1)

(a)If one wished to support an Ari'ach on a Korah of a Tefach, one would simply add a layer (of a thumb-breadth - see Rabeinu Chananel) of cement on either side of the Korah - Consequently, a Korah of one Tefach will suffice to hold an Ari'ach.

(b)According to Rabah bar Rav Huna, the posts that support the Korah do not need to be strong enough to support the Korah and the brick (only the Korah) - since it is the Korah that needs to support the brick, and not the posts which support it.

(c)One may not carry in a Mavoy whose Korah one covered with a mat, which extended downwards, but stopped three Tefachim from the ground - because the Mavoy has neither a Korah (since it is covered), nor a Mechitzah, since it stopped three Tefachim short from the ground, in which case we cannot apply Levud.

2)

(a)If a Korah which extends from one wall across the Mavoy, but falls short of the opposite wall, or two Koros which come out of the two walls, but which do not quite meet in the middle - one is permitted to carry in the Mavoy, provided the Korah in the first case reaches to within three Tefachim of the wall, and in the second, the two Koros are within three Tefachim of each other. Otherwise, a second Korah is required to rectify the fault.

(b)According to the Tana Kama, two parallel poles, neither of which can support a half-brick (i.e. is a Tefach wide) - form a Kasher Eruv, and permit one to carry in the Mavoy, provided their combined width is at least one Tefach, and they are within three Tefachim of each other.

(c)Raban Shimon ben Gamliel holds that the two poles must be able to hold an Ari'ach three Tefachim lengthwise (i.e. they must have a combined width of at least three Tefachim), before carrying in the Mavoy is permitted.

3)

(a)Rebbi Yossi b'Rebbi Yehudah maintains that, if the two poles are at different heights - the Mavoy is nevertheless Kasher, because we consider the top pole to be slightly lowered, and the lower pole to be raised (until they are both of the same height).

(b)We do not say 'Ro'in' - when one of the poles is either above twenty Amos or below ten Tefachim.

(c)Rebbi Yossi b'Rebbi Yehudah holds like his father (in our Mishnah), who says 'Ro'in, though he disagrees with him with regard to a Korah which is higher than twenty Amos, which his father permits, but he invalidates.

4)

(a)When Rav Yehudah quoted our Mishnah as saying 'Rebbi Yehudah Omer, Rechavah, Af al Pi she'Einah Beri'ah' - Rav instructed him to change the text to 'Rechavah u'Beri'ah' (see Tosfos DH 'Asnayah').

(b)This is not the case, when the Korah is four Tefachim wide, because then it does not matter if it is too weak to actually support a brick.

(c)Initially, the Tana quoted Rebbi Yehudah as holding 'Ro'in' by wood for example, which is usually sufficiently strong to support a brick - he then adds the Din of 'Ro'in' by straw, to teach us that Rebbi Yehudah holds 'Ro'in' even by something which is usually not strong enough to hold a brick.

(d)Even if the Korah bends upwards to above twenty Amos, it is obvious that, according to everyone, if less than three Tefachim were to remain if one discounted the excess section, the Mavoy would be Kasher. The Chidush in our Mishnah is if the Korah extends outwards, that, if the two remaining ends of the Korah were less than three Tefachim from each other after that section was removed, the Mavoy is Kasher - to teach us that Chazal did not decree carrying in the Mavoy, for fear that may just follow the section that protruded into the street, and carry there.

5)

(a)The Chidush of 'Agulah, Ro'in Osah Ke'ilu Hi Meruba'as' - lies in the continuation of the Mishnah 'Kol she'Yesh be'Hekefo Sheloshah Tefachim, Yesh Bo Rochav Tefach'.

6)

(a)We learn from the 'Yam shel Shlomo' - the principle that whatever is three Tefachim in circumference, is one Tefach in diameter.

(b)By 've'Ha Ika Sefaso' - the Gemara means to query this principle: The diameter of the Yam shel Shlomoh contends with the space alone, whereas the circumference also includes the thickness of its walls; so how can we connect the two?

(c)The Gemara answers - that the walls of the Yam shel Shlomo were extremely thin, like the petals of a Shushan rose, so they did not really add much to the circumference.

(d)This answer however, is inadequate, because, however thin the Yam was, it must have measured something, which means that the proportion of diameter to circumference is not exactly one to three.

7)

(a)A Mikveh must contain - forty Sa'ah of water (three Amos by one mah by one Amah) in order to be Kasher.

(b)Four hundred and fifty cubic Amos would have sufficed to make a hundred and fifty Mikva'os (whereas the Yam shel Shlomo contained five hundred cubic Amos).

(c)The Kashya is a fallacy however - because the four hundred and fifty cubic Amos would be correct if we were talking about a square Mikveh, whereas the Yam shel Shlomo was circular (meaning a circle contained inside a square).

(d)A square is a quarter more than a circle (i.e. three cubic Amos for every four), in which case, assuming the Yam shel Shlomo to have been totally circular, the 'Yam' should have contained three hundred and seventy-five cubic Amos (making a hundred and twenty-five Mikva'os (and not a hundred and fifty!)

8)

(a)In fact, the Gemara concludes - the bottom three Amos were square, and the top two, round.

(b)It cannot have been the top section of the 'Yam' that was square - since the Pasuk writes "Sefaso Agul".

(c)The bottom three square Amos totaled three hundred cubic Amos - making one hundred Mikva'os), and the top two, round Amos, totaled a hundred and fifty cubic Amos - making fifty Mikva'os.

9)

(a)Two thousand Bas is the equivalent of six thousand Sa'ah which, in turn, is the equivalent of one hundred and fifty Mikva'os - This proves the shape of the 'Yam shel Shlomo, as we just described it, and dismisses the Gemara's suggestion that maybe it was just the top Amah (and not two) that was round.

(b)When the Pasuk in Divrei ha'Yamim gives the volume of the 'Yam' as three thousand Bas - it is referring to the volume of solid contents, which is one third more than the liquid (due to the extra amount of solids that one is able to heap by a solid, but not by a liquid.

(c)A cupboard or a large container holding forty Sa'ah of liquid or sixty Sa'ah of solids - is no longer subject to Tum'ah, because, due to the fact that it is too heavy to be carried full as well as empty, it is no longer considered a Kli.

(d)The Gemara proves from this Mishnah - that one normally heaps one third more of solids into a vessel than one does liquid (as we just stated).

10)

(a)A Lechi must be - at least ten Tefachim tall.

(b)According to Rebbi Yossi - the minimum width of a Lechi is three Tefachim.

(c)No! The fact that our Mishnah speaks of 'Lechayayin' in the plural - does not mean that the author must be Rebbi Eliezer, who requires two Lechayayim for a Mavoy. In fact, the author of our Mishnah could just as well the Chachamim, and 'Lechayayin' refers to Lechayayin in general.

(d)On account of this Kashya (why then, does the Tana refer to 'Korah' and not to 'Koros') the Gemara interprets the Tana's use of the of 'Lechayayin' in the plural in connection with the Machlokes between Rebbi Eliezer and the Chachamim concerning Lechayayin (whether a Mavoy requires one Lechi or two): those Lechayayin over which Rebbi Eliezer and the Chachamim dispute there, must be ten Tefachim tall.

11)

(a)A Chut ha'Sarbol - is the thread with which one ties together the front ends of a coat.

(b)The Chidush of the Beraisa, which permits carrying in the inner of the Mavoy if one placed a Lechi half-way - is that we do not forbid carrying there, for fear than one may forget and carry in the entire Mavoy (even in the section which has no Eruv).

(c)A Lechi that stops between three and four Tefachim short of the ground is not Kasher, even according to Raban Shimon ben Gamliel, who holds Levud up to four Tefachim - even Raban Shimon ben Gamliel agrees that a gap of three Tefachim between the bottom of the Mechitzah and the ground will invalidate a Mechitzah, since it is one which allows kid-goats to get in underneath it.

12)

(a)The Gemara felt inclined to rule like Rebbi Yossi with regard to the width of a Lechi - since Rebbi here follows the opinion of Rebbi Yossi; and also because of the principle 'Rebbi Yossi Nimuko Imo' (meaning that Rebbi Yossi was renowned for his sound reasoning) - though this second reason is not confined to our Machlokes.

(b)We do not, in fact, rule like Rebbi Yossi (neither here, nor by Hilmi - making salt-water on Shabbos, where he is strict) - because of the universal custom to permit a Lechi of any width.

(c)Rebbi Tarfon is referring to a Berachah Rishonah - when he prescribes 'Borei Nefashos' as the Berachah for water.

(d)We do not follow the ruling of Rebbi Tarfon, with regard to the Berachah Rishonah over water, because of the universal custom to recite 'Shehakol' over it