דל ארבע דתחומין וארבע דקרנות כמה הוי תמניא
Subtract four million square cubits of the extended boundary for the area of the open space, which is a thousand cubits by a thousand cubits on each side, and an additional four million square cubits from the corners, a thousand cubits by a thousand cubits in each corner, which are connected to the open space. How much is the sum total? It is eight million square cubits.
תילתא הוו מי סברת ברבועא קאמר בעיגולא קאמר כמה מרובע יתר על העגול רביע דל רביע פשו לה שיתא ושיתא מעשרים וארבע ריבעא הוי
The Gemara asks: According to this calculation, the eight million square cubits of open space are one-third of the total area of the extended boundary, which is twenty-four million square cubits. The Gemara answers as it answered above: Do you think that this halakha was stated with regard to a square city? It was stated with regard to a round city. How much larger is the area of a square than the area of a circle? It is one quarter of the area of the circle. Subtract one quarter from the eight million square cubits of open space, and six million square cubits are left; and six is precisely one quarter of twenty-four.
רבינא אמר מאי רביע רביע דתחומין
Ravina said: What is the meaning of the statement that the open space is one quarter? It is one quarter of the boundary. This halakha was indeed stated with regard to a square city. However, there is open space only along the sides of the city but not at its corners. Accordingly, a city that is two thousand cubits by two thousand cubits has a total extended boundary of thirty-two million square cubits, of which eight million square cubits, two thousand cubits by one thousand cubits on each side, is open space. The open space is thus one quarter of the total.
רב אשי אמר מאי רביע רביע דקרנות
Rav Ashi said the opposite: What is the meaning of the statement that the open space is one quarter of the total extended boundary? One quarter of the corners. Open space is granted only in the corners, and not along the sides. Accordingly, the open space is one thousand cubits by a thousand cubits in each corner, for a total of four million square cubits. The total extended boundary in each corner is two thousand cubits by two thousand cubits, or four million square cubits per corner, which equals a grand total of sixteen million square cubits. Consequently, the open space is one quarter of the total extended boundary.
אמר ליה רבינא לרב אשי והא סביב כתיב
Ravina said to Rav Ashi: Isn’t it written in the verse: “And the open spaces of the cities, that you shall give to the Levites, shall be from the wall of the city and outward one thousand cubits around” (Numbers 35:4)? The verse indicates that the city is provided with open space on all sides and not merely at its corners
מאי סביב סביב דקרנות דאי לא תימא הכי גבי עולה דכתיב וזרקו (בני אהרן) את הדם על המזבח סביב הכי נמי סביב ממש אלא מאי סביב סביב דקרנות הכי נמי מאי סביב סביב דקרנות
Rav Ashi responded: What is the meaning of around? Around at the corners, i.e., an open space of this size is provided at each corner. As, if you do not say so, that the area of the corners is also called around, with regard to the burnt-offering, as it is written: “And they shall sprinkle the blood around upon the altar” (Leviticus 1:5), here, too, will you say that the blood must be sprinkled literally “around” the altar on all sides? The blood is sprinkled only upon the corners of the altar. Rather, what is the meaning of around? Around the corners, i.e., the mitzva is to sprinkle the blood at the corners, and this is considered sprinkling blood “around upon the altar.” Here too, with regard to the open space of the cities of the Levites, what is the meaning of around? Around the corners.
אמר ליה רב חביבי מחוזנאה לרב אשי והא איכא מורשא דקרנתא
The Gemara returns to its previous statement that the open space around a city of the Levites is one quarter of the total extended boundary when the city is round. It questions this statement based upon the mishna’s ruling that the boundaries of a city are always delineated as a square. Rav Ḥavivi from Meḥoza said to Rav Ashi: But aren’t there the protrusions of the corners? How can there be a thousand cubits of open space on each side; when the city is squared, the corners of the square protrude into the open space, thus reducing its area?
במתא עיגולתא והא ריבעוה אימור דאמרינן חזינן כמאן דמרבעא רבועי ודאי מי מרבענא
Rav Ashi replied: We are dealing with a circular city. Rav Ḥavivi responded: But haven’t they squared the city? Rav Ashi responded: Say that we say the following: We view the city as if it were squared. Do we actually add houses and square it? Although for the purpose of calculating the extended boundary we view the city as a square, in actuality the uninhabited sections are part of the open space.
אמר ליה רב חנילאי מחוזנאה לרב אשי מכדי כמה מרובע יתר על העגול רביע הני תמני מאה שית מאה ושיתין ושבע נכי תילתא הוי
Rav Ḥanilai from Meḥoza said to Rav Ashi: Now, how much larger is the area of a square than the area of a circle? One quarter. Therefore, if we calculate how much area a circular city with a diameter of two thousand cubits gains when it is squared, does it add up to these eight hundred cubits mentioned above? The extra area added is only 667 minus one-third cubits.
אמר ליה הני מילי בעיגולא מגו רבוע אבל באלכסונא בעינא טפי דאמר מר כל אמתא בריבוע אמתא ותרי חומשי באלכסונא:
Rav Ashi said to him: This statement applies only to a circle enclosed within a square, as the area of a circle is three-quarters the area of the square around it. However, with regard to the additional diagonal [alakhsona] space added in the corners of the square, more is required. As the Master said: Every cubit in the side of a square is one and two-fifths cubits in its diagonal. Based on this rule, the calculation is exact.
מתני׳ נותנין קרפף לעיר דברי רבי מאיר וחכמים אומרים לא אמרו קרפף אלא בין שתי עיירות אם יש לזו שבעים אמה ושיריים ולזו שבעים אמה ושיריים עושה קרפף את שתיהן להיות אחד
MISHNA: One allocates a karpef to every city, i.e., the measure of a karpef, which is slightly more than seventy cubits, is added to every city, and the two thousand cubits of the Shabbat limit are measured from there; this is the statement of Rabbi Meir. And the Rabbis say: They spoke of the addition of a karpef only with regard to the space between two adjacent cities. How so? If this city has seventy cubits and a remainder vacant on one side, and that city has seventy cubits and a remainder vacant on the adjacent side, and the two areas of seventy-plus cubits overlap, the karpef combines the two cities into one.
וכן שלשה כפרים המשולשין אם יש בין שנים חיצונים מאה וארבעים ואחת ושליש עשה אמצעי את שלשתן להיות אחד:
And likewise, in the case of three villages that are arranged as a triangle, if there are only 141⅓ cubits separating between the two outer villages, the middle village combines the three villages into one.
גמ׳ מנא הני מילי אמר רבא דאמר קרא מקיר העיר וחוצה אמרה תורה תן חוצה ואחר כך מדוד:
GEMARA: The Gemara asks: From where are these matters, that a karpef is added to a city, derived? 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 around. And you shall measure from outside the city on the east side two thousand cubits” (Numbers 35:4–5). The Torah says: Provide a certain vacant space outside the city, and only afterward measure the two thousand cubits.
וחכמים אומרים לא אמרו וכו׳: איתמר רב הונא אמר נותנין קרפף לזו וקרפף לזו חייא בר רב אמר קרפף [אחד] לשתיהן
We learned in the mishna: And the Rabbis say: They spoke of the addition of a karpef only with regard to the space between two adjacent cities. It was stated that the amora’im disagreed with regard to this issue. Rav Huna said: One allocates a karpef to this city and a karpef to that city, so that the two cities together are granted a total of slightly more than 141 cubits. Ḥiyya bar Rav said: One allocates only one common karpef to the two of them.
תנן וחכמים אומרים לא אמרו קרפף אלא בין שתי עיירות תיובתא דרב הונא
The Gemara raises possible proofs for each opinion. We learned in the mishna: And the Rabbis say: They spoke of the addition of a karpef only with regard to the space between two adjacent cities. This appears to be a conclusive refutation of the opinion of Rav Huna, as it states that one karpef is allocated rather than two.
אמר לך רב הונא מאי קרפף תורת קרפף ולעולם קרפף לזו וקרפף לזו
The Gemara answers that Rav Huna could have said to you in response to this difficulty: What is meant here by a karpef ? It means the principle of a karpef. In actuality, one allocates a karpef to this city and a karpef to that city.
הכי נמי מסתברא מדקתני סיפא אם יש לזו שבעים אמה ושיריים ולזו שבעים אמה ושיריים עושה קרפף לשתיהן להיות אחד שמע מינה
The Gemara comments: So, too, it is reasonable to explain the mishna in the following manner: From the fact that it teaches in the latter clause: If this city has seventy cubits and a remainder vacant on one side, and that city has seventy cubits and a remainder vacant on the adjacent side, and the two areas of seventy-plus cubits overlap, the karpef combines the two cities into one. This indicates that an area of seventy cubits and a remainder is added to each city. The Gemara concludes: Indeed, learn from this that this is the correct understanding of the mishna.
לימא תיהוי תיובתיה דחייא בר רב אמר לך חייא בר רב
The Gemara asks: Let us say that this mishna is a conclusive refutation of the opinion of Ḥiyya bar Rav, that two adjacent cities are granted only one karpef. The Gemara answers that Ḥiyya bar Rav could have said to you: