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Today's Daf Yomi

October 4, 2020 | ט״ז בתשרי תשפ״א

Masechet Eruvin is sponsored by Adina and Eric Hagege in honor of our parents, Rabbi Dov and Elayne Greenstone and Roger and Ketty Hagege who raised children, grandchildren and great grandchildren committed to Torah learning.

Eruvin 56

Today’s daf is sponsored by by Rachel Alexander Levy in honor of her mother, Sheila Alexander. When Mom created the Jewish Day School 40 years ago, would she have every thought that I would be learning Daf Yomi at age 47! And by Caroline Ofstein in honor of Yakir’s bar mitzvah and for a refuah sheleima to Sheindel Chaya Bat Leah בתוך שאר חולי ישראל.

Does a Torah scholar needs to live in a place that has vegetables? Are vegetables good for you or bad for you? One should not live in a place with uphills and downhills as it causes premature old age. When one squares off a round city for the purposes of techumim, how does one do it? Each side needs to correspond to the four directions – North, South, etc. The gemara gives advice about how to determine where the directions are – either based on the constellations or on the direction of the sun at different times of the year. How much extra does one get on the angles? A braita is quoted regarding the 1,000 cubits outside each of the Levite cities that are left wmpty as compared to the whole space around the city of 2,000 cubits – a ratio is brought of 1 to 4 and the gemara brings a number of explanations of how this ratio makes sense.

פת קיבר ושכר חדש וירק לא קשיא הא בתומי וכרתי הא בשאר ירקי כדתניא שום ירק כרישין חצי ירק נראה צנון נראה סם חיים

coarse bread, made from coarse flour that has not been thoroughly sifted, new beer, and vegetables. This indicates that vegetables are harmful to one’s well-being. The Gemara answers: This is not difficult. This statement of Rav Huna is referring to garlic and leeks, which are beneficial; that baraita is referring to other vegetables, which are harmful. As it was taught in a baraita: Garlic is a healthy vegetable; leeks are a half-vegetable, meaning they are half as healthful. If radish has been seen, an elixir of life has been seen, as it is very beneficial to the body.

והא תניא נראה צנון נראה סם המות לא קשיא כאן בעלין כאן באמהות כאן בימות החמה כאן בימות הגשמים

The Gemara asks: Wasn’t it taught in a different baraita: If radish has been seen, a lethal drug has been seen? The Gemara answers: This is not difficult. Here, in the baraita that deprecates radish, it is referring to its leaves; there, in the baraita that praises radish, it is referring to the roots. Alternatively, here it is referring to the summer, when radish is beneficial; there, it is referring to the winter, when it is harmful.

אמר רב יהודה אמר רב כל עיר שיש בה מעלות ומורדות אדם ובהמה שבה מתים בחצי ימיהן מתים סלקא דעתך אלא אימא מזקינים בחצי ימיהן אמר רב הונא בריה דרב יהושע הני מולייתא דבי בירי ודבי נרש אזקנון

On the topic of the attributes of different locations, Rav Yehuda also said that Rav said: In any city that has many ascents and descents, which can be taxing to the body, people and animals die at half their days, meaning half of their life expectancy. The Gemara expresses surprise: Can it enter your mind that Rav really meant to say that they die prematurely? Even in such cities they are known to have a regular life expectancy. Rather, say: They grow old at half their days, i.e., they age prematurely due to the strain of climbing up and down the inclines. Similarly, Rav Huna, son of Rav Yehoshua, said: The ascents and descents between Beit Biri and Beit Neresh, my place of residence, made me grow old prematurely.

תנו רבנן בא לרבעה מרבעה בריבוע עולם נותן צפונה לצפון עולם ודרומה לדרום עולם וסימניך עגלה בצפון ועקרב בדרום

With regard to the measurements of a city’s boundaries, the Sages taught the following baraita: If, in order to measure the Shabbat limit, one comes to square a city, i.e., to extend the city’s boundaries to include all of its protrusions within an imaginary square, he squares it so that the sides of the square align with the four directions of the world. He sets the northern side of the square to align with the north of the world, and its southern side to align with the south of the world. And your sign by which you can recognize the directions of the world is as follows: The constellation of Ursa Major is in the north and Scorpio is in the south. The directions of the city are determined by these constellations.

רבי יוסי אומר אם אינו יודע לרבעה בריבוע של עולם מרבעה כמין התקופה כיצד חמה יוצאה ביום ארוך ושוקעת ביום ארוך זה הוא פני צפון חמה יוצאה ביום קצר ושוקעת ביום קצר זה הוא פני דרום תקופת ניסן ותקופת תשרי חמה יוצאה בחצי מזרח ושוקעת בחצי מערב

Rabbi Yosei says: If one does not know how to square the city in alignment with the four directions of the world based upon the constellations, he should square it based upon the seasons, although this is less precise. How so? Where the sun rises and sets on the longest day of the year, the summer solstice, this route of the sun is the face of the north. The sun rises in the northeast and sets in the northwest, and thus travels from east to west across the north side of the world. Conversely, where the sun rises and sets on the shortest day of the year, the winter solstice, this route of the sun is the face of the south. Whereas at the vernal equinox and the autumnal equinox, when day and night are equal in length, the sun rises in the middle of the east and sets in the middle of the west.

שנאמר הולך אל דרום וסובב אל צפון הולך אל דרום ביום וסובב אל צפון בלילה סובב סובב הולך הרוח אלו פני מזרח ופני מערב פעמים מהלכתן ופעמים מסבבתן

As it is stated: “One generation passes away and another generation comes; but the earth abides forever. The sun also rises and the sun goes down, and hastens to its place, where it rises again. It goes toward the south, and turns about to the north; round and round goes the wind, and on its circuits the wind returns” (Ecclesiastes 1:4–6). The verse is understood as describing the sun’s movements, as follows: “It goes toward the south” during the day; “and turns about to the north,” on the other side of the earth, at night. “Round and round goes the wind [ruaḥ];” the word ruaḥ can also mean direction or side. Rabbi Yosei explains that these are the face of the east and the face of the west; sometimes the sun traverses them visibly, and sometimes it turns about them without being seen.

אמר רב משרשיא ליתנהו להני כללי דתניא לא יצאה חמה מעולם מקרן מזרחית צפונית ושקעה בקרן מערבית צפונית ולא יצאה חמה מקרן מזרחית דרומית ושקעה בקרן מערבית דרומית

Rav Mesharshiya said: There is no validity to these rules established by Rabbi Yosei, as it was taught in a baraita: The sun has never risen, even during the summer, at the northeastern corner of the sky and set in the northwestern corner, nor has the sun ever risen, even during the winter, at the southeastern corner and set in the southwestern corner. Therefore, one can establish the directions of the world according to the sun’s path only during the autumn and spring.

אמר שמואל אין תקופת ניסן נופלת אלא בארבעה רבעי היום או בתחלת היום או בתחלת הלילה או בחצי היום או בחצי הלילה

On the topic of the previous discussion with regard to calculating the directions of the world based upon the seasons, Shmuel said: The vernal equinox occurs only at the beginning of one of the four quarters of a day: Either precisely at the beginning of the day, or precisely at the beginning of the night, or at midday, or at midnight.

ואין תקופת תמוז נופלת אלא או באחת ומחצה או בשבע ומחצה בין ביום ובין בלילה ואין תקופת תשרי נופלת אלא או בשלש שעות או בתשע שעות בין ביום ובין בלילה ואין תקופת טבת נופלת אלא או בארבע ומחצה או בעשר ומחצה בין ביום ובין בלילה

Similarly, the summer solstice occurs only at certain times of the day: Either at the conclusion of one and a half hours or seven and a half hours of the day or night. And the autumnal equinox occurs only at certain times: Either at the conclusion of three hours or nine hours of the day or night. And the winter solstice occurs only at certain times: Either at the conclusion of four and a half hours or ten and a half hours of the day or night.

ואין בין תקופה לתקופה אלא תשעים ואחד יום ושבע שעות ומחצה ואין תקופה מושכת מחברתה אלא חצי שעה

And all this is based on the principle that there are only ninety-one days and seven and a half hours between the beginning of one season and the next, as he assumed that a year is exactly 365¼ days. And similarly, each season begins precisely one-half planetary hour past the beginning of the previous season. There are seven heavenly bodies that are each ascendant for an hour at a time in a constant rotation: Mercury, Moon, Saturn, Jupiter, Mars, the Sun, and Venus. Each season begins half an hour later in this rotation than the previous season.

ואמר שמואל אין לך תקופת ניסן שנופלת בצדק שאינה משברת את האילנות ואין לך תקופת טבת שנופלת בצדק שאינה מייבשת את הזרעים והוא דאיתליד לבנה או בלבנה או בצדק:

And Shmuel said: There is no instance when the vernal equinox occurs in the planetary hour of Jupiter and it does not break the trees with its strong winds; and there is no instance when the winter solstice occurs in the planetary hour of Jupiter and it does not dry up the seeds. And this applies only where the new moon appeared either at the hour of the Moon or at the hour of Jupiter.

תנו רבנן המרבע את העיר עושה אותה כמין טבלא מרובעת וחוזר ומרבע את התחומין ועושה אותן כמין טבלא מרובעת

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.

וכשהוא מודד לא ימדוד מאמצע הקרן אלפים אמה מפני שהוא מפסיד את הזויות אלא מביא טבלא מרובעת שהיא אלפים אמה על אלפים אמה ומניחה בקרן באלכסונה

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.

נמצאת העיר משתכרת ארבע מאות אמות לכאן וארבע מאות אמות לכאן נמצאו תחומין משתכרין שמונה מאות אמות לכאן ושמונה מאות לכאן נמצאו העיר ותחומין משתכרין אלף ומאתים לכאן ואלף ומאתים לכאן

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.

אמר אביי ומשכחת לה במתא דהויא תרי אלפי אתרי אלפי

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.

תניא אמר רבי אליעזר ברבי יוסי תחום ערי לוים אלפים אמה צא מהן אלף אמה מגרש נמצא מגרש רביע והשאר שדות וכרמים

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.

מנא הני מילי אמר רבא דאמר קרא מקיר העיר וחוצה אלף אמה סביב אמרה תורה סבב את העיר באלף נמצא מגרש רביע

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.

רביע פלגא הוי אמר רבא בר אדא משוחאה אסברה לי משכחת לה במתא דהויא תרי אלפי אתרי אלפי תחום כמה הויא שיתסר קרנות כמה הויין שיתסר דל תמניא דתחומין וארבעה דקרנות כמה הוי תריסר

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.

נמצא מגרש רביע טפי מתלתא נינהו

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.

אייתי ארבעה דמתא שדי עלייהו אכתי תילתא הוי

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.

מי סברת בריבועא קאמר בעיגולא קאמר

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.

כמה מרובע יתר על העגול רביע דל רביע מינייהו פשו להו תשעה ותשעה מתלתין ושיתא ריבעא הוי

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.

אביי אמר משכחת לה נמי במתא דהויא אלפא באלפא תחומין כמה הוו תמניא קרנות כמה הוי שיתסר

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.

Masechet Eruvin is sponsored by Adina and Eric Hagege in honor of our parents, Rabbi Dov and Elayne Greenstone and Roger and Ketty Hagege who raised children, grandchildren and great grandchildren committed to Torah learning.

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Eruvin 56

פת קיבר ושכר חדש וירק לא קשיא הא בתומי וכרתי הא בשאר ירקי כדתניא שום ירק כרישין חצי ירק נראה צנון נראה סם חיים

coarse bread, made from coarse flour that has not been thoroughly sifted, new beer, and vegetables. This indicates that vegetables are harmful to one’s well-being. The Gemara answers: This is not difficult. This statement of Rav Huna is referring to garlic and leeks, which are beneficial; that baraita is referring to other vegetables, which are harmful. As it was taught in a baraita: Garlic is a healthy vegetable; leeks are a half-vegetable, meaning they are half as healthful. If radish has been seen, an elixir of life has been seen, as it is very beneficial to the body.

והא תניא נראה צנון נראה סם המות לא קשיא כאן בעלין כאן באמהות כאן בימות החמה כאן בימות הגשמים

The Gemara asks: Wasn’t it taught in a different baraita: If radish has been seen, a lethal drug has been seen? The Gemara answers: This is not difficult. Here, in the baraita that deprecates radish, it is referring to its leaves; there, in the baraita that praises radish, it is referring to the roots. Alternatively, here it is referring to the summer, when radish is beneficial; there, it is referring to the winter, when it is harmful.

אמר רב יהודה אמר רב כל עיר שיש בה מעלות ומורדות אדם ובהמה שבה מתים בחצי ימיהן מתים סלקא דעתך אלא אימא מזקינים בחצי ימיהן אמר רב הונא בריה דרב יהושע הני מולייתא דבי בירי ודבי נרש אזקנון

On the topic of the attributes of different locations, Rav Yehuda also said that Rav said: In any city that has many ascents and descents, which can be taxing to the body, people and animals die at half their days, meaning half of their life expectancy. The Gemara expresses surprise: Can it enter your mind that Rav really meant to say that they die prematurely? Even in such cities they are known to have a regular life expectancy. Rather, say: They grow old at half their days, i.e., they age prematurely due to the strain of climbing up and down the inclines. Similarly, Rav Huna, son of Rav Yehoshua, said: The ascents and descents between Beit Biri and Beit Neresh, my place of residence, made me grow old prematurely.

תנו רבנן בא לרבעה מרבעה בריבוע עולם נותן צפונה לצפון עולם ודרומה לדרום עולם וסימניך עגלה בצפון ועקרב בדרום

With regard to the measurements of a city’s boundaries, the Sages taught the following baraita: If, in order to measure the Shabbat limit, one comes to square a city, i.e., to extend the city’s boundaries to include all of its protrusions within an imaginary square, he squares it so that the sides of the square align with the four directions of the world. He sets the northern side of the square to align with the north of the world, and its southern side to align with the south of the world. And your sign by which you can recognize the directions of the world is as follows: The constellation of Ursa Major is in the north and Scorpio is in the south. The directions of the city are determined by these constellations.

רבי יוסי אומר אם אינו יודע לרבעה בריבוע של עולם מרבעה כמין התקופה כיצד חמה יוצאה ביום ארוך ושוקעת ביום ארוך זה הוא פני צפון חמה יוצאה ביום קצר ושוקעת ביום קצר זה הוא פני דרום תקופת ניסן ותקופת תשרי חמה יוצאה בחצי מזרח ושוקעת בחצי מערב

Rabbi Yosei says: If one does not know how to square the city in alignment with the four directions of the world based upon the constellations, he should square it based upon the seasons, although this is less precise. How so? Where the sun rises and sets on the longest day of the year, the summer solstice, this route of the sun is the face of the north. The sun rises in the northeast and sets in the northwest, and thus travels from east to west across the north side of the world. Conversely, where the sun rises and sets on the shortest day of the year, the winter solstice, this route of the sun is the face of the south. Whereas at the vernal equinox and the autumnal equinox, when day and night are equal in length, the sun rises in the middle of the east and sets in the middle of the west.

שנאמר הולך אל דרום וסובב אל צפון הולך אל דרום ביום וסובב אל צפון בלילה סובב סובב הולך הרוח אלו פני מזרח ופני מערב פעמים מהלכתן ופעמים מסבבתן

As it is stated: “One generation passes away and another generation comes; but the earth abides forever. The sun also rises and the sun goes down, and hastens to its place, where it rises again. It goes toward the south, and turns about to the north; round and round goes the wind, and on its circuits the wind returns” (Ecclesiastes 1:4–6). The verse is understood as describing the sun’s movements, as follows: “It goes toward the south” during the day; “and turns about to the north,” on the other side of the earth, at night. “Round and round goes the wind [ruaḥ];” the word ruaḥ can also mean direction or side. Rabbi Yosei explains that these are the face of the east and the face of the west; sometimes the sun traverses them visibly, and sometimes it turns about them without being seen.

אמר רב משרשיא ליתנהו להני כללי דתניא לא יצאה חמה מעולם מקרן מזרחית צפונית ושקעה בקרן מערבית צפונית ולא יצאה חמה מקרן מזרחית דרומית ושקעה בקרן מערבית דרומית

Rav Mesharshiya said: There is no validity to these rules established by Rabbi Yosei, as it was taught in a baraita: The sun has never risen, even during the summer, at the northeastern corner of the sky and set in the northwestern corner, nor has the sun ever risen, even during the winter, at the southeastern corner and set in the southwestern corner. Therefore, one can establish the directions of the world according to the sun’s path only during the autumn and spring.

אמר שמואל אין תקופת ניסן נופלת אלא בארבעה רבעי היום או בתחלת היום או בתחלת הלילה או בחצי היום או בחצי הלילה

On the topic of the previous discussion with regard to calculating the directions of the world based upon the seasons, Shmuel said: The vernal equinox occurs only at the beginning of one of the four quarters of a day: Either precisely at the beginning of the day, or precisely at the beginning of the night, or at midday, or at midnight.

ואין תקופת תמוז נופלת אלא או באחת ומחצה או בשבע ומחצה בין ביום ובין בלילה ואין תקופת תשרי נופלת אלא או בשלש שעות או בתשע שעות בין ביום ובין בלילה ואין תקופת טבת נופלת אלא או בארבע ומחצה או בעשר ומחצה בין ביום ובין בלילה

Similarly, the summer solstice occurs only at certain times of the day: Either at the conclusion of one and a half hours or seven and a half hours of the day or night. And the autumnal equinox occurs only at certain times: Either at the conclusion of three hours or nine hours of the day or night. And the winter solstice occurs only at certain times: Either at the conclusion of four and a half hours or ten and a half hours of the day or night.

ואין בין תקופה לתקופה אלא תשעים ואחד יום ושבע שעות ומחצה ואין תקופה מושכת מחברתה אלא חצי שעה

And all this is based on the principle that there are only ninety-one days and seven and a half hours between the beginning of one season and the next, as he assumed that a year is exactly 365¼ days. And similarly, each season begins precisely one-half planetary hour past the beginning of the previous season. There are seven heavenly bodies that are each ascendant for an hour at a time in a constant rotation: Mercury, Moon, Saturn, Jupiter, Mars, the Sun, and Venus. Each season begins half an hour later in this rotation than the previous season.

ואמר שמואל אין לך תקופת ניסן שנופלת בצדק שאינה משברת את האילנות ואין לך תקופת טבת שנופלת בצדק שאינה מייבשת את הזרעים והוא דאיתליד לבנה או בלבנה או בצדק:

And Shmuel said: There is no instance when the vernal equinox occurs in the planetary hour of Jupiter and it does not break the trees with its strong winds; and there is no instance when the winter solstice occurs in the planetary hour of Jupiter and it does not dry up the seeds. And this applies only where the new moon appeared either at the hour of the Moon or at the hour of Jupiter.

תנו רבנן המרבע את העיר עושה אותה כמין טבלא מרובעת וחוזר ומרבע את התחומין ועושה אותן כמין טבלא מרובעת

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.

וכשהוא מודד לא ימדוד מאמצע הקרן אלפים אמה מפני שהוא מפסיד את הזויות אלא מביא טבלא מרובעת שהיא אלפים אמה על אלפים אמה ומניחה בקרן באלכסונה

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.

נמצאת העיר משתכרת ארבע מאות אמות לכאן וארבע מאות אמות לכאן נמצאו תחומין משתכרין שמונה מאות אמות לכאן ושמונה מאות לכאן נמצאו העיר ותחומין משתכרין אלף ומאתים לכאן ואלף ומאתים לכאן

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.

אמר אביי ומשכחת לה במתא דהויא תרי אלפי אתרי אלפי

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.

תניא אמר רבי אליעזר ברבי יוסי תחום ערי לוים אלפים אמה צא מהן אלף אמה מגרש נמצא מגרש רביע והשאר שדות וכרמים

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.

מנא הני מילי אמר רבא דאמר קרא מקיר העיר וחוצה אלף אמה סביב אמרה תורה סבב את העיר באלף נמצא מגרש רביע

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.

רביע פלגא הוי אמר רבא בר אדא משוחאה אסברה לי משכחת לה במתא דהויא תרי אלפי אתרי אלפי תחום כמה הויא שיתסר קרנות כמה הויין שיתסר דל תמניא דתחומין וארבעה דקרנות כמה הוי תריסר

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.

נמצא מגרש רביע טפי מתלתא נינהו

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.

אייתי ארבעה דמתא שדי עלייהו אכתי תילתא הוי

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.

מי סברת בריבועא קאמר בעיגולא קאמר

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.

כמה מרובע יתר על העגול רביע דל רביע מינייהו פשו להו תשעה ותשעה מתלתין ושיתא ריבעא הוי

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.

אביי אמר משכחת לה נמי במתא דהויא אלפא באלפא תחומין כמה הוו תמניא קרנות כמה הוי שיתסר

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.

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