1) THE SIZE OF A "KARPAF"
OPINIONS: The Gemara explains that both Rebbi Eliezer and Rebbi Yosi in the Mishnah (23a) agree that one is permitted to carry within a Karpaf that was not Hukaf l'Dirah (fenced-in for residential purposes) as long as its length is twice its width. If the length is more than twice the width, one is not permitted to carry there on Shabbos. The Gemara asks what they argue about, and it answers that they argue about "Rivu'a d'Riv'uhah Rabanan" -- "the squaring which the Rabanan squared it." What does this mean?
(a) RASHI explains that according to Rebbi Eliezer, a real Karpaf is one that is twice as long as it is wide, like the dimensions of the Chatzer of the Mishkan. If it is square, then one is permitted to carry there only b'Di'eved. Rebbi Yosi maintains that a real Karpaf is one that is square. If its length is greater than its width, then b'Di'eved one is permitted to carry there as long as its length is not more than twice the width. Rebbi Eliezer and Rebbi Yosi argue over which Karpaf is the "prime" Karpaf mid'Oraisa in which one is permitted to carry l'Chatchilah, and which Karpaf is the secondary Karpaf, in which one is permitted to carry only b'Di'eved.
What, though, does Rashi mean when he says that one is permitted to carry "b'Di'eved"? "Carrying b'Di'eved" does not mean that we advise the person not to carry except under extenuating circumstances, because Rashi would not express this allowance by saying that it is permitted "b'Di'eved," but rather that it is permitted "b'She'as ha'Dechak." If, however, "carrying b'Di'eved" means simply that one is permitted to carry there, then both opinions are saying the same thing!
Apparently, they argue about a case in which a person wants to make a fence around his Karpaf in order to permit one to carry in it on Shabbos. They argue about the best way to fence-in the Karpaf l'Chatchilah.
(b) TOSFOS (DH Ika Beinaihu) and the RIF disagree with Rashi's explanation, because it is unlikely that the argument involves a case of a person who wants to fence-in his Karpaf specifically for the purpose of carrying there on Shabbos. Tosfos explains that the argument between Rebbi Eliezer and Rebbi Yosi is the same argument that we find later (49b) between Rebbi Chanina ben Antignos and the Rabanan. Rebbi Chanina ben Antignos maintains that the 2000-Amah Techum of Shabbos is measured from the point where the person is located when Shabbos begins to 2000 Amos in every direction around him, so that his Techum forms a circle around him. The Rabanan maintain that the 2000-Amah Techum is a square around him, so that he may actually walk more than 2000 Amos, if he walks along the diagonal of the square.
In the Mishnah here, Rebbi Eliezer maintains that the diagonal of a Karpaf may be only twice as long as its width, as Rebbi Chanina ben Antignos says (consequently, the length of the Karpaf must be somewhat less than twice the width). Rebbi Yosi maintains that the diagonal may be longer, as long as the side of the rectangular area is not more than twice the width, as the Rabanan say.