The case of the square inside the circle

Eruvin 76 presents the geometry of a square inside a circle. The math seems badly wrong at the commentators’ first glance, leading to lots and lots of interpretation.

The Mishnah on Eruvin 76a says that for a window to possibly bridge two courtyards, it needs to be at least 4 by 4 tefahim and within 10 tefahim of the ground.

משנה. חלון שבין שתי חצירות, ארבעה על ארבעה בתוך עשרה - מערבין שנים. ואם רצו - מערבין אחד. פחות מארבעה על ארבעה, או למעלה מעשרה - מערבין שנים ואין מערבין אחד.

The Gemara then gives us the ruling of Rabbi Yohanan about a circular window:

אמר רבי יוחנן: חלון עגול צריך שיהא בהיקפו עשרים וארבעה טפחים, ושנים ומשהו מהן בתוך עשרה, שאם ירבענו נמצא משהו בתוך עשרה.

Rabbi Yohanan said: A circular window needs 24 tefahim in its circumference. And two plus a bit from those need to be within ten [tefahim from the ground], so that if you make a square, some of it will be within ten.

This is difficult to understand.

Zedekiah’s Cave

Today was one of those Daf Yomi coincidence days.

Last night I made plans to go to Zedekiah's Cave today, near Damascus Gate in Jerusalem. It's pretty cool. Literally cool, actually – it's a nice temperature down there out of the sun. The inside of the cave has an area of about two football fields ("football" still means "American football" to this oleh hadash) and averages the height of a four-story building. Who knew that such a huge, quiet space exists down there under the Arab markets?

So that's one thing I did today on Jerusalem Day. And guess what? By total coincidence, the cave is also mentioned on today's daf, Eruvin 61b!

Brit milah math

The Mishnah on Shabbat 137a discusses a mohel who mistakenly circumcised a baby on Shabbat when the baby was either seven or nine days old:

משנה. מי שהיו לו שני תינוקות, אחד למול אחר השבת ואחד למול בשבת. ושכח, ומל את של אחר השבת בשבת - חייב. אחד למול בערב שבת, ואחד למול בשבת, ושכח ומל את של ערב שבת בשבת, רבי אליעזר מחייב חטאת, ורבי יהושע פוטר.

The Gemara then presents different versions of this mahloket tanna'im, changing which cases are agreed as hayyav, agreed as patur, or disagreed.

The square root of two

The Gemara on Shabbat 8a discusses a kavveret, a round basket made of reeds.

A drawing of a kavveret from the Steinzaltz Talmud (PDF)

The kavveret can be its own domain:

אמר אביי: זרק כוורת לרשות הרבים, גבוהה עשרה ואינה רחבה ששה - חייב, רחבה ששה - פטור. רבא אמר: אפילו אינה רחבה ששה - פטור. מאי טעמא - אי אפשר לקרומיות של קנה שלא יעלו למעלה מעשרה.

Tractate Shabbat: In and out and in and out...

Massekhet Shabbat discusses, of course, the laws of Shabbat. The beginning of the massekhet is structured chronologically, starting with the laws of Friday afternoon, then candle lighting on Friday evening, then cooking and insulating Friday night. The basics of the 39 melakhot don't come up until the middle of the seventh perek. The rest of the 24 perakim then flow from topic to topic.

But amidst this structure, a surprising topic dominates the first half of the massekhet: hotsa'ah, the melakhah of carrying between a private and public domain. Eight out of the first eleven chapters of the massekhet contain mishnayot about hotsa'ah, and it is the sole topic of five of those chapters: 5, 6, 8, 9, and 11. And most strangely, the very first mishnah of the massekhet is a total non sequitur to the rest of the first perek—a mishnah about, you guessed it, hotsa'ah.

Adding to the surprise, when the Mishnah lists the 39 melakhot in the seventh perek, hotsa'ah is last on the list. And the first Tosafot on the first amud calls it a melakhah geru'ah, an "inferior" melakhah.

So why the focus on hotsa'ah? Why does it get so many chapters? Why is it tacked on to the beginning of the tractate?