Gemara

45a (Mishnah - Chachamim): If one slept on the road and did not realize that it was dark, he has only four Amos.

R. Yehudah says, he can go four Amos in any direction he wants.

R. Yehudah agrees that once he decided, he cannot retract.

If Reuven and Shimon both have four Amos and they overlap, they may bring food to the common area and eat together, as long as one does not take the other's food outside the common area.

48a (Rava): Chachamim permit four in both directions. R. Yehudah permits four in any direction he chooses.

Support (Beraisa - R. Meir): He is permitted eight by eight [Amos].

(Rava): They argue about where he may walk, but all permit Tiltul (moving items while standing still) only within four Amos.

(Beraisa - R. Meir): "Shevu Ish Tachtav" - one must stay in an area equal to what is under himself;

His body is three Amos. We add one Amah for him to stretch his arms and legs.

R. Yehudah says, his body is three Amos, we add one Amah for him to take something from 'under' (next to) his feet [when he lies down] and put it 'above' his head.

R. Meir allows slightly more than four Amos, and R. Yehudah allows [only] precisely four Amos.

49b (Mishnah - R. Chanina ben Antigonus): One gets a Techum of 2000 Amos in every direction, i.e. a circle [of diameter 4000];

Chachamim say, he gets a square [of side 4000]. He gains the corners.

50b (Beraisa): If one saw a tree and said 'my Shevisah is under it', his words have no effect.

The case is, there are 2004 Amos from his feet to the trunk. He did not specify, so perhaps his Shevisah is on the other side, which is outside his Techum.

51a - Question (Rava): If a pillar in Reshus ha'Rabim is 10 tall and four wide, must the width be four and its diagonal [to be Reshus ha'Yachid, according to Chachamim]?

Answer (Rav Papa and Talmidim - R. Chananyah's Beraisa): All laws of Shabbos are 'ka'Zeh' (like cities of the Leviyim. They received 2000 Amos in every direction around their cities, including the diagonal.)

56a (Beraisa): One who squares a city makes it like a square board. He then squares the Techum, and makes it like a square board.

We do not measure 2000 Amos from the middle of the corner (i.e. on the diagonal), for then he would lose [due to] the corners;

Rather, we add a square of 2000 Amos by 2000 Amos in the corner on the diagonal.

Rishonim

Rambam (Hilchos Shabbos 12:15): One may move things in Reshus ha'Rabim within his four Amos. He may move in the entire square.

Rambam (18): One may carry within a square four by four Amos. One may carry along the diagonal, which is five and three fifths Amos. Therefore, one who carries or throws in Reshus ha'Rabim is liable only for this amount.

Magid Mishneh: Rashi, R. Tam and the Rashba explain like the Rambam. This is primary. Some obligate one who carries four Amos straight in front of his face. It seems that the Rambam forbids carrying four Amos, but exempts until he carries four and their diagonal. He learns from "one is not liable." The Gemara did not permit until four and their diagonal, which would have been a bigger Chidush. Also, if Chachamim permitted until four and their diagonal, they did not make a fence for the Torah. Rather, the Torah permits until four and their diagonal, and Chachamim forbade four Amos. This is primary, even though the Rashba connotes that until four and their diagonal is totally permitted.

Rebuttal (Gra OC 349:2, DH v'Yesh, according to Damesek Eliezer): The Gemara said that they argue about walking, but all permit Tiltul only within four Amos. This connotes that Tiltul is like walking (i.e. they argue only about permitting four or eight, but not about adding the diagonal), especially according to the Rambam, who rules like R. Yehudah. Surely walking is without the diagonal, like it says on 50b (his Techum is only 2004 Amos). It seems that also the Rambam permits with the diagonal, but only in the square, and not in every direction, like he wrote in Halachah 15. We measure based on the direction he chose, just like his Heter to walk. The Rambam rules like R. Yehudah. The four Amos are with the diagonal, like the Rema says (408:1). This refers to Tiltul. We cannot say so about carrying four Amos, which is a haphazard occurrence, like Rashi says (DH Hen). (He did not fix his place to carry. Later, he might carry elsewhere!)

Rambam (19): One may carry an item up to four Amos. Between four and five and three fifths, [it is forbidden, but] he is exempt. For more than this he is liable, for it is the diagonal of four Amos.

Ra'avad: One may move less than four Amos only in pressed circumstances. And then, up to five and three fifths is permitted!

Rambam (14:1): Reshus ha'Yachid is a mound at least 10 Tefachim tall and four by four Tefachim wide...

Rashi (45a DH Hayu): If two people are six Amos from each other, two Amos of each's Techum are swallowed in the other's.

Rashi (51a DH Hen): We give the diagonal, i.e. another eight fifths of an Amah, to all Shovsei (cessations of) Shabbos. Regarding carrying, the only way to give corners is to give the diagonal, for square and circle do not apply.

Tosfos (51a DH ka'Zeh): The Rashbam says that if one carries four Amos [in a pure direction, e.g. precisely] from east to west, or from north to south, he is liable. If he carried on a diagonal [northeast, northwest...], he is liable only for four Amos and the diagonal (another Amah and three fifths). If a pillar in Reshus ha'Rabim is 10 [tall] by four, it is Reshus ha'Yachid only if its diagonals face like the world's diagonals (northeast - southwest, and northwest - southeast). If the pillar's diagonals are east - west and north - south, it is not Reshus ha'Yachid [unless it is four and the diagonal]. R. Tam adds the diagonal in every case. The pillar must be five and three fifths Tefachim square. The Gemara says four, for from this we derive the diagonal. I disagree. We learn corners from the Leviyim's cities. We give corners only according to the world's corners.

Rosh (4:13): The Rashbam says that it depends on which direction he carried in. A pillar must have four Tefachim and their diagonal along the world's diagonal to be Reshus ha'Yachid. We say so about the Techum of a city (56a). R. Tam always adds the diagonal, for throwing, carrying, and Techumim. One gets 2000 Amos and their diagonal in every direction. Shabbos 8a discusses a basket six wide. Since it is round, one might have thought that the diagonal is not required.

Rebuttal (Rosh): In Yoma (67a), we say that there were 10 huts [along the path to send the goat to Azazel] and it was 12 Mil. The Shali'ach was able to go through an Eruv (one Mil from the last hut). This shows that Techum Shabbos is only 2000 Amos. Perhaps just like we always call Techum Shabbos 2000 Amos, even though we add the diagonal, there also we say Mil [but it means a Mil and its diagonal]. However, we learn corners from the Leviyim's cities. We give corners only according to the world's corners.

Poskim

Shulchan Aruch (OC 345:2): Reshus ha'Yachid is a place surrounded by walls 10 tall and at least four by four wide.

Rema: Some require four by four and the diagonal, like we will explain in 349:2.

Gra (DH Yesh): This refers to R. Tam, In Siman 349, we hold like R. Tam. There is no proof from there to here. Also the Rambam holds like R. Tam, but here he disagrees. Also the Rosh rejected R. Tam. The Tur mentioned R. Tam only in Siman 349, but not here or regarding Techumim. See the Rema (408:1). Rather, we hold like Rashi. Based on what I wrote there, the Rema is like R. Tam and the Magid Mishneh, but the Halachah does not follow them.

Mishnah Berurah (6): There, all obligate for carrying in Reshus ha'Rabim only for four Amos and their diagonal. If so, here we require four Tefachim and their diagonal. Eliyahu Rabah explains that the Levush omitted this opinion, for it is an individual's opinion. All the Rishonim disagree, and say that a square four by four is Reshus ha'Yachid. This is unlike carrying in Reshus ha'Rabim. Also the Gra rejected this opinion.

Shulchan Aruch (349:2): The four Amos are with the diagonal, so in all it is five and three fifths. Some forbid and exempt from four Amos until five and three fifths.

Beis Yosef (DH v'Elu): Tosfos and the Rosh brought from R. Tam like Rashi, that we add the diagonal for carrying in Reshus ha'Rabim. The Rambam agrees. The Rashbam says that it depends on in which direction he carried. Why did the Tur attribute the Rashbam's opinion to Rashi? Perhaps the text should say PR"SH (Roshei Tevos for Piresh R. Shmuel, i.e. the Rashbam). Mahari Avuhav struggled to explain how the Tur learned this from Rashi. I could not understand it.

Beis Yosef (DH veha'Rambam): In pressed circumstances, the Ra'avad permits carrying four and their diagonal, less Mashehu, but normally he forbids even less than four Amos.

Gra (DH Dalet, according to Damesek Eliezer): Really, this is not like Rashi. Rashi says so only regarding carrying, for there corners do not apply. Here (the four Amos around a person, in which he may move things), one can give corners, so he has only four square Amos, like the Rambam. Also Rashi explained like this on 45a. This is why the Tur said that R. Tam [who always adds the diagonal] is unlike Rashi. (The Beis Yosef did not understand why the Tur said so.)

Mishnah Berurah (7): The Acharonim agreed to this opinion.

Mishnah Berurah (9): The Gra holds that the Rambam agrees that one who picked four Amos may move things along the diagonal. He argues with the first opinion only regarding one who wants to move in a straight line, or did not select his four Amos, and allows him only four Amos.

Bi'ur Halachah (DH Hen): The Shulchan Aruch wrote Stam like R. Tam, who allows four and their diagonal in any direction. Even though Tosfos and the Rosh questioned this, the Rambam, Rashi, Rashba and Ritva agree with R. Tam.

Kaf ha'Chayim (11): When the Shulchan Aruch rules Stam leniently, and then brings 'some say' to be stringent, he holds that the first opinion is primary, but l'Chatchilah he is concerned for the latter opinion.

Bi'ur Halachah (DH shemi'Dalet): The Rambam permits exactly four Amos. If he decrees about carrying on a diagonal lest one carry in a pure direction, he should forbid exactly four!

Note: Perhaps we do not forbid four, for it is not common to carry exactly in a pure direction. If one carries anything but due east (or due north, south or west), the Shi'ur is more than four Amos.