1) THE MATHEMATICAL FORMULAE OF THE JUDGES OF CAESAREA
QUESTION: The Gemara analyzes the statement of Rebbi Yochanan, who says that a circular Sukah must be large enough to seat 24 people around its circumference. The Gemara then mentions the geometrical theorem of the Judges (or "Rabbis") of Kesari.
The Judges of Kesari taught that the circumference of a circle inscribed within a square is 25% smaller than the square's perimeter, and the circumference of a circle circumscribed around the outside of a square is 50% larger than the square's perimeter. Accordingly, the circumference of the circle drawn around the 16-Tefach perimeter of a square is 50% larger than the square's perimeter, or 24 (50% of 16 added to 16 is 24).
The Gemara (8b) concludes that this theorem is incorrect, as empirical observation demonstrates. According to the formula used by the Chachamim (see Insights to Eruvin 14:2), the actual relationship of the perimeter of an inscribed square to the circle around it is 3 X (1.4 X s), where 3 = the value of pi, and s = the length of a side of the square. The ratio that the Chachamim use for the relationship between the side of a square and its diagonal (which is also the diameter of the circumscribed circle) is 1:1.4. Therefore, the circumference of a circle circumscribed around a square with sides of 4 Tefachim is 3 X (1.4 X 4), or 16.8 -- and not 24!
How did the Judges of Kesari make such a mistake? (See also Insights to Eruvin 76:2.)
ANSWERS:
(a) TOSFOS (8b, DH Rivu'a; Eruvin 76b, DH v'Rebbi Yochanan) suggests that the Judges of Kesari were not giving the relationship of the perimeter of the inner square to a circle around it. Rather, they were giving the relationship of the area of the inner square to an outer square drawn around the circle that encloses the inner square. This is what they mean when they say 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 diagram printed in Tosfos.) The area of the inner square is exactly half of the area of the outer square.
(b) RASHI (here and in Eruvin 76a) seems to have no difficulty with the formula of the Judges of Kesari, as he does not explain how to justify their calculation. Perhaps Rashi understands that the Judges of Kesari were proposing a Halachic stringency: When we determine 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). Thus, if the sides of the inscribed square are each 4 Tefachim, then the diagonal is viewed to be 8 Tefachim. Accordingly, the circle around that square must have a diameter of 8 Tefachim, and thus its circumference must be 24 Tefachim and not 16.8 (which is the measure of the circumference based on the actual diameter of the square).
The reason for this is to prevent one from mistakenly using the length of the diagonal in a case in which he is supposed to use the sum of the length of two sides. In addition, physical reality does not allow for the application of pure mathematics (as the actual diagonal of a square is an irrational number; moreover, it is not possible to draw a perfectly exact line or angle). Therefore, the formula given for determining the diagonal of a square for purposes of Halachic applications (such as the size of a circular Sukah around that square) must take into consideration the largest possible diagonal of the angle, which is the sum of the two sides.
(Thus, if the sides of the inscribed square are each 4 Tefachim, then the diagonal is viewed to be 8 Tefachim. The circle around that square has a diameter of 8 Tefachim, and its circumference is treated as 24 Tefachim (and not 16.8, which is the circumference based on the actual diagonal of the square).)
When the Gemara comments on the formula of the Judges of Kesari and says, "This is not so, for we see that it is not that much" (that is, the circle around a square is not as large as the Judges of Kesari posit), it is stating only that their statement is not mathematically precise, but it is not stating that their formula cannot be used in applied Halachah.
If this is the reason why Rashi is not bothered by the apparent inaccuracy of the formula of the Judges of Kesari, then we may suggest that Rashi is consistent with his own opinion as expressed elsewhere (Shabbos 85a, Eruvin 5a, 78a, 94b), where Rashi seems to determine the Halachic length of the diagonal of a rectangle by adding the two sides between the ends of the diagonal. TOSFOS in all of those places argues with Rashi. Perhaps Rashi maintains that this definition of the diagonal for Halachic purposes may be relied upon with regard to rulings that involve Halachos d'Rabanan. (M. KORNFELD)
(c) Perhaps it is possible to propose an entirely new explanation, according to which the Judges of Kesari are entirely 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. His statement is based on the formula of the Judges of Kesari.
The area of a circle is calculated by multiplying pi by the radius squared. The radius of the circle drawn around a square with sides that are each 4 Tefachim long is half of the diagonal (5.6), which is 2.8. Using the Halachic estimate of the value of pi as 3, we arrive at the following calculation: 3 X (2.8)(2.8) = 23.52, or approximately 24.
This is what Rebbi Yochanan means when he says that the circle must have within its circumference an area of 24. (He rounds up to 24 as a stringency.)
(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 by the Rabanan Savora'i.) (M. KORNFELD)
(David Garber and Boaz Tzaban of Bar Ilan University, who have published numerous articles on geometrical themes in Chazal, pointed out to us that the ME'IRI in Eruvin (76b) suggests this solution in the name of 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), the solution for the area of the circle is exactly 24 Tefachim, and not just approximately 24, as we concluded using the equation of pi X r X r (this is because the area of the outer square (32) is exactly double the area of the square drawn inside of the circle (16), and 3/4 of 32 is 24). The Me'iri uses the word "Shibur" or "Tishbores" to refer to the calculation of area.)

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