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Daf Ditty Eruvin 14: Pi, the Ratio of God “Human beings, vegetables, or cosmic dust, we all dance to a mysterious tune intoned in the distance by an invisible player.” – Albert Einstein “The world isn't just the way it is. It is how we understand it, no? And in understanding something, we bring something to it, no? Doesn't that make life a story?” "Love is hard to believe, ask any lover. Life is hard to believe, ask any scientist. God is hard to believe, ask any believer... be excessively reasonable and you risk throwing out the universe with the bathwater." Yann Martel, Life of Pi 1 The mishna continues: If the cross beam is round, one considers it as though it were square. The Gemara asks: Why do I need this clause as well? Similar cases were already taught in the mishna. The Gemara answers: It was necessary to teach the last clause of this section, i.e., the principle that any circle with a circumference of three handbreadths is a handbreadth in diameter. 2 The Gemara asks: From where are these matters, this ratio between circumference and diameter, derived? Rabbi Yoḥanan said that the verse said with regard to King Solomon: And he made the molten sea of ten cubits from 23 גכ את ַַַויּﬠֶשׂ - ָוּ,מצםיַּה ָ:ק ֶֶﬠשׂ ר אָ בּ מַּ הָ הָ מַּ אָ בּ ר brim to brim, round in compass, and the height thereof תוֹפדשּׂﬠ ְִָמַ - ﬠוֹתְשׂפ סלָגָ ִ,יבָבֹ בּשְׁמחו ַהמֵָּאָ ָ ַהמֵָּאָ בּשְׁמחו ִ,יבָבֹ סלָגָ ﬠוֹתְשׂפ was five cubits; and a line of thirty cubits did compass וֹקמ ,וֹתָ וקוה ו( קְ ָ )ו םיִשׁhְ שׁ אָ בּ מַּ ,הָ סָ י בֹ וֹתֹ א וֹתֹ א בֹ סָ י ,הָ מַּ אָ בּ םיִשׁhְ שׁ )ו ָ קְ ו( וקוה ,וֹתָ וֹקמ .it round about ִי.בָסב I Kings 7:23 “And he made a molten sea, 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 roundabout” The Gemara asks: But isn’t there its brim that must be taken into account? The diameter of the sea was measured from the inside, and if its circumference was measured from the outside, this ratio is no longer accurate. 3 Rav Pappa said: With regard to its brim, it is written that the brim is as the petals of a lily, as stated in the verse: And it was a hand-breadth thick; and the brim thereof was 26 וכ ַכּמֲﬠֵשׂה ְַשׂפתוְָּשׂפתוְֹ ַטפח, ְוְָﬠיבוֶֹ - wrought like the brim of a cup, like the flower of a lily; it held כּ סוֹפּ רֶ וֹחַשׁ ָשׁ א;ן לַ פְּ ַ בּםִ י ,תַ,ַבִּיַפּ ַאןָ חשׁר וֹ .two thousand baths ִיָיל.כ }פ{ I Kings 7:26 “And it was a handbreadth thick; and its brim was wrought as the brim of a cup, as the petals of a lily; it contained two thousand bat” i.e. The brim was very thin. The Gemara now calculates how many ritual baths should have been contained in Solomon’s Sea. The volume of the sea was five hundred cubic cubits, as it was ten cubits in length, ten cubits in width, and five cubits in height. The minimum volume of a ritual bath is three cubic cubits. Therefore, three hundred cubic cubits is the volume of a hundred ritual baths, and one hundred and fifty cubic cubits is the volume of another fifty ritual baths. Consequently, four hundred and fifty cubic cubits are enough to contain a hundred and fifty ritual baths; but the volume of the sea was five hundred. The Gemara answers that there is an error in the calculation: These calculations with regard to the volume of the sea would apply to a square, but the sea fashioned by Solomon was round, and its volume was therefore smaller. The Gemara continues to ask: Now, how much larger is a square of ten-by-ten cubits than a circle with a diameter of ten cubits? A quarter. 4 Consequently, four hundred cubic cubits of our original calculation must be reduced to three hundred, which is the volume of one hundred ritual baths; and the remaining hundred cubits must be reduced to seventy-five, which is the volume of twenty-five ritual baths. According to this calculation, Solomon’s Sea was the size of only one hundred and twenty- five ritual baths, not one hundred and fifty as stated above. Tosafos: תופותס ה"ד אכיאהו והשמ (Tosafos infers that the calculation is precise, but challenges this.) משמע בשחהש ו ן צמ ו םצמ Inference: The calculation (the circumference is three times the diameter) is precise. 5 ןכו 'פב ק' 'בד ב' 'ד( די: ) יבג ינש םיחפט ורייתשנש וראב ן םשש רפס רוהת חנומ איהש איהש חנומ רוהת רפס םשש ן וראב ורייתשנש םיחפט ינש יבג ) די: 'ד( ב' 'בד ק' 'פב ןכו הפקיהב השש םיחפט ךירפו יכ ו ן ותיעצמאלד נ ללג שיפנ היל נשמ י םיחפט Support: Also, in Bava Basra (14b) regarding two Tefachim that remained in the Aron, in which the Sefer Torah rested, for its circumference was six Tefachim, the Gemara asks "since it was rolled to its middle, it is more than two Tefachim!" (This implies that when all is on one roller, it is precisely two Tefachim.) ןכו רתב יכה ינשמד רפסב הרזעד ותלחתל לגלנ תךיכרפו ריא ית ירתב יכיה בימשתי עמ עמ בימשתי יכיה ירתב ית ריא תךיכרפו לגלנ ותלחתל הרזעד רפסב ינשמד יכה רתב ןכו מדצ ו םצמ ל ג רמ י Also after this [there], it answers that Sefer Azarah is rolled to its beginning, and asks "still, how can two [Tefachim] fit in two [Tefachim]?!" (When taking it in or out, it scrapes the sides. Parchment can be compressed somewhat, but it is dishonorable to do so to a Sefer Torah. Also, perhaps letters will be rubbed out!) איקוש יאד ן ובשחה ן מ קדודק יפל ימכח :ודתמה Question: The calculation is not precise, according to mathematicians! (Tosfos HaRosh explained that the Gemara asked what is the source to rely on this approximation, even though it is not precise.) Tosafos points out that the ratio of the diameter to the circumference of a circle of 3:1 is not accurate. Tosafos leaves this unresolved.1 Rambam, in his Commentary to the Mishnah, states that the ratio of diameter to circumference is a number which— בוריקב אלא םלועל י גשו אלש —known to be “three and a seventh plus a bit more will never be precise, only approximated.” [This is a clear indication that Rambam, and the Gemara, knew of pi. Even the approximation he gives of 3 1/7 is less than 13 ten thousandths more than pi.] Rambam concludes that because the ratio of pi is an irrational number, and at some point, we have no choice other than to approximate, the Gemara regularly uses the round number of 3:1. This is admittedly a wider approximation than 3 1/7, but the sages deliberately chose to be lenient and not burden the community with measurements of fractions. The Rosh wonders why the Gemara probes to find the scriptural source for the ratio of 3:1, when this is an empirical fact which can be observed by everyone who does the measurement. He answers that we know that the ratio is not exactly three to one, and that it is slightly greater. The question of the Gemara, therefore, is how do we know that we are allowed to calculate halachic guidelines based upon a number which is approximate? 1 Daf Digest 6 In other words, we are technically allowed to use a round beam to fix a mavoi even if its circumference is exactly three tefachim, as the Mishnah states. Yet, the measurement of three is actually too small to yield a diameter of a full tefach and using this round beam will lead to having a korah-beam which is too small. How do we know, nevertheless, that we are allowed to rely upon this ratio? The Gemara answers that the halacha recognizes this approximation and considers measurements based upon it to be valid, based upon a verse found in Melachim 2. It seems that according to the Rosh, we can rely upon this leniency not only in regard to rabbinic injunctions, but also in reference to Torah laws. 2 Circles and Hexagons The Gemara relates that a circle that has a circumference of three will have a diameter of one. Based on this ratio, a koreh – crossbeam — that is three tefachim in circumference is valid since it will have a diameter of one tefach. Tosafos notes that according to mathematicians this ratio is not accurate and the diameter of a circle with a circumference of three will be slightly less than one. Interestingly, Tosafos HaRosh2 explains that the reason the Gemara asks “from where we know these words,” is specifically because the ratio is not precise. If the ratio was exactly three to one there would be no reason for the Gemara to ask for a source. The Gemara’s inquiry was for the source that indicates that even though the ratio is not precise it is still reliable in Halachah. Later authorities question the extent to which one may rely upon this ratio. Mishnah Berura writes that one may certainly rely on this ratio when it comes to Rabbinic matters. Regarding Biblical matters he writes that it may be that this ratio is Halachah L’Moshe M’Sinai and may be relied upon even for Biblical matters.
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