Talmud - Eruvin 56b עירובין נ״ו ב׳

The Sages taught: One who squares a city in order to determine its Shabbat limit renders it like a square tablet, and then he also squares the Shabbat boundaries and renders them like a square tablet. Consequently, after squaring the city, he adds additional squares of two thousand cubits to each of its sides.

And when he measures the Shabbat limit, he should not measure the two thousand cubits diagonally from the middle of each corner of the city, because if he were to do so, he would lose the corners, i.e., the limit would extend only two thousand cubits on the diagonal from each of the corners. Rather, he measures the boundary as though he brought a square tablet that is two thousand cubits by two thousand cubits, and places it at each corner at its diagonal.

As a result, it will be found that the city gains four hundred cubits in this corner and another four hundred cubits in the opposite corner. Assuming that the city itself is round and has a diameter of two thousand cubits, as will be explained below, when the borders of the city are squared, approximately four hundred cubits are added to the city at each corner. When one then squares the Shabbat boundaries, it is found that the Shabbat boundaries gain eight hundred cubits in this corner and eight hundred cubits in the opposite corner. Consequently, by squaring both the city itself and its Shabbat boundaries, it is found that the city and the Shabbat boundaries together gain 1,200 cubits in this corner and 1,200 cubits in the opposite corner.

Abaye said: And you find this projection of the additions to the city’s borders and Shabbat boundaries to be correct in the case of a round city that is two thousand cubits by two thousand cubits.

The Gemara cites a similar discussion with regard to the Levite cities, the forty-eight cities given to the Levites in Eretz Yisrael instead of a tribal inheritance. It was taught in a baraita that Rabbi Eliezer, son of Rabbi Yosei, said: The boundary of the cities of the Levites extends two thousand cubits in each direction beyond the inhabited section of the city. Remove from them a thousand cubits of open space just beyond the inhabited area, which must be left vacant. Consequently, the open space is one quarter of the extended area, and the rest is fields and vineyards.

The Gemara asks: From where are these matters? From where is it derived that the open space surrounding the cities of the Levites measured a thousand cubits? Rava said: As the verse states: “And the open spaces of the cities, that you shall give to the Levites, shall be from the wall of the city and outward a thousand cubits round about” (Numbers 35:4). The Torah states: Surround the city with a thousand cubits on all sides to serve as an open space. Consequently, the open space is one quarter of the area.

The Gemara asks: Is it one quarter? It is one half. One thousand cubits is exactly half of the two thousand cubits incorporated into the boundary of the cities of the Levites. Rava said: Bar Adda the surveyor explained the calculation to me: You will find this in a city that is two thousand cubits by two thousand cubits. How many cubits is the extended boundary of the city itself, without the corners? Sixteen million square cubits. Squares measuring two thousand by two thousand cubits are appended to each of the four sides of the city. The area of each of these squares is four million square cubits, and the total area of all the additional squares is sixteen million square cubits. How many cubits are the corners? Sixteen million square cubits, as additional squares of two thousand by two thousand cubits are appended to the corners of the outer boundaries of the cities. Subtract eight million square cubits from the area of the extended boundary for the open space around the city; the first thousand cubits beyond the inhabited part of the city must be left as open space, which amounts to areas measuring one thousand by two thousand cubits on each of the four sides of the city, for a total of eight million square cubits. And subtract another four million square cubits from the corners, as sections of the corners are parallel to the open spaces. How much is the sum total of the area of the open spaces? Twelve million square cubits.

The Gemara asks: According to this calculation, how is the open space found to be one quarter of the area? It is more than one-third. The entire area of the extended boundary is thirty-two million square cubits and the open space occupies twelve million square cubits, which is more than one-third of the total area of the extended boundary.

The Gemara explains: Bring the four million square cubits of the city itself and add them to the area of the limit, and you will arrive at the correct ratio. The Gemara asks: The opens space is still one-third, as the total area of the city and its extended boundary is thirty-six million square cubits, and the area of the open space is twelve million square cubits.

The Gemara answers: Do you think that this halakha was stated with regard to a square city? It was in fact stated with regard to a round city. The open space beyond the city is also round; however, the total extended boundary is squared, so that the total area of a round city with a diameter of two thousand cubits and its extended boundary is thirty-six million square cubits.

The Gemara explains the calculation: How much larger is the area of a square than the area of the circle? One quarter. Subtract one quarter from the twelve million square cubits of open space, and nine million square cubits are left; and nine is precisely one quarter of thirty-six.

Abaye said: You will also find that the open space is one quarter of the total area in a city that is a thousand cubits by a thousand cubits. How many cubits is the extended boundary of the city without the corners? It is eight million square cubits. Additional areas are appended along each side of the city and extending two thousand cubits beyond the city itself. Each of these areas is two thousand cubits by one thousand cubits, for a total area of two million square cubits. Since there are four of these zones, their total area is eight million square cubits. How many cubits are the corners? They are sixteen million square cubits, as squares of two thousand cubits by two thousand cubits are added to each of the four corners.

Traduction : The William Davidson digital edition of the Koren Noé Talmud, with commentary by Rabbi Adin Even-Israel Steinsaltz.

Eruvin 56b explains how to square a city and its Shabbat boundaries geometrically, calculating cubit gains at each corner using round-city measurements.