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# Happy Hanukkah!

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
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.

### Historical Note

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.

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