On The Rabbinical Exegesis of an Enhanced Biblical Value of $\pi$

Shlomo Edward G. Belaga
C. N. R. S., Université Louis Pasteur
7, rue René Descartes

AMS 1980 Mathematics Subject Classification: 01 A 15, 01 A 17
Key words: history of Pi, Rabbinical exegesis.


We present here a biblical exegesis of the value of $\pi$, $\pi_{\rm
 Hebrew}=3.141509\dots$, from the well known verse 1 Kings 7:23. This verse is then compared to 2 Chronicles 4:2; the comparison provides independent supporting evidence for the exegesis.[*]

1.The Hebrew Bible often speaks the language of numbers and measurements [Feldman 1965]; the Western tradition rarely[*], if at all [Hoyrup 1989], understands this language, and the case of the Biblical value of $\pi$ could be seen as both a remarkable exception of this rule and its striking confirmation.

As a recent publication in The American Mathematical Monthly puts 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, 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, 1 Kings 7:23, giving the dimensions of a tank in the First Temple[*]:

``And he made a molten sea [tank], ten cubits from the one brim to the other: it was round all about, and its height was five cubits: and a 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[*], or do not want[*] [*] , to hide (or are even happy for some ideological reasons, to emphasize[*] ) 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\gt\pi - \pi_{\rm Babylon}\gt.016;\end{displaymath}

\pi_{\rm Egypt}=3\frac{13}{81},\quad
 0.019\gt\pi_{\rm Egypt}-\pi \gt.018\end{displaymath}

Thus, it seems both appropriate and interesting at this point to give a Rabbinical interpretation of the above verse and the way the number $\pi$, implicitly[*] defined in this verse, has to be computed [Max Munk 1962 , 1968] (see also two popular and slightly different accounts in [Posamentier, Gordan 1984] and [Roiter 1993]). We do not claim, however, that the Rabbinical folklore has preserved either the mathematical method which was used in this approximation of $\pi$, or its historical origins: all that was left to us is an extremely natural and concise 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 $\pi_{\rm Egypt}$: `` 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 per se as deserving to be developed, preserved, and disseminated in the written form, as they were not interested (with the Jewish Temple being a notable exception) in creating numerous and splendid monuments of their religion and culture.

2. The key to an alternative reading of the verse 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 differently write (spell) and read some words of the Bible; the reading version is usually regarded as a correct one (in particular, it is always correct from the point of view of the Hebrew grammar, and this is why it could be easily either remembered or reconstructed from the written version), whereas the written version slightly deviates from the correct spelling. (Another approach, involving the comparison between written forms of the same words in 1 Kings 7:23 and Chronicles 4:2 is cited in [Posamentiern, Gordan 1984][*]; see more about this version of the exegesis in 4).

Such a disparity is a common feature for all Books of the Hebrew Bible; and in any such case there exists (or existed: some of this knowledge was definitely lost) a Rabbinical folklore (in fact, strict Rabbinical hermeneutical rules [Steinsaltz 1976, part three: Method], [Britannica 1985], [Banon 1987]) of interpretation of the difference in question.

In our case there is such a disparity for the word ``line'': in Hebrew, it is written as ``QVH (Qof, Vav, Hea)'', but it has to be read as ``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 approximate character of the given length of the line circling around the ``sea''(tank), -- a much more accurate approximation to $\pi$, $\pi_{\rm Hebrew}$, is hidden in the 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 [*] . 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 gematria [Michael Munk 1983, p. 163], [Britannica 1985]. Here are the standard numerical equivalents of the letters of the Hebrew alphabet:

Aleph=1, Beth=2, Gimel=3, Daled=4, Hea=5, Vav=6, Zain=7, CHet=8, Tet=9,
Yod=10, Caf=20, Lammed=30, Mem=40, Noon=50, Samech=60, Aiin=70, Pea=80, TSadik=90,
Qof=100, Reish=200, Shin=300, Tav=400.

In particular, the numerical equivalent of the written version ,``QVH'', is Qof+Vav+Hea=100+6+5=111, whereas the numerical equivalent of the reading version, ``QV'', is Qof+Vav=106.

Using these numerical equivalents, one defines $\pi_{\rm Hebrew}$ as follows:

\pi_{\rm Hebrew}=\pi_0\times\frac{the\ numerical\ equivalent...
 ...sion}{the\ numerical\ equivalent\ of\ the\ reading\ version}



\pi=3.1415926\ldots,\quad \pi_{\rm Hebrew}=3.1415094\ldots,
 \quad \vert\pi_{\rm Hebrew}-\pi\vert<0.000084\ .\end{displaymath}

3. Quantitatively, 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

and here are the convergents (see, e.g., [Khintchine 1963]) corresponding to the first five sections of $\pi$ :

=3;\quad [3;7]=3\frac{1}{7};\quad [3;7,15]=3\frac{15}{106};\end{displaymath}

=3\frac{16}{113};\quad [3;7,15,1,292]=3\frac{4687}{33102}\end{displaymath}

One immediately observes that, firstly, $\pi_{\rm Hebrew}=[3;7,15]$, and, secondly, $\pi_{\rm Hebrew}$ is the second (after $\pi_1=[3;7,15,1]$) best convergent with a denominator under 30,000 ! Notice also that the preceding convergent, [3:7]=22/7, was known to ancient Greeks.

4. It is worthwhile to mention here a remarkable fact, namely, that in the case of the verse 1 Kings 7:23 we have an independent confirmation of the above mentioned written vs. reading disparity.

Namely, it could be easily seen that the verse 2 Chronicles 4:2 of the Hebrew Bible repeats 1 Kings 7:23 almost verbatim [Holy Scriptures, p. 988]. Looking at the Hebrew text, one immediately observes that the Hebrew word translated as line is traditionally spelled (written) here identically to its reading version. Thus, even if somebody would rebuff as irrelevant the problem of interpretation of the disparity written vs. reading version of the word line in 1 Kings 7:23 (because he does not trust the oral tradition of transmission of Biblical texts), he would still have to explain the disparity between two different 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 be this important hint to the enhanced value of $\pi$ 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 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.

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 Jeremiah) actually knew the value $\pi_{\rm Hebrew}$ and intentionally exploited the aforementioned 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 rôle as an approximation which was regarded (and 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, was it conceived by its author) as the [only] textual basis for the following legal definition of $\pi$ : ``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 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, Rambam[*] comments:``...the [exact] ratio of the diameter of a circle to its circumference cannot be known [is irrational]...but it is possible to approximate it...and the approximation used by scientists [Greeks and Arabs] is the ratio of one to three and one seventh... Since it is impossible to arrive at a perfectly accurate ratio, ... 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, 1 Kings 7:23), one level for legal purposes, and another one for ``connoisseurs'', is not only a typical phenomenon in the Rabbinical tradition -- in a sense -- such a multi-level approach to texts is the main methodological legacy of this tradition [Steinsaltz 1976, Part Three: Method], [Banon 87]. As Rabbi Moshe ben Nachman[*] writes: ``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...'' [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 elaborate institutions for such a transmission[*] . 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.

6. With all this understanding, gained thus far, we are, as yet, unable to elucidate the way the exegesis of the verse 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 a priori difficulty in answering these and similiar questions is related to unpleasant two thousand years old legacy of Judaism: as a religion, it invariably remained during this period an underdog, prone to persecutions 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 Rambam, or any other Jewish scholar who lived two hundred years ago, or more, to advance a better approximation of $\pi$, without being able (as we now are) to confirm this value scientifically.

This fundamental 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. cm 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 1 Kings 7:23 from Rabbi Haim Roth, of 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 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 ``the ancient Hebrew regarded $\pi$ as being equal to 3'', citing, of course, the verse 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 Zürich, were most inspiring. Rabbi Dr. Henri Biberfeld, Rabbis Daniel Mund and Arye Posen, of Montréal, suggested several important Talmudic and Halachic sources. Rabbi Dr. Nachum L. Rabinovich, of Maaleh Adumim, read the paper and suggested an important correction. Discussions with Prof. Louis Charbonneau, of Montréal, and his colleagues were helpful in adjusting the presentation to the 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 Carleton, to whom belongs the reference [Zuidhof 1982]. Monsieur Luc Gagnon, the student of Prof. Jacques Lefebvres, Montréal, supplied the reference [Posamentier, Gordan 1984]. Several manifestations of utmost disbelief (in few cases, bordering on ridicule[*]), on the part of colleagues with, apparently, no previous exposure to Jewish studies, helped the author to contain excitement and avoid self-congratulation.

Finally, and miraculously, Prof. Edward Reingold, of Urbana, whose enthusiasm for the subject was most encouraging, introduced the author to Rabbi Dr. Zeharia Dor-Shav, of 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 Montréal, Prof. Reingold has found the articles in 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.


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.

G. Almkvist, B. Bernd 1988: Gauss, Landen, Ramanujan, the Arith- metic -Geometric Mean, Ellipses, $\pi$, and the Ladies Diary, The Amer. Math. Monthly, 95, 585-608.

D. Banon 1987: La lecture infinie: Les voies de l'interprétation midra- chique, Éditions du Seuil, Paris.

P. Beckmann 1971: A History of $\pi$ (Pi), The Golem Press, Boulder.

E. T. Bell 1945: The Development of Mathematics, McGraw-Hill, New-York.

J. Brook 1988: The God of Isaac Newton, in: eds. J. Fauvel, et al., Let Newton Be!, Oxford Univ. Press, pp. 166-183.

J. M. Borwein, P. B. Borwein, D. H. Bailey 1989: Ramanujan, modular equations, and approximations to Pi, or How to compute one billion digits of Pi, The Amer. Math. Monthly, 96, 201-219.

Britannica (The New Encyclopedia) 1985, Vol. 14: Biblical literature and its critical interpretation, Vol 22: Judaism, Chicago.

W. M. Feldman 1965: Rabbinical Mathematics and Astronomy, Hermon Press, New-York.

R. J. Gillings 1972: Mathematics in the Time of the Pharaohs, The MIT Press, Cambridge.

G. Guitel 1975: Historie comparée des numérations écrites, Flammarion, Paris.

J. Hoyrup 1989: The Mathematical Context of the Bible, Technical Report N$^\circ$ 2, university Centre, to appear in Anchor Bible Dictionary.

The Holy Scriptures, Koren Publishers, Jerusalem, 1977.

A. Ya. Khintchine 1963: Continued Fractions, P. Noordhoff, Groningen.

F. Manuel 1974: The religion of Isaac Newton, Clarendon Press, Oxford.

Mishnah, The Artscroll Series, 1983: Seder Moed, Vol. 1(b): Eruvin, Mesorah Publications, New-York.

M(ax) Munk, Rabbi 1962: Three Geometry Problems in Tanach and Talmud (in Hebrew), SINAI (Mossad Harav Kook) 51 (5722) 218-227.

M(ax) Munk, Rabbi 1968: The Halachik Way for the Solution of Special Geometry Problems (in Hebrew), HADAROM (Rabbinical Council of America) 27 (5728) 115-133.

M(ichael) L. Munk, Rabbi 1983: The Wisdom in the Hebrew Alphabet, Mesorah, Brooklyn.

O. Neugebauer 1969: Vorlesungen über Geschichte der antiken mathematischen Wissenschaften, Erster Band: Vorgriechische Mathematik, Spring- er, Berlin.

A. S. Posamentier, N. Gordan 1984: An astounding revelation on the history of $\pi$, The Mathematics Teacher 77, N$^\circ$ 1, pp. 52, 47.

Ramban 1971: Commentary on the Torah, in five volumes. Translated by Rabbi Dr. C. B. Chavel, Shilo Publishing House, New-York.

H. Roiter 1993: La mer d'airain du Roi Salomon et le nombre $\pi$ (PI), Kountrass 7, N$^\circ$ 38, p. 10.

N. Scherman, Rabbi 1980: Measurements from Sinai, an overview to Bircas HaChammah, Mesorah Publications, New-York.

A. Steinsaltz 1976: The Essential Talmud, Basic Books, New-York.

R. S. Westfall 1987: Never at Rest, Cambridge Univ. Press, Cambridge.

Zuidhof 1982: King Solomon's molten sea and ($\pi$), Biblical Archeologist, Summer 1982, 179-184. cm

\vbox{\hrule height1.5pt\hbox{\vrule width1.5pt\hskip0.1cm
 ...pt}\vskip0.1cm}\hskip0.1cm\vrule width1.5pt}\hrule height1.5pt}\end{displaymath}


Robert Israel