646 7 3 6 2 4 7 3 5 1 4 6 2 5
Looking at these tables, one can find that a cycle of 3,857 years (15 cycles of 247 years, each cycle being 13 Metonic cycles of 19 years, plus an additional 8 Metonic cycles of 19 years, for a total of 203 Metonic cycles of 19 years) also repeats the nominal day of the week, and falls back by only 175 parts rather than 905 parts, and is thus a closer repetition, but still not an exact one. The time required for an exact repetition is 689,472 years, as first pointed out, insofar as we have any record, in the year 1,000 A.D. by al-Biruni. This is after 36,288 Metonic cycles.
If we take five cycles of 3,857 years, we lose 875 parts, so if we subtract from that one cycle of 247 years, we get a cycle of 19,038 years in which only 30 parts are gained.
Since the approximation used for the lunar month in this calendar is slightly too large, could dropping these 30 parts and shortening the cycle improve the approximation? As it happens, the difference causes a gain of about 32.1786 parts in only a single 19-year Metonic cycle (which is 235 lunations, so the value used for the length of a lunation in this calendar is still correct to within a small fraction of a part) under the current length of the lunar month, and so there would be better ways to make the improvement. For example, a much closer value of the length of the lunation would result if the calendar were to be changed to repeat every 2,128 years.
However, this would be an improvement to the length of the lunar month only, not to the length of the approximation to the tropical year, so in practice, if a calendar reform were to take place, it would be of a different character.
Since 30 times 6 is 180, six cycles of 19,038 years plus one cycle of 3,857 years make a cycle of 118,085 years in which only 5 parts are gained. And six cycles of 118,085 years less one cycle of 19,038 years, then, as 6 times 5 equals 30, makes up one full cycle of the calendar of 689,472 years.
The current 247 year cycle is the one from 5682 to 5928, and given the starting point provided by the table above, the thirteen 19-year cycles that make it up can be described. From the table above, a 19-year cycle with a starting Molad of 8045 or larger will take 6.940 days, advancing the day of the week by three days, and those with smaller starting Molads will take 6,939 days, advancing the day of the week by two days.
5682 10709 1 2422957 5701 2664 4 2429897 5720 20539 6 2436836 5739 12494 2 2443776 5758 4449 5 2450716 5777 22324 7 2457655 5796 14279 3 2464595 5815 6234 6 2471535 5834 24109 1 2478474 5853 16064 4 2485414 5872 8019 7 2492354 5891 25894 2 2499293 5910 17849 5 2506233
As I write this in the fall of 2007, I may most easily check if my calculations are correct by attempting to determine the calendar for 5768.
That will be in the cycle starting in 5758. As the Molad is 4449, we look at the row under the sequence beginning with Molad 2720.
The nominal lengths of the years are found there, and their nominal starting days may be calculated, as shown in the first three columns:
5758 354 5 5759 354 2 355 2 5760* 384 6 383 5 5761 355 5 355 5 5762 354 3 5763* 384 7 385 7 5764 354 6 353 7 5765* 384 3 384 3 5766 355 2 5767 354 7 355 7 5768* 384 4 383 5 5769 354 3 354 3 5770 355 7 5771* 383 5 385 5 5772 355 3 356 3 354 5 5773 354 1 353 2 353 2 5774* 384 5 385 5 5775 354 4 354 5 5776 384 1 383 2
In the second group of three columns, the result of delaying by one day the start of those years that nominally start on Sunday (day 1), Wednesday (day 4), or Friday (day 6) is shown.
In the third column is shown the result of making the necessary additional changes so that no year is lengthened or shortened by more than one day from the typical values of 354 days for a regular year, and 384 days for a leap year.
Note that as the adjacent cycles are not shown, the starting and ending years of the cycle may be affected by changes in the years of those.
Surprisingly, the start of 5772 is delayed by two days by this rule, which I thought was not ever supposed to happen, so I feared my calculations may be wrong, but indeed, 5771 is a year in which Marheshvan and Kislev are both 30 days long, and starts on a Thursday, I have been able to confirm on the Web.
For 5768, I have a 383-day leap year, starting on Thursday. And I have been able to check that it does start on a Thursday, and that Marheshvan and Kislev are only 29 days long in it on one web page; another, that I used to obtain the starting dates of the years in the current Metonic cycle as shown earlier, confirms that it was a leap year, so at least I worked out that year correctly.
The movie Alien Nation employed, to good effect, a clip of President Ronald Reagan saying "If not us, who? If not now, when?", but he wasn't actually talking about the acceptance of alien refugees from slavery: it was from his Second Inaugural Address, delivered in 1981, and he was talking about cutting taxes.
This was a quotation from the following by the first Rabbi Hillel, who was the chief member of the Sanhedrin:
He (Hillel) would also say: If I am not for myself, who will be for me? And if I am only for myself, what am I? And if not now, when?
from Ethics of the Fathers (Pirkei Avot), 1:14. This is one of the tractates of the Mishnah.
A search on Google Books found a use of "If not us, who? If not now, when?" in 1967, quoted in the September 1967 issue of Boys' Life magazine, from a resolution passed by the youth participants of the "North American Young World Food and Development Seminar", held in Des Moines, Iowa. With the two parts reversed, as "If not now, when? If not us, who?", however, it was used in 1963, but not by John F. Kennedy; rather, it was used by Michigan Governor George Romney, in a speech proposing a 2% income tax.
The Jewish calendar had been traditionally established from direct observation of the New Moon. The current arithmetical system of reckoning the Jewish calendar was established by Rabbi Hillel II on the last occasion when the Sanhedrin was convened, because the ruler of the Eastern Roman Empire, Theodosius II, had instituted a terrible persecution of the Jews, including frightful penalties for the ordination of a rabbi, which meant that observing and promulgating the official date of the New Moon for each year would be too hazardous. For the Jewish calendar to be reformed for greater accuracy, the Sanhedrin would need to be convened again; of course, whether or not a consensus is obtained in the Jewish community on which seventy-one rabbis are the most learned in the Law, since it is clear that the Jewish people are not at present under such persecution as in the time of Theodosius II, it may well be concluded that even a more accurate arithmetical calendar is no longer lawful, and only a return to direct observation would be permitted.
The first Theodosius also has some ill fame, but at least part of this is apparently undeserved. He destroyed the Serapeum, which was at one time an annex to the Museum which housed the main collection of the Library of Alexandria. Apparently, though, it was no longer used as a library when he destroyed it, but only as a temple; also, there was a hostage situation going on at the time. The murder of Hypatia occurred some 24 years later. And, thus, it seems the blame must go to the suppression of the rebellion of Queen Zenobia by the Roman Emperor Aurelian, it appears to me, even though my source for these facts dares only the conclusion that the final fate of the Library of Alexandria was a mystery.
And this means that Caliph Umar, unspeakable as some of his acts may have been, just as he was too late to be the brother of Ur-Nammu, so he was too late to have put the coup de grace to the Library of Alexandria.
I also found out that according to Plutarch's Lives, when Julius Caesar destroyed the Library of Alexandria the first time, it was by accident, and so the movie Cleopatra was unfair to him.