Eruvin, Chapter 5, Mishnah 2. In the previous mishnah we learned that any house or dwelling place located within seventy amos and two thirds of an amah of the city creates what is called the 'ibbur ha'ir' - an extension that widens the boundary of the city.
Our mishnah teaches about an additional type of extension of the city: even if there are no dwelling areas outside the city, and all we have before us is the city wall alone, we still extend the city's limit by an additional seventy amos and two thirds of an amah. This addition is called a 'karpef'.
The opinion of Rabbi Meir:
"Nosen karpef la'ir" - we give the city a karpef, an additional open area of seventy amos and two thirds of an amah. It emerges that even after the city wall ends, and even after those additional dwelling places included in the ibbur ha'ir, we do not begin counting the two thousand amos until after an additional seventy amos and two thirds of an amah. "Divrei Rabbi Meir" - this is Rabbi Meir's view of the function of the karpef.
Rabbi Meir derives this from the verse in Bamidbar: "Mikir ha'ir vachutzah elef amah saviv" - from the wall of the city outward, one thousand amos. The verse is speaking about the migrash, that is, the empty area surrounding the cities of the Levi'im, and from here we learn the rule: give the outside space first and only then measure. First we place an addition outside the city ("from the wall of the city outward"), and only afterward do we begin measuring. From here Rabbi Meir learns that we give this addition, which is seventy amos and two thirds of an amah.
The opinion of the Sages:
The Sages disagree: they too hold that the concept of a karpef exists, but this is not how it is applied. "Lo amru karpef ela bein shtei ha'ayaros" - the law of the karpef was stated only between two cities. That is, if the cities are close enough to one another to fit the measure of each one's karpef, they are considered a single city, and we measure the two thousand amos from the outer side of both of them.
"Im yesh lazo shiv'im amah u'shirayim, v'lazo shiv'im amah u'shirayim" - the 'shirayim' are the addition, two thirds of an amah. It emerges that between the two cities there are one hundred forty one and a third amos. In such a case - "oseh karpef lishteihen lihyos k'achas" - we give the karpef addition to both of them in order to make them one city, and we measure the boundary from the outer side of the cities, since they are within one hundred forty one and a third amos of each other.
In summary: in this mishnah we learned about the addition of the 'karpef' - seventy amos and two thirds of an amah. According to Rabbi Meir, we give a karpef to every single city, and only afterward do we measure the two thousand amos, as derived from the verse "from the wall of the city outward" - give the outside space first and then measure. According to the Sages, the law of the karpef was stated only between two cities: when the distance between them does not exceed one hundred forty one and a third amos, the two of them become like one city, and the measuring is done from their outer side.