Eruvin 76 Circle in the Square Daf Ditty Eruvin 76: Circle in the Square The circle is the set of points in a plane that are equally distant from a fixed point in the plane. The fixed point is called the center. The given distance is called the radius. The totality of points on the circle is called the circumference. “Circle” is from the Latin circulus, which means “small ring” and is the diminutive of the Latin circus and the Greek kuklos, which mean “a round” or “a ring”1 [Squaring the circle] involves constructing an ideal square with an area equal to that of a given circle (where the radius of the circle is one, an area equal to pi) and doing so in a finite number of operations using only a straight edge and a compass. A practically identical problem is the rectification of the circle: Constructing an ideal straight line equal in length to the circumference of the 2 circle. 1 LIDDELL, Henry George and Robert Scott, eds.1889. An Intermediate Greek-English Lexicon. Oxford. Clarendon Press. 2 http://www.song-of-songs.net/Martin_Gardner.html Eruvin 76 Circle in the Square MISHNA: If there is a window in a wall that separates between two courtyards, and the window measures four by four handbreadths and is within ten handbreadths of the ground, the inhabitants of the courtyards establish two eiruvin, one for each courtyard. And if they desire, they may establish one eiruv, thereby merging the two courtyards, as they may be considered as one due to the window. Eruvin 76 Circle in the Square However, if the window measures less than four by four handbreadths, or if it is above ten handbreadths from the ground, it is no longer considered a valid opening, and the two courtyards cannot be considered a single courtyard. Therefore, the residents establish two eiruvin, but they may not establish one eiruv. can choose to חצירות of the ground, the טפחים in size, and within 10 טפחים If the window is 4 by 4 because עירובין because they are connected through the window, or separate עירוב make a single they are separated by the wall. Eruvin 76 Circle in the Square that less ג"רשב is not because we assume like חצירות would connect the טפחים The reason that four If the window .טפחים is always defined as being 4 פתח Rather, it is because a .לבוד is טפחים' ד than or the entire window is more than 10, טפחים is either less than 4 by 4 .עירובין are considered separate and must make two separate חצירות off the ground, the טפחים GEMARA: With regard to the mishna’s determination that the size of the window must be four by four handbreadths, the Gemara asks: Let us say that we learned an unattributed mishna in accordance with the previously cited opinion of Rabban Shimon ben Gamliel, who said: Any gap less than four handbreadths is considered lavud, i.e., two objects are considered connected if the space between them is less than four handbreadths. That would explain why the window must be four handbreadths in size, as otherwise it would be considered as though it were sealed, based on the principle of lavud. Eruvin 76 Circle in the Square The Gemara rejects this suggestion: Even if you say that the mishna is in accordance with the opinion of the Rabbis that only gaps of less than three handbreadths are included in the principle of lavud, the Rabbis disagreed with Rabban Shimon ben Gamliel only with regard to the halakhot of lavud, i.e., what is considered connected. But with regard to an opening, even the Rabbis agree that if there is an opening of four by four handbreadths, it is significant, and if not, it is not significant. אָמַר רבִי יוחָנָן: חַלון עָגול צָריְ שֶיְהֵא בְהֶיקֵפו עֶשְרים וְאַרבָעָה טְפָחִים, ושְנַיִם ומַשֶּהו מֵהֶן בְתוְ .עֲשָרה, שֶאִם יְרבְעֶנו, נִמְצָא מַשֶּהו בְתוְ עֲשָרה Rabbi Yoḥanan said: A circular window must have a circumference of twenty-four handbreadths, with two and a bit of them within ten handbreadths of the ground, so that when he squares the window, i.e., if he forms the shape of a square inside it, it measures four by four handbreadths, and a bit of it is then within ten handbreadths of the ground. .מִכְדי, כֹל שֶיֵש בְהֶיקֵפו שְלשָה טְפָחִים — יֵש בו בְרוחְבו טֶפַח, בִתְריסַר סַגִיא The Gemara poses a question with regard to this calculation: Now, since there is a general principle that any circle with a circumference of three handbreadths is one handbreadth in diameter, then according to this formula, a window with a circumference of twelve handbreadths, meaning that it has a diameter of four handbreadths, should be sufficient to create a window of four by four. .הָנֵי מִילֵי בְעִיגולָא, אֲבָל בְריבועָא בָעִינַן 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 !מִכְדי, כַמָה מְרובָע יָתֵר עַל הֶעָגול — רבִיעַ, בְשִיתְסַר סַגִיא 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 מִילֵי עִיגולָא דְנָפֵיק מִגו ריבועָא. אֲבָל ריבועָא דְנָפֵיק מִגו עִיגולָא, בָעִינַן טְפֵי. מַאי טַעְמָא? .מִשּום מורשָא דקרנָתָא 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. Eruvin 76 Circle in the Square 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. .מִכְדי, כׇּל אַמְתָא בְריבועַ — אַמְתָא ותְרי חומְשֵי בַאֲלַכְסונָא, בְשֵיבְסַר נְכֵי חומְשָא סַגִיא 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? רבִי יוחָנָן אָמַר כִי דַיָינֵי דקיסָרי, וְאָמְרי לַה כְרבָנַן דקיסָרי, דְאָמְרי: עִיגולָא מִגו ריבועָא — ריבְעָא, .ריבועָא מִגו עִיגולָא — פַלְגָא 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 Eruvin 76 Circle in the Square 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. Avrohom Adler writes:3 If between two courtyards there was a window of four tefachim by four, within ten tefachim from the ground, the tenants may prepare two eiruvs, or, if they prefer, they may prepare one (jointly). [The tenants of one courtyard deposit their eiruv in the other, and by joining together, both groups of tenants are permitted the unrestricted use of both courtyards.] If the size of the window was less than four tefachim by four (which cannot be regarded as a valid opening), or higher than ten tefachim from the ground, two eiruvs may be prepared, but not one. 3 http://dafnotes.com/wp-content/uploads/2015/12/Eiruvin_76.pdf Eruvin 76 Circle in the Square [This is because the wall constitutes a solid partition between the courtyards. It is consequently forbidden to move objects between the courtyards either over the wall or through any small apertures or cracks in it.] Our Daf asks: Must it be assumed that we have here learned an anonymous Mishna (when it stated that if the window is less than four tefachim square, it is regarded as closed) in agreement with Rabban Shimon ben Gamliel who ruled that wherever an opening is less than four tefachim, it is considered lavud? [Is it likely, however, that an anonymous Mishna, which usually represents the accepted halachah, would agree with an individual opinion against that of the majority, and the majority maintains that the principle of lavud is stated only when an opening is less than three tefachim?] The Gemora answers: It may be said to agree even with the Rabbis; for the Rabbis differed from Rabban Shimon ben Gamliel only in regard to the laws of lavud. Regarding an opening, however, even they may agree that only if its size is four tefachim by four is it regarded as a valid opening, but otherwise it cannot be so regarded.