הָנֵי מִילֵּי – בְּעִיגּוּלָא, אֲבָל בְּרִיבּוּעָא בָּעֵינַן טְפֵי.

This measurement applies only to a circle and the ratio between its circumference and diameter, but with regard to a square that must fit entirely within that circle, we require a circle with a larger circumference. In order for a square of four by four handbreadths to be entirely contained within a circle, the circumference of the circle must measure more than twelve handbreadths

מִכְּדִי, כַּמָּה מְרוּבָּע יָתֵר עַל הָעִגּוּל – רְבִיעַ, בְּשִׁיתְּסַר סַגְיָא!

The Gemara asks: Now, how much larger is a square than a circle? It is larger by one quarter. If so, a circle with a circumference of sixteen handbreadths at most should suffice.

הָנֵי מִילֵּי – עִיגּוּלָא דְּנָפֵיק מִגּוֹ רִיבּוּעָא, אֲבָל רִיבּוּעָא דְּנָפֵיק מִגּוֹ עִיגּוּלָא – בָּעֵינַן טְפֵי. מַאי טַעְמָא – מִשּׁוּם מוֹרְשָׁא דְּקַרְנָתָא.

The Gemara answers: This statement that a square is larger than a circle by a quarter applies only to a circle circumscribed by a square, but with regard to a square circumscribed by a circle, we require more, and the difference between the square and the circle is greater. What is the reason for this? It is due to the projection of the corners of the square, as the distance from the center of the square to its corners is greater than the distance from the center to its sides.

מִכְּדִי, כָּל אַמְּתָא בְּרִיבּוּעַ אַמְּתָא וּתְרֵי חוּמְשֵׁי בַּאֲלַכְסוֹנָא, בְּשִׁיבְסַר נְכִי חוּמְשָׁא סַגְיָא!

The Gemara further objects: Since every cubit in the side of a square is a cubit and two-fifths in the diagonal, a square of four by four handbreadths has a diagonal of five and three-fifths handbreadths. And since the diameter of a circle equals the diagonal of the square that it encompasses, the circle circumscribing a square of four by four handbreadths has a diameter of five and three-fifths handbreadths. If that measure is multiplied by three to arrive at the circumference of that circle, the result is that a circle with a circumference of seventeen handbreadths minus a fifth is sufficient to circumscribe a square of four by four handbreadths. Why, then, does Rabbi Yoḥanan say that a circular window

רַבִּי יוֹחָנָן אָמַר כִּי דַּיָּינֵי דְּקֵיסָרִי, וְאָמְרִי לָהּ כְּרַבָּנַן דְּקֵיסָרִי, דְּאָמְרִי: עִיגּוּלָא מִגּוֹ רִיבּוּעָא – רִיבְעָא, רִיבּוּעָא מִגּוֹ עִיגּוּלָא – פַּלְגָּא.

The Gemara answers: Rabbi Yoḥanan spoke in accordance with the opinion of the judges of Caesarea, and some say in accordance with the opinion of the Sages of Caesarea, who say: A circle that is circumscribed within a square is smaller than it by one quarter; with regard to a square that is circumscribed within a circle, the difference between them is equal to half the square. According to this explanation, Rabbi Yoḥanan calculated as follows: Since a square of four by four handbreadths has a perimeter of sixteen handbreadths, the circumference of the circle that encompasses it must be fifty percent larger, or twenty-four handbreadths.

“פָּחוֹת מֵאַרְבָּעָה עַל אַרְבָּעָה וכו׳״. אָמַר רַב נַחְמָן: לֹא שָׁנוּ אֶלָּא חַלּוֹן שֶׁבֵּין שְׁתֵּי חֲצֵירוֹת, אֲבָל חַלּוֹן שֶׁבֵּין שְׁנֵי בָתִּים – אֲפִילּוּ לְמַעְלָה מֵעֲשָׂרָה נַמִי, אִם רָצוּ לְעָרֵב – מְעָרְבִין אֶחָד. מַאי טַעְמָא – בֵּיתָא כְּמַאן דְּמָלֵי דָּמֵי.

It was taught in the mishna: If a window is less than four by four handbreadths, or if it is above ten handbreadths from the ground, the residents of the two courtyards may not establish one joint eiruv but must instead establish two independent ones. Rav Naḥman said: They taught this halakha of a window within ten handbreadths of the ground only with regard to a window between two courtyards. But with regard to a window between two houses, even if it is above ten handbreadths as well, if they wish to establish an eiruv , they establish one eiruv . What is the reason for this halakha ? It is that a house is considered as though it were filled, and therefore there is no difference between below and above ten handbreadths with regard to a window in a house.

אֵיתִיבֵיהּ רָבָא לְרַב נַחְמָן: אֶחָד לִי חַלּוֹן שֶׁבֵּין שְׁתֵּי חֲצֵירוֹת, וְאֶחָד לִי חַלּוֹן שֶׁבֵּין שְׁנֵי בָתִּים, וְאֶחָד לִי חַלּוֹן שֶׁבֵּין שְׁתֵּי עֲלִיּוֹת, וְאֶחָד לִי חַלּוֹן שֶׁבֵּין שְׁנֵי גַּגִּין, וְאֶחָד לִי חַלּוֹן שֶׁבֵּין שְׁנֵי חֲדָרִים – כּוּלָּן אַרְבָּעָה עַל אַרְבָּעָה בְּתוֹךְ עֲשָׂרָה!

Rava raised an objection to the opinion of Rav Naḥman from that which was taught in a baraita : A window between two courtyards, and a window between two houses, and a window between two attics, and a window between two roofs, and a window between two rooms are all one and the same to me; they all must be four by four handbreadths and within ten handbreadths from the ground. This directly contradicts Rav Naḥman’s opinion.

תַּרְגּוּמָא אַחֲצֵירוֹת. וְהָא “אֶחָד לִי״ קָתָנֵי! תַּרְגּוּמָא אַאַרְבָּעָה עַל אַרְבָּעָה.

The Gemara answers: Explain that this halakha of ten handbreadths mentioned in the baraita is referring only to courtyards. The Gemara objects: Doesn’t the baraita teach: Are all one and the same to me, indicating that they are all equal in this regard? Rather, explain that they are all equal in that the window must be the size of four by four handbreadths, but not that all must be within ten handbreadths of the ground.

בְּעָא מִינֵּיהּ רַבִּי אַבָּא מֵרַב נַחְמָן: לוּל הַפָּתוּחַ מִן בַּיִת לָעֲלִיָּיה, צָרִיךְ סוּלָּם קָבוּעַ לְהַתִּירוֹ, אוֹ אֵין צָרִיךְ סוּלָּם קָבוּעַ לְהַתִּירוֹ?

Rabbi Abba raised a dilemma before Rav Naḥman: With regard to an aperture that opens from the ceiling of a house occupied by one person to an attic occupied by another, must a permanent ladder be positioned in the opening to render carrying from one level to the other permitted by turning the two into a single residence? Or, is a permanent ladder not necessary to render it permitted?

כִּי אָמְרִינַן בֵּיתָא כְּמַאן דְּמָלֵי דָּמֵי – הָנֵי מִילֵּי מִן הַצַּד, אֲבָל בָּאֶמְצַע – לֹא. אוֹ דִּילְמָא לָא שְׁנָא?

The Gemara clarifies the two sides of the question: When we say that a house is considered as though it were filled, does this apply only to a window positioned on the side, but not to a window in the middle? In that case, the opening would not be viewed as near the full part of the house, and a permanent ladder would be required. Or perhaps there is no difference, and since the house is considered filled, no ladder is necessary.

אֲמַר לֵיהּ: אֵינוֹ צָרִיךְ. סָבוּר מִינָּהּ: סוּלָּם קָבוּעַ הוּא דְּאֵינוֹ צָרִיךְ, הָא סוּלָּם עֲרַאי – צָרִיךְ. אִיתְּמַר, אָמַר רַב יוֹסֵף בַּר מַנְיוּמִי, אָמַר רַב נַחְמָן: אֶחָד סוּלָּם קָבוּעַ וְאֶחָד סוּלָּם עֲרַאי – אֵינוֹ צָרִיךְ.

Rav Naḥman said to him: It is not necessary. The Sages understood from this response that he meant that a permanent ladder is not required, but a temporary ladder is required. However, it is stated in this regard: Rav Yosef bar Manyumi said that Rav Naḥman said: Neither a permanent ladder nor a temporary ladder is required, as the fact that the opening is located within the house is sufficient to render it permitted to carry from the house to the attic.

מתני׳ כּוֹתֶל שֶׁבֵּין שְׁתֵּי חֲצֵירוֹת, גָּבוֹהַּ עֲשָׂרָה וְרוֹחַב אַרְבָּעָה – מְעָרְבִין שְׁנַיִם, וְאֵין מְעָרְבִין אֶחָד.

If a wall between two courtyards is ten handbreadths high and four handbreadths wide, the residents of the courtyard establish two eiruvin , a separate one for each courtyard, but they may not establish one eiruv .

הָיוּ בְּרֹאשׁוֹ פֵּירוֹת – אֵלּוּ עוֹלִין מִכָּאן וְאוֹכְלִין, וְאֵלּוּ עוֹלִין מִכָּאן וְאוֹכְלִין, וּבִלְבַד שֶׁלֹּא יוֹרִידוּ לְמַטָּן.

If there was produce on top of the wall, these residents of one courtyard may ascend from this side and eat from it, and those residents of the other courtyard may ascend from that side and eat from it, provided that they do not lower the produce down from on top of the wall to one of the courtyards.

נִפְרְצָה הַכּוֹתֶל, עַד עֶשֶׂר אַמּוֹת – מְעָרְבִין שְׁנַיִם, וְאִם רָצוּ – מְעָרְבִין אֶחָד, מִפְּנֵי שֶׁהוּא כַּפֶּתַח. יוֹתֵר מִכָּאן – מְעָרְבִין אֶחָד, וְאֵין מְעָרְבִין שְׁנַיִם.

If the wall was breached, the following distinction applies: If the breach was up to ten cubits wide, they establish two eiruvin , and if they desire, they may establish one eiruv , as it is similar to an entrance, like any opening less than ten cubits wide. If the breach was more than this, they establish one eiruv , and they may not establish two, as a breach of this size nullifies the partition and joins the two courtyards into a single domain.

גמ׳ אֵין בּוֹ אַרְבָּעָה מַאי? אָמַר רַב: אֲוִיר שְׁתֵּי רְשׁוּיּוֹת שׁוֹלֶטֶת בּוֹ, לֹא יָזִיז בּוֹ אֲפִילּוּ מְלֹא נִימָא.

The Gemara asks: If this wall is not four handbreadths in width, what is the halakha ? Rav said: In this case, the air of two domains controls it. Since the wall is not broad enough to be regarded a domain of its own, the top of the wall is seen as belonging to both courtyards and is then prohibited to both of them. Accordingly, one may not move anything on top of the wall, even as much as a hair’s breadth.