Gemara

52b (Mishnah): If someone went even one Amah outside the Techum, he may not re-enter;

R. Eliezer says, if he went two Amos outside he may re-enter, but not if he went three Amos outside.

(Mishnah): If one was traveling and he was outside the Techum when it became dark, even if he was only one Amah outside, he may not enter;

R. Shimon says, even if he was 15 Amos outside he may enter, for the Techum marker is not at the full 2000 Amos, due to people who err.

41b (Neharde'ai): (If one has only four Amos, but he needs to eliminate, he may leave them to find a covert place.) If he is clever, he will enter the Techum to eliminate, and once he entered, he may stay

43b: Nechemyah brei d'Rav Chanilai was engrossed in his learning and walked outside the Techum. Rav Chisda told Rav Nachman 'your Talmid is in pain.'

Rav Nachman: Have people stand in lines to make Mechitzos, and he may re-enter the Techum.

Question (Rav Nachman bar Yitzchak): What was Rav Chisda unsure about?

Did he have enough people [to make Mechitzos from Nechemyah until the Techum], but he was unsure whether or not the Halachah follows R. Gamliel [who says that Mechitzos permit even one who was not Shoves inside them]?

Or, did he lack people? (The Mechitzos would reach only to within two Amos of the Techum), and he was unsure whether or not the Halachah follows R. Eliezer?

Answer (Rava): He lacked people. Surely, he knew that Rav rules like R. Gamliel!

Support: Rav Nachman said '...and he may re-enter.' (I.e. from two Amos outside, he may enter without Mechitzos!)

44b: Rav Simi holds that overlap of Techumim is significant. Rabah holds that it is not significant.

Question (Abaye, to Rabah): You must agree that overlap of Techumim is significant, even if he was not Shoves within those Mechitzos!

(Mishnah - R. Eliezer): If one went two Amos outside the Techum, he may re-enter.

This is because his Techum overlaps his original Techum, i.e. overlap is significant!

Rabah bar bar Chanah (to Abaye): Why do you challenge Rabah from R. Eliezer's opinion? Chachamim argue with him!

Abaye: Rabah taught that Chachamim forbid only one who left for Reshus. They agree about one who left for a Mitzvah!

Rishonim

Rif (12a): Abaye proved from R. Eliezer that overlap of Techumim is significant. Rabanan argue only about Reshus, but they agree about a Mitzvah. Rabah did not answer. Therefore, the Halachah follows Rav Simi, that overlap of Techumim is significant.

Rif (15b): The Halachah follows the first Tana.

Ba'al ha'Ma'or (15a DH v'Ha): The Rif says that the Halachah follows the first Tana. Some say that the Halachah follows R. Eliezer, who permits one who went two Amos outside to re-enter. They learn from the episode with Nechemyah. I say that he was permitted because he left due to Ones, i.e. he was engaged in learning, and the wall of people reached only to within two Amos of the Techum. Therefore, Rav Nachman permitted, like R. Eliezer. Without Ones, we do not rely on R. Eliezer. Rav Chanan asked Abaye why he asks from R. Eliezer. This shows that we do not rely on him.

Milchamos Hash-m: The Rif rules like the first Tana who argues with R. Shimon. He did not rule about the argument of R. Eliezer and Chachamim. Even so, surely the Halachah follows Chachamim. The Rif wrote that Rabanan argue only about Reshus. They agree with R. Eliezer regarding Mitzvos. This shows that the Halachah follows Chachamim. The Gemara concluded that Rav Nachman permitted Nechemyah based on R. Eliezer, for if there were enough people, why was Rav Chisda unsure? This is rejected, for Rav Nachman bar Yitzchak challenged Rava regarding a Mechitzah of people, and whether they may be aware that they are making a Mechitzah. This is why Rav Chisda was unsure! [Rav Nachman] said "he may enter" to shows that the Mechitzah helps even though they made it to enable him to reenter. The Rif said "he said 'make a Mechitzah of people, and it is as if he is in a Dir.'" This shows that there were enough people, and the Mechitzah was valid, like R. Gamliel. R. Eliezer and Chachamim argue about overlap of Techumim only when one is in a second Techum, from which he may not go the first Techum, even though they overlap. E.g. if one goes one Amah outside the Techum, Chachamim enacted for him four Amos. He may not leave them, for this is his Techum. Even if it ends within the first Techum, he may not leave [the second Techum]. This is like one who measures [2000 Amos]. Even if his measure ends in the middle of a city, he has only half the city. R. Eliezer holds that since the Techumim overlap, it is as if he never left his Techum. Had he left the latter Techum with Reshus and returned to the first Techum, all would permit. This is like Neharde'ai, who say that if he is clever, he will enter the Techum, and once he entered, he may stay. This is like the Mishnah (41b) says, that if Nochrim returned him, it is as if he did not leave. This is why R. Chananel wrote it, unlike Rashi, who say that Rava argues with Neharde'ai.

Rambam (Hilchos Shabbos 27:11): One who left the Techum, even one Amah, may not enter. His four Amos begin where he stands. Therefore, once he left the Techum an Amah or more, he stays in his place and may move only in four Amos, from where his legs stand and outside.

Rebuttal (Ra'avad): We hold that overlap of Techumim is significant, whether he left for a Mitzvah or due to Ones. If he is clever, he enters the Techum, and once he entered, he entered. Therefore, if he went one Amah outside, he may walk four Amos in any direction, and re-enter!

Magid Mishneh: The Rif rules like Chachamim against R. Shimon, who permits 15 Amos past the Techum. The Acharonim rule that he has four Amos in every direction, and like R. Shimon, who allows another 15 Amos. If so, even if he is 19 Amos outside the Techum, he may enter. The Rashba holds that [nowadays] those who measure are precise, so the law of another 15 Amos does not apply. The Rambam holds like the Ge'onim.

Poskim

Shulchan Aruch (OC 405:1): One who left the Techum, even one Amah, may not enter to be like a resident. He has only four Amos from where his feet stand and outside.

Beis Yosef (DH Mi): The Rashba rules like R. Shimon, but we hold like the Rambam, who rules like the first Tana.

Magen Avraham (1): The Shulchan Aruch addresses the Ra'avad's question, that we hold that overlap of Techumim is significant, if he left for a Mitzvah or due to Ones! The Magid Mishneh admitted that his answer is difficult. I say that one may be lenient if there was Ones. All the Poskim hold like this. However, [if he left] for Reshus, all agree [that we are stringent].

R. Akiva Eiger: Tosfos in our Sugya holds that even regarding Ones, overlap of Techumim does not help. Siman 406 (one who has only four Amos, and needs to eliminate) is different, for he enters a covert place. Since he already left the four Amos that Chachamim gave to him, he should get four other Amos. Since they are in his first Techum, they do not give new Amos, rather, he gets back his first Techum (Tosfos 44a DH Oh). Here, he did not lose his four Amos, therefore he may not enter. Below (407:2, if one left for a Mitzvah, and found out that it was already done, he may return if his new Techum overlaps his initial Techum), we can say that a Mitzvah is different, like we say on 45a. We can say that Ones is unlike a Mitzvah, and when he did not lose his four Amos, he may not enter.

Gra (DH Mi, according to Damesek Eliezer): The Shulchan Aruch is like the first Tana (of R. Eliezer). We say (44a) that Rav Nachman holds like R. Eliezer [since he permitted Nechemyah to reenter]. This is when he left b'Ones, like it says on 45a. 'Why do you challenge Rabah from R. Eliezer' shows that the Halachah does not follow him [if he left intentionally. Ba'al ha'Ma'or explains like this.] Milchamos Hash-m rejects the Sugya on 44a, due to [Rav Nachman bar Yitzchak's questions on] 44a [about Mechitzos of people. We can say that Rav Chisda was unsure about this, but there were enough people]. The Ba'al ha'Ma'or is correct. (The Gemara proved that there were not enough people from the words "he will enter.")

Gra (DH me'Amidas): Rashi explains that he left b'Mezid. He and the Ra'avad explain that we say that a Mitzvah is different (45a), and also Ones is different. The Rambam holds that even if he left b'Ones [he may not reenter].

Mishnah Berurah (1): One might have thought that one Amah is not called leaving, so he is still like people of his city.