OPINIONS: The Gemara (77b) says that a ladder placed against a wall serves to lower the height of the wall to less than ten Tefachim (and, consequently, the top of the wall may be used by the residents of the Chatzer). The Gemara records several opinions as to how long the ladder must be:

Rav Yehudah in the name of Shmuel says that a wall that is ten Tefachim high needs a ladder that is 14 Tefachim long to lower the height of the wall.

Rav Yosef says that the ladder must be 13 Tefachim and a bit.

Abaye says that the ladder may be 11 Tefachim and a bit.

Rav Huna brei d'Rav Yehoshua says that the ladder may be even 7 Tefachim and a bit.

What is the reason for these measurements, and what are the Amora'im arguing about?

(a) RASHI says that the Amora'im are arguing about how close to the top of the wall the ladder must reach.

1. Rav Yehudah maintains that the ladder must reach the top of the wall. According to Rav Yehudah, the ladder must be 14 Tefachim, because not only is it necessary for the ladder to reach the top of the wall (for which a 10-Tefach ladder would suffice), it is necessary for the ladder to be placed at an angle so that one can climb up easily. The foot of the ladder must be on the ground four Tefachim away from the wall in order to achieve an ascendable slope, and thus the length of the ladder must be 14 Tefachim (10 for the height of the wall, and 4 for the distance from the wall to the foot of the ladder).

2. Rav Yosef requires the ladder to reach within a Tefach of the top of the wall, since a Tefach is a negligible distance. He agrees that the foot of the ladder must be 4 Tefachim away from the wall.

3. Abaye requires the ladder to reach within 3 Tefachim of the wall, because within 3 Tefachim, Lavud works to make it as if the ladder is at the top of the wall. Abaye, too, agrees that the foot of the ladder must be 4 Tefachim away from the wall.

4. Rav Huna brei d'Rav Yehoshua agrees with Abaye that the ladder must reach only within 3 Tefachim of the top of the wall. He maintains, however, that it suffices to place the ladder vertically against the wall, and it does not need to be placed 4 Tefachim away.

TOSFOS (DH Tzarich) points out an inaccuracy in the mathematical calculations of Rashi's explanation. Rashi says that according to Rav Yehudah, the ladder must be 14 Tefachim long because it must reach the top of the wall which is 10 Tefachim high, and its foot must be 4 Tefachim away from the wall. Such a ladder, though, would be considerably less than 14 Tefachim high. It would need to be only the length of the diagonal of a rectangle whose sides are 10 and 4 Tefachim, which is approximately 10.8 Tefachim! The foot of a 14-Tefach long ladder, when its top is placed at the top of the 10-Tefach high wall, would reach 10 Tefachim away from the wall, and not 4 (because 14 is the diagonal of a square whose sides are 10 Tefachim long).

How are we to understand the words of Rashi?

Rashi seems to be consistent with his opinion as expressed elsewhere (Shabbos 85a, Eruvin 5a, 76b, 94b). Rashi seems to maintain that the Rabanan established that the length of a diagonal, for any Halachic application that requires measuring the diagonal of a rectangle or square, is the sum of the two sides that create the diagonal, rather than the actual length of the diagonal (see Insights to Eruvin 5:2 and 76:2:c).

(b) TOSFOS (DH Rav Yosef) disagrees with Rashi's explanation. Tosfos cites the RI who explains that the Amora'im argue not only about how high on the wall the ladder must reach, but they also argue about how far away from the wall the foot of the ladder must be placed. They all agree, however, that the ladder must be distanced from the wall the same distance as the height that the ladder must reach. (That is, the ladder forms the hypotenuse of a right triangle.) When each Amora gives a measurement for the length of the ladder, he is not giving the exact length of the ladder; rather, he is giving a number to serve as a reminder for how high on the wall (and how far away from the wall) the ladder must reach.

1. Rav Yehudah requires the ladder to reach the top of the wall, a height of 10 Tefachim. Therefore, he requires that the foot of the ladder be 10 Tefachim away from the wall. The length of the ladder, then, must be 14 Tefachim.

2. Rav Yosef requires the ladder to reach within a Tefach of the top of the wall, and therefore the foot of the ladder must also be just over 9 Tefachim away from the wall. The diagonal of a square with those sides is 12.6 Tefachim. Thus, Rav Yosef requires a ladder whose length is a bit more than 12.6 Tefachim.

3. Abaye requires the ladder to reach within 3 Tefachim of the top of the wall, and therefore the foot of the ladder must also be just over 7 Tefachim away from the wall. The diagonal of a 7 X 7 square is approximately 9.8 (using the calculation of the Chachamim, 1.4 x S), and thus Abaye requires the ladder to be a bit more than 9.8 Tefachim.

(Why, though, does Rav Yosef say that the ladder must be "13 and a bit" when it needs to be only 12.6 and a bit, and why does Abaye say that it must be "11 and a bit" when it needs to be only 9.8 and a bit? The MAHARSHA explains that they are expressing the length of the ladder in terms of the height of the wall which the ladder must reach (as a "Siman b'Alma"). That is, Rav Yosef says that the ladder must reach just less than a Tefach from the top of the wall, and thus 14 (the length of Rav Yehudah's ladder) minus a little less than a Tefach is 13 and a bit. Abaye, too, says that the ladder must reach just less than 3 Tefachim from the top of the wall, and 14 minus just less than 3 equals 11 and a bit.)

4. Tosfos explains Rav Huna brei d'Rav Yehoshua's opinion like Rashi; the ladder needs to reach only within 3 Tefachim of the top of the wall, and it may be placed vertically against the wall.

QUESTION: The Amora'im maintain that a ladder placed against a wall must reach the top of the wall (or within 3 Tefachim of the top) in order to permit the residents in the Chatzeros on each side of the wall to use the area atop the wall, and to permit the two Chatzeros to make one Eruv together.

Why must the ladder reach the top of the wall? The Gemara earlier (77b) says that it is possible to diminish the ten-Tefach height of the wall by placing a platform (Itzteva) on the ground next to the wall. The platform certainly does not need to reach the top of the wall! Similarly, the Gemara earlier (78a) says that a ledge (Ziz) that protrudes from a wall can also decrease the ten-Tefach height of a wall. Why, then, must the ladder reach the top of the wall?

(a) TOSFOS (77b, DH Im) writes that the platform or ledge must be four by four Tefachim; it must be four Tefachim in both length and width. Such a ledge is considered a significant area in itself, and thus it diminishes the height of the wall even when it is not near the top of the wall. A ladder, though, is four Tefachim in only one dimension (width). Therefore, it must reach the top of the wall in order to decrease the height of the wall.

(b) RASHI (77b, DH Im Yesh) does not accept this reasoning. He maintains that a ledge diminishes the height of a wall even when the ledge is four Tefachim in only one dimension. How, then, does Rashi answer our question?

The ROSH cites the MAHARAM MI'ROTENBURG (who follows Rashi's ruling regarding a ledge) who answers that the Gemara earlier is discussing how to decrease the height of the wall so that the Chatzer on the side of the wall where its height has been diminished obtains the exclusive right to use the wall. Once the height has been diminished on that side, the Chatzer on that side is considered to have easy access to and use of the wall. Diminishing the height by placing a platform or a ledge along the wall suffices for this purpose.

In contrast, the Gemara here is discussing making a Pesach, an entranceway, in the wall so that the Chatzeros adjoining the wall on its two sides may make a single Eruv Chatzeros together. In order to be considered a Pesach, the ladders on the two sides of the wall must reach the top of the wall, and not just diminish its height to less than ten Tefachim.