Gemara

13b (Beraisa #1): One may write Torah, Nevi'im and Kesuvim on the same Klaf (parchment). He leaves enough (blank Klaf) at the beginning to wrap around a roller, and enough at the end to surround (the Sefer).

Contradiction (Beraisa #2): He leaves enough at each end to roll.

Question: If he leaves enough to wrap around the pole, this is not enough for the end that must surround! If he leaves enough to surround, this is more than needed for the end wrapped around a pole!

Answer #1 (to both questions - Rav Nachman bar Yitzchak): He leaves what is needed at each end.

Answer #2 (Rav Ashi): Beraisa #2 discusses a Sefer Torah;

(Beraisa): Other Seforim are rolled (on one pole) from the beginning to the end. We leave a Sefer Torah (open) in the middle, and roll from poles at both ends.

(Beraisa): (L'Chatchilah,) the height of a Sefer Torah should equal its circumference when rolled.

(Rebbi): On Gevil (untanned hide), it should be six Tefachim.

Rav Huna wrote 70 Sifrei Torah. For one of them, the height equaled its circumference.

Question (Beraisa - R. Meir): Moshe's Aron was 15 Tefachim long. The two Luchos occupied 12 Tefachim. The thickness of the two walls occupied one Tefach. The (width of the) Sefer Torah fit in the remaining two Tefachim.

If the circumference is six Tefachim, the width is two Tefachim. We roll a Sefer Torah from both ends, so the width is even more!

Answer: The Sefer Torah in the Aron was rolled to the beginning.

Question: Still, if its width is exactly two Tefachim, it does not fit properly in two Tefachim!

Answer (Rav Ashi): A small amount of Klaf was left unrolled (so the width of the rolled part was less than two). It rested above the Torah.

Maseches Yadayim (Mishnah 3:4): The margins at the beginning and end of a Sefer are Metamei the hands;

R. Yehudah says, the end margin is Metamei only there is a pole.

Rishonim

Rif: One leaves empty at the beginning (of a full Tanach) to roll onto a pole, and enough at the end to surround. All (other) Seforim are rolled from the beginning. A Sefer Torah is rolled in the middle. One makes a pole at each end.

Rambam (Hilchos Sefer Torah 9:2): One leaves at the beginning (of a Sefer Torah) and end enough blank hide to wrap onto the pole. This is the Mitzvah. If one left more or less, it is not Pasul.

Rosh (1:52): The circumference is six Tefachim including the poles. Presumably, the poles were in the Sefer in the Aron, and its circumference was six. The thickness of the poles must be fixed, for in only one Sefer of Rav Huna, the circumference equalled the length. If one may make poles of any thickness, in this way he could have made them equal! If they removed the poles before the Sefer was put in the Aron, we can say that the width is two without the poles.

Rashba (14b DH Od): Even if the Sefer was rolled to the end, the width was more than two due to the poles! It seems that the circumference is six with the poles. If so, why did Rav Huna succeed to equate the circumference to the height only once? We must say that average poles must be used.

Note: Seemingly, one must attach a pole in order to roll the Sefer taut to measure its circumference. Perhaps it is improper to change poles just to make the height and circumference equal!

Rashash (14b DH Gemara): The sum thickness of the Klaf equals the area of a circle of diameter 2 when it is rolled to the beginning. It equals the area of two circles of diameter (the square root of 2, about) 17/12 when it is rolled to the middle, so then the total width exceeds 2.

Question (Tosfos Eruvin 14a DH veha'Ika): The width was less than 2, for it is less than a third of the circumference (the ratio is 1/Pi, about 7/22)!

Answer (Chasam Sofer 14b DH Trei): The pole on which it is wound (was about one part in 11 of a Tefach, and) completed the two Tefachim.

Note: The Gemara asked how they could remove the Sefer Torah if its width was exactly two. Perhaps it was a miracle; the entire Aron did not occupy any space (99a)! We can say that we do not rely on miracles.

Rosh and R. Shimshon (Yadayim 3:4): In Bava Basra, we say that we leave at the beginning enough to roll onto a pole, and at the end to surround, i.e. the entire Sefer. R. Yehudah connotes that it is wrapped around a pole at the end! This is more reasonable, for if it were wrapped at the beginning, one who wants to read would need to roll it to the beginning!

Ramban (14b DH ul'Fi): The Rif says that there was enough blank at the end to surround the pole. Writing there would be prone to be erased.

Tosfos (13b DH v'Oseh) and Rosh (ibid): Our text says that (other) Seforim are wound from start to end. It says oppositely in Maseches Sofrim! Also, our Gemara says that the Sefer Azarah was rolled to the beginning. Rashi explains that it was rolled from the beginning; the Gemara connotes otherwise. A Mezuzah is rolled from Echad to Shma, i.e. to the beginning. The Yerushalmi explicitly says that the pole for a Sefer Torah is at the end. Therefore, the Ri says that the correct text is 'we leave enough at the beginning to surround, and at the end to wrap around a pole.' The Rashba defends our text. Normally, the pole is at the end. Here, it would be a disgrace to wrap Torah around Nevi'im and Kesuvim. It would seem like the Torah is a covering to guard them from filth!

Hagahos Maimoniyos (Hilchos Megilah 2:50, citing R. Tam): One leaves at the end of Megilas Esther enough to wrap on a pole, and at the beginning to surround the Megilah.

Rashbash (579): The poles of Seforim are just for beauty. A Sefer Torah is not disqualified due to (lack of) them. Even so, a Megilah for the Rabim should have a pole, due to "Zeh Keli v'Anvehu."

Poskim

Shulchan Aruch (OC 692:2): Megilas Esther must have a roller at one end, and enough blank Klaf at the other end to wrap around it.

Beis Yosef (DH u'Mah she'Chosav): The Tur says that Megilas Esther... needs a pole at the beginning and blank Klaf at the other end to surround it, like a Sefer Torah. 'Like a Sefer Torah' refers to what he said above (the ink and Klaf...); regarding poles, it is like a Tanach (a Torah with Nevi'im and Kesuvim). Our custom is to make a pole only at the end! This is like Tosfos. We roll a Megilah from the end, like other Seforim. Why did the Tur omit the opinion of the Rosh, Tosfos and R. Shimshon? A Megilah is like a Sefer Torah. Since it needs only one pole, it is like a Tanach. The Rosh and R. Shimshon defended our text, which says that a Tanach is rolled to the end. This is difficult. A Tanach is different, lest Torah be a cover for Nevi'im and Kesuvim! This does not apply to Megilah; it should be rolled from the end, like other Seforim. Therefore, the custom is is unlike the Tur.

Bach (DH u'Mah): The Tur merely cited R. Tam. He did not say that he rules like R. Tam against the Rosh and Tosfos! A Sefer Torah that we do not read from regularly, e.g. Sefer Azarah, which was used only for Hakhel and on Yom Kipur, needs only one pole and is rolled to the beginning. The same applies to Megilas Esther, which we read only on Purim. The text of R. Tam in Hagahos Maimoniyos should say '... at the beginning enough to wrap on a pole, and at the end to surround.' The Tur and Maharil cite R. Tam this way. Why did the Tur omit the Rosh's opinion? The Rosh did not mention Megilah. Perhaps he holds that we roll other Seforim from the end, to enable reading from the beginning without unrolling. One who reads the Megilah must spread it open in any case; it is better to roll it to the end, so the end will cover the beginning. However, why did the Tur and Rambam omit this also regarding a Sefer Torah? It would seem that we follow the Rosh, Tosfos, R. Shimshon, and also Sefer ha'Terumos and Semag. However, our custom is not to make a pole at all.

Note: The Tur's Kitzur Piskei ha'Rosh (1:52) says to make a pole for a Tanach at the end.

Rema: The custom is not to make a roller at all at the end.

Taz (2): Because Rishonim argue about whether to make the pole at the beginning or end, the custom is not to make a pole at all.

Gra (20): This is astounding, and requires great investigation.

Question (Mishbetzos Zahav 2): They argue about the poles, and similarly about leaving blank Klaf. If there is a Safek whom we follow, we should leave blank at the beginning and end! Also, we roll from the end, like one opinion. We could make two poles and fulfill both opinions!

Answer (Levushei Serad 8.9): Due to the Safek, we are passive and do not make a pole at all. We leave blank at the beginning to cover the Megilah, for Kavod of Kisvei ha'Kodesh. The Rema connotes like this. We rely on the opinion that it is rolled from the end, but not to make a pole at the end. The argument is only when there is a pole and blank Klaf. If there is only blank Klaf, we are not particular about where it is.

Mishnah Berurah (15): We leave blank Klaf at the beginning, and roll it from the end.

The Steipler initially said that Nevi'im should have one pole. He later said to make two poles, lest people touch the Klaf (Orchos Rabbeinu 3 p.216).

Rema (YD 278:2): If the Klaf was sewn to the pole with silk, if one can use sinews instead, one may not read from it until fixing it.

Question (Dagul me'Revavah): Sefer ha'Azarah had only one pole! The Rambam does not list poles among things that are essential!