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1)

(a) With regard to two courtyards at either end of two adjoining houses, each with a house at the far end, and the residents of each courtyard wished to make an independent Eruv to carry in their own courtyard. To do this, they wanted to place their respective Eruvin in the other house - that did *not* adjoin their courtyard, treating the adjoining house that *did* as a Beis-Sha'ar) - Rachbah's Talmidim answered him that both Eruvin would not be valid. This is because, taking each house separately, if it is to serve as a *Beis-Sha'ar* (for the *one* Chatzer to be able to place their Eruv in the adjoining house), then the other Chatzer cannot place their Eruv in it (as we learn in the Mishnah 85b). On the other hand, if you consider that same house *a house*, then the Chatzer who placed their Eruv in the adjoining house will be carrying in a house which is not part of their Eruv.

(b) If someone placed two Eruvin for two people, one when it was still day, the other during Bein Hashemashos, Rava said that if the first one was eaten during Bein Hashemashos, and the second one, after nightfall - then both Eruvin are valid.

(c) Rava's Chidush is that the two Eruvin are both valid, even though this appears to be a contradiction. Why is that? Because if the first Eruv is valid, that is because we consider Bein Hashemashos to be *Shabbos*. By validating the second Eruv, we are considering Bein Hashemashos to be still *Friday*? The answer must be that, since we are dealing with two individual people, we take each case separately, ruling Safek de'Rabbanan Lekula (regarding Eruv), and ignoring the contradiction. In that case, why should we not do the same in our case, allowing the house to be a *Beis-Sha'ar* for the person who placed his Eruv in the *other* house, and a *house* for the one who placed his Eruv inside it?

(d) The basic difference between the two cases is - the fact that, in Rava's case (of Safek Bein Hashemashos) the source of the Safek is invisible. Consequently, we are able to turn a blind eye to the fact that we are giving contrary rulings to two people (since nobody is able to point at the source of the Safek); whereas in our case (with Rachbah's Talmidim), where the source of the Safek is the status of the two houses), we cannot issue contradictory rulings, because people will point a finger at them and say 'Hey! Is this a house or is it a Beis-Sha'ar'?

2)

(a) A window in a wall between two courtyards must be placed below ten Tefachim to enable them to make a joint Eruv. Its minimum dimensions are four by four Tefachim.

(b) A window that is higher than ten Tefachim is not considered a Pesach in this regard - because above ten Tefachim, the wall itself is superfluous.

(a) The Gemara concludes that the Shiur of Pesach is an independent Din and has nothing to do with Levud.

(b) If it would have, then the author of our Mishnah (which requires a window of four by four Tefachim) would be - Raban Shimon ben Gamliel, who holds that up to four Tefachim we say Levud (but according to the Rabanan, the Shiur Pesach would be three by three Tefachim, like the Shiur Levud according to them).

(c) Having informed us that the window must be within ten Tefachim of the ground, the Mishnah then sees fit to add that if it *above* ten Tefachim, it is invalid - to inform us that it is only invalid if the entire window is above ten Tefachim, but not if some of it is below that.

(d) The Beraisa, having already taught us that a window which is mostly above ten Tefachim is still valid, does not really need to then tell us that if it is mostly within ten Tefachim it is also valid. However, we apply the principle (which the Gemara prefers to avoid whenever it can find a better solution) 'Zu, ve'Ein Tzarich Lomar Zu'.

(a) A circular window must have a circumference of twenty-four Tefachim, according to Rebbi Yochanan - because that is the size circle that is required to accommodate a square of four by four Tefachim.

(b) Two Tefachim and a Mashehu of that circle must be inside the ten Tefachim - to ensure that the bottom of the square will also be inside the ten Tefachim limit.

5)

(a)

1. The circumference of a circle is three times its diameter.
2. The area of a square is a quarter (which we refer to as a third) more than the circumference of the circle that contains it.
3. The diagonal of a square is one and two fifths more than one of its sides.

(b) The circumference of Rebbi Yochanan's circular window ought then to be - 16 4/5 Tefachim (4 x 1 2/5 x 3).

(c) Rebbi Yochanan holds like the Dayanim of Caesaria, who hold that the area of a circle which surrounds a square is a half more than the square (so that the 16 Tefachim area of the square becomes 24). (See Tosfos DH 've'Rebbi').

(a) Rav Nachman rules that a window in a wall above the height of ten Tefachim which is invalid as regards Eruv does not apply to a wall in between two houses - and that *there* a window above ten Tefachim will be considered a Pesach, because a house in a Reshus ha'Yachid is considered as if it was filled in (meaning that the air i.e. the height of the house is of no consequence).

(b) According to Rav Nachman, the Beraisa, which writes that a window must be within ten Tefachim - refers to a window between two *courtyards* (which is also mentioned there), but not to one between two *houses*.

(c) The author of the Beraisa which requires an Eruv for two roofs - must be the Rabbanan of Rebbi Shimon, who say that just as different residents require an Eruv below, so too do they require an Eruv above.

(a) When Rav Aba asked Rav Nachman whether a skylight (that leads to the attic belonging to a second person) in the middle of the ceiling, requires a fixed ladder or not - he was asking him whether the principle of 'Beisa Ke'ma'an de'Malya Dami' extends even to the ceiling. Perhaps it will only apply to the space *inside* the house, but not to the space *above* it.

(b) When Rav Nachman replied that it does not require a fixed ladder - he really meant that it does not require a ladder at all (because 'Beisa Kema'an de'Malya Dami').

(a) Two houses whose garden or courtyard is divided by a wall of ten Tefachim cannot make a combined Eruv.

(b) The wall divides between the two gardens or courtyards irrespective of its thickness. The Mishnah only mentions that the wall is four Tefachim thick - because of the Seifa, which we will deal with immediately.

(c) One may not use the wall to transport food etc. from one courtyard to the other - because it is speaking about a wall which is four Tefachim thick, in which case the top of the wall is an independent Reshus, which may not be used to transport from one Reshus to another.

(a) A breach in the wall would need to be not more than ten Amos wide (the maximum width of a Pesach) in order to retain the right to make an independent Eruv. Once it is in excess of that, the Chazter is considered breached, and one combined Eruv will be required.

(b) If, in the latter case, they nevertheless made independent Eruvin, their Eruvin will not be valid, and all the residents will be forbidden to carry.

(a) According to Rav, if the wall is less than four Tefachim wide, the top may not be used by either Chatzer - because it is Batel to both Chatzeros, in which case, the residents of the one Chatzer forbid those of the other Chatzer to carry, just like they would if there was no Mechitzah dividing them,

(b) Rebbi Yochanan holds - that since the top of the wall is less than four Tefachim wide, it has the Din of a Makom Petur, which (because both courtyards are a Reshus ha'Yachid and the Isur involved is no more than a de'Rabbanan) may be used even as a transit point, to transport from one Chatzer to the other.