Eiruvin, Chapter 5, Mishnah 7. This mishnah deals with several details of making an eiruv techumin when a person is located outside the city.
One who was east of his city and told his son to place the eiruv to the west:
"Mi shehayah bamizrach ve'amar livno: arev li bama'arav, bama'arav ve'amar livno: arev li bamizrach" - a person standing outside the city, on its eastern side, who asks his son to place an eiruv on his behalf on the western side of the city; or the reverse: he is standing on the western side and asks his son to place an eiruv on the east of the city. In both of these situations we must distinguish between two cases:
"Im yesh heimenu ul'veiso alpayim amah ul'eiruvo yoser mikan - mutar l'veiso v'asur l'eiruvo" - he is outside the city and outside his house, but within two thousand amos of his house, while he is more than two thousand amos away from the eiruv he asked his son to place. That eiruv cannot help him, since he has no way of reaching it on Shabbos - it is simply too far. Therefore he is permitted according to the boundary of his house: his central point is not where he is standing but his house, and he is forbidden according to his eiruv, since he cannot get to it and the eiruv is of no use to him.
"L'eiruvo alpayim amah ul'veiso yoser mikan - asur l'veiso umutar l'eiruvo" - he is within two thousand amos of the eiruv his son placed, but he is more than two thousand amos from his house. His house cannot serve as his place of rest and central point, because he cannot reach it on Shabbos; whereas the eiruv, which he can reach, becomes his central place for Shabbos, and from there he measures two thousand amos in every direction.
One point deserves precision: even though he asked his son to place an eiruv, and that eiruv turned out to be of no use, we do not say that by the very request he revealed that he did not wish to rest at his house. The default is that when the eiruv he set up through his son does not take effect, his mind and intention are to rest at his house, and therefore his house is the axis around which the two thousand amos are measured in every direction.
"One who places his eiruv within the outskirts of a city":
"Hanosein es eiruvo b'ibburah shel ir - lo asah v'lo klum" - a person who wishes to place an eiruv outside his city, so that he will be able to travel farther on Shabbos than usual, and he places it within the outskirts of the city. The ibbur, as was mentioned in the previous mishnayos, is any place within seventy and two thirds amos of the perimeter of the city, and the measurement of the two thousand amos begins only after it. Since he placed his eiruv within the ibbur, he is still in the city, and he has not made an eiruv outside his boundary.
"Nesano chutz lat'chum, afilu amah achas - mah shenischar hu mafsid" - the intent here is not literally outside the boundary, but outside the ibbur. He placed it outside the ibbur, as one who places an eiruv should, even by just one amah - whatever he gains on one side of the city he loses on the other side.
To illustrate the matter, let us take an example of one thousand amos (even though the mishnah speaks of a single amah, and the law holds true even then): one who places his eiruv one thousand amos outside the city, his new central point is neither the city nor his house, but that spot one thousand amos outside the city. From that eiruv he can walk another two thousand amos - a total of three thousand amos from the city in that direction, instead of only two thousand amos. But in the opposite direction, where beforehand he could have walked two thousand amos, he can no longer do so; only one thousand amos remain to him on that side.
The Gemara: the area of the city is not measured:
The Gemara points out an important idea, which will also be relevant to the next mishnah: the city itself is not considered as taking up any space in the measurement. The entire city is like his four amos, and therefore he can count one thousand amos on its other side - three thousand amos on one side of the city, one thousand amos on its other side, and the area of the city itself is not measured at all but is entirely absorbed.
However, this law holds true only when the measurement of the two thousand amos from the eiruv toward the city brings the person out beyond the far side of the city; then the ruling of the mishnah is precise, that he has an additional one thousand amos outside the city.
And what if the two thousand amos from the eiruv end in the middle of the city?
The Gemara says: in such a case he may walk only up to that point, and he may not pass it. Only when the two thousand amos from the eiruv extend beyond the city is the city not counted in the measurement; but if they end in the middle of the city, he may walk within the city itself only up to that point.
The one exception to this, as Tosafos notes: if he actually slept inside the city, he is permitted to walk through the entire city, but he cannot walk anything beyond the city, since the two thousand amos of his eiruv end within the city. But if he does not sleep in the city, the law we stated applies, that he may not pass the point of two thousand amos from his eiruv.
In summary: in this mishnah we learned the details of an eiruv made outside the city: one who stands on one side of the city and asks his son to place an eiruv on the other side - if the eiruv is more than two thousand amos away from him, he is permitted according to his house and forbidden according to his eiruv, since the default is that his intention is to rest at his house; and if his house is more than two thousand amos away from him, he is forbidden according to his house and permitted according to his eiruv, and the eiruv is his place of rest. We also learned that one who places his eiruv within the outskirts of a city has accomplished nothing, and one who places it outside the ibbur - whatever he gains on one side he loses on the other side, since the area of the city itself is not measured, provided that the two thousand amos from the eiruv extend beyond the city.
This principle - that the area of the city is not counted in the measurement, and under what condition - will be relevant to the next mishnah.