Sukkah 8a — the Talmud in English

The Gemara answers:

These matters:

the rule stating, “Anyone whose circumference is three handbreadths has a width of one handbreadth,” applies only

to a circle.

For only the width-line of a circle [called “diameter”] is one-third of the circumference of the circle. Indeed, in a circle, if the diameter is four cubits, the circumference of the circle will be twelve cubits.

But with a square, more is required.

That is, in order to enclose a certain width in the shape of a square, a greater circumference is needed than the circumference needed to enclose that same width in the shape of a circle.

For the width of the circle is only at its middle, whereas the square has width-lines on all its sides. And according to Rabbi, the sukka must be four cubits wide on each of its four sides, and therefore a circumference of twelve cubits is not sufficient.

But the Gemara still asks: even if it is as you say, that a circumference of twelve cubits is insufficient, it is still difficult: why is a circumference so large as twenty-four cubits required?

Granted,

there is a rule stating:

how much

is the circumference of

a square greater than that of a circle?

One-quarter.

That is, a circle around a width of one cubit will have a circumference of three cubits.

And the circumference of a square around a width of one cubit is one-quarter greater than the circumference of the circle.

Therefore, according to Rabbi, since the width of the sukka must be four cubits in the shape of a square,

with sixteen

cubits of circumference [which are one-quarter more than twelve] alone

is it sufficient?!

A circumference of only sixteen cubits around a width of four cubits is sufficient, and a circumference of twenty-four cubits is not necessary!

And the Gemara explains:

These matters:

this rule—that the circumference of a square is greater than that of a circle only by one-quarter [and according to this, a circumference of only sixteen cubits is sufficient]—was stated only

with regard to a circle that emerges from a square.

That is, with a square enclosing a circle, a circumference of sixteen cubits is sufficient to enclose a width of four cubits, because there the rule was stated that the circumference of the outer square exceeds the circumference of the inner circle by one-quarter. Therefore, if the sukka were square, a circumference of sixteen cubits would be sufficient.

But a square that emerges from a circle requires more, because of the “projections of the corners.”

That is, in a round sukka, the circumference of the circle must be of such a measure that it can enclose within it a square of four cubits by four cubits.

For the diameter of the circle enclosing the square is much greater than the width of the square itself.

For the width of the square is measured from side to side, whereas the diameter of the circle enclosing the square is measured according to the diagonal, extending from one corner to the corner opposite it.

Therefore, in order for the inner square to be four cubits wide by four cubits wide, the circumference of the circle must be made greater than sixteen cubits, because the four corners of the square must be brought within the circumference of the circle [that is, as stated, the diameter must be calculated according to the diagonal of the square, and therefore the circumference increases, since the diagonal of the square is longer than the width of the square].

But even if we say so, it is still difficult: why does Rabbi Yohanan require a circumference so large as twenty-four cubits?

Granted,

there is

every cubit in a square:

for every cubit in the width of the square, when you come to measure its diagonal, you find

a cubit and two-fifths

of a cubit

in the diagonal.

That is, the length of the diagonal of a square is always forty percent greater than the width of the square [two-fifths].

Accordingly, the measure of the diagonal of a square whose width is four cubits by four cubits is 5.6 cubits, and this diagonal is also the diameter of the circle enclosing the square.

Therefore, in order for there to be within the circle enclosing the sukka a square of four cubits by four cubits, the circumference of the circle must be calculated according to a diameter of 5.6 cubits.

And since the rule is, “Anyone whose circumference is three handbreadths has a width of one handbreadth,” then

with seventeen cubits less two-fifths

of a cubit, that is, with a circumference of 16.8 [according to the calculation of three multiplied by five and six-tenths]

is it sufficient!

And why did Rabbi Yohanan require twenty-four cubits?!

The Gemara answers:

He was not precise.

Indeed, a circumference of seventeen cubits less two-fifths is sufficient. Rather, Rabbi Yohanan was not precise in his words so as to reduce the measure of the sukka, and he stated a greater measure of circumference.

But the Gemara rejects this:

Granted that we say, “He was not precise,” a little,

when the difference is small. But

a great deal,

when the difference is so large,

do we say, “He was not precise”

in his words?!

The Gemara answers:

Mar Kashisha, son of Rav Hisda, said to Rav Ashi: Do you think that a person sits in a cubit,

and do you think that the place for a person’s sitting is the measure of one complete cubit, and therefore you were perplexed by Rabbi Yohanan’s words: why did he require so great a circumference for the sukka, the place of sitting of twenty-four people?

It is not so!

Rather, according to Rabbi Yohanan,

three people sit within two cubits.

The place for the sitting of three people is only two cubits. Therefore, in order for the sukka to have a width of four cubits by four cubits, one must make a sukka whose circumference has room for twenty-four people.

The Gemara asks: according to this calculation, that three people sit within two cubits, it follows that the place of twenty-four people

how much does it amount to? Sixteen,

sixteen cubits.

But

we,

according to the calculation explained above in the Gemara, in order to make a round sukka with a square of four cubits by four cubits inside it, require a greater measure, for

we require seventeen cubits less two-fifths,

the circumference of the circle must be seventeen cubits less two-fifths, and sixteen cubits is insufficient?!

The Gemara answers: indeed, according to Rabbi Yohanan, the circumference of the circle must be seventeen cubits less two-fifths, and

he was not precise

in his words in saying that sixteen cubits and four-fifths are required, and he stated only sixteen cubits.

The Gemara rejects this:

Granted that we say, “He was not precise,”

in Rabbi Yohanan’s words

stringently,

when he adds to the required measure; but

leniently,

when he subtracts from the required measure,

do we say, “He was not precise”?

Surely he certainly should have been precise in his wording, so that his words would not cause people to stumble?!

The Gemara answers:

Rav Asi said to Rav Ashi: Actually,

it is indeed as was explained initially,

a person sits in a cubit, and

as for what perplexed you—why did Rabbi Yohanan require so large a sukka of twenty-four cubits—

Rabbi Yohanan does not count the place of people

as part of the sukka. Rather, his intention is to say that if twenty-four people are seated in a circle, the area formed within the circle, apart from the place of their sitting, is the area required for the size of the sukka. Therefore, since the place of a person is one cubit, when twenty-four people sit in a circle, the diameter of the circle will be eight cubits [which are one-third of twenty-four], and two cubits must be subtracted from this measure [one cubit from each side] to deduct the place of their sitting; therefore, a diameter of six cubits remains.

But it is still difficult, for this measure is not precise, since [if we multiply this diameter of six cubits by three],

how many

cubits

do they amount to

within this circle?

Eighteen,

eighteen!

With seventeen cubits less two-fifths,

seventeen cubits less one-fifth,

is it sufficient?!

As was explained above.

Rather,

that is,

this is what we said:

Rabbi Yohanan was not precise

in his words.

And he was not precise stringently,

for he added a little to the required measure. And this is sometimes the way of the amoraim: they are not precise in their measures when the change is stringent, and they prefer to state a whole-number calculation. Therefore, Rabbi Yohanan said that the diameter of the sukka is six cubits.

But

the Rabbis of Caesarea—and some say, the judges of Caesarea—said

that Rabbi Yohanan’s words should be explained differently:

Actually, Rabbi Yohanan counts the place of the people’s sitting as part of the sukka, and the circumference of the sukka must be twenty-four cubits.

And one should not ask why Rabbi Yohanan required so large a circumference, for indeed

a circle that emerges from a square is one-quarter,

when the square encloses the circle, the circumference of the square is greater than the circumference of the circle by only one-quarter.