Explanation #1 (Rashi): Regarding carrying [four Amos in Reshus ha'Rabim] we cannot give corners. [We give] only in a case like this [Techum], that we give a Shi'ur for the diagonal. We cannot say about carrying] a square and a circle.

Explanation #2 (R. Shmuel): We square four Amos in Reshus ha'Rabim according to the directions of the world. If one carries or throws from east to west or from north to south, he is liable for four Amos alone;

If he [carries or throws] along the diagonal of the world (e.g. northeast to southwest, or northwest to southeast), he is not liable unless he carries or throws four Amos and their diagonal (another Amah and three fifths).

Also a pillar in Reshus ha'Rabim 10 [Tefachim] tall and four [Tefachim] wide, we require four by four square (to make Reshus ha'Yachid), i.e. when you stand the diagonal of the pillar along the diagonal of the world;

However, if he stood the diagonal of the pillar from east to west or from north to south, four by four square do not suffice [to make Reshus ha'Yachid].

Also, a circular city, we say below (56a) that we square it according to the directions of the world.

Explanation #3 (R. Tam): We give the diagonal in every direction (no matter how it is aligned) - for throwing, carrying, Techum Shabbos and also a pillar, it must be six Tefachim less two fifths (i.e. like the diagonal of a square of four, to make Reshus ha'Yachid].

Implied question: Why does it mention four Amos everywhere? (Really, the Shi'ur is five and three fifths!)

Answer: From four Amos we derive [five and three fifths, i.e.] the diagonal.

Implied question: The hive in Shabbos (8a) is cylindrical. It does not have a diagonal. Why does it say that it is six wide? (Everywhere else we say four, and it means five and three fifths!)

Answer: [Because it is cylindrical and has no diagonal,] had it mentioned four, this would connote that we do not require the diagonal [of four].

Rebuttal: We learn corners of Shabbos from the cities of the Leviyim. There, we give corners only according to the corners of the world, like is proven below (56b). It says that it turns out the Migrash (area left vacant around a city) is a quarter (of the city's area)!

Implied question: Why didn't he say that this refers to one who came on the road and it got dark, and it is like everyone?

Answer: It is proven that our Mishnah discusses what R. Meir and R. Yehudah argue about.

Question: According to R. Yehudah, how is this a leniency? Even in his house, he can say "my Shevisah is in the place Ploni!"

Answer: It is a leniency when there is no tree, fence or specific place in the place where he wants to acquire Shevisah.

Question (Ri): Since he calls him an Ashir, this connotes that he wants to return and lodge in his house. How can he return? He has only 2000 Amos to the direction of his city!

Even if his house is the first when he enters on the side of the city, he will have in his house only four Amos. Do we discuss a fool, that he will lodge in his house and have only four Amos?!

Answer #1 (Ri): "Outside the Techum" is outside the Ibur of the city.

Support: We find like this below (60a).

Answer #2 (Ri): "An Ashir may go outside the Techum" does not means that he wants to acquire Shevisah outside the Techum. Rather, he wants to return and acquire Shevisah inside the Techum;

It taught this due to R. Meir, who taught that an Ashir may not leave the Techum. The Chidush is, even though he left the Techum, and now he is like one who comes on the road, and even so he cannot be Me'arev with his feet, since the sole reason he left was only to be Me'arev;

Since it mentioned R. Meir, it mentioned also R. Yehudah. (Really, all agree to this.)