Gemara

(Beraisa): "El Mul Pnei ha'Menorah Ya'iru Shiv'as ha'Neros" teaches that the six lamps of the branches are tilted towards the middle lamp.

R. Noson says, this teaches that the middle is most praiseworthy.

Megilah 21b (Mishnah): On Monday, Thursday and Shabbos Minchah, three people read (at least 10 verses) from the Torah.

(Rava): Whichever of the three people reads four verses is praiseworthy.

We find a preference for the first regarding the three boxes of (half-) hekalim of Terumas ha'Lishkah;

(Mishnah): The boxes are labeled so we will know which is first, for the Mitzvah is to buy Korbanos Tzibur from the first box.

We find a preference for the middle regarding the branches of the Menorah;

(Beraisa): "El Mul Penei ha'Menorah Ya'iru Shiv'as ha'Neros" teaches that the six lamps of the branches are tilted towards the Ner Ma'aravi, and the Ner Ma'aravi is tilted towards the Shechinah (in the west).

(R. Yochanan): This shows that the middle is preferable.

There is a preference for the last because we ascend in Kedushah, and do not descend.

A case occurred in which the first person read four verses, and Rav Papa praised him.

Yoma 37a (Beraisa): When three walk together, the Rav is in the middle, the next biggest on his right, and the smallest on his left.

When the three angels visited Avraham, Micha'el was in the middle, Gavri'el was on his right, and Uri'el was on his left.

(Rav Shmuel bar Papa): The ones on the sides cover the Rav (they are in back of him).

Question (Beraisa): One who walks even with his Rebbi is crude. One who walks in back of his Rebbi is haughty.

Answer: The ones in back are to the sides. They cover the Rav only slightly, on the sides.

Shabbos 51b: Rabah bar Rav Huna and Levi brei d'Rav Huna bar Chiya were riding together on donkeys. Levi's donkey went in front of Rabah's. Rabah (who was a bigger Chacham) was upset. Levi sought to (show that it was unintentional to) appease him.

Berachos 47a: Ravin and Abaye were traveling. Ravin's donkey went ahead of Abaye's, yet he did not tell Abaye to go first (even though Abaye was greater). When they came to the entrance to the Beis ha'Keneses, Ravin asked Abaye to enter first.

Abaye: Until now, you did not consider me to be above you!

Ravin: R. Yochanan taught that we honor only at an opening with a Mezuzah.

Question: If so, we do not honor at the entrance to a Beis ha'Keneses or Beis Midrash!

Correction: Rather, we honor only at an opening worthy to have a Mezuzah.

Rishonim

Rambam (Hilchos Talmud Torah 6:5): When three go on the road, the Rav is in the middle, the next biggest on his right, and the smallest on his left.

Hagahos Maimoniyos (2): In Berachos we say that we honor only at an opening worthy to have a Mezuzah. R. Tam answered that this is when everyone goes on his own way. One may rush to go on his way. When they go for one matter in one group, we honor everywhere, like it connotes here. Also Shabbos 51b connotes like this.

Rashi (98b DH mi'Kan): R. Noson teaches that when three people read 10 verses from the Torah, the middle reads four, and the others read three each, because the middle is most praiseworthy.

Tosfos (DH sheha'Emtza'i): This is difficult, for in Megilah we bring also other Beraisos showing that the first and last are praiseworthy! Rather, here R. Noson teaches that when three walk together, the Rav is be in the middle, like it says in Yoma. Megilah brings the Beraisa #1 because since we find that the middle is praiseworthy for other matters, it is good that he read four verses.

Rashi (21b DH Rishon): If the first did not read four (verses), and the second read four, or if neither of them read four, and the third read four, he is praiseworthy. If there is room in the Parshah and all of them read four, all of them are praiseworthy.

Poskim

Shulchan Aruch (OC 137:2): Everyone reads at least three verses. Two read three, and one reads four. Whichever reads four is praiseworthy.

Magen Avraham (1, citing Maharshal): Nowadays that a Shali'ach Tzibur reads for everyone, he does not read four for the middle, lest he embarrass the last.

Rebuttal (Bach 2): The Gemara says that whoever reads four is praiseworthy. The Tur wrote 'we read with each...' to teach that even though in the days of the Gemara everyone read for himself, also nowadays that a Shali'ach Tzibur reads for everyone, he can read four for anyone. I do not know a source for the Maharshal.

Eliyahu Rabah (1): The Prishah brought the Maharshal's actual words - 'he reads four only for the first or last, but not for the middle, lest he embarrass others.' Others cited him incorrectly. Rashi connotes that l'Chatchilah, the first should read four. If so, nowadays that a Shali'ach Tzibur reads for everyone, we should not embarrass the first (and read only three for him). Similarly, if he did not read (four) for the first, he should read for the second, but not (wait) for the third, for the second has an attribute like the first.

Magen Avraham (2): Regarding Ner Chanukah we say that we ascend in Kedushah, and do not descend. We do not say that the first is praiseworthy! Regarding one person, he should ascend in Kedushah. Here, the three people do not depend on each other.

Chasam Sofer: When there are three, we had a reason to say that the first r middle is praiseworthy. Without a reason, we increase in Kedushah even when there are many people! In Parshas Ha'azinu (on Shabbos morning), the first three read six verses each, and the fourth reads 10, to fulfill 'we increase in Kedushah', and the middle (fourth) reads the most.

Eshel Avraham (1): Eliyahu Rabah explains that the Maharshal forbids intentionally waiting to read four for the middle. If one (reads three for each of the first two, and) waits for the third, he embarrasses the first two. The Magen Avraham said that Maharshal forbids only if the middle reads four. This is like the Magen Avraham's own opinion, that the last is more esteemed, like we find regarding Ner Chanukah. Also the first is (more) esteemed. Even though the Kohen is more esteemed, the Gemara said that whoever reads four is praiseworthy. Why did the Gemara say that it is praiseworthy for the last to read four? In any case he must read four, so that there will be a total of 10 verses! It seems that the difference is if someone else (e.g. a Shali'ach Tzibur) reads. (I.e. l'Chatchilah he may choose to read four for the last - PF.) This is unlike the Magen Avraham says in the name of Maharshal. It seems that the custom is to read more for the third than for the Kohen and Levi, like the Magen Avraham says (regarding one person, i.e. a Shali'ach Tzibur who reads for everyone).

Machatzis ha'Shekel (1): The Maharshal holds that people know the attributes of the first and last, so there is no disgrace if a Shali'ach Tzibur reads more for the first or last. Not all know the attribute of the middle, so people will think that it is due to the honor of the Oleh (the person called to the Torah), so this disgraces the others. When the Oleh himself reads four, people know that he honors himself, so it does not disgrace the others.

Magen Avraham (2): Whoever read more is more praiseworthy.

Eshel Avraham (2): At first I understood the Magen Avraham to say that if there are 12 verses in the Parshah, if one read six (and left only three for the others), this is most praiseworthy. This is unlike Rashi, who says that it is best for all of them to read four! Later I realized that the Magen Avraham merely explains that it is more praiseworthy to read more than three, but there is no disgrace to read only three.

Prishah (3): When there is a double Parshah, some are particular that the Kohen read only one Parshah, for we do not change the beginning. This is wrong. Our Gemara says that it is praiseworthy also if the first reads four! There is no reason to minimize what the Kohen reads. He should read two Parshiyos (i.e. what is normally two Aliyos).

Eliyahu Rabah (1): The Prishah learned from the end of Rashi (when there is room, if all read four, all are praiseworthy).

Mishnah Berurah (4): Nowadays that a Shali'ach Tzibur reads, some say that he reads three for the first two, and four for the last. Others disagree. Nowadays the custom is that on Shabbos (at Minchah), Monday and Thursday, we read the entire first Aliyah of the coming Shabbos morning. Sidurim say where each Aliyah ends. One may deviate, as long as everyone reads at least three verses.

Kaf ha'Chayim (4): The Shulchan Aruch connotes that there is no difference whether everyone reads for himself or a Shali'ach Tzibur reads. Anyone may read four.

Shulchan Aruch (YD 242:17): When three walk together, the Rav is in the middle, the next biggest on his right, and the smallest on his left.

Beis Yosef (DH veha'Rambam): The Rambam brought the Beraisa, but not what the Gemara said about it. This is because it is obvious; everyone fulfills it due to Derech Eretz. He needed to teach how three walk, lest one think that they walk one after the other in a line. It is better to say that he relied on what he wrote regarding prayer, that one may not pray in front of (or even with or directly in back of) his Rebbi. He holds that the Isur to pray in back of one's Rebbi is due to Derech Eretz (and not because it looks like he bows to him), therefore we can learn from there to here.

Rema: The Gemara says that we honor on the road only at an opening appropriate for a Mezuzah. This is when everyone goes n his own way. When they are one group (traveling together), we honor on the road. If there is danger, we do not honor at all.

Gra (58): (The last words of) the Rema answer(s) Tosfos' question (why Ravin did not honor Abaye on the road).

Birkei Yosef (24, citing Teshuvas ha'Ge'onim): The greatest is first to leave or enter. After he leaves or enters, those after him need not honor each other.

Shiyurei Berachah (13): Ha'Rav ha'Ashkenazi questioned this. It is unreasonable. Chachamim distinguished between a greater and lesser Talmid, even regarding where they sit to eat! I say that Teshuvos ha'Ge'onim are concise, and do not explain the reason. It discusses when none of the others has an attribute over the others. If there is no Gadol, they honor each other. If there is a Gadol, it is the Gadol's honor that they honor only him, just like we do not honor a Talmid in front of his Rebbi.