Eruvin 57a — the Talmud in English

And for the calculation of the open space:

subtract

from this area

four

squares of one thousand by one thousand

designated for the city’s open space all around.

And

further

four

squares for filling

the

four

corners,

which are between the city’s open spaces all around.

How many are there

squares of the open spaces in total?

Eight,

eight.

And we challenge this: If so, eight out of twenty-four—

they are a third,

a third

they are,

and not a quarter.

And we answer:

Do you maintain

that in a city made

in a square Rabbi Abaye speaks

of?

That is not so.

Rather,

in

a city made like

a circle Rabbi Abaye speaks,

and as above: the entire תחום of the city is measured in a square, whereas the open space around it is measured in a circle.

And this is the calculation:

By how much is the square greater than the circle—

a quarter.

Subtract

a

quarter

from the eight squares of the open space [for this is how we calculated it had the city been square].

There remain

six

for it,

six squares.

And six out of twenty-four—

it is a quarter.

And the Gemara brings a dispute between Ravina and Rav Ashi as to which quarter Rava and Abaye intended.

And both establish it as at first, that the matter concerns a square city having two thousand cubits by two thousand cubits, but they disagree about the calculation of the quarter.

Ravina said: What is the quarter—

we calculate the open space without its corners, which is eight squares, two squares on each side, which is

a quarter of the תחומין

all of them together with their corners.

And as we calculated above, that altogether there are thirty-six squares, and therefore the eight squares of the open space are a quarter of them.

Rav Ashi said: What is the quarter—

the quarter of the corners.

That is, the open space at the corners, which is one square of one thousand by one thousand, is a quarter of one corner of the תחום, which is four squares.

Ravina said to Rav Ashi: But

with regard to the open space

“around”

it is written,

and how do you calculate only the account of the quarter at the corners?

Rav Ashi answered him:

What

is

“around”?

Around

the corners

if you do not say so,

that if you do not hold as I do, that the corners themselves are also considered “around,” it will be difficult for you:

With regard to the burnt offering, for it is written

concerning it

“And they shall throw the blood upon the altar around”—

is

this also

to be said, that one must throw the blood of the burnt offering

around

the entire altar

literally?

But it is necessary to throw the blood only at the corners of the altar.

Rather, what

is meant by

“around”?

Around the corners

of the altar.

So too

it is said:

What

is the open space that is

“around

the cities”?

Around the corners

of the cities.

Rav Havivi of Mehoza said to Rav Ashi: According to what we learned above, that an open space is provided in a circle around the city, while, on the other hand, we also learned that the city is squared [with regard to the תחומין], the area of the city’s open space will consequently be reduced.

For

there is

the projection of the corner,

that is, the corner of the city’s square cuts into the circular open space surrounding the city.

Rav Ashi said to him: Here,

we are dealing with a circular city,

a circular city, which has no corners in its squaring.

And we wonder:

But they square it

for the city even when it is circular, and it follows that the square cuts into the circular open space surrounding it.

And we answer:

I can say that we say

that

we see

the circular city

as though it were squared,

only with regard to measuring the city’s תחומין for walking on Shabbat.

But

actual squaring,

to square it in reality, so that the square would cut into the Levite open space—

would I square it?

Rav Hanilai of Mehoza said to Rav Ashi: How did they say above that through the squaring of the city they gained eight hundred cubits?

Now,

indeed,

it is so,

by how much is the square greater than the circle—

a quarter.

And this quarter is a quarter according to the relation of “from outside,” that is, within the circle there are three additional parts like the part outside the circle. It follows that if we measure the parts within the circle together with the part outside the circle, the part outside the circle is a quarter of the whole together.

But according to the relation of “from within,” the section outside the circle is a third of what is inside the circle.

And if so,

these eight hundred,

those eight hundred cubits, which they said are thereby gained in the length of the diagonal outside the circle—are not eight hundred cubits! Since the diagonal section outside the circle is only a third [from within!] of the two thousand cubits [according to the relation of a quarter from outside].

For the diameter of the circle is two thousand cubits. And if so, the diagonal of the square, which exceeds it by a third, is only

six hundred and sixty-seven less a third

cubits,

it is.

He said to him,

Rav Ashi: This principle that they said, that the square is greater than the circle by a quarter from outside, which is a third from within—

this applies to

the area or circumference of

a circle within a square.

But with regard to

the measure of the

diagonal—

I need more.

For the Master said: Every cubit in a square—

a cubit and two fifths in the diagonal

of the square. It follows that the diagonal exceeds the side of the square of two thousand cubits by forty percent, which is eight hundred cubits.

Mishna:

They provide

a “karpef”

for a city,

they add to the area of the city around it seventy cubits and remnants, which are considered a “karpef” used by the city for open air [but it is not the karpef used as a storehouse], and it is part of it.

And they measure the city’s תחום from the end of the karpef,

the words of Rabbi Meir.

And the Sages say: They said

that they provide

a karpef

for a city in order to expand its area, and measure its תחום from there, only—

rather,

they provide a karpef only

between two towns

in order to join them into one city with regard to measuring the תחומין.

For if there are two neighboring cities, then

if this one has seventy cubits and remnants

around it,

and that one has seventy cubits and remnants

around it—

it makes

the

karpef

provided for each of them, and thereby brings together

the two of them—

to be

both one city

one.

And likewise three villages arranged in a triangle,

one at a distance,

if there is between the two

outer villages

one hundred forty-one and one third,

the measure of two karpefs—

it makes

the

middle village all three of them to be one.

And this matter will be explained in the Gemara.

Gemara:

We learned in the Mishna: They provide a karpef for a city, the words of Rabbi Meir.

From where are these words?

Rava said: For the verse says

concerning the תחום of the Levite cities [Numbers 35]: “And you shall measure

from the city wall outward

two thousand cubits.”

The Torah said: Provide

first

“outward”

to the city, that is, a karpef, which is designated for the uses of the city that must be performed outside the city,

and

only

afterward measure

the two thousand cubits from it.

We learned in the Mishna:

And the Sages say: They did not say

that they provide a karpef for a city except in order to join two towns.

It was stated: Rav Huna said:

According to the words of the Sages,

they provide a karpef for this one and a karpef for that one.

Hiyya bar Rav said:

They provide

one karpef for both of them

in order to join them.

And we challenge Rav Huna:

We learned

in our Mishna:

And the Sages say: They did not say

that they provide

a karpef except between two towns.

And the wording “a karpef” implies that they provide only one karpef.

And

a refutation of Rav Huna!

Rav Huna could say to you: What is

“a karpef”?

The status of a karpef. And in fact,

they provide

a karpef for this one and a karpef for that one.

And the Gemara proves:

It is also reasonable

according to Rav Huna.

For from what it teaches in the latter clause: If this one has seventy cubits and remnants and that one has seventy cubits and remnants—

it makes one karpef for both of them to be one.

Learn from this

that two karpefs are provided in order to join them.

And we inquire about it:

Let it be

our proof from the Mishna

a refutation of Hiyya bar Rav!

And we answer:

Hiyya bar Rav could say to you: