HIDE/SHOW IMAGE
24a
{Eruvin 75b continues}
For anyone who sees that he is carrying, if he {the one who sees} knows that he is an individual {=that the Israelite is the only occupant} then he knows that any individual is permitted to carry. And if he does not know {that the one he sees is an individual} would say that he is carrying because they jointly made an eruv. If so, for the gentile {as the only occupant} as well, if he {who sees an Israelite carrying} knows that he {the gentile} is an individual {resident of the inner courtyard}, he knows that with a single occupant {of the inner courtyard} it is permitted to carry, and if he does not know, he says that this one is carrying because he {=the Israelite} rented from him {from the gentile the rights}! The average gentile, if he rents, makes a noise about it, and since he {the gentile} did not make a noise about it, we {who see} say that he is carrying while there is a gentile {occupant}. And therefore it is forbidden.
END PEREK SIX  hadar im hagoy
BEGIN PEREK SEVEN  chalon
{Eruvin 76a}
Mishna:
A window between two courtyards, four by four, within ten – they make two eruvs, and if they wish, they make one eruv;
less than four by four, or above ten – they make two eruvs and they do not make one eruv.
Gemara:
The Sages learnt {in a brayta}: If {almost} all of it was above ten {handbreadths} and a portion of it was within ten, or {almost} all of it was within ten, and a portion of it was above ten, they make two eruvs, and if they wish, they make one eruv.
And if the window is circular, it must have a circumference of 17 minus 1/5 handbreadths, for it is established for us {that Pi is approximately 3, and so} anything that has in its circumference 3 handbreadths has a width {=diameter} of 1 handbreadth.
{Eruvin 76b}
Therefore we need the circumference of this window to be 17 minus 1/5 handbreadths, such that the width will be 5 and 4/5 handbreadths.
{And now the calculation of how this is so, which is straightforward:
First, 17  1/5 = 16 and 4/5.
(5 + 3/5) * 3 = 15 + (3/5 * 3) = 15 + 9/5.
Now, 9/5 = 1 and 4/5.
15 + 1 + 4/5 = 16 and 4/5.
}
For when we subtract 8/5 from it, which is 2/5 of 4 handbreadths, for it is established for us that any square which is 1 cubit {in length and width} is 1 cubit and 2/5 cubits in its diagonal (that is, sqrt(2)). Therefore, the width of the sqaure inscribed in the circle will be 4 X 4 handbreadths, no more and no less.
{To explain, why do we need the diameter of the circle to be 5 and 4/5 handbreadths? We do not care about the circle, but rather we care about square measurements for the window, and this window needs to be 4 X 4 handbreadths. We need to inscribe such a square inside our circle. Now, the inscribed square hits the circle at each of the square's four corners, and so the diagonal of the square must be = to the diameter of the circle. If the square needed to be only one handbreadth, the diagonal of the square (and thus the diameter of the circle) need be sqrt(2). Since the square needs to be 4 X 4 handbreadths, the diagonal needs to be 4 * sqrt(2). Since they did not work in quare roots, they knew that the diameter of a 1 handbreadth wide square was 1 and 2/5 cubits.
For a four handbreadth square:
4 * (1 + 2/5) = 4 + 8/5. Now, 8/5 = 1 + 4/5, for a total of 5 and 4/5.
}
"less than 4 X 4 {handbreadths}...":
Rav Nachman said: They only learnt this regarding a window between two courtyards, but for a window between two houses, even if it is higher than ten {handbreadths} also, if they want they may make separate eruvs.
What is the reason?
A house is reckoned as if it is filled.
And so is the halacha.
Rava inquired of Rav Chisda {our gemara: R' Abba inquired of Rav Nachman}: If an aperture {=trapdoor} led from {a room in} the house to an attic {,the two rooms being occupied by different tenants}, does it require a {permanent} ladder to permit it or no? When we say that a house is reckoned as if filled, are these words intended only to the side {and thus rooms on the side}, but in the center {of the room}, no? Or perhaps it does not matter.
He said to him: It {=the ladder} is not necessary.
He thought from his words that a permanent ladder was not necessary, but a temporary ladder was necessary.
Rav Yosef bar Minyumi cited Rav Nachman: Neither a permanent nor a temporary ladder is required.
Mishna:
If a wall between two courtyards is ten {handbreadths} high and four {handbreadths} thick – they make two eruvs and they may not make one eruv.
If there was produce at its top – and these may go up from here and eat and these may go up from here and eat, provided that they do not bring down.
If the wall is broken through up to ten cubits, they make two eruvs, and if they wish, they make one eruv, because it is a doorway; more than this, they make one eruv, and they do not make two eruvs.
Bereishit: Natural Consequences

I've written before extensively on reading parshat Bereishit, and the
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