Sukkah, Chapter 1, Mishnah 10. In Mishnah 6 we mentioned the concept of dofen akumah (a bent wall), a halachah given to Moshe at Sinai which teaches that when invalid sechach is found at the side of the sukkah, at the point where the wall meets the sechach, the sukkah remains valid for up to four amos of invalid sechach. One may not sit beneath the invalid sechach itself, since we view the wall as though it bends over and meets the valid sechach that is set away from it, separated from it by the invalid sechach.
Our Mishnah brings three examples of this law:
The first case - "Bayis shenifchas v'sichech al gabav":
A house in which a breach was made, a hole in the middle of the ceiling, and he placed sechach over that breach. Says the Mishnah: "Im yesh min hakosel lasichuch arba amos - pesulah" - if there are four amos from the wall to the sechach, it is invalid. That is, if the distance from the valid sechach to the wall is four amos or more, the sukkah is invalid; and if it is less than four amos it is valid by virtue of dofen akumah, since we view the wall as meeting the valid sechach, and one may sit beneath it. This case is very common nowadays, among those who install a roof that opens in the middle of the living room in order to sit in a sukkah: the sechach is in the center, and on each side there is less than four amos of ceiling, and whoever sits beneath the sechach is sitting in a valid sukkah.
The second case - "V'chen chatzer shehi mukefes achsadrah":
A courtyard surrounded by a portico on at least three sides, and the portico has an ordinary roof. He placed sechach over the open part of the courtyard, and the sechach rests on the roof of the portico on all three sides. The result is that the valid sechach is in the middle, and around it on three sides is invalid sechach, namely the roof of the portico. Therefore, as long as the portico section is less than four amos, it is valid; four amos or more, it is invalid.
The third case - a large sukkah covered at its edges with something that is not fit for sechach:
A sukkah that is ordinary in every respect, except that he covered its edges with an item that is not valid for sechach. In the words of the Mishnah: "Im yesh tachtav arba amos - pesulah" - if there are four amos beneath it, it is invalid. If the area covered by the invalid sechach is four amos, the sukkah is invalid, and if it is less than that it is valid.
The order of the novelty in the three cases:
The Gemara explains that each of these three cases represents a greater novelty than the one before it:
A house in which a breach was made: the roof is attached to the wall, and therefore it is easy to understand the extension of the wall and to view the roof as a continuation of the wall. This is the smallest novelty of them.
A courtyard surrounded by a portico: here we view the roof of the portico and the walls as a single unit, and this is a less natural fit.
A large sukkah: the invalid sechach is not connected to the walls at all, and nevertheless the law of dofen akumah is applied to it. This is the greatest novelty of all.
In summary: the three cases in our Mishnah - a house in which a breach was made, a courtyard surrounded by a portico, and a large sukkah covered with something that is not fit for sechach - teach us the scope of the law of dofen akumah: invalid sechach adjacent to the wall for a span of less than four amos does not invalidate the sukkah, but one may not sit beneath it.