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!
(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.)