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\author{Shlomo Edward G.~Belaga \\
C.~N.~R.~S., Universit\'e Louis Pasteur \\
7, rue Ren\'e Descartes \\
F-67084 STRASBOURG Cedex FRANCE \\
}
\title{On The Rabbinical Exegesis of an Enhanced Biblical Value of $\pi$}
\begin{document}
\maketitle
\centerline{{\bf AMS 1980 Mathematics Subject Classification:} 01 A 15,
01 A 17}
\centerline{{\bf Key words:} history of {\bf Pi}, Rabbinical exegesis.}
\begin{abstract}
We present here a biblical exegesis of the value of $\pi$, $\pi_{\rm
Hebrew}=3.141509\dots$, from the well known verse {\bf 1 Kings 7:23}.
This verse is then compared to {\bf 2 Chronicles 4:2}; the comparison
provides independent supporting evidence for the exegesis.\footnote{An
earlier version has appeared in the Proceedings of the {\sl XVIIth
Canadian Congress of History and Philosophy of Mathematics}, Queen's
University, Kingston, Ontario, May 27-29, 1991, pp. 93-101.}
\end{abstract}
{\bf 1.}The {\bf Hebrew Bible} often speaks the language of numbers and
measurements [Feldman 1965]; the Western tradition rarely\footnote{One
of those rare cases is Isaac Newton's {\sl ``obsession with the {\rm
[King Solomon's]} temple's plan and dimensions\dots Being the man he was,
he plunged into an extensive program of reading in Josephus, Philo,
Maimonides, and the Talmud scholars''} [Westfall 1987, pp. 346-348].
Newton's inspirations were conjectured by Frank Manuel [Manuel 1974] in
the following form:{\sl ``The temple of Solomon was the most important
embodiment of a future extramundane reality, a blueprint of heaven; to
ascertain every last fact about it was one of the highest forms of
knowledge, for here was the ultimate truth of God's kingdom expressed in
physical terms''} (quoted in [Brooke 1988], p. 177.)}, if at all [Hoyrup
1989], understands this language, and the case of Biblical value of
$\pi$ could be seen as both a remarkable exception of this rule and its
striking confirmation.
As a recent publication in {\it The American Mathematical Monthly}
puts it, ``{\it the ancient Hebrews regarded $\pi$ as being equal to
3}'' [Almkvist, Berndt 1988, p. 599]. This claim (as several identical
claims made by both working mathematicians [Borwein, {\it et al.}1989]
and historians of science [Bell 1945], [Beckmann 1971]) is based on the
plain meaning of the following verse of the Hebrew Bible, {\bf 1 kings
7:23}, giving the dimensions of a tank in the First
Temple\footnote{Built by the King Solomon, the ninth century BCE; the
water of the tank was used by priests for ritual ablutions. {\sl ``The
molten sea was a large, bronze water reservoir set on backs of twelve
bronze oxen and placed in the court of Solomon's temple\dots The
diameter was about 5 m (16 feet), the height about 2.5 m (8 feet), and
the volume ammounted to roughly 45,000 litres (12,000 U.S. gallons).
There can be little doubt that it was one of the greatest engineering
works ever undertaken in the Hebrew nation. Its size is comparable to
some of the largest church bells cast in modern times''} [Zuidhof 1982,
p. 179].}:
``{\it And he made a molten sea} [tank], {\it ten cubits from the one
brim to the other: it was round all about, and its height was five
cubits: and a \underline{line} of thirty cubits did circle it round
about}''. [Holy Scriptures, p. 412]
As a matter of fact, after mentioning this verse, people either can
not\footnote{{\sl ``but several difficulties complicate the analysis of
the design of the vessel, its dimensions and the volumetric capacity
\dots The sea apparently was not the typical straight-walled
mathematical cylinder\dots a brim and a lily has outward curving
petals\dots The biblical account mentions first the brim to brim
diameter of ten cubits. A line streched across the top would easily have
measured this\dots It is then reasonable to conclude that the 30-cubit
circumference was measured below the brim''} [{\sl loc.cit.}, pp.
179-181].}, or do not want\footnote{{\sl ``It has been suggested, perhaps
by someone who believes that `God makes no mistakes', that `round' and
`depth' are to be interpreted loosely, and that the tank was elliptical
in shape''} [Almkvist, Berndt 1988, p. 599].} \footnote{{\sl ``Not all
ancient societies were as accurate, however - nearly 1500 years later
the Hebrews were perhaps still content to use the value 3''} [Borwein,
{\sl et al.}1989, p. 204].} , to hide (or are even happy for some
ideological reasons, to emphasize\footnote{{\sl ``The inaccuracy of the
biblical value of $\pi$ is, of course, no more than amusing curiosity.
Nevertheless, with the hindsight of what happened afterwards, it is
intresting to note this little pebble on the road to the confrontation
between science and religion''} [Beckmann 1971, p. 13-14].} ) their
surprise by such a low accuracy of the Biblical approximation,
$\pi_0=3$, especially in the light of well-documented evidence that the
ancient Babylonians and Egyptians used for $\pi$ much better
approximations [Neugebauer 1969], [Gillings 1972] many hundred years
before this part of the Hebrew Bible was written:
$$\pi_{\rm Babylon}=3\frac{1}{8},\quad
0.017>\pi - \pi_{\rm Babylon}>0.016;$$
$$\pi_{\rm Egypt}=3\frac{13}{81},\quad
0.019>\pi_{\rm Egypt}-\pi >0.018$$
Thus, it seems both appropriate and intresting at this point to give a
Rabbinical interpretation of the above verse and the way the number
$\pi$, implicitly\footnote{{\sl ``Also, the ratio between circumference
and diameter ($\pi$) of the circular vessel is not mentioned in the
Bible\dots ''} [Zuidhof 1982, p. 180]} defined in this verse, has to be
computed [Max Munk 1962 , 1968] (see also two popular and slightly
diffrent accounts in [Posamentier, Gordan 1984] and [Roiter 1993]). We
do not claim, however, that the Rabbinical folklore has preserved either
the {\it mathematical method} which was used in this approximation of
$\pi$, or its {\it historical origins}: all that was left to us is an
extremely natural and concise {\it mnemonic rule of the reconstruction of
$\pi_{\rm Hebrew}$} (see more about it in [Max Munk 1962, 1968]).
Such an absence of mathematical justification is, of course, well known
to historians; as, e.g., a researcher writes about the value of
$\pi_{\rm Egypt}$: {\it `` Just how this remarkably close approximation
was found, we do not know, but we can offer a suggestion on examining
the diagram of RMP 48''} (cited in [Gillings 1972, p. 142]). In our
case, no diagrams were preserved; one could even doubt that such
diagrams ever existed: ``ancient Hebrews'' have never regarded
mathematical or, for that matter, any other scientific knowledge {\it
per se } as deserving to be developed, preserved, and disseminated {\it
in the written form}, as they were not intrested (with the Jewish Temple
being a notable exception) in creating numerous and splendid monuments
of their religion and culture.
{\bf 2.} The key to an alternative reading of the verse {\bf 1 Kings
7:23} is to be found in the very ancient Hebrew tradition (see, e.g.,
[Britannica 1985], [Banon 1987, pp. 52, 53]) to {\it differently write}
(spell) and {\it read} some words of the Bible; the {\it reading}
version is usually regarded as a correct one (in particular, it is
always correct from the point of view of the Hebrew {\it grammar}, and
this is why it could be easily either remembered or reconstructed from
the written version), whereas the {\it written} version slightly
deviates from the correct spelling. (Another approach, involving the
comparison between {\it written} forms of the same words in {\bf 1 Kings
7:23} and {\bf Chronicles 4:2} is cited in [Posamentiern, Gordan
1984]\footnote{Who attribute their exegesis to Rabbi Eliyahu of {\sl
Vilna}, alias {\sl Gaon-mi-Vilna}, the famous Talmudic scholar of the
late eighteenth century; unfortunately, the author was unsuccessful in
locating the related reference to works of {\sl Gaon-mi-Vilna}}; see
more about this version of the exegesis in {\bf 4}).
Such a disparity is a common feature for all Books of the Hebrew Bible;
{\it and in any such case} there exists (or existed: some of this
knowledge was definitely lost) a Rabbinical folklore (in fact, {\it
strict Rabbinical hermeneutical rules} [Steinsaltz 1976, part three:
{\it Method}], [Britannica 1985], [Banon 1987]) of interpretation of the
diffrence in question.
In our case there is such a disparity for the word ``\underline{line}'':
in Hebrew, it is written as ``{\it QVH (Qof, Vav, Hea)}'', but it has to
be read as ``{\it QV (Qof, Vav)}'' (the reader is advised to look at any
edition of the Hebrew Bible with the Hebrew text and its translation;
all disparities are either marked by an atersik, or the reading version
is written on the margins).
Tradition asserts that not only does this disparity testify to an {\it
approximate} character of the given length of the {\it line} circling
around the ``sea''(tank), --- a much more accurate approximation to
$\pi$, $\pi_{\rm Hebrew}$, is hidden in the {\it choice} of the written
version!
The letters of the Hebrew alphabets were traditionly used (well before
the building of the First Temple [Guitel 1975]) for numerical purposes
and, thus, have had numerical values \footnote{Analogous numeric
systems were used later, and, without doubt, following the Hebrew
tradition, in the Arabic, Greek, and Cyrillic texts [Guitel 1975]} .
Using these values, one can calculate values of words (as sums of values
of letters, but also in several other, less obvious and/or more involved
ways); these methods became later known as {\bf gematria} [Michael Munk
1983, p. 163], [Britannica 1985]. Here are the standard numerical
equivalents of the letters of the Hebrew alphabet:
\vskip 0.5 cm
\font \st = cmti8 scaled \magstep 0
\centerline{{\st Aleph=1, Beth=2, Gimel=3, Daled=4, Hea=5, Vav=6,
Zain=7, CHet=8, Tet=9,}}
\centerline{{\st Yod=10, Caf=20, Lammed=30, Mem=40, Noon=50, Samech=60,
Aiin=70, Pea=80, TSadik=90,}}
\centerline{{\st Qof=100, Reish=200, Shin=300, Tav=400.}}
\vskip 0.5 cm
In particular, the numerical equivalent of the {\it written} version
,``{\it QVH}'', is {\it Qof}+{\it Vav}+{\it Hea}=100+6+5=111, whereas
the numerical equivalent of the {\it reading} version, ``{\it QV}'',
is {\it Qof}+{\it Vav}=106.
Using these numerical equivalents, one defines $\pi_{\rm Hebrew}$ as
follows:
$$\pi_{\rm Hebrew}=\pi_0\times\frac{the\ numerical\ equivalent\ of\ the\
written\ version}{the\ numerical\ equivalent\ of\ the\ reading\ version}
=$$
$$=3\times\frac{111}{106}=\frac{333}{106}=3\frac{15}{106}$$
Thus, $$\pi=3.1415926\ldots,\quad \pi_{\rm Hebrew}=3.1415094\ldots,
\quad |\pi_{\rm Hebrew}-\pi|<0.000084\ .$$
{\bf 3.} Quantatively, this is quite a remarkable approximation!
However, it is even more remarkable qualitatively. Here is a finite
section of the (infinite) continued fraction of the number $\pi$:
$$\pi=3+{1\over 7+{1\over 15+{1\over 1+{1\over 292+{1\over
1+{1\over\cdots}}}}}}$$
and here are the {\it convergents} (see, e.g., [Khintchine 1963])
corresponding to the first five sections of $\pi$ :
$$[3;]=3;\quad [3;7]=3\frac{1}{7};\quad [3;7,15]=3\frac{15}{106};$$
$$[3;7,15,1]=3\frac{16}{113};\quad [3;7,15,1,292]=3\frac{4687}{33102}$$
One immediately observes that, firstly, $\pi_{\rm Hebrew}=[3;7,15]$,
and, secondly, $\pi_{\rm Hebrew}$ is the {\it second} (after
$\pi_1=[3;7,15,1]$) {\it best convergent} with a denominator under
30,000 ! Notice also that the preceding convergent, [3:7]=22/7, was
known to ancient Greeks.
{\bf 4.} It is worthwhile to mention here a remarkable fact, namely,
that in the case of the verse {\bf 1 Kings 7:23} we have an {\it
independent confirmation} of the above mentioned {\it written} vs. {\it
reading} disparity.
Namely, it could easily be seen that the verse {\bf 2 Chronicles 4:2} of
the Hebrew Bible repeats {\bf 1 Kings 7:23} almost verbatim [Holy
Scriptures, p. 988]. Looking at the Hebrew text, one immediately
observes that the Hebrew word translated as \underline{line} is
traditionally {\it spelled} (written) here {\bf identically} to its {\it
reading} version. Thus, even if somebody would rebuff as irrelevant the
problem of interpretation of the disparity {\it written} vs. {\it
reading} version of the word \underline{line} in {\bf 1 Kings 7:23}
(because he does not trust the {\it oral} tradition of transmission of
Biblical texts), he would still have to explain the disparity between
two different {\it written} versions of the same word (with only one
version being grammatically correct) in two almost identical verses of
the Bible! This last disparity is chosen as the point of departure for
the Rabbinical exegesis in [Posamentier, Gordan 1984].
One could ask, why would this important hint to the enhanced value of
$\pi$ be omitted from the Books of Chronicles? An answer might be that
the Books of Chronicles were written more than four hundred years after
the Books of Kings, and the author of the Chronicles (traditionally
identified with the Scribe Ezra) was much more preoccupied with
rebuilding the Temple and preserving the spirit of the Torah, than with
the ``correct'' value of $\pi$ hidden in the descriptions of dimensions
of the sacred objects in the First Temple; still, Ezra has faithfully
reproduced these dimensions in his book.
A methodological remark: whereas the exegesis based on comparison of
{\it written-vs.-reading} versions of a verse is a very general method
in the Rabbinical tradition [Munk 1962, 1968], [Banon 1987], the above
exegesis exploits a more rare event: the existence of two almost
identical verses.
{\bf 5.} The following question arising from the above analysis has to
be, at least briefly, touched upon: if the author of the first Book of
Kings (traditionally identified with Prophet Jeremia) {\it actually
knew} the value $\pi_{\rm Hebrew}$ and {\it intentionally exploited} the
aforementioned {\it written-vs.-reading} disparity to encode it, why
couldn't he simply write this value down in his text?
The answer might be that the value $\pi_0=3$, implicitly given in the
text, plays an important r\^ole as an approximation which was regarded
(and {\it is still regarded}) as best suited for all ritual purposes in
the everyday life of a common practitioner (possibly, mathematically
illiterate) of the Jewish law. Thus, our verse serves, in fact, (and so,
we conjecture, {\it was it concieved} by its author) as the [only] {\it
textual basis} for the following {\it legal} definition of $\pi$ : {\it
``Any [circle] which has a circumference of three fists has a diameter
of one fist''} [Mishnah 1983, p. 23] (this important dictum is
encountered in at least four different places of the Babylonian Talmud
[Max Munk 1962, 1968]).
Still, all legal texts thoroughly investigate the problem [Max Munk
1962, 1968], [Scherman 1980], [Mishnah 1983, p. 22] and confirm that the
{\it real} value of $\pi$ is ``slightly bigger'' than 3, with some
commentators advancing an almost modern point of view on irrational
nature of $\pi$ (the irrationality of $\pi$ was strictly proved only in
the late eighteenth century); thus, {\it Rambam}\footnote{ A Rabbinical
authority, codifier, philosopher, and royal physician, Rabbi Moshe ben
Maimon (1135-1204), known by his acronym, {\it RAMBAM}, and as {\sl
Maimonides}, was one of the most illustrious figures in Judaism of all
time.} comments:``\dots {\it the [exact] ratio of the diameter of a
circle to its circumference cannot be known [is irrational]\dots but it is
possible to approximate it\dots and the approximation used by scientists
[Greeks and Arabs] is the ratio of one to three and one seventh\dots
Since it is impossible to arrive at a perfectly accurate ratio, \dots
they [the Jewish Sages] assumed a round number and said: `Any [circle]
which has a circumference of three fists has a diameter of one fist'.
And they relied on this for all the measurements they needed''} [Mishnah
1983, p. 22].
It should be stressed that the purposed interpretation of the two-level
semantical structure of a Biblical verse (in our case, {\bf 1 Kings
7:23}), one level for legal purposes, and another one for
``connaisseurs'', is not only a typical phenomenon in the Rabbinical
tradition, - in a sense, such a multy-level approach to texts is the
main methodological legacy of this tradition [Steinsaltz 1976, Part
Three: {\it Method}], [Banon 87]. As Rabbi Moshe ben Nachman\footnote{A
Rabbinical authority, codifier, philosopher, physician, and poet; born
in 1195, died circa 1270; known by his acronym, {\it RAMBAN}, and as
{\sl Nachmanides}} writes: {\it ``Everything that was transmitted to
Moses our teacher through the forty-nine gates of understanding was
written in the Torah explicitly or by implication in words, in the
numerical value of the letters or in the form of the letters, that is,
whether written normally or with some change in form, such as bent or
crooked letters, and other deviations\dots ''} [Ramban 1971, Vol.1, p.
10].
Of course such an approach makes sense only if applied to texts which
are faithfully transmitted from generation to generation; in fact,
Judaism possesses elaborated institutions for such a
transmission\footnote{A historian comments: {\sl Josephus, writing not
long after 70 CE boasts of the existence of a longstanding fixed text of
the Jewish Scriptures''} [Britannica 1985, vol.14, p. 760].} . In this
sense, it is (and always was) similar to modern science, with its
elaborated institutions of training and supporting professionals, whose
duty is to discover, accumulate, and transmit knowledge.
{\bf 6.} With all this understanding, gained thus far, we are, as yet,
unable to elucidate the way the exegesis of the verse {\bf 1 Kings 7:23}
has come to us: was it rediscovered by Rabbi Matityahu Hakohen Munk on
his own [Max Munk 1962, 1968], or was it transmitted to him? Is there
another source in the Rabbinical literature for the exegesis?
A formidable {\it a priori} difficulty in answering these and similiar
questions is related to unpleasent two-thousands years old legacy of
Judaism: as a religion, it invariably remained during this period an {\it
underdog}, prone to presecutions and derision. This external pressure,
together with related to it scarcity of social resources, explain why
Rabbis have strictly separated legal matters (as, e.g., the legal
definition of $\pi_0$) from ``esoteric'' knowledge available to them
(our exegesis possibly included). In fact, it would be a nightmare
scenario for {\it Rambam}, or any other Jewish scholar who lived two
hundred years ago, or more, to advance a better approximation of $\pi$,
{\it without being able} (as we now are) {\it to confirm this value
scientifically.}
This fundemental difficulty still remains the main obstacle to
scientific ``customization'' of the vast body of esoteric knowledge
accumulated, commented upon, and faithfully transmitted by Jewish
scholars. The author hopes to be able to contribute more to our better
understanding of this precious intellectual and spiritual heritage.
\vskip 0.3 cm
{\bf Acknowledgements.} Any acknowledgements would be both incomplete
and difficult to appreciate without some rather personal remarks about
the history of the writing of the present paper.
The author has acquired the knowledge of the Rabbinical exegesis of the
verse {\bf 1 Kings 7:23} from Rabbi Haim Roth, of {\it Mevasseret
Yerushalaim}, eleven years ago (the winter of 1979-1980); since then,
several scholars in Talmudic studies have confirmed the existence of the
exegesis, however, no sources for it were ever mentioned.
The author decided to publicize the exegesis, in the fall of 1990, after
he stumbled upon two recent papers in {\it The American Mathematical
Monthly} (written for a wide mathematical audience and devoted to new
methods of computation of $\pi$), which claimed, in a matter-of-fact
manner, that ``{\it the ancient Hebrew regarded $\pi$ as being equal to
3}'', - citing, of course, the verse {\bf 1 Kings 7:23 !}
The first draft of the paper appeared in October 1990, with a very
gratifying reponse from both the Talmudic and scientific communities.
The comments of Rabbi Naftali Gut, of {\it Z\"urich}, were most
inspiring. Rabbi Dr. Henri Biberfeld, Rabbis Daniel Mund and Arye Posen,
of {\it Montr\'eal}, suggested several important Talmudic and Halachic
sources. Rabbi Dr. Nachum L. Rabinovich, of {\it Maaleh Adumim}, read
the paper and suggested an important correction. Discussions with Prof.
Louis Charbonneau, of {\it Montr\'eal}, and his colleagues were
helpfull in adjusting the presentation to tastes of practitioners of
history of mathematics; the references [Feldman 1965], [Hoyrup 1989]
belong to Prof. Charbonneau. Later, he introduced the author to Prof.
Roger Herz-Fischler, of {\it Carleton}, to whom belongs the reference
[Zuidhof 1982]. Monsieur Luc Gagnon, the student of Prof. Jacques
Lefebvres, {\it Montr\'eal}, supplied the reference [Posamentier, Gordan
1984]. Several manifestations of utmost disbelief (in few cases,
bordering on ridiculous\footnote{As an anonymous reviewer has written on
the third draft of the present paper (which went in all through a dozen
of drafts), {\sl ``Il n'auirait pas \`a adh\'erer \`a un acte de foi,
comme celui d\'ecrit en p.2 ni comme en p.3-4:`(\dots) Ezra has
faithfully reproduced these dimensions in his book' ''}.
The present author does not remember now what exactly has the reviewer
referred to on the page 2 (nor was it clear to the author immediately
after he has received the reviewer's text), but the author's statement
about the {\sl ``faithfullness of Ezra''}\ \ has survived all changes
(see the end of {\bf 4}), to testify that no {\sl ``act of faith''} \ is
needed to compare two verses and to conclude that the second one is
a faithful copy of the first one.}),
on the part of colleagues with, apparently, no previous exposure to
Jewish studies, helped the author to contain excitement and avoid
self-congratulations.
Finally, and miraculously, Prof. Edward Reingold, of {\it Urbana}, whose
enthusiasm for the subject was most encouraging, introduced the author
to Rabbi Dr. Zeharia Dor-Shav, of {\it Bar-Ilan}, who, by sheer
coincidence, has just become aware about the existence of an exegesis
and started to look for its source. In a week or so, the crucial
references [Max Munk 1962, 1968] were found and transmitted to the
author, - and all this has happened in the last week of April 1991,
after eleven years of unsuccessful search for such a source! After
hearing about the author's difficulties to locate the (Hebrew)
references in {\it Montr\'eal}, Prof. Reingold has found the articles in
{\it Urbana} and sent the copies to the author.
Still, with all the aforementioned interest and encouragement, the risky
endeavor to bridge the gap between the Rabbinical tradition and modern
history of science would be impossible without the steadfastness and
support of the author's family.
\newpage
\centerline{\bf REFERENCES}
\vskip 0.3 cm
{\bf Note:} The Rabbinical literature on the subject which are dealt
with (or only briefly mentioned) in this paper is enormous. However, the
present author has intentionally restricted his choice to such English
(and, in three cases, French) references which are widely available in
modern libraries. The only (and, unfortunately, unavoidable) exceptions
are the original papers of Rabbi Max Munk, written in Hebrew and never
translated in any of Western languages.
{\bf G. Almkvist, B. Bernd 1988:} Gauss, Landen, Ramanujan,
the Arith- metic
-Geometric Mean, Ellipses, $\pi$, and the {\it Ladies Diary,
The Amer. Math. Monthly}, {\bf 95}, 585-608.
{\bf D. Banon 1987:} {\it La lecture infinie: Les voies de
l'interpr\'etation midra- chique}, \'Editions du Seuil, Paris.
{\bf P. Beckmann 1971:} {\it A History of $\pi$ (Pi)}, The Golem Press,
Boulder.
{\bf E. T. Bell 1945:} {\it The Development of Mathematics},
McGraw-Hill, New-York.
{\bf J. Brook 1988:} The God of Isaac Newton, {\it in:} eds. J. Fauvel,
{\it et al., Let Newton Be!}, Oxford Univ. Press, pp. 166-183.
{\bf J. M. Borwein, P. B. Borwein, D. H. Bailey 1989:} Ramanujan,
modular equations, and approximations to {\bf Pi}, or How to compute one
billion digits of {\bf Pi}, {\it The Amer. Math. Monthly}, {\bf 96},
201-219.
{\bf Britannica (The New Encyclopedia) 1985}, Vol. {\bf 14}: Biblical
literature and its critical interpretation, Vol {\bf 22}: Judaism,
Chicago.
{\bf W. M. Feldman 1965:} {\it Rabbinical Mathematics and Astronomy},
Hermon Press, New-York.
{\bf R. J. Gillings 1972:} {\it Mathematics in the Time of the
Pharaohs}, The MIT Press, Cambridge.
{\bf G. Guitel 1975:} {\it Historie compar\'ee des num\'erations
\'ecrites}, Flammarion, Paris.
{\bf J. Hoyrup 1989:} The Mathematical Context of the Bible, Technical
Report N$^\circ$ 2, university Centre, {\it to appear in Anchor Bible
Dictionary}.
{\bf The Holy Scriptures}, Koren Publishers, Jerusalem, 1977.
{\bf A. Ya. Khintchine 1963:} {\it Continued Fractions}, P. Noordhoff,
Groningen.
{\bf F. Manuel 1974:} {\it The religion of Isaac Newton}, Clarendon
Press, Oxford.
{\bf Mishnah, The Artscroll Series, 1983:} {\it Seder Moed}, Vol. {\bf
1(b)}: {\it Eruvin}, Mesorah Publications, New-York.
{\bf M(ax) Munk, Rabbi 1962:} Three Geometry Problems in Tanach and
Talmud {\it (in Hebrew), SINAI} (Mossad Harav Kook) {\bf 51} (5722)
218-227.
{\bf M(ax) Munk, Rabbi 1968:} The Halachik Way for the Solution of
Special Geometry Problems {\it (in Hebrew), HADAROM} (Rabbinical Council
of America) {\bf 27} (5728) 115-133.
{\bf M(ichael) L. Munk, Rabbi 1983:} {\it The Wisdom in the Hebrew
Alphabet}, Mesorah, Brooklyn.
{\bf O. Neugebauer 1969:} {\it Vorlesungen \"uber Geschichte der antiken
mathematischen Wissenschaften}, Erster Band: {\it Vorgriechische
Mathematik}, Spring- er, Berlin.
{\bf A. S. Posamentier, N. Gordan 1984:} An astounding revelation on the
history of $\pi$, {\it The Mathematics Teacher} {\bf 77}, N$^\circ$ 1,
pp. 52, 47.
{\bf Ramban 1971:} {\it Commentary on the Torah, in five volumes}.
Translated by Rabbi Dr. C. B. Chavel, Shilo Publishing House, New-York.
{\bf H. Roiter 1993:} La mer d'airain du Roi Salomon et le nombre $\pi$
(PI), {\it Kountrass} {\bf 7}, N$^\circ$ 38, p. 10.
{\bf N. Scherman, Rabbi 1980:} Measurements from Sinai, an overview to
{\it Bircas HaChammah}, Mesorah Publications, New-York.
{\bf A. Steinsaltz 1976:} {\it The Essential Talmud}, Basic Books,
New-York.
{\bf R. S. Westfall 1987:} {\it Never at Rest}, Cambridge Univ. Press,
Cambridge.
{\bf Zuidhof 1982:} King Solomon's molten sea and ($\pi$), {\it Biblical
Archeologist}, Summer 1982, 179-184.
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