Tosfos opening words indicate that the Kashya from Pi applies only to the Gemara, and not to the Mishnah. The Mishnah does not bother with the extra seventh and prefers to give the round figure of three. The Tosfos Yom-Tov on the Mishnah, cites the Rambam who says this. The Rambam queries this on the grounds that taking Pi as three (instead of three and a seventh) creates a leniency, since the diameter will then be less than a Tefach (and the Gemara in Sukah poses a similar Kashya on the case there). He answers that since the Din of Koreh is purely mi'de'Rabbanan, the Tana is not perturbed by that.

As Tosfos writes, the problem lies with the Gemara, which queries the 3-1 theory from the Mashehu thickness of the walls of the Yam shel Sh'lomoh (see also Rashash).

The Bi'urei ha'Daf (at the side of the Shas) cites the Sha'ar ha'Tziyun who rules that one can rely on the statement of our Mishnah even in matters which are d'Araysa, such as a round Sukah. He also cites the Chazon Ish, who explains that Halachah le'Moshe mi'Sinai (of 3-1, and other similar Halachos) was given in this way (in full knowledge and in spite of Pi). It was given as an approximation, with the intention that Chazal present the Shi'ur of the diameter as three (and not as 3.14), both le'Kula and le'Chumra.

be'Virchas Kol Tuv

Eliezer Chrysler

The following is copied from the Kollel's Insights to the Daf, Eruvin 14:2

QUESTION: The Gemara quotes the Mishnah (end of 13b) which says that the circumference of a circle is three times greater than its diameter (if the circumference of the circle is 3, then the diameter is 1). How do we reconcile this statement with the mathematical fact that the ratio of the circumference of a circle to its diameter is slightly more than three (pi = 3.14159...)?

ANSWERS:

(a) The TOSFOS HA'ROSH explains that the Gemara itself addresses this issue. The Gemara asks "from where do we learn" that the circumference of a circle is three times greater than the diameter. Why does the Gemara need a source to teach the ratio of the circumference of a circle to its diameter? We do not need a verse to teach us a mathematical, observable fact! It must be that the Gemara is asking for the source that teaches us that we may use a slightly inexact value to determine the circumference of a circle. The Gemara answers that the verse that describes the circumference of the pool that Shlomo ha'Melech constructed (the "Yam Shel Shlomo") as three times its diameter (Melachim I 7:23) teaches that for all Halachic purposes we may use the approximate ratio of three to one.

Similarly, the RAMBAM (Perush ha'Mishnayos; see also Hilchos Tum'as Mes 12:7) points out that pi is an irrational number, and "the exact relationship of the diameter to its circumference cannot be known and it is not possible to speak of it... its actual value cannot be perceived." He writes that the value which is commonly used in calculations is 3 1/7 (3.142857...). The Tana'im of the Mishnah rounded this number and expressed it in terms of the nearest whole integer (3).

(c) A fascinating insight regarding the value of pi is attributed to the Vilna Ga'on. (Actually, there is no source to substantiate the claim that the Vilna Ga'on said it. The actual source for the insight may be credited to Matityahu ha'Kohen Munk (Frankfurt-London), who published the thought in the journals "Sinai," Tamuz 1962, and "ha'Darom," 1967.) In the verse that the Gemara cites as the source for the ratio of the circumference to the diameter (Melachim I 7:23), there is a "Kri" and a "Kesiv" -- a word that is pronounced differently than it is spelled. The word in the verse is written "v'Kaveh" (with the letter "Heh" at the end), but it is pronounced "v'Kav" (with no "Heh" at the end). The Gematriya of the word "Kav" is 106, and the Gematriya of the word "Kaveh" is 111. The ratio of the Kesiv (111) to the Kri (106), or 111/106, is 1.0471698. This value represents the ratio of the value for pi to 3 (3.1415094/3 = 1.0471698).

Hence, the difference between the actual value of pi and its practical, Halachic value is expressed by the difference between the Kesiv (the actual, but unread word) and the Kri (the word as we read it) of the verse that discusses the value of Pi!