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Eruvin 76
ERUVIN 76  dedicated by Dick and Beverly Horowitz and sons Sam and Mike,
in memory of Beverly's mother, Miriam Kurtz.

1) MAKING AN ERUV DURING "BEIN HA'SHEMASHOS"
OPINIONS: The Gemara cites Rava's teaching regarding one who made two
Eruvin for two different people at different times, which both became
lost. One Eruv was made while it was still day (Erev Shabbos), and one was
made during Bein ha'Shemashos. The first Eruv became lost during Bein
ha'Shemashos, and the second one become lost after Bein ha'Shemashos, when
it was definitely night. Rava teaches that both Eruvin are valid. What
type of Eruv is Rava discussing?
(a) RASHI explains that this case refers to Eruv Techumin. According to
Rashi, then, Rava here is apparently ruling like Rebbi Yosi (35a) who says
that an Eruv Techumin which is in doubt is nevertheless a valid Eruv.
(b) TOSFOS (DH Sh'neihem) says that Rava is not talking about Eruv
Techumin, because in this case even Rebbi Yosi would agree that such an
Eruv would not be valid. Here there is no Safek about the Eruvin; it is
known for certain when the Eruvin were made and when they were lost. The
only doubt is whether Bein ha'Shemashos is day or night. In such a case of
uncertainly, Rebbi Yosi would not be lenient and validate the Eruv.
Rather, Rava is discussing Eruvei Chatzeros, the laws of which are
generally more lenient that those of Eruvei Techumin, and even if it was
made during Bein ha'Shemashos it will be valid. (See also Insights to
Eruvin 35b)
76b
2) THE MATHEMATICAL FORMULAE OF THE RABBIS OF CAESAREA
QUESTION: The Mishnah says that a window in the wall between two Chatzeros
must be at least four by four Tefachim in size, and must be within the
first ten Tefachim of the height of the window, in order to be considered
a Pesach (opening) and allow the Chatzeros the choice of joining together
with one Eruv.
What do the dimensions of the window have to be if the window is *round*?
Rebbi Yochanan made a statement that if the window is round, it "must be
24 Tefachim in its circumference, and two Tefachim (plus 4 Tefachim) and a
bit of the window must be under ten Tefachim in the wall, so that if a
square was inscribed in the circle a part of it would be within ten
Tefachim of the ground." That is, Rebbi Yochanan is asserting that a
circle drawn around a square with sides of 4 Tefachim (which has a
perimeter of 16 Tefachim) has a circumference of 24 Tefachim.
The Gemara concludes that Rebbi Yochanan's mathematical calculations were
based on the theorem of the Rabbis of Kesari. They said that the
circumference of an 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 16Tefach
perimeter of a square is 50% larger, or 24 (that is, take 50% of 16 and
add it to 16).
As the Gemara in Sukah (8a) points out, this theorem is clearly incorrect,
as can be seen with a cursory glance. The actual relationship of the
perimeter of an inscribed square to the circle around it, according to
Chazal, is 3 * (1.4 * s), when 3 is used for pi (Eruvin 13a) and s = 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, and why did Rebbi
Yochanan follow them?
ANSWERS:
(a) TOSFOS (DH v'Rebbi Yochanan) answers 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 the *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 picture printed in Tosfos in our Gemaras, which is
slightly misleading; in the picture that appears in the TOSFOS HA'ROSH,
reproduced in our Graphics section, the inner square is shifted so that
its sides are at a diagonal to the sides of the outer square. This is more
demonstrative of Tosfos' point). The area of the inner square is exactly
half of the area of the outer square.
According to Tosfos, Rebbi Yochanan misunderstood the Rabbis of Kesari and
made his Halachic statement regarding the relationship of the
circumference of a circle to the perimeter of a square based on his
misunderstanding.
(b) The RITVA explains that the Rabbis of Kesari and Rebbi Yochanan are
correct. When he mentioned a "round" window, Rebbi Yochanan did not mean a
circular window with an imaginary square inscribed within it. Rather, he
was referring to a window made in the shape of a fourleaf clover; that
is, a square with four semicircles protruding from each side (see Graphic
section). In such a case, the perimeter of the window (i.e. the arcs of
the four semicircles) indeed add up to 50% more than the perimeter of the
square around which they are drawn. In order to make sure that the square
inside the clovershaped window reaches to within a height of ten Tefachim
from the ground, at least 2 Tefachim and a bit of the *radius* of the
bottom semicircle must be within ten Tefachim (since the radius of each
semicircle is 2, or half of one side of the square, which is four).
Alternatively, 2 and a bit Tefachim plus four Tefachim of the perimeter of
the semicircle must be under 10 Tefachim from the ground (as Rashi
explains on bottom of 76a), since the total perimeter of each semicircle
is 6 Tefachim.
(c) RASHI does not explain how to justify the formula of the Rabbis of
Kesari and how to understand Rebbi Yochanan. He seems not to have any
difficulty with them. 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, we
Halachically 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. (Thus, if the sides of inscribed square are each 4
Tefachim, then the diagonal is viewed to be *8* Tefachim. The circular
window 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.)
If this is why Rashi is not bothered by the formula of the Rabbis of
Kesari, then 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. TOSFOS in *all* of those places argues with Rashi,
but 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.
(d) Perhaps it is possible to propose an entirely new explanation. The
Rabbis of Kesari and Rebbi Yochanan are perfectly correct.
Perhaps Rebbi Yochanan's statement that there "must be 24 Tefachim in its
circumference," does not mean that the *circumference* must be 24
Tefachim, but that there must be 24 Tefachim *inside* the circumference 
in other words, the *area* of the circle must be 24 Tefachim!
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 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)!
What did Rebbi Yochanan mean that there must be 2 and a bit within a
height of ten? 24 Tefachim is the area of the circle. Within that area is
an inscribed square of 4 by 4, which has an area of 16 Tefachim. What is
the area of the four arcs that are outside of the square? Since they are
the difference between the area of circle and the square, altogether they
add up to 2416=8, and thus each one has an area of 2 Tefachim. That is
exactly what Rebbi Yochanan meant when he said that in order to get the
inscribed square of 4 by 4 Tefachim below a height of ten Tefachim, at
least 2 Tefachim and a bit of the *area* of the circular window must be
below ten Tefachim! (According to this approach, it is no longer necessary
to say, as Rashi (76a) suggests, that when it says "two and a bit" it
means two and a bit in addition to *four*)
(M. Kornfeld)
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