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<P> From: Elozor Reich</P><B><FONT size=4>
<P>Tisha-Asar Mi Yode'a </B></FONT><I><FONT size=3>or</FONT></P></I>
<P><U><FONT size=5>A</FONT>nniversaries, <FONT size=5>B</FONT>irthdays &
<FONT size=5>C</FONT>ycles</U> </P>
<P>Many are aware that Hebrew and Civil birthdays don't usually correspond in
most years, but that they often </P>
<P>do coincide or come near to each other on any 19<SUP>th anniversary or on any
multiple of 19 years. This note explains this phenomenon and more.</P>
<P>The Hebrew calendar attempts to reconcile the astronomical length of the
lunar month of over 29.5 days </P>
<P>with the solar year of under 365.25 days and its four seasons. In use it
ensures that <I>Rosh Chodesh</I> is </P>
<P>always near the time of the astronomical new moon and that <I>Pesach</I>
occurs in Spring and <I>Rosh </P>
<P>Hashonoh</I> in Autumn. This is done by arranging a 19 year cycle, known as
<I>Machzor Koton. </I>This 19 </P>
<P>year cycle consists of 12 year of 12 months, and 7 years of 13 months, known
as <I>Shnois Ha'Ibur</I>, </P>
<P>during which we have an extra <I>Adar</I>. </P>
<P>The longer or <I>Ibbur</I> years are those which have a remainder of 3, 6, 8,
11,14, 17 & 19, when dividing the </P>
<P>year number by 19, so giving us its position in the <I>Machzor</I>. The 19
year cycle or <I>Machzo</I>r 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 </P>
<P>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 </P>
<P>of 12 – which is not one of the <I>Shnois Ha'Ibur</P></I>
<P>(<I><FONT size=3>Although all the classical works use a time system of hours
being divided into 1080 "Parts" (Chalokim) and each </FONT></P>
<P><FONT size=3>"Part" (Chelek) being divided into 76 "Moments" (Rego'im),
</FONT><B><FONT size=3>we shall here make things more familiar by using
</FONT></P>
<P><FONT size=3>minutes and seconds</FONT></B><FONT size=3>. A minute has 18
Chalokim; one Chelek equals </FONT></SPAN><SPAN lang=EN><FONT
size=3>3⅓</FONT></I><FONT size=3> </FONT></SPAN><I><SPAN lang=""><FONT
size=3>seconds and one Rega is equivalent to </FONT></P>
<P><FONT size=3>five parts of a one hundred and fourteenth of a second ,
<B>5/114 sec.)</P></B></FONT></I>
<P>The accepted astronomical length of a lunar month for <B>fixed</B> calendar
purposes is an average; individual </P>
<P>months can vary considerably. This average, known as <I>Molad Ho'emtzo'i</I>,
is 29 days, 12 hours, 44 Minutes </P>
<P>& 3⅓ seconds. Multiply this by the above mentioned 235 gives us the
</SPAN><I><SPAN lang=EN>astronomical</I> length of a <I>Machzor Koton.
</I><B>6939</B> days, 16 hrs, 33 minutes & 3⅓ seconds.</P>
<P>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 </P><I>
<P>true </I>solar year, which is known to astronomers as the tropical year) is
the basis of the Jewish Calendar and </P>
<P>is commonly called <I>Tekufas Rav Ada. </I>The solar year of 365 days and
exactly 6 hours is called <I>Tekufas Shmuel. Tekufas Shmuel</I> is equivalent to
the old civil Julian Calendar which was replaced by the Gregorian </P>
<P>one, which is shorter by three days in a 400 year period.</P>
<P> </P>
<P>From an <I>Halachic</I> viewpoint <I>Tekufas Shmuel</I> has only two uses. It
decides the date of the start of the <I>Tefiloh </I>for rain (Tal U'Motor in
<I>Chutz Lo'oretz</I> and is also the basis for the 28 year <I>Machzor
Godol</I>, whose </P>
<P>completion we commemorated this year. (It is also used by some in
<I>Minhagim</I> connected with the avoidance of drinking water at the change of
seasons.) However, <I>Tekufas Rav Ada </I>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.</P>
<P>Now a <I>Machzor Koton</I> starts on <I>Rosh Hashonoh</I> of year 1 of the
cycle and ends on the last day of <I>Ellul</I> 19 years later. Astronomically
this is about <B>6939.7 </B>days later. Since we can't split days in the real
calendar, one might think that the <B>calendrical </B>length of a <I>Machzor
</I>would be either <B>6939</B> or <B>6940</B> days. In fact it can also </P>
<P>be <B>6941</B> and even, very rarely, <B>6942</B> days. The cause of this
wide range is the fact that <I>Rosh Hashonoh</I> is decided by the <I>Molad
Ho'emtzo'i</I> of the 1st of <I>Tishri</I>. By the basic rules of the Jewish
Calendar <I>Rosh Hashonoh</I> can land on the same day as the <I>Molad</I> but
can also be postponed one or two days (<I>Molad Zokon, Lo Adu Rosh etc</I>). If
one <I>Machzor</I> starts on the day of the <I>Molad</I> and the following one
is postponed by a day or two, then the calendrical length of the <I>Machzor</I>
is extended. </P>
<P></P>
<P>Furthermore, 19 Civil Years can include either 4 or 5 leap years, i.e. 6939
or 6940 days. The 4 year </P>
<P>Civil Leap Year cycle is not linked to the <I>Machzor Koton</I> pattern,
hence another contributor to the 'discrepancies' in the 19th anniversaries. </P>
<P>To summarise; take any Hebrew Calendar date, move on to its 19th Hebrew
anniversary, and you have, </P>
<P>in effect, completed a <I>Machzor Koton </I>, but it may be 6939, 6940, 6941
or 6942 days later. Take the </P>
<P>Civil Date of the same starting point, move on to its 19th Civil anniversary
and it may be 6939 or 6940 </P>
<P>days later. This means that 19th anniversaries can show a difference of two
(& very rarely three) days </P>
<P>in their Hebrew and Civil dates. </P>
<P>Let us now look at the <I>Machzor Koton </I>more closely. We start the cycle
with the Hebrew and Civil years </P>
<P>being level. Since the Civil Year of 365+ days is approximately 11 days
longer than the average Hebrew </P><I>
<P>Shono Peshuta </I>(a non-leap year) of 354+ days, by the next <I>Rosh
Hashonoh</I> the Hebrew year lags by </P>
<P>11 days. After two years it is 22 behind. After three years it would 33
behind had we not made it a <I>Shono Me'uberes </I>(adding an extra
<I>Adar</I>), which "pays off" 30 of the 33 days, but still leaving us three
days "in </P>
<P>arrear". This slipping and correcting continues throughout the 19 year cycle.
The addition of an extra <I>Adar</I> </P>
<P>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 </P>
<P>end of the final leap year in year 19.</P>
<P>If you do this addition and subtraction for the whole of the cycle, you will
discover that although we only </P>
<P>get back exactly to par at the end of the 19th year, we come quite near to it
at the end of years <B>8</B> and <B>11</B>. </P>
<P>After 8 years into the cycle, the Civil Calendar will have counted the
passing of 2922 days; the </P>
<P>corresponding Hebrew count (5 years of 12 months and 3 of 13) comes to
<B>around</B> 2923 (<FONT size=2>non-leap years </P>
<P>can be 353, 354 or 355 days and leap years 383, 384 or 385</FONT><FONT
size=3>), so we are not far out. A similar exercise for the </FONT></P>
<P>first 11 years shows 4017 or 4018 days in the Civil Calendar and around 4016
(7 years of 12 months </P>
<P>and 4 of 13) in the Hebrew one.</P>
<P>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 </P>
<P>either be spot on or not be far out.</P>
<P>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 </P>
<P>Civil dates and an additional third within one day, leaving a small fraction
more than one day out. On both </P>
<P>19th + 8 and on 19th + 11 anniversaries around one eighth coincide precisely
and over one half of the</P>
<P>remainder land within one or two days.</P>
<P>Readers interested in arithmetic might wish to explore the following. In
examining the <I>Machzor Koton </I>it </P>
<P>has been pointed out that we nearly reach equilibrium after 8 years with
their 3 leap years, and after 11 </P>
<P>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 </P>
<P>1/88<B>.</B> 7/19, which lands between the two is 1/152 less than 3/8 and
1/209 more than 4/11. </P>
<P> </P><FONT size=1>
<P>Elozor Reich</FONT><FONT size=1>)</P></FONT>
<P> </P>
<P> </P>
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