[Avodah] Fw: Tisha-Asar Mi Yode'a

[L Reich]

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From: L Reich
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Sent: Friday, May 08, 2009 5:16 PM
Subject: Tisha-Asar Mi Yode'a


　From: Elozor Reich
Tisha-Asar Mi Yode'a or
Anniversaries, Birthdays & Cycles
Many are aware that Hebrew and Civil birthdays don't usually correspond in 
most years, but that they often
do coincide or come near to each other on any 19th anniversary or on any 
multiple of 19 years. This note explains this phenomenon and more.
The Hebrew calendar attempts to reconcile the astronomical length of the 
lunar month of over 29.5 days
with the solar year of under 365.25 days and its four seasons. In use it 
ensures that Rosh Chodesh is
always near the time of the astronomical new moon and that Pesach occurs in 
Spring and Rosh
Hashonoh in Autumn. This is done by arranging a 19 year cycle, known as 
Machzor Koton. This 19
year cycle consists of 12 year of 12 months, and 7 years of 13 months, known 
as Shnois Ha'Ibur,
during which we have an extra Adar.
The longer or Ibbur years are those which have a remainder of 3, 6, 8, 
11,14, 17 & 19, when dividing the
year number by 19, so giving us its position in the Machzor. The 19 year 
cycle or Machzor lasts for 235 months since 19 X 12 + 7 = 235, or 12 X 12 + 
7 X 13 = 235. Currently (Year 5769) we are in the
12th year of Cycle 304. The arithmetic for this is straight forward, e.g. if 
we divide 5769 by 19 we get a dividend of 303 – showing that we have 
completed 303 cycles and are now in cycle 304 – and a remainder
of 12 – which is not one of the Shnois Ha'Ibur
(Although all the classical works use a time system of hours being divided 
into 1080 "Parts" (Chalokim) and each
"Part" (Chelek) being divided into 76 "Moments" (Rego'im), we shall here 
make things more familiar by using
minutes and seconds. A minute has 18 Chalokim; one Chelek equals 3⅓ seconds 
and one Rega is equivalent to
five parts of a one hundred and fourteenth of a second , 5/114 sec.)
The accepted astronomical length of a lunar month for fixed calendar 
purposes is an average; individual
months can vary considerably. This average, known as Molad Ho'emtzo'i, is 29 
days, 12 hours, 44 Minutes
& 3⅓ seconds. Multiply this by the above mentioned 235 gives us the 
astronomical length of a Machzor Koton. 6939 days, 16 hrs, 33 minutes & 3⅓ 
seconds.
We can now divide the last figure by 19 and get a very near approximation of 
the astronomical solar year. It come to 365 days 5 hours 55 minutes and 25.4 
seconds. This year length (about 7 minutes longer than the
true solar year, which is known to astronomers as the tropical year) is the 
basis of the Jewish Calendar and
is commonly called Tekufas Rav Ada. The solar year of 365 days and exactly 6 
hours is called Tekufas Shmuel. Tekufas Shmuel is equivalent to the old 
civil Julian Calendar which was replaced by the Gregorian
one, which is shorter by three days in a 400 year period.
　
>From an Halachic viewpoint Tekufas Shmuel has only two uses. It decides the 
date of the start of the Tefiloh for rain (Tal U'Motor in Chutz Lo'oretz and 
is also the basis for the 28 year Machzor Godol, whose
completion we commemorated this year. (It is also used by some in Minhagim 
connected with the avoidance of drinking water at the change of seasons.) 
However, Tekufas Rav Ada is the one which matters in settling the 19 year 
cycle. It governs all calendar dates and this is the one which we will 
continue to explore.
Now a Machzor Koton starts on Rosh Hashonoh of year 1 of the cycle and ends 
on the last day of Ellul 19 years later. Astronomically this is about 6939.7 
days later. Since we can't split days in the real calendar, one might think 
that the calendrical length of a Machzor would be either 6939 or 6940 days. 
In fact it can also
be 6941 and even, very rarely, 6942 days. The cause of this wide range is 
the fact that Rosh Hashonoh is decided by the Molad Ho'emtzo'i of the 1st of 
Tishri. By the basic rules of the Jewish Calendar Rosh Hashonoh can land on 
the same day as the Molad but can also be postponed one or two days (Molad 
Zokon, Lo Adu Rosh etc). If one Machzor starts on the day of the Molad and 
the following one is postponed by a day or two, then the calendrical length 
of the Machzor is extended.
Furthermore, 19 Civil Years can include either 4 or 5 leap years, i.e. 6939 
or 6940 days. The 4 year
Civil Leap Year cycle is not linked to the Machzor Koton pattern, hence 
another contributor to the 'discrepancies' in the 19th anniversaries.
To summarise; take any Hebrew Calendar date, move on to its 19th Hebrew 
anniversary, and you have,
in effect, completed a Machzor Koton , but it may be 6939, 6940, 6941 or 
6942 days later. Take the
Civil Date of the same starting point, move on to its 19th Civil anniversary 
and it may be 6939 or 6940
days later. This means that 19th anniversaries can show a difference of two 
(& very rarely three) days
in their Hebrew and Civil dates.
Let us now look at the Machzor Koton more closely. We start the cycle with 
the Hebrew and Civil years
being level. Since the Civil Year of 365+ days is approximately 11 days 
longer than the average Hebrew
Shono Peshuta (a non-leap year) of 354+ days, by the next Rosh Hashonoh the 
Hebrew year lags by
11 days. After two years it is 22 behind. After three years it would 33 
behind had we not made it a Shono Me'uberes (adding an extra Adar), which 
"pays off" 30 of the 33 days, but still leaving us three days "in
arrear". This slipping and correcting continues throughout the 19 year 
cycle. The addition of an extra Adar
in years 3,6,8,11,14,17 and 19 makes up for all the slippage and gets us 
back to our starting point at the
end of the final leap year in year 19.
If you do this addition and subtraction for the whole of the cycle, you will 
discover that although we only
get back exactly to par at the end of the 19th year, we come quite near to 
it at the end of years 8 and 11.
After 8 years into the cycle, the Civil Calendar will have counted the 
passing of 2922 days; the
corresponding Hebrew count (5 years of 12 months and 3 of 13) comes to 
around 2923 (non-leap years
can be 353, 354 or 355 days and leap years 383, 384 or 385), so we are not 
far out. A similar exercise for the
first 11 years shows 4017 or 4018 days in the Civil Calendar and around 4016 
(7 years of 12 months
and 4 of 13) in the Hebrew one.
All this arithmetic shows us that 19th anniversaries and their multiples 
will be spot on or near, and that multiples of 19 with the addition of 8, 
e.g. 27, 46 & 65, or with addition of 11, e.g. 30, 49 & 68 will
either be spot on or not be far out.
Although I have not attempted a rigorous mathematical analysis of the 
probability ratio, an inspection of a sizeable sample shows as follows. On 
19th anniversaries around one half land on the same Hebrew and
Civil dates and an additional third within one day, leaving a small fraction 
more than one day out. On both
19th + 8 and on 19th + 11 anniversaries around one eighth coincide precisely 
and over one half of the
remainder land within one or two days.
Readers interested in arithmetic might wish to explore the following. In 
examining the Machzor Koton it
has been pointed out that we nearly reach equilibrium after 8 years with 
their 3 leap years, and after 11
years with their 4 leap years, and reach a true balance after 19 years with 
their 7 leap years. Consider the relationship of this to the three fractions 
3/8, 4/11 and 7/19. The difference between 3/8 and 4/11 is only
1/88. 7/19, which lands between the two is 1/152 less than 3/8 and 1/209 
more than 4/11.
　
Elozor Reich)