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 circle.2

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.

The Gemora asks on the Mishna’s wording: Isn’t it obvious? For, since it was said that the window (if it is to be regarded as a valid opening that enables the tenants of both courtyards to join in a single eiruv) must be four tefachim by four, within ten tefachim, would I not naturally understand that if it was less than four and higher than ten it is not valid opening?

The Gemora answers: It is this that we were taught: The reason (that the window above ten tefachim is invalid) is because all of it was higher than ten tefachim from the ground, but if a part of it was within ten tefachim from the ground, the tenants may prepare two eiruvs, or, if they prefer, they may prepare one. [This could not have been inferred from the first clause of our Mishna which might have been taken to imply that the entire window must be within ten tefachim from the ground; and since ‘higher than ten tefachim’ has to be stated, it incidentally states also ‘less than four.’]

The Gemora notes: Thus we have learned in a Mishna what the Rabbis taught elsewhere in a braisa: If (almost) the entire window was higher than ten tefachim from the ground, but a part of it was within ten tefachim from it, or if (almost) all of it was within ten tefachim and a part of it was higher than ten tefachim, the tenants may prepare two eiruvs, or, if they prefer, they may prepare one.

The Gemora asks: Now then, where (almost) the entire window was higher than ten tefachim from the ground, but a part of it was within ten tefachim from it, you ruled that the tenants may prepare two eiruvs, or, if they prefer, they may prepare one; was it also necessary to mention the case where almost) all of it was within ten tefachim and a part of it was higher than ten tefachim? Eruvin 76 Circle in the Square

The Gemora answers: The Tanna taught it using the following format: This, and there is no necessity to say that. Rabbi Yochanan ruled: A round window must have a circumference of twenty-four tefachim (and then they can join together in one eiruv), and (the) two (lower tefachim) and a fraction more must be within ten tefachim from the ground, so that, when it is squared (and thus reduced on each side of the square by two tefachim, leaving a square window of the size of 8 — (2 + 2) by 8 — (2 + 2) = 4 X 4 tefachim; this is based upon the assumption that the area of a square constructed within a circle is half the area of the circle itself), a fraction remains within the ten tefachim from the ground.

The Gemora asks: Let us consider: Any circle that has a circumference of three tefachim is one tefach in diameter. Shouldn’t then twelve tefachim be sufficient?

The Gemora answers: This applies only to a circle, but where a square is to be inscribed within it, a greater circumference is required. [As the window under discussion must be four tefachim square, the diameter of the circle in which such a square can be inscribed must have, as ruled by R’ Yochanan, a minimum circumference of twenty-four tefachim.]

The Gemora asks: Let us consider: By how much does the perimeter of a square exceed that of a circle? It is by a quarter; shouldn’t then a circumference of sixteen tefachim be sufficient? The Gemora answers: This applies only to a circle that is inscribed within the square, but where a square is to be inscribed within a circle, it is necessary for the circumference of the circle to be much bigger.

The Gemora explains the reason for this: It is in order to allow space for the projections of the corners (of the square). [A circular window with a circumference that is less than twenty-four tefachim would not contain the area that is required.]

The Gemora asks: Let us consider: Every cubit in the side of a square corresponds to one and two fifths cubits in its diagonal; shouldn’t then a circumference of sixteen and four fifths tefachim be sufficient?

The Gemora answers: Rabbi Yochanan holds the same view as the judges of Caesarea, or, as others say, as that of the Rabbis of Caesarea, who maintain that the perimeter of a circle that is inscribed within a square is one-quarter less than the circumference of the square, while the perimeter of the square that is inscribed within that circle is one-half less than the circumference of the circle. Eruvin 76 Circle in the Square

THE

GEOMETRICAL FORMULAE OF THE JUDGES OF CAESAREA

Rav Mordechai Kornfeld writes:4

The Mishnah (76a) teaches that in order for a window in the wall between two Chatzeros to be considered a Pesach (opening) between the Chatzeros and provide the Chatzeros with the choice to join together with one Eruv, the window must measure at least four by four Tefachim and be within ten Tefachim of the bottom of the wall.

What must be the dimensions of the window if it is not square but round? Rebbi Yochanan (76a) states that a circular window "must have 24 Tefachim in its circumference, and two Tefachim and a bit of the window must be within ten Tefachim of the bottom of the wall, so that if a square would be inscribed in the circle a part of it would be within ten Tefachim of the ground." Rebbi Yochanan maintains that a circle drawn around a square with sides of four Tefachim (and a perimeter of 16 Tefachim) has a circumference of 24 Tefachim.

Our Daf concludes that Rebbi Yochanan's geometrical calculations are based on the formula of the Judges of Kesari. The Judges of Kesari taught that the circumference of a circle inscribed

4 https://www.dafyomi.co.il/eruvin/insites/ev-dt-076.htm Eruvin 76 Circle in the Square inside of a square is 25% smaller than the square's perimeter, and the circumference of a circle circumscribed around the outside of a square is 50% larger than the square's perimeter. Accordingly, the circumference of the circle drawn around the 16-Tefach perimeter of a square is 50% larger than the square's perimeter, or 24 (50% of 16 added to 16 is 24).

As mathematics demonstrates, and as the Gemara itself in Sukah (8a) points out, this formula is clearly incorrect. According to the formula used by the Chachamim (see Insights to Eruvin 14:2), the actual relationship of the perimeter of an inscribed square to the circle around it is 3 X (1.4 x S), where 3 = the value of Pi, and S = the length of a side of the square.

The ratio that the Chachamim use for the relationship between the side of a square and its diagonal (which is also the diameter of the circumscribed circle) is 1:1.4. Therefore, the circumference of a circle circumscribed around a square with sides of 4 Tefachim is 3 X (1.4 X 4), or 16.8 -- and not 24!

How did the Judges of Kesari make such a mistake, and why did Rebbi Yochanan follow them?

TOSFOS (DH v'Rebbi Yochanan) (see below) suggests that the Judges of Kesari were not giving the relationship of the perimeter of the inner square to a circle around it. Rather, they were giving the relationship of the area of the inner square to an outer square drawn around the circle that encloses the inner square.

This is what they mean when they say that "when a circle is drawn around the outside of a square, the outer one's (i.e., the outer square's) perimeter is 50% larger than the inner one's." (The picture printed in Tosfos in the Vilna Shas is slightly misleading. In the picture that appears in the TOSFOS HA'ROSH (see Graphic), the inner square is rotated 45 degrees from the orientation of the outer square. This is a clearer demonstration of Tosfos' point.) The area of the inner square is exactly half of the area of the outer square.

Tosfos concludes that Rebbi Yochanan misunderstood the intention of the Judges of Kesari, and he made his statement regarding the relationship of the circumference of a circle to the perimeter of a square based on his misunderstanding.

However, the VILNA GA'ON (Hagahos ha'Gra here and in OC 372) takes issue with the conclusion of Tosfos and asserts that even Rebbi Yochanan is referring to the perimeter of a square with a circle inscribed (which, in turn, circumscribes a square of four by four Tefachim). That perimeter is indeed nearly 24 Tefachim. (According to the formula used by the Chachamim's to calculate the diagonal of a square, it is 22.4 Tefachim. The geometric calculation yields 22.6 Tefachim.) Eruvin 76 Circle in the Square

When Rebbi Yochanan says that two Tefachim of the window must be within ten Tefachim of the ground, he is referring to the length of the arc that begins from the lowest point of the circle that was drawn around the square window (four by four), to the point where it intersects with the bottom of the square window. (Actually, as the Vilna Ga'on points out, a bit more of the arc must be within ten Tefachim of the ground -- 0.1 Tefach according to the formula of the Chachamim, or 0.121 Tefach according to the geometric calculation.)

RITVA explains that the calculations of the Judges of Kesari and Rebbi Yochanan are accurate. When Rebbi Yochanan mentions a "round" window, he does not mean a circular window with an imaginary square inscribed within it. Rather, he is referring to a window made in the shape of a square with four semi-circles protruding from the four sides (like a four-leaf clover; see Graphic).

In such a case, the perimeter of the window (i.e., the arcs of the four semi-circles) indeed is 50% larger than the perimeter of the square around which the arcs are drawn. In order to ensure that the square inside the clover-shaped window reaches within ten Tefachim from the ground, at least two Tefachim and a bit of the radius of the bottom semi-circle must be within ten Tefachim (since the radius of each semi-circle is two, or half of one side of the square, which is four). Alternatively, two and a bit Tefachim plus four Tefachim of the perimeter of the semi-circle must be within ten Tefachim from the ground (as Rashi explains, end of 76a), since the perimeter of each semi-circle is six Tefachim.

RASHI does not explain how to justify the formula of the Judges of Kesari or how to understand Rebbi Yochanan's calculation. He seems to have no difficulty with them. Perhaps Rashi understands that the Judges of Kesari were proposing a Halachic stringency: When we determine a value (such as the circumference of a circle) by using the diagonal of a square for the purpose of a practical application in Halachah, we consider the diagonal to be equal to the sum of two consecutive sides of the square or rectangle (since the two sides connect one end of the diagonal to the other). Thus, if the sides of the inscribed square are each 4 Tefachim, then the diagonal is considered to be 8 Tefachim. Accordingly, the circle around that square would have a diameter of 8 Tefachim, and thus its circumference would be 24 Tefachim and not 16.8 (which is the length of the circumference based on the actual diameter of the square).

The reason for this is to prevent one from mistakenly using the length of the diagonal in a case in which he is supposed to use the sum of the lengths of two sides. In addition, reality does not allow for the application of pure mathematics (as the ratio of the diagonal to the sides of a square is an irrational number; moreover, it is not possible to measure an angle exactly). Therefore, the formula for determining the diagonal of a square for purposes of Halachic applications is the sum of the lengths of two sides.

If this is the reason why Rashi is not bothered by the apparent inaccuracy of the formula of the Judges of Kesari, then we may suggest that Rashi is consistent with his own opinion as expressed elsewhere (Shabbos 85a, Eruvin 5a, 78a, 94b), where Rashi seems to determine the Halachic Eruvin 76 Circle in the Square length of the diagonal of a rectangle by adding two consecutive sides. TOSFOS in all of those places argues with Rashi. Rashi may hold that this computation of the length of the diagonal may be relied upon for rulings that involve Halachos d'Rabanan. (M. KORNFELD)

(d) Perhaps it is possible to propose an entirely new explanation, according to which the Judges of Kesari and Rebbi Yochanan are entirely correct.

When Rebbi Yochanan says that a circular window "must have 24 Tefachim in its circumference," he does not mean that the circumference must be 24 Tefachim, but that there must be 24 square Tefachim inside the circle. In other words, he means that the area of the circle must be 24 square Tefachim!

The area of a circle is calculated by multiplying Pi by the radius squared. The radius of the circle drawn around a square with sides that are each 4 Tefachim long is half of the diagonal (5.6), which is 2.8. Using the Halachic estimate of the value of Pi as 3, we arrive at the following calculation: 3 X (2.8)(2.8) = 23.52, or approximately 24.

This is what Rebbi Yochanan means when he says that the circle must have within its circumference an area of 24. (He rounds up to 24 as a stringency.)

What does Rebbi Yochanan mean when he says that there must be two and a bit within a height of ten from the ground? 24 square Tefachim is the area of the circle. Within that area is an inscribed square of 4 by 4, which has an area of 16 square Tefachim. What is the area of the four sectors that are outside of the square? They represent the difference between the area of the circle and the area of the square, which is 24 - 16 = 8, and thus each sector has an area of 2 Tefachim. This is what Rebbi Yochanan means when he says that in order to get the inscribed square of 4 by 4 Tefachim below a height of ten Tefachim, at least 2 Tefachim and a bit of the area of the circular window must be below ten Tefachim!

David Garber and Boaz Tzaban of Bar Ilan University5, who have been printing articles on geometrical themes in Chazal for a number of years, pointed out to us that the ME'IRI here suggests this solution, citing it in the name of the BA'AL HA'ME'OR. It can be traced further back to a responsum of the RIF in Temim De'im #223.

An Acharon, TESHUVOS GALYA MASECHES #3, offers this solution as well. Note, also, that according to the mathematics of Chazal, the ratio of the area of a circle to the area of the square in which the circle is inscribed is a ratio of 3:4.

The outer square is double the area of the inner square (which is 4 by 4 Tefachim). Thus, the outer square has an area of 32 square Tefachim. Accordingly, the calculation for the area of the

5 ON THE RABBINICAL APPROXIMATION OF 7r: BOAZ TSABAN AND DAVID GARBER, Department of Mathematics, Bar-Han University, Eruvin 76 Circle in the Square circle is exactly 24 Tefachim (or 3/4 of the outer square), and not just approximately 24, as we concluded using the equation of Pi X r X r (this is because the area of the outer square (32) is exactly double the area of the square drawn inside of the circle (16), and 3/4 of 32 is 24). The Me'iri uses the word "Shibur" or "Tishbores" to refer to the calculation of area.)

RAMBAM Commentary to Mishna Eruvin I:5

You need to know that the ratio of the circle’s diameter to its circumference is not known and it is never possible to express it precisely. This is not due to a lack in our knowledge, as the sect called Gahaliya [the ignorants] thinks; but it is in its nature that it is unknown, and there is no way [to know it], but it is known approximately. The geometers have already written essays about this, that is, to know the ratio of the diameter to the circumference approximately, and the proofs for this.

This approximation which is accepted by the educated people is the ratio of one to three and one seventh. Every circle whose diameter is one handbreadth, has in its circumference three and one seventh handbreadths approximately. As it will never be perceived but approximately, they [the Hebrew sages] took the nearest integer and said that every circle whose circumference is three fists is one fist wide, and they contented themselves with this for their needs in the religious law Eruvin 76 Circle in the Square

Various ancient Greek writers, including Hero, Eutocius, and Simplicius, understand the difficulty of finding an exact value for the ratio, and seem to realise the possibility of its being irrational, yet it appears that none of their extant statements are as strong as Maimonides’.

As for medieval mathematicians preceding Maimonides, we have the following: Yusuf al-M u’tam an (11th century ), in the Isiikmal (which was revised and taught by Maimonides) cites 7r in the chapter dealing with irrationality. However, he does not explicitly assert such suspicions. The only explicit statement concerning the irrationality of w in the earlier extant literature is to be found in al-Blmnl’s Masudic Canon (ca. 10.30 c e ) ( Qanun al-MascUdi, Book III, Chapter 5): “and the number of the circumference has also a ratio to the nu ber of the diameter, although this (ratio) is irrational”.

It is not known whether Maimonides knew the Masudic Canon, anyway, the irrationality of Pi was proved (by Lambert) only in the eighteenth century.

It is therefore still a mystery what made Maimonides so sure about the irrationality of Pi. Victor J, Katz has noted that this is similar to Ptolemy’s claim (in Almagest I, 10) that one cannot trisect an angle using a straightedge and a compass: “The chord corresponding to an arc which is one- third of the previous one cannot be found by geometrical methods”.

Tosafos

תוספות ד"ה ורבי יוחנן אמר כדייני דקיסרי כו '

Tosfos concludes that R. Yochanan erred about their words.

דקסבר אמתא בריבועא תרי אמתא באלכסונא (a) Explanation: He holds that a square Amah has a diagonal of two Amos וליתא להך דדייני דקיסרי כדאמר בפ''ק דסוכה (דף ח:) דהא קא חזינא דלאו הכי הוא שכל האורך והרוחב לא הוי אלא תרי אמה (b) Observation: The judges of Kisari (who say so) are wrong, like it says in Sukah (8b), for we see that this is not so. The entire length and width combined is only two Amos! (The diagonal must be shorter.) ואע''ג דהתם מפרש שפיר מילתיה דר' יוחנן הכא לא מצי לאוקומי אלא כדייני דקיסרי וכי היכי דאמר התם דליתא לדייני דקיסרי ה''נ ליתא לדרבי יוחנן דהכא 1. Eruvin 76 Circle in the Square

Even though there R. Yochanan's teaching is explained properly, here we can establish it only like the judges of Kisari. Just like it says there [in Sukah] that the judges of Kisari are wrong, likewise R. Yochanan's teaching here is wrong. וקשה היאך טעו דייני דקיסרי הא קא חזינן דלאו הכי הוא (c) Question #1: How could the judges of Kisari err? We see that it is not so! ועוד דכי היכי דקאמרי (כן נראה להגיה) עיגולא מגו ריבועא ריבעא דהיינו מכל הריבוע הכי נמי הוה להו למינקט ריבועא מגו עיגולא תילתא מכל העיגול שהוא פלגא מן הריבוע שבפנים (d) Question #2: Just like they said that a circle in a square is a quarter [less], i.e. than the entire square, they should have said that a square in a circle is a third [less] than the entire circle, which is half of the square inside! או הוי להו למינקט עיגולא מגו ריבועא תילתא מן העיגול שבפנים 1. Or they should have said that a circle in a square is a third [less], i.e. than the circle inside (just like they said that a square in a circle is a half [less] than the square inside). וי''מ דדייני דקיסרי לא דברו אלא לענין קרקע שבתוך הריבוע והעיגול דלענין זה דבריהם אמת (e) Answer: Some say that the judges of Kisari discuss only area in the square and circle. Regarding this, their words are true; שכשתעשה ריבוע ב' אמות על ב' אמות ותעשה עיגול בפנים ב' על ב' ועוד ריבוע בתוך העיגול תמצא בריבוע החיצון ארבע חתיכות אמה על אמה 1. When you make a square two Amos by two Amos, and make a circle inside two by two, and another square inside the circle, it turns out that the outer square is four pieces of an Amah by an Amah; ובעיגול מתוך ריבוע שלש חתיכות של אמה על אמה דמרובע יותר על העיגול רביע 2. In the circle inside the square there are [the area of] three pieces of an Amah by an Amah, for a square is a quarter more than [of its own area] than the circle [inside]; ובריבוע הפנימי אין בו כי אם ב' שהרי הוא חציו של חיצון דהיינו תילתא פחות מן העיגול 3. The inner square has only two [square Amos], for it is half of the outer [square], i.e. a third less than the circle. אלא שהש''ס בסוכה ור' יוחנן דהכא טעו בדבריהם והיו סבורים שעל ההיקף אמרו (f) Eruvin 76 Circle in the Square

Answer #1 (cont.): The Gemara in Sukah and R. Yochanan here erred about their words (of the judges of Kisari). They thought that they discuss perimeter. והשתא אתי שפיר מה דנקטו (כן נראה להגיה) פלגא דהכל קאי אריבוע החיצון כלומר עיגולא מגו ריבועא ריבעא כלומר פחות רביע מריבוע החיצון (g) Support: Now it is fine that they mentioned half, for everything refers to the outer square. I.e. a circle in a square is a quarter, i.e. a quarter less than the outer square; ריבועא מגו עיגולא פלגא ממה שנשאר בריבוע החיצון על ריבוע הפנימי דהוא נמי חציו של פנימי 1. A square in a circle is half, from what remains in the outer square over the inner square (i.e. it is less than the circle one square Amah, which is half of [two Amos, i.e.] the excess of the outer square over the inner), which is also half the inner [square].

Halacha

[In a case of] a window between two courtyards:

If it has four handbreadths by four handbreadths or more and it was within ten handbreadths of the ground — even if [almost] all of it was above ten and a little of it was within ten or [almost] all of it was within ten and a little of it above ten — the residents of the two courtyards have permission to make one eruv for all of it and make it into one courtyard so that they can carry Eruvin 76 Circle in the Square from this one to that one, if they wanted. And if they wanted, they could [also] make two eruvs, these for themselves and those for themselves.

[But if] the window was less than four or it was all above ten, they may [only] make two eruvs, these for themselves and those for themselves. RAMBAM Hil Eruvin 4:1

Orach Chayim 372:4

Eruvin 76 Circle in the Square

At the end of the sugya, our daf concludes that the window in the wall between the chatzeros must have a circumference of 16 and 4/5 tefachim, and the Shulchan Aruch (372:4) rules accordingly.

Mishnah Berura (note 30), and Sha’ar Hatziun (note 18) state that this circumference allows a square of 4 by 4 tefachim to be inscribed within it. Actually, using the rule of Tosafos (Sukka 8a, that every unit along the edge of a square results in one and two-fifths along the” (כל ה ד diagonal, the measure of the circle would have to be slightly larger to accommodate a square of a full 4 by 4.”

[The Gemara uses pi as 3. Using this, a circumference of 16.8 translates to a diameter of 5.6, and an inscribed square of 4. However, pi is actually slighty more than 3 .14, and a circle of 16.8 would not allow for a full 4 by 4 as needed.]

Nevertheless, the halachah recognizes this discrepancy, by still allows the measurement to be used, albeit being aware that it is approximate, and even too low. In fact, it may be that the tradition to rely upon these rounded numbers dates back to Sinai itself, and that we may therefore use these measurements not only in rabbinic laws, but in Torah law calculations as well.

Chazon Ish (138:2) also writes along these lines, and the truth is that this opinion has already been recorded in Tosafos HaRosh (in his commentary, earlier to 12a).

This diagram presents Tosafot’s proof that the area of a square circumscribed by a circle is exactly half of the area of a square circumscribing a circle. The Eruvin 76 Circle in the Square

four small squares formed by the lines within the circle demonstrate that the inner square is comprised of four triangles, each of which measures half a small square.

Ratio between a square circumscribed by a circle and a square circumscribing the same circle

Steinzaltz (OBM) writes:6

The sixth chapter of Massekhet Eiruvin dealt with situations where two courtyards joined together and became one unit for the rules of eiruv and carrying on Shabbat.

The seventh perek, which begins on our daf, opens by discussing cases where hatzeirot (courtyards) that are next to one another either cannot join each other, are obligated to join each other, or are permitted to do so.

The first Mishna deals with courtyards that are divided by a wall which has a window in it. If the window is within ten tefahim (handbreadths) of the ground and is minimally four tefahim square in size, then the courtyards can choose whether or not to join as one. If the window is higher than ten tefahim or smaller than four by four, the hatzeirot are considered separate and need to make their own eiruvin.

6 https://www.steinsaltz-center.org/home/doc.aspx?mCatID=68446 Eruvin 76 Circle in the Square

Rabbi Yohanan in the Gemara introduces the possibility of a round window, arguing that it would need to be 24 tefahim in circumference in order to ensure that a square inscribed in that window would be at least four by four. In the ensuing discussion about the relationship between circles and squares, the Gemara explains that Rabbi Yohanan’s position is based on the rule taught by the judges of Caesarea that “a square inscribed within a circle is half of the square.”

Later on in the Gemara, Rabbi Yohanan’s position is rejected in its entirety as being based on an error. Rabbi Ya’akov Kahane in his Ge’on Ya’akov argues that Rabbi Yohanan knew that his figures were not accurate, but chose to present a larger than necessary rule so that, in case of a Eruvin 76 Circle in the Square mistake, there would be well over the minimum four by four, leaving room for error. Nevertheless, many attempts have been made to try and explain the mathematical positions presented by these Sages.

In explaining the rule of the judges of Caesarea, Tosafot argue that they are discussing the case of a square inscribed in a circle, which, itself, is inscribed in a square.

By drawing lines that bisect the outer square, the circle and the inner square, it becomes clear that the inner square is half the size of the outer one. The inner square is made up of four triangles, each of which is half of the four smaller squares that together make up the outer square.

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Jeremy Brown writes:7

Our Daf is discussing the necessary dimensions of a window that would allow two private domains separated by a wall to merge as one. Rabbi Yochanan rules that a round window must have a circumference of at least twenty-four tefachim. Then we learn of a mathematical rule stated by the “Rabbis of Caesarea.”

ערובין עו,ב

עיגולא מגו ריבועא ריבעא ריבועא מגו עיגולא פלגא

A circle that comes out of a square - one fourth; a square that comes out of a circle- one half

Rashi explains the second part of this cryptic statement: One half of the square’s perimeter is what you loose by going from a circle’s perimeter to that of a square that fits inside of it. Consider a circle, says Rashi, with a circumference (i.e. a perimeter) of 24 (inches, meters, amot, whatever). Inside of this circle is a square. The perimeter of the square will be 16. Half of 16 is 8, and the circumference of the circle is 8 more than the perimeter of the square. Another way of thinking about this is that in this particular geometric setup, the circumference of the circle will be always be 1.5 larger than the perimeter of the square.

7 talmudology.com Eruvin 76 Circle in the Square

But this is not correct. In the circle below, the value of the diameter d is also the same as the value of the hypotenuse of an isosceles triangle of length 4. And as you can see, following the simple math, it isn’t. According to Rashi, d=8, but in fact d=5.65.

The same discussion appears in the tractate Sukkah (8a) and Tosafot in both places notes that the Rabbis of Caesarea, together with Rabbi Yochanan who cites the rule in their name, were wrong. There is also an error in Rashi, whose explanation we have just seen..

תוס עירובין עו, ב

דהא קא חזינא דלאו הכי הוא שכל האורך והרוחב לא הוי אלא תרי אמה…וקשה היאך טעו דייני דקיסרי הא קא חזינן דלאו הכי הוא

It is not true that an [isosceles] triangle with sides [one unit long] has a hypotenuse of two…how could the Rabbis of Caesarea have made such an error, for we can measure and see that this is not correct… Eruvin 76 Circle in the Square

“A SQUARE INSIDE A CIRCLE INSIDE A SQUARE”

To save Rabbi Yochanan, the Rabbis of Caesarea and Rashi from this embarrassing error, Tosafot suggests that the Rabbis of Caesarea were never discussing the circumference; instead, they were discussing the area. And they were discussing a square that is inside a circle that is inside a square. Like this:

עיגולא מגו ריבועא) ”Tosafot suggests that phrase “A circle that comes out of a square - one fourth .means that the area of the inner circle is one fourth less than the outer square (ריבעא

ריבועא מגו) ”Tosafot also suggests that the phrase “a square that comes out of a circle- one half means that the area of an inside square is one half that of the outside square. Here is (עיגולא פלגא the demonstration, assuming that π=3. Eruvin 76 Circle in the Square

ANOTHER RASHI WITH WRONG MATH

In a couple of pages, (Eruvin 78a) we read

עירובין עח, א

אמר רב יהודה אמר שמואל כותל עשרה צריך סולם ארבעה עשר להתירו רב יוסף אמר אפילו שלשה עשר ומשהו

Rav Yehuda said that Shmuel said: If a wall is ten handbreadths high, it requires a ladder fourteen handbreadths high, so that one can place the ladder at a diagonal against the wall. The ladder then functions as a passageway and thereby renders the use of the wall permitted. Rav Yosef said: Even a ladder with a height of thirteen handbreadths and a bit is enough, [as it is sufficient if the ladder reaches within one handbreadth of the top of the wall].

Here is Rashi’s explanation:

עירובין עח, א רשי

סולם ארבעה עשר - שצריך למשוך רגלי הסולם ארבעה מן הכותל לפי שאין סולם זקוף נוח לעלות

A ladder that is 14 handbreadths high: The feet of the ladder need to be placed four handbreadths from the wall

Here is a diagram of the setup, and the problem with Rashi: Eruvin 76 Circle in the Square

This error is noted by Tosafot (loc. cit.)

צריך סולם ארבעה עשר להתירו - פי' בקונטרס שצריך למשוך רגלי הסולם ארבעה מן הכותל ולא דק דכי משיך ליה י' טפחים נמי מן הכותל שהוא שיעור גובה הכותל יגיע ראש הסולם לראש הכותל דארבעה עשר הוא שיעור אלכסון של י' על י' דכל אמתא בריבועא אמתא ותרי חומשי באלכסונא

A ladder 14 handbreadths heigh is needed - Rashi explains that the feet of the ladder need to be placed 4 handbreadths from the wall. This is not accurate. For when the feet of the ladder are placed ten handbreadths from the wall, (a larger measure) the ladder will still reach the top of the wall. Because the length of the diagonal [i.e.the hypotenuse] is 14 handbreadths in a 10x10 [right angled] triangle. Because for every unit along the sides of square, the diagonal will be 1.4

What Tosafot is getting at is that in a 1x1 right angled triangle, the hypotenuse must be √2. But √2 is an irrational number, meaning the calculation never ends. However, an irrational number is not useful for our real word measurements, and so Tosafot rounds √2 to 1.4, just as in the Talmud the value of π, another irrational number, is rounded to 3. 8

8 (Shalom Kelman, a loyal Talmudology reader, sent us another explanation of the errors which you can read here. Eruvin 76 Circle in the Square

WHAT TO MAKE OF THESE ERRORS?

We have reviewed two mathematical errors made by the Rabbis of Caesarea and Rashi, but how should we view them? As ignorance, mistaken calculations, or something else?

In his classic 1931 work Rabbinical Mathematics and Astronomy, (p58) W.M Feldman is firmly in the "Rashi was ignorant” camp.

It is however, most remarkable that although Rashi displayed great genius in mathematical calculation, he was quite ignorant of the most elementary mathematical facts. He was not aware that the sum of two sides of a triangle is greater than the third, for he says if a ladder is to be placed 4 spans from the foot of a wall 10 spans high so as to reach the top, then the ladder must be 14 spans high, (i.e. the sum of the two lengths), which of course is absurdly incorrect, the real minim length of the ladder must be only √(4x4)+(10x10)= √116=10.7 spans.

Judah Landa, in his book Torah and Science suggests that the Rabbis of Caesarea (and Rashi too, I suppose) had mistaken calculations. They did not give mathematics “the serious attention it deserved and that as a consequence their knowledge of it suffered.” Later commentators, like Tosafot and Maimonides knew that π was slightly greater than three, and Tosafot “demands to know why the Rabbis of Caesarea made statements without attempted verification by measurement and experiment.”

NO ERROR HERE

Rabbi Moshe Meiselman believes that the rabbis of the Talmud were incapable of making an error. Of any sort. In his book Torah, Chazal & Science he dedicates eight pages and copious footnotes to explain why, in fact, there were no errors in any of the math found on today’s page of Talmud. Among the sources he cites is a commentary on the Talmud written by the fifteenth century Rabbi Simeon ben Tzemach Duran, better known as the Tashbetz. In his Sefer Hatashbetz, he addresses the parallel discussion in the tractate Sukkah:

ובתוס' תרצו כי התלמודיים טעו בדבריהם של רבנן דקסרי …וכל זה חיזוק וסמך שאין בכל דברי רז"ל דבר שיפול ממנו ארצה כי הם אמת ודבריהם אמת

Tosafot explains that the Talmud believed that the Rabbis of Caesarea were mistaken…but there is not a single error in all the words of the Sages, for they are true and their words are true… Eruvin 76 Circle in the Square

Rabbi Meiselman concludes that not one of the commentaries on the Talmud “suggests that any of the Chachamim [Sages in the Talmud] made elementary errors in calculation or were ignorant of basic principles.”

Rabbi Meisleman aside, Rashi certainly seems to have made an error in his calculations. But why should this bother us? Of the hundreds of thousands of words written by Rashi, commenting on and explaining the entire Hebrew Bible and Babylonian Talmud, an error or two is hardly unexpected. (Just ask any author who has proof-read her manuscript a dozen only to find a typo in the published book.) Did Rashi misunderstand, or was he ignorant of Pythagorus and his Theorem? In the end it doesn’t matter much. We all make mistakes. I only hope I make as few of them as Rashi did. Now that would be an accomplishment.

Back to our Bar Ilan colleagues:9

Matityahu Hacohen Munk10 suggests a mystical explanation: some of the geometrical rules did not hold in King Solomon’s temple, according to Hebrew ancient traditions11. In the temple, the ratio of the circumference of a circle to its diameter was exactly Pi.

“In our reality this fails, but in order to join our reality with the “world of truth ,”

the temple’s values should be used in calculations for religious purposes. Of course, applying the halachic Pi naively to our reality would yield circles which do not satisfy the halachic requirements. For example, in order for a circle (in our reality) to circumscribe a certain square, its circumference must be Pi times the diagonal of the square. Using Pi yields a circle too small.

Nevertheless, even in our reality, it is possible to “experience” Pi in a manner of speaking. This is accomplished if one computes the circumference not of the circle but rather of the regular hexagon inscribed in it. Then, the circle circumscribing this hexagon will satisfy the halachic requirements; it will circumscribe the square in the above example.

Rabbi Haim David Z. Margaliot12 noted this possibility more than two decades before Munk. He suggests that the reason for this was that the circumference of the circle was measured from inside using a stick of length equal to the radius of the needed circle. In his interpretation, one edge of the stick was placed at an arbitrary point A on the circle, and the other edge was used to find the point B on the circle. Then the edge was put on B in order to find C , etc. (see Fig. 1).

9 https://www.jct.ac.il/media/3494/boaz-tsaban-and-david-gardner-on-the-rabbinical-approximation-of-pi.pdf

10 Matityahu Hacohen Munk, Three Geometric Problem s in the Bible and the Talmud, Sinai 51 (1962), 218-227. [Hebrew]

11 see for instance, Talmud Megilla 10b; Yoma 21a; Baba B atra 99a [7]; [8];[28]

12 Rabbi Haim David Zilber Margaliot, Dover Yesharim (Moadim section), 1938, reprint ed,: Givatayim : Kulmus 1959, 102-105. [Hebrew] Eruvin 76 Circle in the Square

If, after six iterations, the stick’s edge returned to the point A. then the “circumference” of the circle was six times the length of the stick or Pi times the diameter.

Rabbi Shimon Ben Tsemah (1361-1444) suggests another explanation in The Tashbets (part I, responsa 165): in fact, more precise values for Pi were known to the Talmudic Rabbis, but in order for their students to understand, they used the less precise value — “One should always teach his student in the easiest way” (Talmud Pesahim 3b; 63b). However, de-facto they used more precise values.

In summary, the following are the major approaches to the understanding of the Biblical and Talmudic value for Pi:

1. The rational-religious approach of Maimonides holds that, since we cannot know the exact values, the Bible tells us that we do not have to worry about this and that it suffices to use the value 3.34 2. The mystical approach of Munk contends that 3 was indeed the ratio of the circumference to the diameter in King Solomon’s temple: This value is used in order to bridge the gap between our world and the “world of truth .” For the sake of consistency, the halachic conditions are applied to the suitable regular polygons. 3 . The practical approach of Rabbi Shimon Ben Tsemah asserts that the rough approximations are used when teaching the students, but, when it comes to practice, the calculations are to be done by the experts.

——————————————————————————————————————— Eruvin 76 Circle in the Square

Sara Wolf writes:13

On our daf, the mishnah provides measurements for a window that is considered large enough to halakhically connect two courtyards for the purposes of making just one eruv:

If there is a window measuring four by four handbreadths between two courtyards, and it is within ten handbreadths of the ground, the residents make two eruvim — or if they wish, they can make one eruv.

We are next presented with a math puzzle. The Talmud wants to know: if a standard, square window needs to measure at least 4x4 to be large enough to potentially “connect” the two courtyards for the purposes of eruv, what is the minimum circumference of a circular window that would also fit the mishnah’s requirements — that is, a circular window that would encompass the minimum size square? If you’re so inclined, give it a try now!

OK, do you have an answer? Let’s see how your math compares to the rabbis’!

According to Rabbi Yohanan, the circle would need to have a circumference of 24 handbreadths. The anonymous voice of the Talmud, however, points out that Rabbi Yohanan has overestimated.

13 myjewishlearning Eruvin 76 Circle in the Square

The Talmud offers a better answer: a circle with a circumference of 16.8 handbreadths would be large enough to circumscribe a 4x4 square.

This second answer is much closer, but relies on approximating pi by a value of three instead of using its proper value of 3.1415… and rounding the square root of two down from 1.4142… to 1.4 (those ellipses indicate that both pi and the square root of 2 are irrational numbers, and the numbers after the decimal points continue infinitely without repeated pattern). If we use more precise values for pi and √2, we discover that the sages’ suggestion of 16.8 is actually a bit too small to circumscribe a 4x4 square. So while Rabbi Yohanan grossly overestimated, the other sages were off too and dictated a round window slightly too small.

If you are disappointed at the imprecision of the sages’ math, you are not alone.

Medieval Jewish thinkers grappled with the Talmud’s inexact calculation as well, especially since it seems to lead to a halakhic mistake — a window that is a bit too small. Both the Tosafot on Eruvin 14a and Maimonides’ commentary on the Mishnah struggled with the fact that the rabbis round pi down to 3, but they couldn’t come up with any real way of resolving this mathematical error without contradicting the rabbinic ruling.

The rabbis of the Talmud were creative and brilliant legal and exegetical minds, but rabbinic culture did not prize all forms of intellectual pursuit to the same degree. Although they weren’t exact about the values of these mathematical constants, they were at least pretty close.

Incidentally, the right answer is as follows: the diameter of the circle that would circumscribe a square is arrived at by multiplying pi times the length of the diagonal of the square (which is the side length times the square root of two). In this case, it comes to approximately 17.77 handbreadths.

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Circle in the Square: A Mystical Metaphor

URMI CHANDA-VAZ writes:14

1. SYMMETRY IN NATURE AND THE BEGINNINGS OF GEOMETRY

Once chaos gave way to creation, life and matter aligned themselves into grand patterns. Everywhere you look in nature, symmetry makes itself evident. From the smallest crystals to the largest trees, from delicate snowflakes to massive animals, the ordering principle of nature is manifest. Simple repetitions and consistent spaces give rise to symmetrical designs. While there

14 URMI CHANDA-VAZURMI CHANDA-VAZ:(PDF) Mysticism and Symbolism | Urmi Chanda - Academia.edu Eruvin 76 Circle in the Square are cases of seeming randomness in nature too, these are exceptions rather than rules. The larger picture is always methodical and hence awe- inspiring.

In its simplest forms, symmetry can be classified in terms of rotation and reflection15. These are called point symmetries, where a simple element rotated around a point forms a symmetrical design. Many flowers d i s p l a y t h i s k i n d o f symmetry. Yet another element, when duplicated and placed as its reflection forms the second kind of symmetry. A butterfly's wings are a perfect example in this case. Radial symmetry is another kind of point symmetry, and the structures are finite. Replications may occur in two or three dimensions, as in the case of snowflakes or dandelions. Another kind of symmetry is that of self similarity. In this kind, the basic structure remains the same despite regular amounts and periods of growth. Tree rings, shells and horns are examples of this kind of spiral symmetries.

15 David Wade, Symmetry: The Ordering Principle, (Wooden Books, 2006), p 4 Eruvin 76 Circle in the Square

Another common symmetrical manifestation in nature is that of the sphere. A sphere is one of nature's primary ordering mediums. From the smallest droplet of water to fruits and eggs to galactic bodies, a sphere is seen everywhere. While the smallest spherical bodies assume this shape due to surface tension, the larger bodies owe their shape to the power of gravity. In fact, the sphere was considered so sacred in the ancient times that the Greek philosopher Xenophanes16 replaced the whole pantheon on Greek gods with 'the sphere' and declared it a divine entity. With man coming across such instances in nature all the time, it was natural that he would study and revere the principles of symmetry and consequently, geometry.

The observation of symmetry gave birth to the discipline of geometry and its first reflections were religious rather than scientific. How some of the earliest cultures were inspired by and adopted these shapes in their religious practices will be dealt with in the following sections. For now, we turn our attention to the earliest references to geometry, particularly sacred geometry.

The earliest instances of geometrical reflection occur in cave paintings or etchings. Simple geometrical patterns have been found depicted in the cave art of cultures across the world, right back to the Palaeolothic age. Holme17 says that these paintings 'may very likely have served as a magic vehicle for gaining control over nature, for casting a spell on the game thus ensuring a successful hunt.

16 Jonathan Barne, The Presocratic Philosophers, (Routledge, 2002), p 76

17 Audun Holme, Geometry: Our Cultural Heritage, (Springer Science & Business Media, 2010), p 4 Eruvin 76 Circle in the Square

From the hexagonal cell of a beehive to the aligned movement of celestial bodies, nature offered wondrous examples of perfect geometry. Science has long established that symmetry has a biological significance and that it affects humans' perception of health and beauty. It therefore stands to reason that when man began to worship nature, it included the veneration and imitation of its geometrical aspects. This is evident from the earliest engravings on seals and cave paintings in different parts of the world how geometrical shapes have been a part of man's artistic and eventually, religious journey.

Coleman asserts that “... art and religion... – though not interchangeable or identical – frequently parallel each other or converge4 .” With a crossover of precepts from art to religion, geometry gains a special place in religious art and symbolism. Nearly all religions and cults of the world have some form of symbolism that is geometric. Simple points, circles and triangles become more than just shapes. They become aids of ritual and then of meditation within a religious framework. Now, where there is a question of religion, there is a question of faith; and where there is a question of faith, there is an element of mysticism.

2. Sacred geometry

Sacred Geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions. It is associated with the belief that a god is the geometer of the world. Eruvin 76 Circle in the Square

The geometry used in the design and construction of religious structures such as churches, temples, mosques, religious monuments, altars, and tabernacles has sometimes been considered sacred. The concept applies also to sacred spaces such as temenoi, sacred groves, village greens, pagodas and holy wells, and the creation of religious art.

Nautilus shell's logarithmic growth spiral Eruvin 76 Circle in the Square

According to Stephen Skinner,18 the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein. Many forms observed in nature can be related to geometry; for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape. Also, honeybees construct hexagonal cells to hold their honey. These and other correspondences are sometimes interpreted in terms of sacred geometry and considered to be further proof of the natural significance of geometric forms.

Paul Calter writes:19

A dynamic branch of mathematics, geometry also serves as a creative tool for engineers, artists, and architects. Squaring the Circle: Geometry in Art and Architecture is the first textbook to cover both art and geometry extensively.

3. Squaring the Circle in the Great Pyramid

"Twenty years were spent in erecting the pyramid itself: of this, which is square, each face is eight plethra, and the height is the same; it is composed of polished stones, and jointed with the greatest exactness; none of the stones are less than thirty feet." -Heroditus, Chap. II, para. 124.

18 Skinner, Stephen (2009). Sacred Geometry: Deciphering the Code. Sterling

19 https://math.dartmouth.edu/~matc/eBookshelf/art/SquaringCircle.html Eruvin 76 Circle in the Square

Giza

Pyramids and Sphinx as depicted in 1610, showing European travelers Tompkins, Peter. Secrets of the Great Pyramid. NY: Harper, 1971. p. 22 Eruvin 76 Circle in the Square

The problem of squaring the circle is one of constructing, using only compass and straightedge;

(a) a square whose perimeter is exactly equal to the perimeter of a given circle, or

(b) a square whose area is exactly equal to the area of a given circle.

There were many attempts to square the circle over the centuries, and many approximate solutions, some of which we'll cover. However it was proved in the ninteenth century that an exact solution was impossible.

Squaring of the Circle in the Great Pyramid The claim is:

The perimeter of the base of the Great Pyramid equals the circumference of a circle whose radius equal to the height of the pyramid.

Does it? Recall from the last unit that if we let the base of the Great pyramid be 2 units in length, then

pyramid height =

So:

Perimeter of base = 4 x 2 = 8 units

Then for a circle with radius equal to pyramid height .

Circumference of circle = 2 7.992 So the perimeter of the square and the circumference of the circle agree to less than 0.1%.

An Approximate Value for in Terms of Since the circumference of the circle (2 ) nearly equals the perimeter of the square (8)

2 8 we can get an approximate value for , Eruvin 76 Circle in the Square

4 / = 3.1446 which agrees with the true value to better than 0.1%.

Area Squaring of the Circle The claim here is:

The area of that same circle, with radius equal to the pyramid height equals that of a rectangle

whose length is twice the pyramid height( ) and whose width is the width (2) of the pyramid.

Area of rectangle = 2 ( ) ( 2 ) = 5.088

Area of circle of radius = r 2 ( ) 2 = 5.083 an agreement within 0.1% Eruvin 76 Circle in the Square

Rachel Fetcher writes:20

From the domed Pantheon of ancient Rome, if not before, architects have fashioned sacred dwellings after conceptions of the universe, utilizing circle and square geometries to depict spirit and matter united. Circular domes evoke the spherical cosmos and the descent of heavenly spirit to the material plane. Squares and cubes delineate the spatial directions of our physical world and portray the lifting up of material perfection to the divine. Constructing these basic figures is elementary. The circle results when a cord is made to revolve around a post.

The right angle of a square appears in a 3:4:5 triangle, easily made from a string of twelve equally spaced knots. But "squaring the circle”—drawing circles and squares of equal areas or perimeters by means of a compass or rule—has eluded geometers from early times. The problem cannot be solved with absolute precision, for circles are measured by the incommensurable value pi (S = 3.1415927…), which cannot be accurately expressed in finite whole numbers by which we measure squares.

At the symbolic level, however, the quest to obtain circles and squares of equal measure is equivalent to seeking the union of transcendent and finite qualities, or the marriage of heaven and earth. Various pursuits draw from the properties of music, geometry and even astronomical measures and distances. Each attempt offers new insight into the wonder of mathematical order. In this column, we consider methods for achieving circles and squares of equal perimeters, focusing on geometric approaches conducive to design applications and setting aside for now the problem of achieving circles and squares of equal areas.

Circle in the Square in Kabbalah

20Squaring the Circle: Marriage of Heaven and Earth Eruvin 76 Circle in the Square

But who is this Shabbat? The Donkey Driver [replies]: This is the one that the Holy One, blessed be He, included [with the other], saying: "I am the Lord" (Ibid.). And I heard this from my father, who emphasized [that the word] et symbolizes the Shabbat limits. "My Sabbaths", denotes a circle and with a square within, which are two [different shapes that, yet, are now one]. According to these two, there are two hallowed prayers that we should recite. One starts with Vaychulu [Beresheet 2:1-3], and the other is the Kiddush. The passage of Vaychulu consists of 35 words, and in the Kiddush that we perform there are also 35 words. They add up to seventy names, with which the Holy One, blessed be He, and the Congregation of Israel adorn themselves.

"My Sabbaths" ... denotes the circle and the square within and corresponds to Genesis 2:1-3 (commencing Vayikhulu and to kiddush). Each contains 35 words together making 70 and corresponds to the 70 names of the Holy One, blessed be He.

This construction is particularly ironic, given that, according to Rabbi Shimon ben Gamliel, since the sixth day of Creation, there is no square shape in creation, as squares imply abrupt changes of direction. Rather, life on earth is symbolized by the circulation of the heavenly bodies orbiting elliptical paths (Talmud Yerushalmi, Nedarim 3:2)

According to the Zohar, the point is acknowledged as the mystical symbol of the world. For it is the point that receives all light to illuminate the body and provide everything: "the central point of a circle on which the whole circle depends" (Zohar, 1:229a). The point is in the middle of the world, and the world expanded from there: to the right and to the left and upon all sides. The world is sustained by this central point.21

21 Elliot R. Wolfson, Circle in the Square: Studies in the Use of Gender in Kabbalistic Symbolism (Albany, N.Y.: State University of New York Press, 1995), Eruvin 76 Circle in the Square

Circle and Squares in Marriage In some Chasidic circles, the wedding band is round on one side and square on the other

Simon Jacobson writes:22

Circle and squares are found in every aspect of life – both on earth and in the cosmos, in our psyches, our physics and our metaphysics. The purpose of existence is to fuse the two; to join both structure (the linear square) and beyond structure (the circle), the finite and the infinite.

Can a circle marry a square?

Two questions remained lingering: How is it possible to synthesize two opposites – the square and the circle? And perhaps even more pertinent: Can a circle marry a square?

22 https://www.chabad.org/kabbalah/article_cdo/aid/2236526/jewish/Circle-and-Squares-in-Marriage.htm Eruvin 76 Circle in the Square

These questions will be answered by first gaining a better understanding of the circle/square dynamic. One place where circles play a vital role is in a Jewish wedding ceremony. Two circles mark the ceremony: The Chupah (canopy), which encircles and encompasses both bride and groom in its all-surrounding embrace. And the spherical wedding ring placed on the bride’s index finger. [In some Chasidic circles, the wedding band is round on one side and square on the other!

—Ed]

According to Torah law marriage entails two distinct stages: The first stage, called kidushin or eirusin, is the engagement and betrothal, when a couple commits to each other, establishing a formal and exclusive connection. They have sanctified their union and bound themselves to each other. However, the couple is still two distinct individuals. The second stage, nisuin, marriage, is when they actually become one entity – they become elevated

(nisuin from the word elevated) to a higher state of being; instead of two they are now one.

Stage one is traditionally achieved through the wedding ring. Stage two is accomplished through the chupah, encircling both bride and groom.

The wedding ring is a legal commitment, represented by a monetary exchange. But it still is only a ring on a finger; a small gesture, not an all encompassing commitment. The chupah, on the other hand, covers and surrounds bride and groom completely – embracing their entire bodies and beings. Eruvin 76 Circle in the Square

The third Chabad Rebbe, known as the Tzemech Tzedek, wonders about the difference between these two circles marking a wedding ceremony: In mystical terminology the circles represent the level of makif, a transcendent energy that surrounds, but does not fully enter. Since both the wedding band and the chupah are circular, what, asks the Tzemech Tzedek, distinguishes between them? (Ohr HaTorah Berocho p. 1845)

The first initial circle proceeds to evolve into a “line"...

He explains the difference with the imagery of the square and the circle. Though in the general cosmic structure the circle precedes the square, the energy first surrounds then permeates, yet when we break it down into finer detail, the "circle-square" structure repeats itself continuously throughout the process. The first initial circle proceeds to evolve into a "line" (yoshor), which in turn conceives a new "circle," followed by another square and circle, ad infinitum.

By way of example, think of the transmission process from teacher to student. Initially the ideas are "over the student’s head," surrounding but not yet fully entering his grasp. Then as the student acclimates himself to the ideas, he assimilates, integrates and internalizes them, the circle becomes, in effect, a square, entering the "box" of the students mind. But these same ideas

(which have been internalized in a square for the advanced student) remain over the head of a less advanced student. And so it goes, level after level, in which the "square" (internalized) energy on a higher level, remains a removed "circle," hovering above the level below. "The internal (penimiyiut) of a higher level becomes the transcendent (makif) of the lower level." Eruvin 76 Circle in the Square

We thus have two types of circles: A circle that precedes and is higher than a square. And a circle that is lower and follows a square. Visualize a large circle, which contains a square within it, stretched to the edges of the circumference, and then a smaller circle inside the square, and another smaller square inside the circle. Keep going as far as you imagine. Like reflecting mirrors you’ll have a certain picture of the inner workings of existence and of our beings.

Using this imagery, the Tzemech Tzedek explains the difference between the two stages in marriage: The first circle is a "relative circle" – it is only a circle compared to the levels beneath it. This is the circle of the wedding band. The circle of the chupah is an "absolute circle" – one that totally encompasses and equalizes all those that stand under the chupah, i.e. everything inside the circle.

What is the psychological and personal application of this concept? Why do we need these two stages and what is the difference between them?

These two stages – two circles and the box in between – capture the two components necessary in a healthy and enduring relationship:

Love actually consists of two overriding dimensions:

1) Closeness and intimacy – internalization of the relationship. as well as requiring: Eruvin 76 Circle in the Square

2) a dimension of mystery and awe – a surrounding type of aura of your partner that remains beyond you.

Eliminate (or compromise) one of these two dimensions and the relationship will wither (we’re not talking about a relationship of convenience, but a one of passion and life).

A relationship requires both the circle and the square (line) – the transcendent energy and the internal one.

Breaking it further down, the circle itself divides into two dimensions: If a relationship consisted only of the two elements, closeness and awe, the two would possibly never converge. Ultimately, the goal in a full relationship is the unification not just of the two people but also of their

"circular" and "square-like" dimensions. And this is achieved by the so-called "relative circle.

From one (the higher) perspective the "relative circle" is actually a square, but from another (the lower) perspective it is a "circle." In psychological terms: You sense mystery in your beloved, but in time you gain entry and can somewhat internalize it and grow in the process, only to discover, like the shedding petals of a flower, new and hitherto deeper mysteries lying within.

The wedding ring serves this role, to remind you of the ungraspable "circle" in your bride, but also to tell you that with devotion you can access its power. The chupah serves the role of reminding you that there are always new mysteries, and that ultimately there is a dimension that transcends both of you. Eruvin 76 Circle in the Square

We see from this that the concept of marrying "circle" and "square" is not only possible but actually a necessary component in every marriage.