The above devaluation of mathematics and astronomy/calendar in m. Avot 3:18 and b. Hullin 95b are by no means the only examples of mathematics being demoted and
undermined in rabbinic literature. In this section, I will analyze m. Ohalot 12:6, b. Eruvin 13b-14a, b. Eruvin 76a, and b. Sukkah 7b, all of which demonstrate the exercise of deontic authority over empirical knowledge – in this case, within the domain of mathematics. In many ways, mathematics may be considered to be emblematic of the empirical.
Theorems, formulae and calculations are not subject to the vagaries of opinion; rather, outcomes are replicable again and again and there should be no room for interpretation of the numerical. As such, mathematics should be immune from the exercise of
authority.
However, as exemplified by the following rabbinic texts, even the calculation of π, the ratio of the circumference of a circle to its diameter, was subject to the exercise of deontic authority. As described in chapter one, while scholars have taken note of the rabbinic approximation of π in these texts, this is generally explained as being due either to the possibility that the rabbis were unfamiliar with the more accurate, and earlier calculations made by Babylonian, Egyptian, and Greek mathematicians, or that
approximation was sufficient for practical, halakhic use. However, the pattern present in the texts suggests a more deliberate practice, particularly when set against the broader context of the recurrent tendency toward the devaluation of mathematics and astronomy found in m. Avot 3:18 and b. Hullin 95b.
Example 1: m. Ohalot 12:6
םאו .הלוכ תחת האמוטה תא האיבמ.חפט חתופ הב שי םא.היתחת האמוטו לתוכל לתוכמ הנותנ איהש הרוק ש ןמזב.חפט חתופ הב אהיו הפקהב אהי המכו .תדרויו תעקובו .הלועו תעקוב האמוט.ואל הפקה.הלוגע איה
[Regarding] a beam placed across from one wall to another wall, and has impurity
underneath it, if it is one handbreadth wide, the impurity is transferred to everything beneath it. If it is not (that width) the impurity attaches up and down. How much does its
circumference need to be so that its width must be a handbreadth? In a time that it is round, the circumference must be three handbreadths. In a time that it is square, four
(handbreadths), since a square is greater (in circumference) by one quarter over a circle’s
This example is straightforward, with π being clearly described as equal to three, given its circumference of three with a width of one חפט if the beam is round.
Example 2: b. Eruvin 13b-14a Mishnah (b. Eruvin 13b)
שי םיחפט השלש ופיקיהב שיש לכ .תעבורמ איה וליאכ התוא ןיאור ־ הלוגע ,הטושפ איה וליאכ התוא ןיאור חפט בחור וב. (If) round it is seen as if it were square. . . Whatever has a circumference of three
handbreadths is one handbreadth in diameter. Gemara: ותפש דע ותפשמ המאב רשע קצומ םיה תא שעיו :)׳ז ׳א םיכלמ( ארק רמא ,ןנחוי יבר רמא ־ ?ילימ ינה אנמ תפש ותפש :אפפ בר רמא ־ ִותפש אכיא אהו .ביבס ותא בסי המאב םישלש וקו ותמוק המאב שמחו ביבס לגע ש חרפ סוכ השעמכ ותפשו חפט ויבעו )׳ז ׳א םיכלמ( ביתכד ,היב ביתכ ןשוש חרפ אכיאהו .ליכי תב םיפלא ןשו הוקמ םישמחו האמ קיזחמ היה המלש השעש םי :אייח יבר אינת .בישח אק יאוגמ ־ בישח אק יכ ־ ִוהשמ From where are these calculations deduced? R. Yohanan said: our scripture (1 Kings 7) stated: And he made the molten sea of ten cubits from lip to lip, round in all around, and its height was five cubits. And a line of thirty cubits encompassed it all around. But there was the thickness of its lip (brim). R. Papa said: Regarding its lip, it is
written in scripture (that it was like) the flower of a lily. As it is written: (1 Kings 7) And it was a handbreadth thick, and the lip of it was made like the brim of a cup,
like the flower of a lily. It contained two thousand baths. But was there yet at least a fraction? When (the circumference) was reckoned it was the inner circumference. A teaching of R. Hiyya: The sea that Solomon made contained (the volume of) one hundred and fifty ritual baths.156
156 The biblical source here is 1 Kings 7:23: הָמַאָב שֵׁמָחְו ,ביִבָס לֹגָע וֹתָפְש-דַע וֹתָפְשִמ הָמַאָב רֶשֶע :קָצוּמ ,םָיַה-תֶא שַעַיַו
ביִבָס וֹתֹא בֹסָי ,הָמַאָב םיִשׁלְֹשׁ )וָקְו( הוקו ,וֹתָמוֹק. “And he made a molten sea of ten cubits from lip to lip, round in all around, and its height was five cubits. And a line of thirty cubits encompassed it all around.”
The Mishnah here in 13b resembles that of its counterpart in m. Ohalot 12:6. Of greater interest is the interpretation of the Gemara, in which 1 Kings 7:23 is cited as the source text for a calculation that the text itself suggests is not sufficiently accurate. ?ילימ ינה אנמ (“From where are these calculations deduced?”) is met with the biblical citation,
suggesting that here too, deontic authority provided the format for this discussion, with the focus being the biblical source text. The discussion of the brim, or lip, is aptly
highlighted by a peek at the mathematical awareness behind the biblical exegesis when R. Papa refers to the inner circumference.
We note similar discussions in b. Eruvin 76a and b. Sukkah 7b – the former addressing the circumference of a round window and the latter, the validity of a round sukkah:
Example 3: b. Eruvin 76a רמא יבר ןנחוי : ןולח לוגע ךירצ אהיש ופקיהב םירשע העבראו םיחפט , םינשו והשמו ןהמ ךותב הרשע , םאש ונעברי אצמנ והשמ ךותב הרשע . ־ ידכמ לכ שיש ופקיהב השלש םיחפט ־ שי וב ובחורב חפט
R. Yohanan said: A round window needs to have a circumference of twenty-four handbreadths. Two and a fraction of which must be within ten (handbreadths from the ground), so that when it is squared, a fraction remains within the ten (handbreadths from the ground.) Consider that any item with a circumference of three handbreadths is
approximately a handbreadth in diameter.
Example 4: b. Sukkah 7b רמא יבר ןנחוי : הכוס היושעה ןשבככ , םא שי הפיקהב ידכ בשיל הב םירשע העבראו ינב םדא ־ הרשכ , םאו ואל ־ הלוספ . ןאמכ ־ יברכ , רמאד : לכ וס ארבג ,ידכמ .הלוספ ־ תומא עברא לע תומא עברא הב ןיאש הכ חפט בחור וב שי םיחפט השלש ופיקהב שיש לכ ,ביתי אתמאב R. Yohanan said: A sukkah that was like a furnace (round), if twenty-four
men can be seated around its circumference, it is kosher, otherwise it is invalid. According to whom? According to Rabbi who says that any sukkah that is not
four cubits square is invalid. But consider: A man is in the space of one cubit.
Wherever there is a circumference (of a circle) that is three handbreadths, its diameter is one handbreadth.
—
Ilana Wartenberg is by no means alone in her opinion that the rabbis may have known a more accurate value for π, but that “they may have considered 3 good enough for Talmudic ‘working purposes.’” (Wartenberg, 1212) Indeed, this reading was
expressed as early as the 1878, by Benedict Zuckerman, mentioned earlier, who assumed that the rabbis in the Bavli knew the correct calculations. (Zuckerman, 23) We also note this viewpoint expressed by W.M. Feldman, Tsaban and Garber, as well as Noah Efron, who comments on talmudic discussions of mathematics serving practical purposes such as calculating the dimensions of a sukkah, calendrics, or calculating the volume capacity of a mikvah. According to Efron, these needs “rarely demanded great accuracy and, as a result, many of the conclusions reached by the rabbis were inexact.” (Efron, 45) Turning to π as presented in b. Eruvin 14a, above, Efron underscores the following about the rabbis’ rough estimate of 3, terming the opening question, ?ילימ ינה אנמ “blithe” in its hint of disingenuousness:
Valuing the circumference of a circle at three times its diameter is less accurate than the figures calculated two millennia earlier by Babylonian (who arrived at figure of 3 1/8) and Egyptian mathematicians (who reached the formula 8 divided 9 [8/9] squared multiplied by 4, or the equivalent of today’s 3.1605). And of course it is less accurate than the figures arrived at for π by Archimedes (less than 3 1/7 but greater than 3 10/71, which by today’s notation would translate to 3.1429) centuries before. The rabbis of the Gemara ask, “Whence are these calculations deduced?” (a blithe question that led commentators reasonably to infer that these rabbis knew the number was off, and were essentially asking, “Why were these inaccurate calculations deemed acceptable?”). They answer that the calculations are based on a biblical passage in 1 Kings 7:23 that describes the “Sea of Solomon” as ten cubits across and thirty around.” (Efron, 46)
Here, the rabbis appear to be questioning their own calculation, hinting at this awareness in their discussion of their figure for π.157 As such, where it might have been possible to brook a slightly less extreme approximation of π in the rabbinic texts, the gross
approximation of 3 calls out for additional scrutiny.
Once again, in keeping with the matter of transmission and significant historical and textual evidence of the rabbinic knowledge of mathematics, we are presented with hints at awareness, in the Bavli, of more accurate calculations, even if they were not brought into the rabbinic calculations in the text for all to see. As such, it would appear that the argument for lack of knowledge cannot hold. Moreover, I must reiterate my point that the argument regarding the need for mere halakhic utility with respect to π is undercut by the devaluation of mathematics and astronomy found in the rabbinic texts m. Avot 3:18 and b. Hullin 95b, when set against other halakhic topics deemed central. At the very least, we note a possible motive for the ways in which we see mathematics used and recorded in these rabbinic texts – an argument that respects the intellect of the rabbis while also bracketing their apparent demotion of these sciences for our closer examination.
The apparent obfuscation in the texts, then, suggests not mere mathematical utility or calculations sufficient for halakhic uses, but a reasoned decision to record and use
157 As noted in chapter one, Efron, Feldman, Tsaban and Garber, and Deakin and Lausch all point out that
the Babylonians, and even the Egyptians, possessed more accurate estimations of Pi than the rabbis appear to reveal hundreds of years later.
gross approximations.158 When viewed in concert with the evidence presented above for the transmission and rabbinic awareness of more accurate values and calculations, the context of the rabbinic devaluation of mathematics and astronomy as described in m. Avot 3:18 and b. Hullin 95b, and the priority placed upon Torah and halakhic truth, the deliberate lack of precision in the rabbinic usage of mathematics is strongly underscored.