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Sukah 8


QUESTION: After analyzing the statement of Rebbi Yochanan, who said that a Sukah built in the shape of a circle must be large enough to seat 24 people around its circumference, the Gemara mentions the geometrical theorem of the Rabbis of Kesari. The Rabbis of Kesari said that the circumference of a circle inscribed inside of a square is 25% less than the square's perimeter, and the circumference of a circle circumscribed around the outside of a square is 50% more than the square's perimeter. Accordingly, the circumference of the circle drawn around the 16-Tefach perimeter of a square is 50% larger, or 24 (that is, take 50% of 16 and add it to 16).

The Gemara concludes (8b) that this theorem is incorrect, as one can see. We know that the actual relationship of the perimeter of an inscribed square to the circle around it, according to Chazal, is 3 * (1.4 * s), where 3 is used for pi (Eruvin 13a) and "s" equals the length of a side of the square. (The relationship between the side of a square and its diagonal -- which is also the diameter of the circumscribed circle -- is 1:1.4, according to Chazal). If so, the circumference of a circle circumscribed around a square with sides of 4 Tefachim is 3(1.4 * 4), or 16.8 -- and not 24!

How did the Rabbis of Kesari make such a mistake?


(a) TOSFOS (8b, DH Rivu'a; Eruvin 76b, DH v'Rebbi Yochanan) suggests that the Rabbis of Kesari were not giving the relationship of the *perimeter* of the inner square to the *circle* around it. Rather, they were giving the relationship of the *area* of the inner square to an *outer square* that is drawn around the circle which encloses the inner square. This is what they meant by saying that "when a circle is drawn around the outside of a square, the outer one's (i.e., the outer *square's*) perimeter is 50% larger than the inner one's." (See the second picture printed in Tosfos.)

According to Tosfos, Rebbi Yochanan (both here and in Eruvin 86a) misunderstood the Rabbis of Kesari.

(b) The Gemara comments that we can see that the circle around a square is not as large as the Rabbis of Kesari posit. Based on the comments of Rashi elsewhere, though, we might suggest that Gemara is commenting only about the mathematical correctness of their statement; however, when considering the actual Halachic applications, we do take into account their formula. In fact, we find in Eruvin (76a) that Rashi seems to have no difficulty with the statements of the Rabbis of Kesari and Rebbi Yochanan. Perhaps Rashi held that the Rabbis of Kesari were proposing a Halachic stringency: when determining a value (such as the circumference of a circle) by using the diagonal of a square for the purpose of a practical application in Halachah, we consider the diagonal to be equal to the sum of the two sides of the square or rectangle between the ends of the diagonal (since the lines of those two sides go from one end of the diagonal to the other). The reason for this is to prevent people from confusing the diagonal and the sum of two sides. In addition, physical reality does not permit for the application of puristic mathematics (for one reason, the actual diagonal of a square is the length of the side times the square root of two, which is an irrational number; second, it is not possible to draw a perfectly exact line or angle in the physical reality), and therefore the figure given as the diagonal of a square for purposes of determining Halachic applications (such as the size of a circular Sukah around that square) must take into consideration the largest possible diagonal of the right angle, which is the sum of the two sides. (Thus, if the sides of inscribed square are each 4 Tefachim, then the diagonal is viewed to be *8* Tefachim. The circle around that square, then, must have a diameter of 8 Tefachim, which means that its circumference must be *24* Tefachim, and not 16.8 which is what it would be based on the *actual* diameter of the square.)

It could be that Rashi is consistent with his opinion elsewhere (Shabbos 85a, Eruvin 5a, 78a, 94b), where Rashi seems to count the diagonal of a rectangle as the sum of the two sides between the two ends of the diagonal. Rashi may hold that such a Halachic definition is applied and may be relied upon entirely, both as a leniency and a stringency, with regard to Rabbinic rulings. (M. Kornfeld)

(c) Perhaps it is possible to propose an entirely new explanation for the statement of the Rabbis of Kesari. The Rabbis of Kesari and Rebbi Yochanan are perfectly correct. Perhaps Rebbi Yochanan's statement that "the circumference of the Sukah must be large enough to seat 24 people in it" does not mean that the *circumference* must be 24 Amos, but that there must be 24 Amos *inside* the circumference -- in other words, the *area* of the circle must be 24 square Amos!

The area of a circle that is drawn around a square which is 4 by 4 is calculated by multiplying pi by the radius squared. The radius of the circle around a square which is 4 by 4 is half of the diagonal (5.6), which is 2.8. Let us use the Halachic estimate of pi=3. Then: 3 * (2.8)(2.8) = 23.52, or ~24.

This is what Rebbi Yochanan meant when he said that the circle must have within its circumference an area of 24 (he rounded up to 24 as a Chumra)! (According to this explanation, we may accept the Ritva's suggestion that the words "v'Lo Hi..." do not belong in the Gemara and were added mistakenly by the Rabanan Savora'i.) (M. Kornfeld)

(David Garber and Boaz Tzaban of Bar Ilan University, who have been printing articles on geometric themes from Chazal for a number of years, pointed out to me that the ME'IRI in Eruvin 76 suggests this solution for the Rebbi Yochanan's statment there, citing it from the Ba'al ha'Me'or. It can be traced further back to a responsum of the RIF in Temim De'im #223. An Acharon, Teshuvos GALYA MASECHES #3, offers this solution as well. Using the mathematics of Chazal to project the area of the circle based on the area of another square that is drawn *around* it (3:4 -- note that the outer square is exactly double the square drawn *inside* of the circle in both perimeter *and* area), the solution for the area of the circle is *exactly* 24 Tefachim, and not just approximately, as I concluded using the equation of pi*r*r. The Me'iri uses the word "Shibur" or "Tishbores" to refer to the calculation of area.)


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