Analysis of a Disagreement:
The Month in Science and Torah
by Morris Engelson
My book The Heavenly Time Machine
is based on the premise that both science and the Bible are true.
We would prefer to have no disagreements, of course. But we do
not consider disagreements between science and Torah as something
to avoid. Rather we look at these as a possible source of new
learning and knowledge as we explore means to a common understanding.
I would like to demonstrate this process by exploring the implications
of a statement that I make in my book. I state that Torah Sages
knew the time duration of the month to a very high degree of
accuracy. My intent was to show that these people were not ignorant
of science, as some would have us believe. Having made this point,
I then proceed to other matters. But there is more that can be
said here.
Statement of the problem
Quoting from pages 140-141 of my book. "We
have a statement on the length of the lunar month on page 25a
in Tractate Rosh Hashanah. This would be the cycle of
the phases of the moon, otherwise known as the synodic month.
Rabban Gamliel says that he has it from the traditions of his
grandfather's house that the length of the month is 29 days plus
half a day plus two-thirds of an hour plus 73 parts of an hour.
An hour is divided into 1080 parts in Talmudic measure. Addition
of these numbers yields a lunar month that is 29.530594 days
long. Current measurements, as reported in the Encyclopedia Britannica,
show a mean time period of 29.530588 days."
What is the problem? Rabban Gamliel of Yavneh,
who made this statement, died in the year 110. His grandfather,
Rabban Gamliel the Elder, lived almost two thousand years ago,
in the early part of the first century. He claimed a time period
for the month that is within half a second of the best result
that current science can achieve. We can only marvel at such
an accurate result. But that does not make for a problem. The
problem comes from a part of the statement, made by Rabban Gamliel,
that I did not quote in my book. The full Talmudic statement
is that the time duration of the month can not be less than 29.530594.
But modern science says that the time is, in fact, less. That
is the source of the problem.
Resolving the problem
We have four ways to get rid of our problem.
- There is always an uncertainty error in every
measurement. Is it possible that the half second difference is
within the measurement uncertainty? No, that is not possible
for two reasons. One reason is that the best current value is
given as 29.5305888531. This is orders of magnitude beyond the
number we are using; truncated after 88. This level of error
is equivalent to a 10 meter measurement uncertainty when using
a centimeter-based ruler. It is impossible. A second possibility
is that the theory upon which the measurement is based is flawed.
But the measurement is based on Newton's laws of motion. We would
have to scrap pretty much all of physics if Newton's laws were
wrong to this extent. No, the science-based number is rock solid.
However, I will adjust the number I use in the rest of this discussion
to end at 89, and not 88, as I previously stated. The next value
is an 8, hence rounding the previous 8 to a 9, is more appropriate
for accurate computations.
- Is it possible that the number determined
from the Bible is wrong? No, because we stipulate as part of
the logic of the process, that the Bible is true. Nevertheless,
it is possible that a statement made by Rabban Gamliel is not
correct, providing he did not get his information from the Torah.
It is well known that various people, including Rabban Gamliel,
made astronomical observations and measurements. Could Rabban
Gamliel (or rather his grandfather) have obtained his value by
measurement? Secular Bible scholars claim that that is how the
result was obtained.
Here is how it could have been done. We determine, as accurately
as possible, the day, date, and time of a conjunction at the
time of an eclipse. The information is maintained for hundreds
of years, so that we accumulate thousands of lunar months. Let
us say that we know the time and date of two conjunctions separated
by 3600 months, to an accuracy of 30 minutes. Half an hour yields
an error of 1800 seconds. But that is for all of the 3600 months.
The average error per month is only 1800 / 3600 = 0.5 seconds.
We no longer have an issue between Torah and science, but rather
between modern science and ancient science. My statement that
the ancient Torah Sages were not ignorant of science still holds,
and all is well. But this solution does not work, because Rabban
Gamliel was, in fact, dealing with Torah, as discussed next.
- You will recall that Rabban Gamliel gave
the time between two new moons as 29 days, plus half a day, plus
2/3 of an hour plus 73 parts of an hour. Half a day is 12 hours,
and 2/3 of an hour is 720 parts of an hour, at 1080 parts per
hour. The total number of days is 29, individual hours is 12
and the number of remaining parts is 720 + 73 = 793. Hence it
is common in Talmudic discourse to identify the time period of
the month as 29-12-793. We have a firm tradition that this number,
29-12-793, was handed down from Moses at Sinai. Not only that,
but we hold that the formulas for calculating the months, some
of which are shorter and some longer, how to determine the Rosh
Hashanah (New Year), which may fall on certain days of the week,
but not on others, and a number of other related matters came
to us from Moses. Yes, Rabban Gamliel did make measurements.
But the authority for making the statement that he made is based
on Torah and not on science. Rabban Gamliel, as you will see
later, made the statement as he overruled the central court who
were about to announce a new month. It is inconceivable that
anybody would do such a thing except on the authority of Torah.
I refer to the article, Calendar Configurations, by Rabbi Moishe
Kimelman, on page 85/86 of the April 20, 2001 issue of Hamodia,
for those interested in a summary explanation respecting the
numerous calculations, formulas and numbers that we hold as a
tradition from the time of Moses. Rabbi Kimelman cites the Gaon
of Vilna, who notes that the number 793 came before the number
1080. The hour could have been divided simply into 15 portions,
of which 11/15 is equal to 792 parts of an hour. Instead we are
forced to use the large number, 1080, so as to permit the more
precise 793. The number 793 is important not only for measurement
accuracy but also, other, more mysterious reasons. Thus, Rabbi
Avrohom Chaim Carmel points out the following. Multiply 29 days
by 24 to get hours. Add the additional 12 hours and multiply
by 1080 to get parts of an hour. Finally, add 793 to the result
to obtain all the parts of an hour. This yields the descending
order 765,432 + 1. This number has kabbalistic implications.
But I do not know what it means.
- The next conclusion to this issue is to agree
that we have no answer. Not having an answer does not hurt the
original premise that both Torah and science are true. There
are many things that people do not know. But that does not make
them untrue. Fermat's last theorem was shown to be true just
a few years ago. But it was just as true for the previous centuries
when nobody knew the answer, as it is true now that we do know
the answer. There are certain things that are beyond our knowledge
and we need to let these go into the future. Fortunately, that
is not the case respecting the matter of the month.
- We now come to the process that will resolve
the puzzle as to how a statement that some value can not be "less
than" is just as true as a statement that the value is in
fact "less than." The resolution involves a deeper
look into what the Torah-based statement is really about. Combined
with a more sophisticated science-based analysis, will show that
science and Torah are in full agreement. Not only that, but we
will uncover in the process some relationships, that we might
otherwise miss.
What did Rabban Gamliel say, and what did
Rabban Gamliel mean?
I refer to the following sources for what
follows.
The primary reference is the Talmud Tractate
Rosh Hashanah. Rosh Hashanah means the New Year. A new
year begins at the beginning of a month. Hence, to understand
the timing of Rosh Hashanah, we need to understand the timing
of Rosh Chodesh (new month). This explains why the Tractate Rosh
Hashanah has much information about Rosh Chodesh. Talmud
pages have the same number front and back. The front is identified
as "a", and the back is identified as "b."
Rosh Hashanah, 25a refers the reader to the front side
of page 25 in Tractate Rosh Hashanah.
Another reference is the classic commentary
Bais HaBechirah authored by 13th century commentator Rabbi
Menachem HaMeiri (Meiri). I also refer to Maimonides' discussions
respecting the month in Hilchos Kiddush Hachodesh (rules
respecting the sanctification or announcement of the month).
Finally, we have the previously referenced article by Rabbi Kimelman.
The time of the new month has been determined
by calculation for the last 1640 years (since the year 359).
This was necessitated by the exile and dispersion of the Jews.
Calculations could have been done previously, as we hold that
formulae come to us in the oral Torah from Moses. However, the
preferred method is for the court to announce the new month on
the basis of testimony from witnesses who saw the new moon. This
was the process at the time of Rabban Gamliel, head of the community
in the late first century.
We are told (Rosh Hashanah, 25a) that
the sky was overcast, and some people thought that they saw the
appearance of the new moon in the clouds. They testified before
the court, and the court was about to announce the new month.
But Rabban Gamliel stopped the process. He explained that the
witnesses could not have seen the moon yet, because he had a
tradition from his grandfather that the time from one new moon
to the next can not be less than, etc. Clearly Rabban Gamliel
overruled the testimony of the witnesses on the basis of a calculation.
Is that permitted? Yes. We have a discussion on this matter in
Rosh Hashanah, 20b, where Rav Ashi explains that, "By
calculating the conjunction, we are able to refute false witnesses."
Indeed, calculation is the only way to know the precise moment
of the conjunction, because the moon is not visible at that time.
We know from the foregoing that Rabban Gamliel
was interested in a specific practical case. He did not set out
to expound on the information that he had from his grandfather.
There is more such information that Rabban Gamliel mentions at
other times. Meiri explains that Rabban Gamliel simply provided
the information needed to refute the testimony of the witnesses.
He did not give less, and here also did not give more than needed
for his purpose. Meiri tells us that the full tradition is that
the time period for the month can not be less than 29-12-793,
and it also can not be more than 29-12-793. "Not less than
and not more than." Does that mean that the time period
is equal to 29-12-793? No it does not. We know today that each
month lasts a slightly different time period, and the science-determined
value is an average. This was also known from Torah. Rabban Gamliel
tells us that he has it from his grandfather, that the period
varies and that "sometimes it (the moon) travels by the
long route, and sometimes by the short route." Maimonides
explains the meaning of "not less than, and not more than."
He notes that we use an average value of 29-12-793, and the deviation
from this value is sufficiently small that this is the number
that we should use.
We now have a much better understanding of
the information from the Torah. But it does not eliminate our
problem. The fact is that Rabban Gamliel invoked the "not
less than" rule. Could he do that when science says that
it really is less? Furthermore, we know that the moon is gradually
slowing down due to frictional drag. It took less time for the
moon to go around the Earth two thousand years ago than it does
today. If the actual month (this is not any particular month,
but rather the average value) is less than the Rabban Gamliel
number today, then it had to be even less than that in the past.
How much less, and can we justify the difference? This calls
for a deeper understanding of the science, which comes next.
What is a month?
The solar year is divided into 12 months.
These, solar months, are arbitrary constructs. We could divide
the year into ten months, or fifteen months, or any other number
of months. We just happen to use twelve because of our experience
with the moon. The moon moves around the Earth roughly 12 times
a year. The elapsed time between two successive new moons, or
other phase of the moon, is one lunation. The lunar month is
a time measure of a lunation. These, lunar, months are not arbitrary
constructs, because we can measure the time duration of a month
by astronomical observation. How the measurement is made will
determine what type of (lunar) month we are talking about.
We will ignore the more obscure months, such
as the tropical month or nodical month. This leaves two well
known lunar months: the sidereal month and the synodic month.
The sidereal month lasts a bit over 27 days, and is the time
of one rotation of the moon as measured against the position
of the stars. The synodic month lasts about 29.5 days, and is
the time of one rotation of the moon as measured with respect
to the sun. The time difference between these two type months
is accounted for by the movement of the Earth with respect to
the stars, as the Earth rotates around the sun. The Torah deals
with the synodic month, and that is the subject of our discussion.
The current average value of the synodic month, as determined
by the exact conjunction of the Moon and Sun (as observed from
the Earth), has been determined to be 29.5305888531 days. I will
round this to 29.530589 days to provide the same decimal precision
as the number given by Rabban Gamliel, which is 29.530594 days
(converted to modern units of measure from 29-12- 793). The difference
between these two numbers is 0.000005 days, or 0.432 seconds.
Currently the month lasts 0.43 seconds less time than noted by
Rabban Gamliel. Therein lies our original question, because Rabban
Gamliel says that the month can not be shorter than his value.
In fact, the situation appears to be even
worse than noted above. This is because the moon is slowing down
its rate of movement around the Earth at the rate of about 0.18
seconds per millenium. Hence it took 0.4 seconds less time for
the moon to go around the Earth 2000 years ago when Rabban Gamliel
made his pronouncement. Combining with the current difference,
it appears that the month was 0.8 seconds less than the Rabban
Gamliel number, two thousand years ago. I say "it appears",
because that is all it is. This is the appearance, but not the
actual situation. The discrepancy comes from the choice of a
timing reference.
We have numerous measures and definitions
for the passage of time. Here are a few: atomic time, apparent
time, universal time, dynamical time, sidereal time..., and others.
The current standard for time measurement, which has been in
effect since 1967, is the IAT (international atomic time) second.
The IAT second is equal to 9,192,631,770 cycles of the transition
between two hyperfine levels of the ground state of cesium 133.
This reference second is a fixed number that does not change
with the date. The cesium atom behaved the same way 2000 years
ago as it does today. When I say that 2000 years ago the month
was 0.8 seconds less than Rabban Gamliel claimed, I mean 0.8
IAT seconds. But Rabban Gamliel was not talking about IAT seconds,
or any other kind of seconds. He was talking about days and fractions
of a day. A day is one rotation of the Earth on its axis, and
it need not be precisely 86,400 seconds in duration, at 24 hours
and 3600 seconds/hour. Here the day is the reference standard,
and its atomic time duration is of no consequence. A day is a
day. There are 24 hours in one day, whatever time duration that
happens to be. There are 1080 parts to each hour. The month,
according to Rabban Gamliel is 29-12-793, and not 29.530594 days
consisting of 86,400 seconds each. This brings in an additional
factor that we had not previously considered.
The Earth is slowing down its rotation, and
the length of the day is increasing at about 2.3 milliseconds
per century. Days were shorter 2000 years ago than now. Hence
the month lasted more days then, than it does today. We have
two opposite effects here. The slowing of the moon means that
the IAT time for the month was shorter in the past. But the slowing
of the Earth means that the observed month was longer in the
past. We need to determine the net result of these two effects.
The values for the slowing of the moon, the
slowing of the Earth and formulas for determining the impact
of these, is not simple to determine. I am pleased that I was
spared any of this fundamental work by a thorough presentation
on the topic, Lunar Calendars, written by Peter Meyer. The reader
will find this reference at URL serendipity.magnet.ch/hermetical/cal_stud/lunarcal/luncal.htm.
Meyer provides formulas from which the time duration of, what
he calls, the IAT synodic month and the observed synodic month,
can be calculated for any calendar date. We are interested in
the observed month, as that is what Rabban Gamliel was talking
about. The result is not exactly constant, or linear, with change
of date. But I am not looking for a very high level of precision.
Hence I can use a single number for all that I need to accomplish.
Using the formulas provided by this reference, and taking into
account both the slowing of the moon (the month was shorter in
the past) and the slowing of the Earth (which makes the month
longer in the past), I find that the observed month changes at
the rate of 0.51 seconds per millenium. The month was 0.51 seconds
longer 1000 years ago than it is today.
At this time, the month is 0.43 seconds less
than the Rabban Gamliel number. Hence, 2000 years ago, the month
was 0.51 x 2 – 0.43 = 0.59 seconds longer than the value
cited by Rabban Gamliel. Rabban Gamliel was correct when he overruled
the witnesses. The month was, indeed, longer and not shorter
than his number. This situation lasted 1156 years (roughly 850
years ago), when the difference became zero. The observed month
has been less than the Rabban Gamliel value since that time.
But being less, and not more, is perfectly legitimate given the
comments by Meiri and Maimonides, and the fact that we no longer
rely on witnesses.
Going further
The above pretty much settles the differences
between science and Torah. But we have information here that
can go further than that. Recall that Maimonides says that while
the value of the month can be lesser or greater than 29-12-793,
the deviation will be such that we will choose 29-12-793 as the
result. We call this rounding to some level of precision. Here
the rounding is to the nearest part of an hour. There are 1080
such parts in one hour, hence each part lasts 3 and 1/3 seconds.
The deviation needs to be kept to half a part, or 1.66 seconds,
in order to round to 793. How many years ago does a deviation
of 1.66 seconds represent? (1.66 + 0.43) / 0.51 = 4100 years
ago. This is the time of the flood. We know that the world and
the people were different before the flood and after the flood.
Apparently there was also an impact on the month.
Going forward in time is not an issue, as
we are now in year 5762 and we need to go to only year 6000.
Six thousand is the period of a celestial time cycle, as I discuss
in my book. Hence we stop at this point. At that time the month
will be shorter than the Rabban Gamliel value by 0.43 + [(6000
– 5762) / 1000] x 0.51 = 0.55 seconds. But one part of an
hour is 3.3 seconds. Hence the deviation in year 6000 is 1/6
of a part of an hour. What is the significance of this 1/6 fraction?
I do not know. Finally, one more interesting item. The change
in the time of the month is 0.51 per 1000 years. This makes the
total deviation from the time of creation to the year 6000 equal
to 0.51 x 6 = 3.3. But 3.3 seconds is one part of an hour.
Is the fit to the time of the flood a coincidence?
Is the deviation of one part of an hour, over all of time for
one celestial cycle, a coincidence? Is the integer fraction,
1/6, for the end of the cycle a coincidence? Is the agreement
of modern science and ancient Torah a coincidence? You decide.
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