Talmud - Eruvin 76b עירובין ע״ו ב׳

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

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.

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.

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 must have a circumference of twenty-four handbreadths?

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.

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.

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.

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.

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?

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.

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.

MISHNA: 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.

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.

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.

GEMARA: 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.

Traduction : The William Davidson digital edition of the Koren Noé Talmud, with commentary by Rabbi Adin Even-Israel Steinsaltz.

Eruvin 76b examines the geometric relationship between a square window and its circumscribing circle, debating precise circumference calculations for halachic measurements.