molad reference meridian

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molad reference meridian

Irv Bromberg
I had previously estimated the reference meridian of the traditional
molad of the Hebrew Calendar as 5 3/4 degrees east of Jerusalem, or
midway between Israel and Babylonia, after finding that in the era of
Hillel II the average moments of the molad were 23 mean solar minutes
ahead of the mean lunar conjunctions of the era.

The new rectified molad arithmetic is more accurate, and yields a
difference from the traditional molad of 9 3/4 mean solar minutes in
the era of Hillel II, suggesting that the traditional molad reference
meridian is only 2 7/16 degrees east of Jerusalem.  This meridian is
too far west to be considered "midway" between Israel and Babylonia,
yet too far east to be considered merely inclusive of the Israeli
tribes that were on the east bank of the Jordan River.

Searching around on maps, the only explanation that I could come up
with is that that meridian is midway between Israel and Mecca!  How's
that for inexplicable?

I can't imagine that there was any intentional choice for such a
meridian -- it must be that the intention was to align with the
meridian of Jerusalem and the outcome was very close but not perfect.

On the other hand, taking the era of Hillel II as corresponding to Rosh
HaShanah 4119 = Julian September 19, 358 AD, my approximation of Delta
T for that day is 1h53m57s, so that Delta T approximation would only
have to be in error by less than 9% to account for the 9 3/4 mean solar
minutes.  This is quite plausible, so it is likely that the original
molad reference meridian was correctly assigned to Jerusalem.

Any other ideas?


-- Irv Bromberg, Toronto, Canada

<http://www.sym454.org/hebrew/>
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Re: molad reference meridian

Tom Peters-2
Op 30-nov-2005, om 19:06 heeft Irv Bromberg het volgende geschreven:

> I had previously estimated the reference meridian of the  
> traditional molad of the Hebrew Calendar as 5 3/4 degrees east of  
> Jerusalem, or midway between Israel and Babylonia, after finding  
> that in the era of Hillel II the average moments of the molad were  
> 23 mean solar minutes ahead of the mean lunar conjunctions of the era.
>
> The new rectified molad arithmetic is more accurate, and yields a  
> difference from the traditional molad of 9 3/4 mean solar minutes  
> in the era of Hillel II, suggesting that the traditional molad  
> reference meridian is only 2 7/16 degrees east of Jerusalem.  This  
> meridian is too far west to be considered "midway" between Israel  
> and Babylonia, yet too far east to be considered merely inclusive  
> of the Israeli tribes that were on the east bank of the Jordan River.
>
> Searching around on maps, the only explanation that I could come up  
> with is that that meridian is midway between Israel and Mecca!  
> How's that for inexplicable?
>
> I can't imagine that there was any intentional choice for such a  
> meridian -- it must be that the intention was to align with the  
> meridian of Jerusalem and the outcome was very close but not perfect.
>
> On the other hand, taking the era of Hillel II as corresponding to  
> Rosh HaShanah 4119 = Julian September 19, 358 AD, my approximation  
> of Delta T for that day is 1h53m57s, so that Delta T approximation  
> would only have to be in error by less than 9% to account for the 9  
> 3/4 mean solar minutes.  This is quite plausible, so it is likely  
> that the original molad reference meridian was correctly assigned  
> to Jerusalem.

My favourite expression yields 1h50m55s +/- 15m for DeltaT on 9-Sep-358

> Any other ideas?

Are you sure that the molad would fit Hillel's epoch?  Normally an  
expression is based on a longish period of earlier observations, so  
iff Hillel established the values, the mean epoch could fall well  
before his time.
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Re: molad reference meridian

VictorEngel
In reply to this post by Irv Bromberg
Dear Irv and Calendar People,

I'm having trouble picturing how astronomical observations can be precise
enough to make such precise measurements of longitude. What would you
estimate the precision to be based upon observing techniques of the time?
And how do you arrive at such a figure?

Victor

> -----Original Message-----
> From: East Carolina University Calendar discussion List
> [mailto:[hidden email]]On Behalf Of Irv Bromberg
> Sent: Wednesday, November 30, 2005 12:07 PM
> To: [hidden email]
> Subject: molad reference meridian
>
>
> I had previously estimated the reference meridian of the traditional
> molad of the Hebrew Calendar as 5 3/4 degrees east of Jerusalem, or
> midway between Israel and Babylonia, after finding that in the era of
> Hillel II the average moments of the molad were 23 mean solar minutes
> ahead of the mean lunar conjunctions of the era.
>
> The new rectified molad arithmetic is more accurate, and yields a
> difference from the traditional molad of 9 3/4 mean solar minutes in
> the era of Hillel II, suggesting that the traditional molad reference
> meridian is only 2 7/16 degrees east of Jerusalem.  This meridian is
> too far west to be considered "midway" between Israel and Babylonia,
> yet too far east to be considered merely inclusive of the Israeli
> tribes that were on the east bank of the Jordan River.
>
> Searching around on maps, the only explanation that I could come up
> with is that that meridian is midway between Israel and Mecca!  How's
> that for inexplicable?
>
> I can't imagine that there was any intentional choice for such a
> meridian -- it must be that the intention was to align with the
> meridian of Jerusalem and the outcome was very close but not perfect.
>
> On the other hand, taking the era of Hillel II as
> corresponding to Rosh
> HaShanah 4119 = Julian September 19, 358 AD, my approximation
> of Delta
> T for that day is 1h53m57s, so that Delta T approximation would only
> have to be in error by less than 9% to account for the 9 3/4
> mean solar
> minutes.  This is quite plausible, so it is likely that the original
> molad reference meridian was correctly assigned to Jerusalem.
>
> Any other ideas?
>
>
> -- Irv Bromberg, Toronto, Canada
>
> <http://www.sym454.org/hebrew/>
>
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Re: molad reference meridian

Irv Bromberg
On Nov 30, 2005, at 14:45, Engel,Victor wrote:
> I'm having trouble picturing how astronomical observations can be
> precise
> enough to make such precise measurements of longitude. What would you
> estimate the precision to be based upon observing techniques of the
> time?
> And how do you arrive at such a figure?

BROMBERG replies:

This is explained on my "Molad of the Hebrew Calendar" and "Rectified
Hebrew Calendar" web pages, see <http://www.sym454.org/hebrew/>.

Briefly, I calculated the differences between the molad moment for each
month and the corresponding actual lunar conjunction moment (referred
to the meridian of Jerusalem) using the algorithms from "Astronomical
Algorithms" 2nd edition, by Jean Meeus.  This yielded a scatter of
points that fit a least-squares parabolic regression curve.  That curve
reached a minimum in the era of Hillel II, which corresponded to a mean
difference of 23 minutes.  After conversion to a longitude difference,
this suggested that the molad reference meridian was midway between
Israel and Babylonia, which seems highly plausible because in the era
of Hillel II most of world the Jewish population was in Babylonia, but
there was still an appreciable population in Israel.

I then built a rectified molad function and compared its difference
against the corresponding actual lunar conjunctions, tweaking the
coefficients until the regression line was a horizontal line with a
slope and intercept were as close to zero as possible.

Then I subtracted rectified from traditional molad moments for the
first 12000 years of the Hebrew Calendar, and again found a minimum in
the era of Hillel II, although with this more accurate calculation the
minimum difference is 9 3/4 minutes.  The molad web page details the
conversion of minutes of mean solar time to degrees of longitude.  The
revised reference meridian is sufficiently close to the meridian of
Jerusalem that in all likelihood the original calculations were correct
for that meridian, but my 9 3/4 degree discrepancy is due to a
slightly-less-than-9% error in my Delta T approximation.

On Nov 30, 2005, at 14:42, Tom Peters wrote:
> Are you sure that the molad would fit Hillel's epoch?  Normally an
> expression is based on a longish period of earlier observations, so
> iff Hillel established the values, the mean epoch could fall well
> before his time.

I'm not sure of anything, of course, that's why I'm asking for
suggestions!

However both my original calculation and the revised more accurate
calculation yielded parabolic curves with a minimum in the era of
Hillel II, suggesting that somehow he managed to make it fit.  I
believe that the explanation is that from that era they back-calculated
the molad epoch using the constant mean lunation interval that was
appropriate for their own era (or very nearly their own era), so that
in the same way that Delta T reaches a minimum around the Gregorian
year 1900, the molad minus actual lunar conjunctions differences are
minimal in the baseline era.

-- Irv Bromberg, Toronto, Canada
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Re: molad reference meridian

Amos Shapir
In reply to this post by Irv Bromberg
AFAIK Hillel did not invent the algorithm, only published an algorithm which
had been in use before -- even centuries earlier.  Since this meridian has
been drifting eastwards, its value at the time of Hillel may be the result
of using an algorithm which was formulated according to earlier
observations, made at the meridian of Jerusalem.


>From: Irv Bromberg <[hidden email]>

>Date: Wed, 30 Nov 2005 13:06:40 -0500
>
>I had previously estimated the reference meridian of the traditional molad
>of the Hebrew Calendar as 5 3/4 degrees east of Jerusalem, or midway
>between Israel and Babylonia, after finding that in the era of Hillel II
>the average moments of the molad were 23 mean solar minutes ahead of the
>mean lunar conjunctions of the era.
>
>The new rectified molad arithmetic is more accurate, and yields a
>difference from the traditional molad of 9 3/4 mean solar minutes in the
>era of Hillel II, suggesting that the traditional molad reference meridian
>is only 2 7/16 degrees east of Jerusalem.  This meridian is too far west to
>be considered "midway" between Israel and Babylonia, yet too far east to be
>considered merely inclusive of the Israeli tribes that were on the east
>bank of the Jordan River.
>
>Searching around on maps, the only explanation that I could come up with is
>that that meridian is midway between Israel and Mecca!  How's that for
>inexplicable?
>
>I can't imagine that there was any intentional choice for such a meridian
>-- it must be that the intention was to align with the meridian of
>Jerusalem and the outcome was very close but not perfect.
>
>On the other hand, taking the era of Hillel II as corresponding to Rosh
>HaShanah 4119 = Julian September 19, 358 AD, my approximation of Delta T
>for that day is 1h53m57s, so that Delta T approximation would only have to
>be in error by less than 9% to account for the 9 3/4 mean solar minutes.  
>This is quite plausible, so it is likely that the original molad reference
>meridian was correctly assigned to Jerusalem.
>
>Any other ideas?
>
>
>-- Irv Bromberg, Toronto, Canada
>
><http://www.sym454.org/hebrew/>

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Re: molad reference meridian

Lance Latham
RE:
> AFAIK Hillel did not invent the algorithm, only
> published an algorithm which
> had been in use before -- even centuries earlier.
> Since this meridian has
> been drifting eastwards, its value at the time of
> Hillel may be the result
> of using an algorithm which was formulated according
> to earlier
> observations, made at the meridian of Jerusalem.

Lance replies:
If the meridian HAS been drifting in an eastward
direction, then I like this explanation a lot.

Question then is, what is the rate of the drift? It
seems to me that the rules, once determined, would be
unlikely to be subjected to constant tinkering; hence,
one might reasonably expect to get a handle on the
date of their origin, given Jerusalem; or
contrariwise, get a handle on location (longitude),
given a rate of drift and a date of publication.

-Lance


Lance Latham
[hidden email]
Phone:    (518) 274-0570
Address: 78 Hudson Avenue/1st Floor, Green Island, NY 12183
 





               
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Re: molad reference meridian

Irv Bromberg
On Dec 1, 2005, at 11:08, Amos Shapir wrote:

> AFAIK Hillel did not invent the algorithm, only published an algorithm
> which had been in use before -- even centuries earlier.  Since this
> meridian has been drifting eastwards, its value at the time of Hillel
> may be the result of using an algorithm which was formulated according
> to earlier observations, made at the meridian of Jerusalem.

On Dec 1, 2005, at 12:22, Lance Latham wrote:
> If the meridian HAS been drifting in an eastward
> direction, then I like this explanation a lot.

BROMBERG says:

Forget the "eastward drift" theory -- we are not talking about an
"algorithm" here, just a fixed amount added to an accumulator for each
elapsed month = 29 days 12 hours 44 minutes 3 1/3 seconds.

> Question then is, what is the rate of the drift? It
> seems to me that the rules, once determined, would be
> unlikely to be subjected to constant tinkering; hence,
> one might reasonably expect to get a handle on the
> date of their origin, given Jerusalem; or
> contrariwise, get a handle on location (longitude),
> given a rate of drift and a date of publication.

BROMBERG adds:

I have shown exactly why there is a drift, not in the reference
meridian but rather in the timing of the molad moments -- it is because
the traditional molad uses a fixed interval for the lunation, whereas
the mean lunation interval has been steadily getting shorter due to the
tides.  Therefore with respect to the actual secular mean lunar
conjunctions, the average molad moment is drifting later, to date
having accumulated almost an average error of 2 hours late.  My point
was that in the era of Hillel II my calculations show that the average
molad was only 9 3/4 minutes late, assuming that the reference meridian
was Jerusalem, and that "error" is small enough that it could easily be
accounted for by a less than 9% error in my DeltaT approximation for
that era.

Please see my web pages on the Molad of the Hebrew Calendar and on the
Rectified Hebrew Calendar (in the latter especially see the section on
the TraditionalMolad and RectifiedMolad functions) at
<http://www.sym454.org/hebrew/>, where these matters are thoroughly
(hopefully) explained.

-- Irv Bromberg, Toronto, Canada.
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Re: molad reference meridian

Irv Bromberg
In reply to this post by Lance Latham
I'm pleased and relieved to report that the molad reference meridian
indeed appears to have been set where I said so in the first place,
that is midway between Israel and the former Babylonia.

 From your comments I realized that the Delta T 9% error was not a good
explanation, especially when I tried Robert van Gent's javascript Delta
T calculator (near the end of his web page at
<http://www.phys.uu.nl/~vgent/astro/deltatime.htm>) and found that most
of the Delta T estimates were very close to mine and did not support
anywhere near as much as a 9% error.  I even tried the most recent
Chapront lunar acceleration factor that I could find: -25.858 "/cy/cy.

My previous calculation was based on a statistical analysis using Excel
for linear regression, and on a sampling of every 8th year over a very
long span of years.
OK, yes, it was an absurdly long span of years (20,000).

My new calculation is based on statistical analysis by GraphPad Prism
v4 for linear regression, and analyzed EVERY month of EVERY year from
Hebrew year 4500 through 7000, which is about plus or minus 1250 years
from today (Hebrew year 5766), and which spans the most accurate region
of the Meeus astronomical algorithms that I am employing.

Now the minimum difference between the traditional and rectified molad
moments, still found in the era of Hillel II, is 25 minutes 54 seconds,
which would place the molad reference meridian midway between Jerusalem
and the meridian of the modern Iraq / Iran border region nearest the
Gulf of Persia.  This is fully consistent with my thesis that the molad
reference meridian was selected to split the difference between Israel
and Babylonia.

Please see the discussion and arithmetic for the TraditionalMolad and
RectifiedMolad functions on the Rectified Hebrew Calendar web page:

<http://www.sym454.org/hebrew/>

-- Irv Bromberg, Toronto, Canada
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Re: molad reference meridian

Lance Latham
RE:

> Now the minimum difference between the traditional
> and rectified molad
> moments, still found in the era of Hillel II, is 25
> minutes 54 seconds,
> which would place the molad reference meridian
> midway between Jerusalem
> and the meridian of the modern Iraq / Iran border
> region nearest the
> Gulf of Persia.  This is fully consistent with my
> thesis that the molad
> reference meridian was selected to split the
> difference between Israel
> and Babylonia.

Lance queries:
Doesn't this place the meridian somewhere in the
vicinity of current Baghdad? If so, might we then
consider the Babylonian menologies as a possible
source of data?

-Lance


Lance Latham
[hidden email]
Phone:    (518) 274-0570
Address: 78 Hudson Avenue/1st Floor, Green Island, NY 12183
 





               
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Re: molad reference meridian

Irv Bromberg
On Dec 2, 2005, at 11:09, Lance Latham wrote:
> Doesn't this place the meridian somewhere in the
> vicinity of current Baghdad? If so, might we then
> consider the Babylonian menologies as a possible
> source of data?

BROMBERG replies:

To paraphrase from my web page "The Molad of the Hebrew Calendar" at
<http://www.sym454.org/hebrew/>:

During subsequent development of the rectified molad of the Rectified
Hebrew Calendar, a more accurate calculation of the traditional molad
reference meridian based on its difference from the rectified molad in
the era of Hillel II suggested that it was 25 minutes and 54 seconds of
mean solar time east of Jerusalem, for which the corresponding
longitude difference is 360? * (25+9/10 minutes of time) / (1440
minutes of time per day) = 6 degrees and 28+1/2 arcminutes, placing the
traditional molad reference meridian at 41? 42' 34" E, still essentilly
halfway between the meridian of Jerusalem and Babylonia (in the region
of the modern Iran / Iraq border extending up from the Persian Gulf).

The meridian of the modern Iran / Iraq border extending up from the
Persian Gulf is near the 48? 11' 4" E that one obtains by taking the
meridian of Jerusalem and then go eastward by doubling the 6 degrees
and 28+1/2 arcminutes.  So the traditional molad reference meridian is
about halfway between Israel and Babylonia.

-- Irv Bromberg, Toronto, Canada
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Re: molad reference meridian

Irv Bromberg
In reply to this post by Lance Latham
Dear Calendar People:

This weekend I completed a more accurate and more direct calculation of  
the traditional molad reference meridian based on least-squares  
second-order polynomial regression of its difference from the actual  
lunar conjunctions (referred to the meridian of Jerusalem) for all  
30933 lunations of Hebrew years 4500 through 7000 (these are the ±1250  
= 2500 years centered on the present era), using Mathematica 5.0.

The regression yielded a quadratic equation, and Mathematica found the  
minimum difference was 25 1/4 minutes of time in Hebrew year 4091. That  
year was about 28 years or one generation before Hillel II — this may  
be a tantalizing hint that Shmuel "the Astronomer" originally  
established the molad calculation (he lived in Babylonia).  The Talmud  
does mention that Shmuel developed a fixed arithmetic Hebrew calendar  
which was used for a generation prior to Hillel II internally by the  
Sanhedrin as a means to check their calendar decisions.

The corresponding longitude difference is 360° x (25 1/4 minutes) /  
(1440 minutes per day) = 101/16° = about 6° 18 3/4', placing the  
traditional molad reference meridian at 41° 42' 49" E, which is halfway  
between Israel and the former Babylonia (in the region of the modern  
Iran / Iraq border extending up from the Persian Gulf), or about 2  
hours and 46 1/4 minutes ahead of Universal Time.

The following lines contain long URLs so if the email system has  
wrapped the URL you may have to fix it manually before pasting it into  
your web browser.  Alternatively you can follow the same links from my  
"Molad of the Hebrew Calendar" web page at  
<http://www.sym454.org/hebrew>.

This MapQuest URL will show the apparent molad reference meridian.  The  
red star will be at the same latitude as Jerusalem but the longitude is  
6° 18 3/4' east of Jerusalem:
http://www.mapquest.com/maps/map.adp?
searchtype=address&formtype=latlong&latlongtype=decimal&latitude=31.7778
&longitude=41.7136&zoom=2

This MapQuest URL will show the point that is double that distance away  
from Jerusalem, in the former Babylonia.  The red star will be at the  
same latitude as Jerusalem but the longitude will be 12° 37 1/2' east  
of Jerusalem:
http://www.mapquest.com/maps/map.adp?
searchtype=address&formtype=latlong&latlongtype=decimal&latitude=31.7778
&longitude=48.0428&zoom=2

-- Irv Bromberg, University of Toronto, Canada

<http://www.sym454.org/>
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